Loughborough University

Institutional Repository

A novel method for the

design of induction heating
work coils
This item was submitted to Loughborough University's Institutional Repository

by the/an author.

Additional Information:

• A Doctoral Thesis. Submitted in partial fulfilment of the requirements

for the award of Doctor of Philosophy at Loughborough University.

Metadata Record: https://dspace.lboro.ac.uk/2134/27353

Publisher:
c Ali Kamil Makki Al-Shaikhli

Rights: This work is made available according to the conditions of the Creative
Commons Attribution-NonCommercial-NoDerivatives 2.5 Generic (CC BY-NC-

ND 2.5) licence. Full details of this licence are available at: http://creativecommons.org/licenses/by-

nc-nd/2.5/

Please cite the published version.

This item was submitted to Loughborough University as a PhD thesis by the
author and is made available in the Institutional Repository
(https://dspace.lboro.ac.uk/) under the following Creative Commons Licence
conditions.

For the full text of this licence, please go to:

http://creativecommons.org/licenses/by-nc-nd/2.5/
 ' 'I·'

LOUGHBOROUGH
UNIVERSITY OF TECHNOLOGY
LIBRARY
AUTHOR/FILING TITLE 1

________ JJ.~_Slt1LirHb_Lr __(J__/s__fi ___ ~ __ ·1.

--------- ------~
--..\ccessl(iNico;;v-rio~---------- - I
I
, o o 'L4 "J.tb o I.
-----------------
VOL. NO. CLASS MARK
1
--------------------------- ------ I
I
II
··--------_,_________,_________ .
LoAN Cof\j
'
I
I
I.

I
I

.. I
...... '

II
.J I
I
·~'-­
~
I
!
3o~9 I

I
'
I I
t
I
I
I
I

, I
I
I' .

!
I
I
i
ooo 2498 01

Jl~@!!!lllm@lllmi/IWIIIIIII!III~IM
'
I
This book was bound by
Badminton Press
18 Half Croft, Syston, Leicester, LE? 8LD
Telephone: Leicester 10533) 602918.
 A NOVEL METHOD FOR THE DESIGN OF

INDUCTION HEATING WORK COILS

by

ALI KAMIL MAKKI AL-SHAIKHLI

A DOCTORAL THESIS

Submitted in partial fulfilment of the requirements

for the award of Ph.D. of the Loughborough
University of Technology 1985

Supervisor: Dr L Hobson
Department of Electronic and Electrical Engineering

c by Ali Kamil Makki Al-Shaikhli, 1985

 j,i

Synopsis

Induction billet heating is a well established industrial process

for preheating prior to forging, rolling and extrusion. In many

cases the application of induction heating techniques has been

hindered by the inability of existing methods of work coil design

to easily produce a design which will give a non-uniform power

density along the surface of a workpiece.

Existing methods of work coil design almost universally assume

that the load is subjected to a uniform magnetic field, and hence

it is very difficult to adapt these methods of design to tackle

problems of non-uniform power density distributions.

In order to simplify the problem of work coil design for a non-uniform

power density along a workpiece a more flexible approach was required.

In this project a novel design technique is described in which the

surface current density distribution produced by a single conductor

has been investigated and equations governing this distribution

established. The superposition principle was then used to predict

the power density distribution produced by a number of conductors.

The method was then successfully applied to typical industrial loads

of different materials and shapes. The simplicity of this method

means that it can be used on desk top computers and even to some

programmable hand calculators. The investigations have also included

the effect of workpiece material, size and shape, the existence of

lamination packs and the current distribution within the work coil

conductor itself.
 iii

ACKNOWLEDGEMENTS

I would like to acknowledge Dr L Hobson for his encouragement

and guidance throughout this work.

I wish to thank the Head and Staff of the Electronic and Electrical

Engineering Department of Loughborough University of Technology

for the facilities provided throughout this research.

I also wish to thank Miss Anne Newton for the translation of

the French and German references and Mrs Ann Hammond for the typing

of this thesis.

Finally, many thanks to my sisters in Baghdad for their

support and help through the period of study.

The work was supported by the ORS awards scheme.

 iv

CONTENTS

page no.
Title Page
i
Synopsis
ii
Acknowledgements
iii
Contents
iv
List of Princip~l Symbols VJ

1. INTRODUCTION
1

2. METHODS OF ANALYSIS FOR INDUCTION HEATING WORK COILS 5

2.1 The Equivalent Circuit Method 5

2.1.1 Comments on the Equivalent Circuit ll

2.2 Numerical Methods of Work Coil Design 14

2.2.1 Mutually Coupled Circuit Method 15

2.2.2 Finite Element Method 18

2.2.3 Finite Differences 21

2.3 Conclusion 26
3. INITIAL INVESTIGATIONS 27

3.1 Investigations into the Equivalent Circuit Method 28

3.1.1 Uniform Load 28

3.1.2 Nonuniform Load 37

3.2 Investigations into Numerical Techniques 49

3.2.1 The Finite Difference Technique 51

3.3 Conclusions and Recommendations 61

4. THE SUPERPOSITION THEORY
64
1)
4.1 The Derivation of the Superposition Theory 64

4.2 Verification of the Superposition Theory 70

4.3 The Use of the Superposition Theory 72

4.4 Discussion and Conclusions 88
 V

5. THE SUPERPOSITION THEORY-PRACTICAL EXAMINATION 90

5.1 Preliminary Investigations 90

5.1.1 Single Conductor 91

5.1.2 Number of Conductors 123

5.1.3 Discussion of the Preliminary Investigations 129

5.2 Ferromagnetic Materials 132

5.2.1 Ferromagnetic Workpiece 133

5.2.2 The Lamination Packs 151

5.2.3 Conclusions from the Work on Ferromagnetic 173

Materials

5.3 Summary of Results and Suggestion 174

6. SUPERPOSITION AS A COIL DESIGN METHOD FOR 176

) . HEATING ALUMINIUM PRIOR TO EXTRUSION

6.1 The Superposition Method with a Cylindrical Workpiece 178

6.2 The Temperature Distribution in a Solid Cylinder 186

6.3 A Coil for Power Density Taper 193

6.4 The Construction of a Versatile Coil 197

6.5 A Simplified Coil Design Program 201

6.6 Conclusion 204

7. CONCLUSION AND SUGGESTIONS FOR FURTHER RESEARCH 207

7.1 Conclusion 207

7.2 Possible Areas of Further Research 210

Al. Heating a Cylindrical Workpiece by Induction 213

A2. The Reluctances of the Equivalent Circuit 221

A3. Illustration of Mutually Coupled Circuit Method 225

A4. Illustration of Finite Element Methods 231

AS. Current Density Probes 235

References 238

The Listing of the Programs and the Publications 244

 vi

LIST OF PRINCIPAL SYMBOLS

q, : Magnetic flux (Wb)

R Reluctance (A/Wb)
m
N Nwnber of turns

I Current (A)

V Voltage (V)

21ff Angular velocity (rad/s)

"'
~

f Frequency (Hz)

11 ~
llollr Absolute permeability (H/m)

110 Free-space permeability (H/m)

11 Relative permeability
r
z Impedance ((l)

R Resistance (n)

X Reactance W>

r Radius (m)

9. J,ength (m)

k Empirical factor-equation (A2.13)

k Empirical factor-equation (A2.10)

r
k Empirical factor-equation (2.30)
c
k Empirical factor-equation (2.31)
e
p Dimensionless constant-equation (Al.36)

Q Dimensionless constant-equation (Al.37)

0 ~~
or
Penetration depth (m)

p Resistivity ((lm)

VA Volt-ampere (VA)

FA Constant of Frohlich formula-equation ( 2. 32)

FB Constant of Frohlich formula-equation ( 2. 32)

FC Constant of Frohlich formula-equation (2.32)

p. F. Power factor
 vii

7 Efficiency (%)

H Magnetic field strength (A/m)

H* Conjugate of magnetic field strength

E Electric field strength (V/m)

2
J Current density (A/m )

J* Conjugate of current density

a Conductor radius (m)

h Distance from the centre of the conductor to the

surface of the load (m)

z Distance along the load measured from a point

directly beneath the conductor (m)

h Effective height (m)

e
A Current per unit length (A/m)
2
PD Power density (W/m )

T Temperature (OC)

d Coil pitch (m)

a Dimensionless constant - Fig. 5.8

Dimension less constant - Fig. 5.9

Suffix

r Return

c Coil

t Total

w Workpiece

g Air gap

Other symbols are defined as they occur.

 l

CHAPTER 1

INTRODUCTION

Induction heating was first used in industry to nelt metals but now

deals efficiently with many other applications in the general field

of metal heating, tube welding, brazing, soldering, surface hardening,

and through heating for rolling, forging and extrusion.

The assembly of the basic induction neater is shown in Fig. 1.1.

The principle of induction heating is illustrated by Fig. 1.2. A

water cooled copper coil with thermally insulated lining surrounds

the metal to be heated. The coil is supplied with an alternating

current, thus creating a pulsating magnetic field which interacts

with the billet and induces eddy currents within it. The eddy currents
2
in turn produce heat as a result of the Joule heating or I R losses.

The amount of heat generated in the workpiece is dependent upon the

physical properties of the billet material, and the magnitude of the

induced current, which is proportional to the workmil turns and

current.

Induction heating has a number of inherent advantages over its fuel

fired competitive processes. Very fast heating rates can be achieved

and the ease of control and the ability to repeat the working cycle

make induction heating ideal for a fully automated process. When hot

metal is not required the heater can be switched off thus eliminating
 PYROMETER
CTION
COIL.

RETRACTABLE
STOP.
REJECTS CHUTE

TABLE
UNSCRAMBLER.
-.........
HOT PIECES
10 FORMI~
BASE UNIT/ PROCESS.
!MAY HOUSE TRANSFORMERS
AND CAPACITORS.!

Fig. 1.1 Basic 1nduction through heater

Induced currents Watgr cooled coi I

in billet

I
I '
''
A.C. Supply

Fig. 1.2 Induction billet heating

 3

virtually all standby losses. Induction heating equipment usually

provides a clean and relatively pleasant working environment with

little extraneous heating and a low noise level.

These process advantages have ensured that·induction heating has

achieved a substantial market share of the heating installations

within certain parts of the metal forming industry [1.1]. Other areas

however, have so far remained virtually untouched, especially those

requiring a non-uniform surface power density along the length of the

workpiece such as off-the-bar forging or the extrusion of aluminium.

Existing methods of work coil design almost universally assume that

the load is subjected to a uniform magnetic field, and hence it is

very difficult to adapt these methods of design to tackle problems of

non-uniform power density distributions. In industrial practice the

design of induction billet heaters is largely based on the equivalent

circuit method devised by Baker [1.2, 1.3] and developed by Reichert

[1.4] and vaughan and Williamson [1.5, 1.6). The method assumes a

uniform magnetic field strength along the length of the workpiece and

many empirical factors are incorporated to take into account such

things as a shortness of coil and spacing between turns. Numerical

solutions using finite difference, finite element and mutually

coupled circuit techniques are in existence [1.7-1.9] but they also

assume a uniform magnetic field strength and hence a surface power

density along the length of the workpiece. They require specialist

knowledge and powerful computational facilities not normally available

to UK induction heating manufacturers.

 4

In order to deal realistically with problems involving non-uniform

surface power density along the length of a workpiece, a more flexible

approach is required. A novel technique is presented in which the

surface current density distribution produced by a single conductor

has been investigated and equations governing this distribution

established. The superposition principle to find the power density

produced by a number of conductors has been verified and a work coil

design to produce a particular power density distribution in a

billet has been carried out. The method is very simple and can

be used on some programmable hand calculators.

 5

CHAPTER 2

METHODS OF ANALYSIS FOR INDUCTION HEATING

WORK COILS

Industrial designers of induction heating work coils depend

largely on relatively simple equivalent circuit techniques

supplemented by empirical data accumulated over many years. On

the other hand, academics and other research workers have

developed highly sophisticated numerical techniques to solve

particular problems.

This chapter explains the principles behind each form of work

coil design.

2.1 The Equivalent Circuit Method

The basic assembly of an induction billet heater consists of a water

cooled copper coil surrounding a metallic workpiece. A relatively

large air gap between the coil and the workpiece is required so

as to permit the free movement of the billet through the heater.

In addition to this, thermal insulation is provided so that the

radiated heat losses from the billet to the coil can be reduced.

The magnetic flux produced by the work coil, ~t' has alternative parallel

paths through either the workpiece 4w , the coil ~

c
or the air gap ~
g
,
as shown in Fig. 2.1. These paths have the following respective
 6

WORKPIECE

AIR GAP

DD DD DD COIL

Fig. 2.1 The flux paths in induction heating system

magnetic reluctances R
mw' Rme and Rmg The return flux <fJ has a reluctance
r
of R
mr
. The effect of <fJ on the coil exterior is usually
c
.·.
ignored [2.l,2.2] because it is numerically far less than <fJ • The
r
effect of <fJ on the coil interior is represented by R The
c me
assembly total reluctance equals R in series with the parallel
+ mr
connection of R , R and R The position of R relative to R
mw mg me me mr
does not .alter the performance of the magnetic circuit seriously

because, the coil reluctance is generally far higher than the external

reluctance. The magnetic circuit of Fig. 2.2 is formulated to

simplify the calculations.

 I

An expression for the total ampere-turn requirement is derived

using the reluctances of each of the magnetic circuits and Ampere's

law:

N I = ~t Rmt
c c
R R
R (R mg mw
+
me mr R + I{
mg mw
= ~t [ R R J (2 .l)
R + R + m9: mw
me mr R + R
mg mw

The work coil voltage, Vc , is related to the total flux by Faraday's

law:

V = jwNc~t (2.2)
c

Hence an expression for the total circuit impedance Z can be obtained

from
V
c
z I (2.3)
c

The electrical equivalent circuit is, therefore, derived and

shown in Fig. 2.3, where

X N2 w
= (2. 4)
r c R
mr

N2 w
X = (2. 5)
g c R
mg
'N2 w
zw = J c R ( 2. 6)
mw

w
zc = 'N2
J c --
R ( 2. 7)
me
 8

R
mr
_j
..
<j>c <j>g <j>w

mmf R R R
me mg mw

Fig. 2.2 The equivalent magnetic circuit

"

z
c
I
I
I
c
R
c
X
c
' I

r---' T T T

1 R
w
..,
V
• X
c . r f-Z w
.
• X
w
•
I
•
X
g
T T

I
Fig. 2.3 The equivalent electrical circuit ' I
 ')

The reluctances are derived in detail in appendices 1 and 2

and are given by:

.. -,
2(0.45+k}
Rmr = (2. 8}
llo" re

t
c
R
mg
= 2 2 (2. 9}
11 7r(r.-r}
0 c w

t
c
R
mw
= (2.10}

t
R
c
me (2.11}
11 rrk
o
or
r c c
(1-j}

The components of the electrical circuit are derived by substituting

equations (2.8-2.11} respectively into equations (2.4-2.7} so as

to give:

r
X
c
r ~ (2.12}
2 (0. 45+k} ' '
(r2 - r 2 }
X =
c w
g ~ t
c
(2.13}

2
11 r
zw = R + ·x = KR r w
w J w (Q + jP} (2.14}
i
w
..•
k or
zc = R + jX = KR r c c
c c (1 + j} (2.15}
1-
c

where 2
KR = WIJ 1TN
0 c (2.16}

--·----.
 10

The equivalent circuit of Fig. 2.3 can be solved by circuit

analysis to give work coil turns, the current carrying capacity

and the efficiency and power factor. As the coil number of turns

is not known, the circuit components are calculated for a single

turn coil. In these calculations the components of the single turn

coil are represented by adding the suffix s to the symbol.

The single turn circuit impedance, z , is:

s

·x [·x
J rs J gs
+Z]
ws
zs = z +-=C-_,2.::...__ _;.:.,=-- (2.17)
CS

z is of a complex value;
s

Z = R + jX (2.18)
s s s

it~s absolute value is;

2
iz s I = + l(
s (2.19)

The coil efficiency is;

1
WS
= (2.20)
R
s

and the power factor is;

R
§
P.F. = (2. 21)
TZ'.T
s

The volt-ampere, VA, of the coil can be calculated from the power input

to the coil P :
c
 ll

VA = - - (2.22)
P.F.

The voltage of the single turn coil V can be calculated from

er
the volt-ampere.as follows:

V= Z I (2.23)
c c c
2
multiply by the voltage and divide by N ,
c
zc
V I (2,24)
c c 7
. c
take the square root
v = r'{vA) z (2.25)
CS S

As the coil voltage V is known, then the number of turns is;

c
V
c
N = (2.26)
c V
CS

Since the number of turns is now derived, the impedance of the coil is;

2
z = Z N (2. 27)
s c

and the coil current is

V
I =-
c (2.28)
c z

2.1.1 Comments on the Equivalent Circuit

In general, the aforementioned equivalent circuit method is adopted

widely by many researchers. The literature, however, does tend to

differ in this domain.

 12

Reichert (2.2] expressed the factor kr as:

0.92
f.J ~ --=- k = (2.29)
r
lkc
N t
=
c c
where k (2.30)
c t
c

t is the coil conductor width

c
while Baker [2.1] suggested that 1.5 ~ k ~ 1.0 with a typical
r
value of 1.15. Instead of taking kr , Vaughan and Williamson [2.3, 2.4]

used the space factor, which is approximately the inverse of

According to Baker [2.1} any single layer coil can be classified

as being either a short or a long coil. The equivalent circuits

are shown in Figs. 2.4 and 2.5 and the difference between them is

the shunt reactance X . The justification of the position of this

r
reactance in the circuit can be considered insufficient. It s

value was given as:

2
W1J11r N
• c c
X = (2. 31)
r 0.9 k
e

where k is an empirical factor with a typical value of unity.

e
The above expression is similar to that of equation (2.12) where k = 0.

Reichert [2.2] adopted the equivalent circuit shown in Fig. 2.3.

while Vaughan and Williamson [2.3, 2.4} used the circuit shown

in Fig. 2.5. The methods of Baker and Vaughan and Williamson were

discussed in detail and worked examples were supplied in

reference [2.5] .
 l3

Fig. 2.4 The equivalent circuit of the short coil

zc
I
I
c /R
c \
~T

1 R
I

V
c l• X
r zw

) w
I
X
g
••

Fig. 2.5 The equivalent circuit of the long coil

 14

In the equivalent circuit method, the workpiece material is

assumed to be uniform and the calculation is based on a single

value for the resistivity and the permeability. The resistivity

is a function of temperature, while the ferromagnetic material

permeability, below the Curie point, is a function of the magnetic

field strength and can be reasonably represented by Frohlich formula:

FA
+ FC (2.32)
FB + H

The equivalent circuit method suffers from three main imperfections:

1. The use of empirical data. The values of these factors are

not necessarily known for every application.

2. The assumption of uniform material properties is incorrect,

as the resistivity is a function of the temperature, and the

permeability is a function of both temperature and magnetic

field strength.

3. The assumption of uniform magnetic field along the length

of the workpiece hinders any application of this method

where non-uniform power densities must be induced· in the load.

2.2 Numerical Methods of Work Coil Design

Three distinct numerical approaches have been applied to induction

heating problems: finite difference, finite element and mutually coupled

circuit methods. In the finite difference and finite element methods

the workpiece is divided into subregions and the relevant non-linear

 lS

partial differential equations are replaced by a set of algebraic

equations to be solved in each subregion by means of iterative

v,>'
procedures. The mutually coupled circuit method [2~6] divides

the system into a number of subconductors mutually coupled with each

other and applies Kirchoff's Laws to obtain a system of linear

equations describing the problem which in turn are solved by

procedures similar to those used in other"numerical methods. The

application of these numerical methods to induct£on heating problems

will now be discussed.

2.2.1 Mutually Coupled Circuit Method

This method was developed to solve problems in induction heating

and melting applications by Kolbe and Reiss [2.6] and later

developed by Dudley and Burke ""

[2.7]. The region to be investigated

is subdivided into subconductors and using known expressions [2.8)

the resistance and self inductance of each subconductor, together with

the mutual inductances between each subconductor are determined. The

expressions are usually presented in the form of an impedance matrix

and the application of Kirchoff's Laws lead• to a system of linear

equations with complex coefficients of the form,

[z] [I] = [v]

where [z] is the square matrix comprising all of the coil and billet

self and mutual impedances, [r] is the column matrix of unknown coil

and billet segment currents and [v] is the column matrix of driving

voltages. A more detailed illustration of the use of the mutually

coupled method is given in Appendix 3.

 16

The mutually coupled circuit method can be used to determine the

induced currents, power distribution and mechanical forces produced by any

two dimensional, linear and axissymmetric induction heating problem.

The limitations of the method are that magnetic materials cannot be

included in the problem geometry and that the number of elements

into which the conductors can be divided is limited by the size

of the matrix which may be solved. This number is of the order of

300 unknowns for the IBM 3033 computer [2.9) which has a memory

capability of approximately one megabyte. Difficulties also arise

when dealing with curved surfaces and when the penetration depth is

small in comparison with the load dimensions. The number of unknowns

can partly be reduced by finely subdividing only the regions of

greatest interest, typically between a distance 0.1 to 0.33 times

the skin depth 6[2.9) from the surface and using larger subdivisions

elsewhere. This, of course, requires more complicated software.

Kolbe and Reiss [2.6] employed the mutually coupled circuit method

in order to determine:

l. Current density and power distributions in an inductively heated

load of rotational symmetry."

2. Temperature distribution as a function of space and time.

These problems were firstly treated separately and then coupled

together by the temperature dependence of the electrical material's

values. This technique is characterized by calculating the current

density distribution for initial temperature distribution. Once

this has been determined one can then calculate the temperature
distribution at the time t = 6t, and with it the new material's

values. The derivation of the current density distribution for

the time t = 26t and the corresponding materials values are then

possible. This sequence of calculations is continued until the

desired temperature or heating time is reached.

The application of this procedure to a specific case in the paper

showed that the assumed inductor arrangement is unsuitable for the

intended objective. This, of course, requires a re-run of the program;

this is an expensive operation and requires the use of a powerful

computer, even though the thermal conductivity of the material was ignored.

Dudley and Burke [2.7] used the mutually coupled circuit method to

determine the current distribution in three induction heating

configurations. The first is a coil and billet where the coil

current was assumed to be distributed uniformly along the coil length;

see Appendix 3. The second case is that of a sheet coil and billet;

the coil current in this case was assumed to be uniform. This

required the single turn and the billet to be divided into a number

of segments, all of which are coupled to each other. This, of course,

increases the number of equations to be solved. The third case

involves a coil without a workpiece, where each turn is to be

divided into a number of segments.

This method has been used in the solution of other problems and

the various formulations differ, primarily in terms of the expressions

used to represent the circuit elements. These expressions are usually

complicated and the calculations very lengthy; see Appendix 3.

2.2.2 Finite Element Method

The finite element method has become an important and practical

numerical analysis tool. It has found application in almost all areas

of engineering and applied mathematics.

In this method, the field region is subdivided into a finite number

of discrete sized subregions or finite elements. The unknown

quantities at each element are presented by suitable interpolation

__....,_ ------ ---
'

functions that contain the node values of each element.

From the mathematical point of view the finite element method is

based on integral formulations. By way of comparison the older

finite difference methods are usually based on differential

formulations. Finite element integral formulations are obtained

by two different procedures: variational formulations and weighted

residual formulations. Both of these techniques generate the final

assembly of algebraic equations that must be solved for the unknown

nodal parameters.

The earliest mathematical formulations for finite element models

were based on variational techniques [2.10-2.13]. Variational techniques

still are very important [2.14, 2.15] in developing elements and in

solving practical problems. Variational models usually involve

finding the nodal parameters that yield a stationary (maximum or

minimum) value of a specific integral relation known as a functional.

The solution that yields a stationary value of the integral

functional and satisfies the boundary conditions is equivalent

to the solution of an associated differential equation, known as

the Euler equation. If the functional is known, then it is

relatively easy to find the corresponding Euler equation. Most

engineering and physical problems are initially defined in terms

of a differential equation. The finite element method requires

an integral formulation. Thus, one must search for the functional

whose Euler equation corresponds to the given differential equation

(and boundary conditions). Unfortunately, this is generally a

difficult, or impossible task. Therefore, there is increasing

emphasis on the various weighted residual techniques [2.16] that

can generate an integral formulation directly from the

original differential equations.

The generation of finite element models by the utilization of

weighted residual techniques is a relatively recent development.

The weighted residual method starts by the assumption of an

approximate solution for the governing differential equation and

avoids the often tedious search for a mathematically equivalent

variational statement. This method leads to a set of algebraic

equations that can be solved for the unknown coefficients in the

approximate solution. The variational and the weighted residual

techniques, both, are not easily applied by any one other than

the specialist numerical analyst. A more detailed illustration of

the use of the finite element method is given in Appendix 4.

 20

In conclusion, it must be borne in mind, that when employing the

finite element methods to solve induction heating problems, one

must (2.9]

(i) Include an "exterior element" to represent the region outside

the coil

(ii) Use a sparse matrix solution routine

Typically, induction heating problems involving the computation of the

electromagnetic and thermal fields at 500 nodes requires approximately

0.75 megabytes of computer memory (2.9] even when using a sparse

matrix solution routine. Problems involving 500 to 2000 unknowns

can be run at a reasonable cost on an IBM 3033 which compares very

favourably with the mutually coupled circuit method mentioned

previously.

Nemoto and Tabuchi [2.15] used the finite element method to analyse

induction heating systems in two computer programs. The first was

to solve Maxwell's equations so as to determine the eddy current

losses which were employed in the second program. as heat sources

to predict the temperature distribution in the workpiece. The

reduction of the computing time necessitated the representation of

the relations among the elements by first degree equations. On the

other hand, the accuracy derived from these equations will be minimal.

When applying these programs to a particular problem, the heat sources

and the material properties were assumed to be constant. In practice,

they change simultaneously and the princip~l advantage of the finite

 21

element method, until now overlooked, is the facility to take

this change partially into account. Moreover, this method can not

be considered a trial and error procedure, as it was recommended

by Nemoto and,Tabuchi, because of the high cost of running

the program.

Sabonnadiere [2.17] employed a weighted residual technique into a

finite element program in order to predict the distributions of

the flux and the power in an induction heating system during the heating

cycle. The boundary conditions and the elements physical properties

were required as an input data to the program. This program, like

other numerical methods programs, is characterized by a lengthy

running time. Recent communication with Sabonnadiere suggest!that

the work is still in progress.

2.2.3 Finite differences

This was the first numerical technique applied to induction heating

'/•, -. -·.}

problems [2.18] and is still in use today. The technique provides

a relatively straightforward means of· formulating and solving

two dimensional problems.

The basis of the finite difference-method is the conversion of the

governing differential equations of the problem into a set of

algebraic equations by the use of the Taylor series approximations.

The equations are solved at every point on a grid constructed over

the required space domain. The mathematical operations are performed

not upon continuous functions but rather in terms of equations about

one discrete point. By this approach operations such as differentiation

and integration may be reduced to simple arithmetic forms and can

then be conveniently solved using digitial computers.

Finite difference equations are derived by approximating the partial

or total derivatives in a differential equation with algebraic

expression, usually based on Taylor series expansion of the

unknown function. Given the function $(x), the value of the

function $(x + Ax} and $(x- Ax} can be calculated from:

$(x + Ax} = $(x) =-

+ 6x ""' + -(6x)
1 i"'
2 .:::...:r. + -(Ax}
1 3 (2.33a}
dx 2 dx2 6

$(x- Ax} $(x} - !('

6 <.>X }3 u3 3 + (2.33b}
dx

4
By omitting terms from (6xl onwards the simple expression for
d2$
is obtained by adding equations (2.33a and b)
2
dx

1
[$(x + Ax} - 2$(x) + $(x- Axl] (2.34)
2
(Ax)

In the same way the following approximations can be obtained for

2
the first derivative by ignoring (Ax} and above

(2.35a}

(2. 35b}
Equations (2.35a and b) are known respectively as the forward

and backward difference approximation. The central difference

approximation is defined as the average of the two given expressions,

i.e.

dx - _!_
d<P- 2llx [<P(x + llx) - <P(x- llxl] (2.35c)

If <P is a function of two independent variables x and y, subdivide

the x-y plane into sets of equal rectangles of sides llx and lly,

as shown in Fig. 2.6, and let the coordinates (x, y) of the

representative mesh point p be;

x = illx and y = jlly

where i and j are integers.

Denote the value of <P at p by;

<P = <P(illx, jlly) =<Pi .

p ,J

Then by equation (2. 34)

2
= (u>
2
p ax i .
,J

(2.36)

1
= [ <Pi+l ,]. - 2 <P·l.,J. + <P._l ·] (2.37)
l. ,]
 24

i,i+l

P(iAX jl\y)
i-l,j i+i ,j

i,j-1

Fig. 2.6 Finite difference computation grid

1 (2.38)
Similarly, =
(t.y)2
i,j

With this notation the forward-difference approximation for

aay4> at p
•
1s

= (2. 39)

~i
Reichert (2.19] developed a finite difference computer program to
determine the field and the temperature distribution in an inductively

heated workpiece. The program takes the variation of the physical

properties during the heating cycle into account. A considerable

data input base is one of this program•s basic requirements

and it is ~n this area that the program can create problems

for the inexperienced user. Also, the computer used was not

sufficient to the program which meant sacrificing in the accuracy.

 25

Finally, the program's running time was too long. This might not

be considered too important a factor, if the sole purpose of the exercise

was to examine an existing system. However, if the aim is to find

an optimum design criteria which require several re-runs of the

program, this factor does become more significant.

Gibson [2.20] developed a finite difference computer program to

determine the temperature profiles of a metal workpiece heated by

induction or other heating methods. The program was complicated

by the fact that it could be used for a wide range of applications

and with various conditions, such as constant power or current

and other such constraints. The changes in the workpiece's permeability

were taken into account, by employing a functional representation

based on Frdllich formula, rather than a large amount of data,

as in reference [2.19]. This does, of course, reduce the user's

work load.

In general, the main features of the numerical methods are:

l. Over the heating cycle time span, the change in the physical

properties of the workpiece can be taken into account. This,

of course, increases the running time and a greater memory

facility is essential.

2. The program's main objective is to analyse existing configurations.

The optimum design criteria can be obtained by a trial and error

approach, which necessitates several re-runs of the program. This

is an expensive method, as the program is, usually, quite long.

"
3. The problem of non-uniform power density induced along

the workpiece has not been tackled as there is currently no

relevant research on this subject.

2. 3 Conclusion

For many years, very simple models, based on the electrical equivalent

circuit, were adopted together with empirical data to design

induction heating equipment. During the last decade, various

research workers have begun to employ numerical techniques to aid

the design and development process of induction heating systems.

Almost all of the existing work coil design methods are based on

the assumption that the workpiece is subjected to a uniform

magnetic field. It is, therefore, not easy to employ these methods,

when designing a coil capable of inducing a non-uniform power

density along the load. These methods, too, either assume a uniform

workpiece material and require the use of empirical data as in the

equivalent circuit method or they need considerable computational

facilities when adopting' numerical techniques.

 27

CHAP'l'ER 3

INITIAL INVESTrGATlJ:ONS

Any attempt to develop or improve a design method for a particular

piece of equipment naturally requires a study of the existing

methods of analysis of that equipment. This applies to induction

heating work coils as well as to other' devices.

The aim is to gain experience in the use of work coil design methods,

to assess their limitations and to examine the possibility of

introducing any improvement.

This chapter describes initial investigations into the equivalent

circuit and the numerical methods. The study of the equivalent

circuit method includes the examination of the empirical factors'

effects, the reactance of the external flux path and the nonuniformity

of the workpiece.

This method has, also, been improved by developing an equivalent

circuit of a multilayer load of differing physical properties

so as to take the nonuniformity of the workpiece into account.

The use of the finite difference method, a numerical technique, has

been investigated to assess the typical limitations and difficulties

 28

which occur, when solving induction heating problems '<ith

these methods.

The finite difference method was implemented to predict the

magnetic field and power distributions in an inductively heated

workpiece.

3.1 Investigations into the Equivalent Circuit Method

Three computer programs were developed to assess the influence of

the external reactance, the empirical factors and the nonuniformity

of the workpiece physical properties. These programs "EQUIV",

"SHORT" and "LONG" are based on the equivalent circuits of

Figs. 2.3, 2.4 and 2.5 respectively.

The calculations of the program "EQUIV" is as was shown in

section 2.1. The calculations of the programs "SHORT" and "LONG"

are compiled according to Baker's equations [3.1], see Section 2.1.1.

3.1.1 Uniform Load

The previously mentioned computer programs were developed in the

PRIME 400 system and employed to assess the effect of the external

reactance and the empirical factors on the design and performance

of a coil to heat a uniform load.

 2')

A file entitled "BESSEL" was developed independently to evaluate

the Bessel functions. The polynomial expressions of these functions

were obtained from reference [3.2]. This file was inserted into

each of the three programs so as to obtain the evaluations of the

Bessel functions required for the calculations.

The program "EQUIV" is based on the equivalent circuit of Fig. 2.3

and the expressions for the components of the circuit are those

given in Section 2.1.

The program "SHORT" is based on Baker's method [3.1} for the

design of a short coil, Fig. 2.4. The expressions for the

components of the equivalent circuit, are those given by Baker and

mentioned in the previous chapter.

The program "LONG" is similar to SHORT with the absence of

the external reactance - that is the equivalent circuit of long coil,

Fig. 2. 5.

The three computer programs differ only on the position of the

external reactance X • The flow chart of them is identical and

r
is given below.
 30

( START )

L Read the input data

/
Calculate 0 ' 11 ' P, Q, K /N 2
w c R c

Calculate the circuit components

in terms of K/N 2
c

Calculate P.F. , , , VA, I N , N , I

c c c

,....,. . . . . I
I
Write

Nc ' Ic'/, P.F.

( STOP

The Flow Chart of the Programs

 31

The three computer programs were employed to assess the effects of:

1. The external flux path reactance

2. The empirical factors

Three cases were considered:

1. A nonmagnetic load with 50 Hz power supply

2. A nonmagnetic load with 3 kHz power supply

3. A magnetic load with 50 Hz power supply.

The following values of the parameters were used in the calculations:

Coil power = 200 kW

Coil voltage = 440 V

Load diameter = 0.08 m

Coil diameter = 0.14 m

Coil length c Load length = 0.5 m

Coil resistivity -8
= 2 X 10 rlm

Coil relative permeability = 1.0

The values used for each of the following factors are as follows unless

it·s effect was examined

k = 1.15 k = 1.0 k = o. 8 k = 0.0

r e c
'

The nonmagnetic load was assumed to be aluminium with surface

temperature of about 600°C and 20°C at the centre. This corresponds

 >'>
-'•

to resistivities of 10 x 10-8 nm and 2 x 10-8 nm respectively [ 3.3 • J

An average value of Pw = 6 x 10-8 nm was adopted.

The magnetic load was assumed to be EN3 mild steel with a surface
0 0
temperature of 600 C and a centre temperature of 20 C. This corresponds

to resistivities Of 75 X 10-8 nm and 15 X 10 -8 nm respectively [ 3.3 ] •

The permeability can be calculated from Frohlich formula r-4]

with the following constants: AF = 2.08, BF = 1380 and

CF = 1.2566 X 10- 6 . These will give a maximum relative per~ility

of 1200 at zero magnetic field and a minimum of 1 at very high

magnetic field. Average values of pw = 45 X 10- 8 nm and

~ = 600 were employed in the calculations.

rw

The results obtained from the three computer programs are shown

in Tables (3.1-3.7).

Although the numerical values of the results apply to particular

cases, they do show the main features of the relevant methods.

Table 3.1 shows that when dealing with a nonmagnetic load the three

circuits give values which are all within 10% of each other for

the required coil turns and current. The difference in

the estimated performance of the system i.e. efficiency and power

factor is more than 15%. The long coil, Fig. 2.5, tends to produce

better results than the short coil, Fig. 2.4, when compared with

the circuit of Fig. 2. 3 "EQUIV" .

 Jj

Nonmagnetic Load, 50 Hz Nonmagnetic Load, 3 kHz Ferromagnetic Load, 50 Hz

EQUIV SHORT LONG MAX % EQUIV SHORT LONG MAX% EQUIV SHORT LONG MAX %
DIFF DIFF DIFF
N 151 156 145 7.6 10 11 10 10 93 65 62 49
c
I 1709 1636 1678 4.5 8401 8189 8280 2.6 674 1589 674 136
c

l 40 35 40.4 15.4 46.2 41.3 45.5 11.9 96.5 89.4 98.3 10

p. F. .266 .28 .27 5.3 .054 .056 .055 3.7 .675 .286 .674 136

Table 3.1: Results from the three different circuits with

optimum values of the empirical factors

Short Coil Long Coil

k = 1.0 k = 1.5 Max % k = 1.0 k = 1.5 Max %

r r r r
Diff Diff

N 153 161 5.2 143 149 4.2

c
I 1742 1449 20.2 1773 1507 17.7
c

P.F.
7 38.3

o. 26 0.31
29 32.1

19.2
43.8

.256
34. 2

o. 302
28.1

18

Table 3.2: The effect of k on a nonmagnetic load at 50 Hz

r

Short Coil Long Coil

k = 1.0 k = 1. 5 .Max % k = 1.0 k = 1.5 Max %

r r r r
Diff I Diff

N 10 12 20 10 11 11
c
II
I 8832 7014 25.9 8876 7171 23.8
c

7 44.8 35 28 49 39 25.6

p. F. 0.051 0.065 27.5 0.051 0.063 23.5

Table 3.3: The effect of k on a nonmagnetic load at 3 kHz

r
 34

Short Coil Long Coil

k
r = 1.0 k
r = 1.5 Max % k
r = 1.0 I k
r = 1.5 Max %
Diff ' I Diff
i
N 64 66 3.13 i 62 62 0
c
le 1607 1549 3.75 674 674 0

7 90.7 86.5 4.86 98.5 97.7 0.8

P.F. 0.28 0.294 5.0 0.674 0.674 0

Table 3.4: The effect of k on a ferromagnetic load at 50 Hz

r

Nonmagnetic Load at 50 Hz Nonmagnetic Load at 3 kHz Ferromagnetic Load at SOH

k
e
=· 0.'8 k
e
= 1.2 %Diff k = 0. 8
e. k
e
= 1.2 %Diff k. =
e.:
o. 8 ke = 1.2. %Diff

N 153 158 3. 3. 10 11 lO 63 66 4.8

c
I 1646 1626 1.2 8212 8165 0.6 1418 1747 23.1
c

7 36 34 5.9 42.2 40.5 4.2 92 86 6.5

P.F. .276 .28 1.5 .0554 .0557 0. 5 0. 32 o. 26 23.1

Table 3.5: The effect of k

e

Nonmagnetic Load at SO Hz Nonmagnetic Load a.t 3 kHZ Ferromagnetic Load at 50 Hz

k = -.22 k = 0.1 %Diff k = -.22 k = 0.1 %Diff k = -.22 k = 0.1 %Diff

N 146 153 4.8 10 11 10 79 98 24

c
I 1742 1694 2.8 8572 8325 3.0 674 674 0
c

7 41.1 39.5 4.0 47.3 45.7 3.5 97.4 96.1 1.4

P.F. 0. 261 0.268 2.7 0.053 .055 3.8 . 674 .675 0. 2

Table 3.6: The effect of k

 35

Nonmagnetic load I Nonmagnetic load Ferromagnetic load

at SO Hz i at 3 kHz at SO Hz
k =0.7 k =1.0 % Diff i' k =0. 7 k =1.0 %Diff k =0. 7 k =1.0 %Diff
c c ' c c c c

N 1S2 149 2.0 ! 11 10 10 93 93 0

c
I 1663 1787 7.S 8117 8878 9.4 674 674 0
c
38.4 42.7 11.2 44.S 49 10.1 96.3

P.F.
l 0.274 0.2S4 7.9 .OS6 .OSl 9.8 .67S
96.9

.675
0

0
I
Table 3. 7 The effect of k
c

The magnetic load results obtained from the three circuits have

di veiS e characteristics. 'lh.e number of tuns calculated J:y

the snort and long coil circuits are equal, but a different number

is produced b[ the circuit of Fig. 2. 3 "EQUIV". 'lhE! required

current calculated l:y the former 2 circuits again equal in both

cases·, but different in the latter case of the short coil circuit.

'
The effect of the empirical factor k , which allows for the spacing
r

between the turns, on the short and long circuits is shown in

Tables 3.2-3.4. The effect on magnetic materials, for the case under

consideration, is very small when adopting the short coil circuit

and negligible when adopting the long coil circuit. However, the

difference increases with the frequency for nonmagnetic materials and

 --- ---------------------------------

3G

it becomes more noticeable when applying the short coil circuit.

The estimation error in the number of turns can be anything up

to 20%, while in the current can reach 26% and the efficiency 32%.

Changing the value of k , the external reactance factor, from 0.8

e
to 1.2 does not affect the results by more than 10% except when

estimating the current for magnetic materials when they will vary

by 23% (Table 3. 5).

The effect of the factor k, which is for the return flux, on the

circuit of Fig. 2.3 "EQUIV" is shown on Table 3.6. When this constant

was varied between the maximum and minimum values (0.1 and -0.22),

the results did not fluctuate by more than 10%, apart from when

estimating the number of turns for a magnetic load, in this case

the variation was 24%.

Table 3.7 shows k, which is for the spacing between turns,

c
did not influence the magnetic load results. The effect on the

case of the nonmagnetic load was within 10% on the number of turns

and current, and to 11% on the efficiency.

The above work showed that the three circuits may give different answers.

The discrepancy between these answers depends on the case under

consideration and there is no general rule to assess these differences.

Also, the empirical factors have an influence on the results produced

by these circuits, and here again, this influence depends on the case
 37

under consideration and cannot be assessed ·in gen·eral ·terms, hence

it is difficult to the inexperienced person to choose the right

circuit and the right values of the empirical factors suitable for

a given configuration.

The other limitation of the equivalent circuit method is that it does

not take the nonuniformity of the workpiece into account. This

is to be discussed in the next section.

3.1.2 Nonuniform Load

The workpiece, during the induction heating cycle is not

uniform, as the resistivity of any part is a function of the

local temperature, while the permeability depends on the

magnetic field strength in that part. The load can be

considered as consisting of a number of concentric cylinders,

each one of different physical properties. This approach

improves the equivalent circuit method by overcoming one

of its imperfections.

Fig. 3.1 shows a typical billet heating application, whereby

a coil of radius r surrounds a billet of radius r • The billet

c w
is shown as being divided into n concentric cylinders.

Following the procedure outlined in Section 2./,, the magnetic and

electrical equivalent circuits are shown in Figs. 3.2 and 3.3

 38

respectively. The derivation of X , X and Z

r g c
are as in Section 2.1
and they are represented below:

r
c
X K (3.1)
r R
2 (0. 45 + k)

2 r2)
(r -
c w
X = KR ( 3. 2)
g R.
c

Reichert's expression [3.5] for zc will be drawn upon to represent

the coil impedance

0.92
oc r
c
(1 + j) (3. 3)
,.'k R.
c
c

2
where W)l TTN (3.4)
0 c
 39

Fig. 3.1 The flux paths in a load of n concentric cylinders

The impedance of the composite load can be determined by employing

Maxwell's equations and the complex Poynting vector. The total

complex power S consumed by a cylinder of radius r and length i

w
is:

S = (E (r)x H* (r)). 21rr R. (3.5)

r w

where E(r) and H*(r)are the r.m.s. values of the electric field

strength and the conjugate of the magnetic field strength on r

respectively. E(r) is a function of the current density J(r) and

the resistivity p at radius r;

r

E(r) =- p
r
J{r) ( 3. 6)
 ·lO

R
··m,.
t r

$c $g $n $2 $1

mmf R
me
R
mg ~-mn Rm2 ~ ml

Fig. 3.2 The magnetic circuit of a coil-multilayer

workpiece

zc

\
X
I c I
c w

X
V r
c

X
g

Fig. 3.3 The equivalent electrical circuit of

a coil-multilayer workpiece
 41

Since the electric and magnetic field strengths are perpendicular

to each other; equation (3.5) can be rewritten as;

S =-27rrR. p J (r) H* (r) ( 3. 7)

r w r

The apparent power consumed within the cylinder of radius r,

according to Fig. 3.3, is;

sr ( 3. 8)

where Z is the impedance at radius r. If the coil is a long

r
solenoid then;

I=NI=HR. (3. 9)
w c c 0 w

2
H H*R- (3.10)
0 0 w

where H is the magnetic field strength and H* is the magnetic

0 0

field strength's conjugate on the workpiece surface.

Substitute equation (3.10) into equation (3.8) and equate

it with equation (3.7) so as to yield

-27rrp J (r) H* (r)

zr = -::---=-
R.
r
H H*
(3.11)
w 0 0

The cylinder's magnetic field strength and it's current density

can be evaluated by solving Maxwell's equations, see Appendix 1,

and they are:

 42

(3.12)

(3.13)

where C and B are constants.

The solution in the inner cylinder, r ~ r 3 0, which consists

1
of a homogenous material is:

(3.14)

(3.15)

In the other cylinders, B is not necessarily zero, hence the complete

solution is essential.

In the second cylinder, r >- r ~ r

2 1

(3.16)

(3.17)

In the third cylinder, r >- r >- r

3 2

(3.18)
 43

(3.19)

where 61, 62 and 63 are the skin depths in the 1st, 2nd and 3rd

cylinders respectively, and this continues to be the case until

the outer cylinder is reached.

( 3. 20)

J (r)
n
=? [c 2un n- 2
.r'(/2j; ) + c
o un 2n- 1
.K'(fi); ) ]
o un
(3. 21)

The constants c to c n-l can be obtained by considering the boundary

1 2
conditions at the junctions between the concentric cylinders and also at

the junctions between the surface of the billet and the air gap.

It is known that the magnetic and electric field strengths are both

continuous across the boundary. Once having defined H as the magnetic

0

field strength at the surface of the billet, it is possible to determine

(C /H) to (C /H), and therefore to solve for H(r)/H and J(r)/H too.
1 O· n o o o
By substitution into equation (:LU), the impedances can be found. The

following method of solution can be applied for any number of concentric

cylinders. For simplicity consider a two layers load.

At the billet surface:

r = r (3.22)
w
and H (r) H ( 3. 23)
0
r r
i .. e. H = c2.Io(ffj
0
6;) + c3.Ko (/2] 6;) (3.24)
 44

At the boundary between the two layers:

(3.25)

and (3.26)

(3.27)

i.e. ( 3. 28)

and
p 12 r
1
c r· <mJlil
"l"o ....!:.,
01

(3.29)

From equations (3.24) ,(3.28) and (3.29) it is possible to solve

The results are quoted below.

Defining,
r.
X ..
1,J 12jrl (3.30)

and (3.31)

then

(3.32)

_c_2 = A.Io(x2,l).I~(xl,l)
(3.33)
H 11
0
 45

(3.34}

where

(3.35}
At this point all the electric circuit components were defined.

The specifications of the coil can be found by solving the circuit

following the method outlined previously in Section 2./.

The computer program "EQUIV" was improved to cater for this derivation.

The program's objective is to design and predict the performance

of a work coil suitable to heat a multilayer cylindrical workpiece and

each cylinder's material is not necessarily identical. The equivalent

circuit was that shown in Fig. 3.3 and the calculation was as stated

previously. If concentric cylinders of similar physical properties

were to be considered, 'the equivalent circuit would resemble

Fig. 2.3;- that being a uniform load.

In the program, the workpiece was divided into n concentric

cylinders, where n ~ 100. The resistivity of the inner (1}

and outer (n} cylinders, correspond to minimum and maximum temper-

atures to be supplied by the user. The resistivity of any other

 46

cylinder, say x, is then calculated from' the linear interpolation

of these resistivities.

(3.36)

Frohlich formula, equation (3.37), was employed to calculate the

permeabilities of a ferromagnetic workpiece, as a function of the

magnetic field strength in each cylinder.

u = ___AF~--+ CF (3.37)
X BF + H
X

The appropriate workpiece material constants AF, BF and CF should

be provided.

The'Gaussian elimination method was adopted for the solution of

the constants c , c , ••. , en.

1 2

The flow chart of this program is shown below.

The parameters employed in the calculations were identical to those

in Section 3.1.1. The resistivities of the aluminium load were

8
taken to be: 10 X 10-S nm and 2 X 10- nm on the surface and the centre
-8
respectively. The resistivities of the mild steel were 75 x 10 nm
-8
and 15 x 10 Qm on the surface and the centre respectively while the

permeability was represented by Frohlich formula with the constants

which were supplied in Section 3.1.1. The operating frequency was

assumed to be 50 Hz in every case. The results are shown in Tables

3.8 and 3.9 below.

 47

START )
/ Read the Input Data
/
Divide the billet into number of cylinders

Set TTE=O
I

Calculate Skin Depths

Solve the simultaneous equations

to determine the constants
cl, .... ,en
,,,

Calculate (H) and (J) at each cylinder and

re-calculate (u) for each cylinder from
Frohlich formula as function of (H)

? NO
TTE= 1 I TTE 1

YES

Calculate: Reluctances and the

circuit components (X , X , Z )
r g c

Complex Power to be calculated

by Poynting's vector

Calculate other circuit components

zl' z2' • • • 1 z n

/ Calculate and write, N

c' I c . ; , P . F /

( STOP

The flow chart of "EQUIV 11

 48

~
% Diff between
f
0 1 2 5 8 1 and 8
inders
p

N 151 167 143 123 23

c
I 1709 1390 1865 2360 28
c

1P.F.
40 51.7 36.5

0.266 0.327 0.244 0.193

24.0 67

38

Table 3.8: The effect of dividing the nonmagnetic

load into a number of concentric cylinders

~"''"""
% Diff between
1 2 5 8 1 and 8

N 93 84 84 67 39
c
I 674 663 663 833 19
c

7 96.5 97.2 97.2 97.2 0. 7

P • F. 0.675 0.686 0.686 0.546 24

Table 3.9: The effect of dividing the ferromagnetic

load into a number of concentric cylinders
 49

The above tables showed the differences between a uniform load and,

when dividing it into 2, 5 and 8 layers. If by increasing the number

of concentric cylinders it was assumed that the similarities between the

equivalent circuit and the actual situation increased, then if one

were to ignore the nonuniformity of the load the results might

be greatly distorted. This effect can be as much as 23% of the

number of turns required to heat a nonmagnetic load and up to 39%

when the load is a ferromagnetic material. The estimating performance

of the coil also varies: the difference in the efficiency is up to

67% and the power factor difference is up to 38%· Hence the correct

choice of the physical properties values is very important.

3.2 Investigations into Numerical Techniques

The adoption of the numerical techniques by many research workers

necessitates one gaining experience in their use in the induction

heating domain. It is only then, that their limitations and the

difficulties associated with them can be assessed. It was impossible

to investigate all the three numerical methods reviewed in the

previous chapter, as this would require much time and effort. The

time and· effort are better to be devoted towards developing a method

to design a coil, which is capable of inducing a nonuniform power

density along the load.

One numerical technique, the finite difference method, was implemented

to determine the distributions of the magnetic field and the power

in a uniform long rectangular workpiece, subjected to a uniform magnetic

 50

field. The finite difference method was chosen because it is well

documented when compared with the more recent finite element

and mutually coupled circuit methods. The finite difference method

is sufficient when a solution is required to this particular problem,

as both the uniformity of the material and the simple geometry of the

region do not require the more complicated finite element or mutually

coupled circuit methods.

A computer program was developed to determine the distributions of the

magnetic field and the power in a rectangular billet of nonmagnetic

material. It is a simple task to amend the program, if one wishes to

consider magnetic material with constant permeability. The program

was written for any user, as one simply needs to run the program and answer

the required questions.

The program was based, mainly, on the computer program "BIEEDY" [3. 4 1·
This piece of research has been selected because it is well documented

and explained.

The derivation of the necessary magnetic field equation and the

equivalent finite difference equation, together with a description

of the computer program is supplied below:

 51

3. 2.1 The Finite Difference Technique

The eddy current in a conducting medium is governed by Maxwell's

equations:

as
~E.dl\ =
w
s
J ( -at) ds (3.38)

VxH = Jc + Jd (3. 39)

where Jc and Jd are ~he conduction and displacement current densities

respectively and B is the magnetic flux density. In induction heating

Jd can be ignored, as it is far less than Jc' since

( 3. 40)

where £ is the permlltivity of the material. Equation (3.39)

becomes:

VxH = J (3.41)
c

As: E pJ (3.42)
and B = IJH (3.43)

Substitute equation (3.41) into equation (3.42) then substitute

the resultant equation together with equation (3.43) into equation (3.38)

so as to yield:

[p(VxH).dl\ J- ~t(IJH)ds (3 •. 44)

s

If the direction of the applied magnetic field is along the length

of the workpiece ll 2 , as it is in billet heating, then

 52

H ~ H ~ 0 (3.45)
y X

and H ~ H (3.46)
z

The curl of H is:

oH aH 3H oH oH 3H
(- -z- - - Jy)_. + ( :x
VxH - - - - -z)] +
. (__x - ---2!.) k (3. 47)
ay az az ax ax ay

As the change of the magnetic field occurs only within the depth of

the workpiece, then

(3. 48)

oH .
VxH ~ - a;{J (3.49)

Substituting equation (3.49) into equation (3.44), yields

CJlp aH d£. ~
ax
s
f ;t (JlH)ds (3.50)

As the magnetic field is sinusoidal,

a
at can be replaced by jw:

CJlp oH di ~ f jW)lHds (3.51)

ax s

The magnitude of the current density J, according to

equations (3.41) and (3.49) is:

J ~-
aH
(3.52)
ax
3
The average volume power density (W/m) at any point [3.3) when
sinusoidal magnetic field is in operation is:

p pJ J* (3.53)
V
 53

A program "slab" has been written to predict the magnetic field and

power density distributions in a homogeneous non-magnetic rectangular

workpiece subjected to a uniform surface magnetic field strength. The

program was written in Fortran 77 and to reduce the computing time

only one quarter of the cross-section was considered. The region is

divided into MX by MY meshes. The quarter section will then have

(MX + 1) by (MY + 1) nodes, Fig. 3.4. In finite difference method

it is essential that a good mesh is selected as this can affect the

results obtained to considerable degree. Where the magnetic field

changes rapidly the mesh nodes should be close together, i.e. near the

surface. On the other hand, where the magnetic field is changing

more gradually the nodes need to be spaced further apart, i.e. towards
the centre of the load.

The surface and line integrals of equation (3.51) are taken over the

region about the i, j node extending halfway to the neighbouring

nodes of the finite difference mesh in Fig. 3.5.

If the load is of non-uniform physical properties then the particular

resistivity and permeability of each mesh should be included in the

finite difference equation. In the case under consideration the load is

of uniform material. Following Gibson [3.4] who considered the more gen-

eral case of non-uniform load, the finite difference equation for a uniform
load will be
 54

2:.2 SLX

i= 1 2 3 4 5 6 7 8 MX

. ~· -p r, -
2 3 4 ~
j=l 1

2
10 11 12 13 fl4 fl5 fL6 7 8

3
"19 20 21 22 03 04 b5 6 7
.
.!_ SLy
2
1
28 29 30 31 b2 ~3 ~4 5 6
I

5
p7 38 39 40 ~1 ~2 ~3 5
1'14

6 46i 47 48 49 ISO 151 r>2 4

r>3

7
55 56 57 58 159 ~0 ~1 r>2 3
8
64 65 66 67 tsB 169 no r,l 2
9
173 74 75 76 77 78 179 ~0 1
MY 182 83 84 85 ~6 187 ~8 ~9 0

Fig. 3.4 Finite difference mesh for rectangular billet

 55

I I
I i ,j+l I
i I

i I
I
I
I
- Y2 I

i-
I
I ---- __ TI _____
------
I
f.-- -- -
. I I
- Y2 I I
I I
i-l,j I i,j I ~+l,j
I
I
flYl I I
-2- I
I

t-
flY 1
-2-
I
-----,.----- ----- -+- ----
I
I
I
I
---

I I
~ I
I i,j-1
I I
- fl;l
I t. xl tJ.'x2 I flx2
-2-
I
2 2
·~·
I I
I I

Fig. 3.5 Finite difference mesh for magnetic field equation

 5(,

H - H
+ [< i,j-1 i,j) (llxl +
llyl 2

(3.54)

The boundary conditions are

aH
as X = 0 -= 0
ax

aH
as y = 0 -= 0
ay

1
as X = - SLX H= H
2 0

as y = !_SLY H = H
2 0

Equation (3.54) was applied to each node of the grid and the set of

the algebraic equations are written in matrix form to be solved by

Gaussian elimination technique (Subroutine GAUSS). As a result of

this the magnetic field strength at each node of the quarter section

load is obtained. These will be used to calculate the power

density within each mesh, the power generated at each node and the

total power induced in the load which is four times the total power

in the quarter of the load.

 57

To illustrate how the power density over a mesh element is calculafed

in subroutine (SETUP) consider the mesh shown in Fig. 3.6.

Jx =
(3.55)

<lJx Hll - HlO

Jx + -·- dy =
3¥
-=:.=...,--=-=-
dx (3.56)

similarly, JY =
(3.57)

and Jy +
(3. 58)

where H is the magnetic field strength at each specific node and

J is the current density. dx and dy are the horizontal and vertical

mesh spacings for the mesh shown.

Jx + [ Jx + ~ dy J
Average Jx = AvJx = ( 3. 59)
2

H2 - Hl + Hll - HlO
=
2dx (3. 60)

Jy +[Jy + ~ dx)
Average J·y = AvJy
2 (3. 61)

HlO - Hl + Hll - H2
=
2dy (3.62)

3
The volume power density p (W/m ) was calculated by the expression (3.63).
 58

Jx

Jy + ~d
8X X
dy Jy

19 Jx
I
Jx + - - dy
ay ~1
I I
X

Fig. 3.6 Calculation of the Average Power Density Over a Mesh

P = Bd~ dyp dx dy [2 Av Jy 2 Av Jy* + 2 Av Jx 2 Av Jx*] (3. 63)

1
=-
8 p 4dxdxdydy (Av Jy Av Jy* + Av Jx Av Jx* ) (3.64)

1
=z- p(Av Jy Av Jy* + Av Jx Av Jx*] ( 3. 65)

1 3
p = 2 p J J* W/m (3.66)
 5~

The program consists of four sections:

1. The main program:- sets up the geometry of the load, declares

the arrays and checks for mesh consistency.

2. The subroutine (SETUP):- carries on the calculations and sets

up the finite difference matrix.

3. The subroutine (GAUSS) :- a call is made to this subroutine

by (SETUP). The subroutine (GAUSS) solves the matrix by the

Gaussian elimination method. The program then returns to the

subroutine SETUP.

4. After the program has returned to SETUP, the subroutine

calculates and prints the power density within each mesh,

the power generated at each node and the total power induced

in the load. The subroutine also goes on to print the magnetic

field strength at each node.

The user has the choice of printing only the values of the outputs

stated above or to print all the results of calculations.

The flow chart of the program is shown below.

 Go

( START

Main program - sets up the geometry

of load, declares arrays and checks
.
for mesh consistency

,,
<

Subroutine SETUP: carries the calculations

f.--'
and sets up the finite difference matrix

l' 1- - - - - - - --- - - - - - - -
I_ Calculate and print the total power in
the slab, power.at each nOde, power
density within each mesh and the magnetic
field strength at each node

( STOP

Subroutine GAUSS
solves the matrix by
!._.__
the Gaussian elimination '-

method

The flow chart of "slab"

 61

The magnetic field and volume power density distribution

within
an aluminium workpiece of cross-section 500 mm by 150
mm subjected
to a surface magnetic field strength of lo 5 Am -1 at a
frequency
of 50 Hz are shown in Figs. 3.7 and 3.8.

The program was relatively simple to implement and gave satisfactory

results when applied to a simple two dimensional problem.

3.3 Conclusions and Recommendations

The design and performance of a coil obtained by adopting the

equivalent circuit method depend largely on the values of the empirical

factors employed in the calculation; the wrong choice of these factors

could affect the results quite considerably. The assumption of a

uniform workpiece material can also lead to incorrect results which

depend largely on the resistivity and permeability values.

An accurate knowledge of the power density and temperature

distribution in an inductively heated load is of great importance.

Analytical expressions are available for simple one dimensional

geometries such as a cylinder or a slab [3.3] but for the solution

of even the simplest two dimensional geometries numerical techniques

are required.
 62

'~'

0·8

0·6

0·4

Fig. 3. 7 Variation of magnetic field distribution (as a fraction of

the surface field) with depth y along the centre
line of the cross-sectional face.

POWER (W)

140

120

100

80

60

40

20

OL---~1----~2----3~~~4~~;---~6----~7--L-~8------+
y ( X10- mJ

Fig. 3.8 Power distribution down the centre line of the cross-sectional face
 63

The finite difference grid must be suited to the problem. The

program should, therefore, be amended each time the dimensions of

the load, it''s physical properties or the frequency change.

Induction billet heating is a long-standing process in the metal

forming industry. There are areas, however, in which induction heating

has not achieved the market share that perhaps one might expect

e.g. off-the-bar forging or the extrusion of aluminium. In both

applications the ability to create a non-uniform surface power

density along the billet or bar is required. Existing methods of

work coil design which are based on the assumption of an infinitely

long solenoid and/or the use of a great amount of empirical data

are not easily adapted to the design of non-uniform surface power

densities. Further investigations to develop a new technique for

work coil design are required which can be adapted to solve problems

requiring a non-uniform surface power density. This technique should

be based on the fact that the coil consists of a number of individual

turns. The distribution of the power density induced on the load can

be determined by applying the superposition principle.

 64

CHAPTER 4

THE SUPERPOSITION THEORY

The objective is to develop a flexible work coil design method

which will enable the design of a coil capable of producing a nonuni-

form surface power density along a workpiece. The achievement of

this aim necessitated the abandonment of the assumption whereby

a uniform magnetic field is produced by the coil and the adoption

of a new approach, based on the superposition principals.

In order to deal realistically with applications requiring nonuniform

surface power density to be induced along the length of the load,

an equation has been derived for the surface magnetic field distribution

produced by a single conductor. Then the superposition principle.

has been applied to determine the power density produced by a number

of conductors. The theoretical derivation will be shown in the

ensuing chapter.

4.1 The D=livation of the Superposition Theory

Fig. 4.1 shows a filament carrying current I at a distance h

from the surface of a semi-infinite metal slab. Given that the slab

is a good conductor, the magnetic field strength at any point P

along the surface of the slab can be calculated by assuming an

imaginary filament within the slab at a distance h from the surface

and carrying current -I. Both the conductor and the image will produce

a magnetic field H on the surface, according to Ampere's law, equal to

 G5

7 7 7 /stab 7 7 ~I 7 7 7 7 7 ~7 7
I 0rp
I
Qymage

(a l

X....
' ...... H Hr
''
' ......
2Hz --...:
_...,...-; p
/
/
/
/
/
H 'Hr
...... •
(bl

Fig.4.1 (a) Conductor near a semi- infinite slab

(b) Themagnetic field intensity on

point P.
 66

( 4. l)

where z is the distance of point P along the load measured from a

point directly beneath the conductor, see Fig. 4.l(a).

The axial components of the conductor's magnetic field strength H

z
and it's image will aid each other, while the radial components H
r
will cancel each other. The resultant field on point P 1dll be:

HE = 2H z (4.2)

but H = H cos a ( 4. 3)
z

I h
= 2
~ 2 >1,2 i (4.4)
211;{ + 'z'i + z

I h
Hp = 1T- ( 4. 5)
h2 + z 2

It can be shown [4.1] that the magnetic field intensity at the

surface of the slab is equal to the linear current density, A,

induced per unit length of the slab

(4.6)

( 4. 7)

If the filament is replaced by a circular conductor of radius a and

the current penetration depth into the slab and the conductor is small

compared with the height h and the thickness of the slab, the case
 67

becomes similar to the electrostatic case of charged cylinder

near a metal plate [4.2]. The solution of this problem (4.3, 4.41

results on re-writing equation (4.7) in the following form:

(4.8)
2 2 2
h - a + z

This equation can be written in general form as:

(4.9)
2
+ z

Where the effective height, h , for a circular conductor is:

e

2
h - a (4.10)
e

According to Davies and Simpson [4.1] it is valid to treat a

circular conductor around a cylinder as a line current at a distance

from a flat plane and the above derivation may be applied to a single

turn coil surrounding a cylinder, provided that the distance between the

conductor and the cylinder, h, is:

l. Small.in comparison with the cylinder radius and,

2. Large in comparison with the current penetration depth

in the cylinder.

Consider next, the case illustrated in Fig. 4.2 which comprises of

N identical conductors parallel to each other and parallel to a metal

slab at a distance h from it's surface. Each conductor carries a

current I amperes.
 Gfl

1 2 ~ N
,...
I
I
$I
<P <P .... 0 <P
I I

.
.... z1~
I
I
Ze.
•'
I
I
I
I
I

1- Za .,
I
I

ZN
I
I
I

~77777/metallic slab 77777777

Fig 4.2 Number of identical conductors at a specific air cpp

from a metallic load . ·

The current per unit length A induced by these conductors at a

point P on the slab, according to superposition principi:ds, is the

sum of the current per unit length induced by each individual conductor.

This can be written in mathematical form as:

(4.11)

Where APl' AP 2 , Ap)' .•• , ~N are the current per unit length induced

at point P from conductors 1, 2, 3, .•. , N respectively.

If the performance of each conductor complies \dth equation (4. 9)

then

(4.12)
 69

If the distances between the conductors are constant and equal

to d, i.e. a uniform coil, then

zl = lzll

' = lzl + dl
z2
(4.13)

z3 = lzl + 2dl

z = lz + (N-l)dl
cl 1

and ( 4. 14)

I h X=N-1
e 1
1T I (4.15)
X=O

In induction heating, it is more convenient to deal with surface

power density PD which is a function of the surface current density J.

The latter is given for a semi-infinite slab at point P by [4.1] as:

JP = /2 HP exp J. ( wt + 4
") (4.16)
0

fi 1T 1T
i.e. JP=~ HP {cos(wt + 4> + jsin(wt +
4)} (4.17)

( 4.18)

where o is the penetration depth and is given by:

(4.19)
 70

Substitute equation (4.6) into equation (4.18) and rearrange:

0
= -J (4.20)
ffp

The power density at point Pis given by [4.1]:

(PD) p = f poJ~ (4. 21)

Equation (4.21) gives the surface power density at any particular

point P on the load as a function of the surface current density

at that point.

4.2 Verification of the Superposition Theory

Before applying the superposition theory to any particular problem

it must be verified. The theory is to be assessed in two successive

stages. The first stage is to investigate the accuracy of the theoretical

derivation. If this is successful; the second stage is to examine

this theory as a work coil design technique.

In order to verify the theoretical derivation, the results from the

superposition theory were checked with the equations given by Callaghan

and Maslen [4.5] for the case' of large cylinder. The agreement was very

good and the difference did not exceed 2%. Callaghan and Maslen .•

ha_ve checked their·equations thoroughly with practical measurements.

 72

The next stage in testing the superposition theory, after having

verified the theoretical derivation, is to check if it can be

used as a work coil design technique. This is to be done by

predicting the coil number of turns required for a given specification.

A computer program "PROXIMITY", based on the superposition princip.._/.s,

was developed to calculate the number of turns, in a uniform coil,

required to produce a pre-determined power per unit width of the load.

As a result of adopting this program, the number of turns required

for a large cylindrical load and high frequency nearly equals that

obtained from the equivalent circuit method. The two results

are within 10% difference from each other.

4.3 The Use of the Superposition Theory

Having verified the superposition theory, a computer program "W-P-POWER"

was developed, based on equations (4.15}, (4.20} and (4.21}, to cal-

culate and plot the current per unit length and the surface power

density distributions induced on the load. The coil was assumed

to be of uniform pitch and the load was assumed to be of nonuniform

resistivity; so that the change in the physical properties can be

taken into account.

The surface power density induced at any point P on a nonmagnetic

or linear magnetic load was given by equation (4.21} which can be

 START )

I
Read: a, h, t, f,
p' u
r'
I
c'
p
w J
Set N = 2

Zero the index of

the dimension

Calculate Coil Pitch

and IJ.z = 9./200

z - 0.0
p = 0.0

Incremented the index

by 1

Calculate A ' (PD)

z z

p = !J.z* (PD)
z z

p = p + p
z

?
YES
z+ IJ.z<t

NO
?
NO
L----..j N N+l p ;: p

YES

/ Write N _/

STOP

The Flow Chart of the Program "PHOXIMITY"

 74

rewritten, by substituting equation (4.20) as:

(PD) p (4.22)

The power density induced on the load is a function of the resistivity

which is not necessarily uniform along the load if the temperature

is not uniform. To examine the applicability of the superposition

theory to tackle the problem of the nonuniform load; two types of

calculations were included in the program.

First, consider a scanning system of a billet moving inside an induction

coil. When the billet is inside the coil, it·s starting edge is at

maximum temperature and the other end remains at minimum temperature.

The temperatures of the rest of the load ranges between these two

limits. A gradual change of the resistivity along the load was

assumed by dividing the load into (X) equal divisions and taking

the change in resistivity ~p to be

pf .-p.~
~p ( 4. 23)
X - l

where pi the resistivity at minimum temperature

p ~ the resistivity at maximum temperature

f

and the resistivities of the individual segments will be:

The resistivity of first segment ~ p ~

l
The resistivity of second segment ~ p pi + llp
2
The resistivity of nth segment p. + (n-l)~p (4.24)
~
 75

The penetration depth of the nth segment is

(4. 25)

Since the current per unit length induced at any point in the load

can be calculated by the superposition theory, then the surface

power density of any segment can be found from equation (4.26)

below

(PD) = (4.26)
n

Second, consider a billet heated by the "single-shot" technique.

The billet, stationary inside the coil is subjected to a constant

magnetic field. At first the temperature is at a minimum, as are

the resistivity and the induced power density, the latter being

a function of temperature, the resistivity and the power density

will gradually increase until they reach a maximum. This case was

included in the program and the result is to be shown in a number

of curves representing different values of the resistivity.

The computer program "W-P-POWER" was used to study different cases

and the results are shown in graphs which relate to theoretical

situations only and therefore do not, as yet, represent a practical

application. They are presented, simplY to show the different

parameter's effects. The flow chart of the program "VI-P-POVIER"

is shown below.
 ( STAR'!'

I
Read LM, a., h , R..,

I Fi' f\i'
1 1
p2~' llri'Ici'Ni'
i=l, LM
1

Zero the index of

the dimension

Calculate Coil Pitch

and t.z = t/[l6(N-l) J

z- 0

I
Incremented the index by
l

Calculate A (PD)
z ' z

'I' Store z, A, PD in arrays r

z = z - t.z I

?
YES
z + R,i ~ o.

NO
?
NO
,.. i ~ LM

YES

draw the x-axis and y-axis for (A) against

(z) and (PD) against (z)
I
I
plot Ai and PDi against z

?
NO
i ~ LM

YES

( STOP

The Flow Chart of the Program "W-P-POWER"

 77

Figs. 4.3 and 4.4 show that increasing the number of turns will increase

the uniformity, the magnitude of the current per unit length

and the power density induced on the load. To be more precise,

the increase in the number of turns from 15 to 30 turns, doubles

the current per unit length induced in the load and increases the

power density by a factor of 4. The reason for the increase in the

power density by this factor being that it is a function of the current

squared. This also applies to the coil of 45 turns, where the

power density is increased by a factor of 9 when compared with the coil

of 15 turns. The above result agrees with the fact that the current

induced in the load is a function of the coil current multiplied by

the number of turns. If the number of turns is increased, then the

coil pitch decreases thus causing it to resemble a current sheet

which will result in a very uniform power density induced in the load.

Figs. 4.5 and 4. 6 were plotted, so as to show the. effect of changing

the air gap. Decreasing the air gap increases the current per unit

length and the power density induced on the load. However, it

decreases the uniformity as can be seen from the upper curves on these
graphs.

Although the distributions of the current per unit length and the power

density induced in the load due to single conductors are functions

of the conductor diameter; this diameter, in the case of multiturn

coil is not an important factor in the control of the amplitude of

the current per unit length and the power density induced in the load;

this can be read off Figs. 4.7 and 4.8. The curves represent 3 very
 78

similar coils differing only in the size of the conductor. The air gap

was taken to be constant in all three coils (g =h-a= 30mm).

Although the size of the conductor is not important when considering

the induced power, the thinner the conductor the more the non-

uniformity of the induced power, and the less the efficiency. This

is a result of the increase in the losses.

The effect of the resistivity in the power density is shown in

Figs. 4.10 and 4.11, while the current per unit length induced from

the coil is constant: this is illustrated on Fig. 4.9. Fig. 4.10

shows the change of the local resistivity as against time, while

Fig. 4.11 shows the axial change of resistivity. The power density

induced in the load is a function of the square root of it·s resistivity;

see Figs. 4.10 and 4.11.

 7'J

45

'10

35

a
';g 30
..;

.c'
<-1
25
""<:
Q)
.-<
N=45

<-1
·.-!
<:
"
'"'
Q)
20
0.
<-1 30
<:
Q)

'"' 15
'"'
"
u
'0
Q)
u
"
'0
<: 15
H
10

Distance along the load, z, (m)

Fig. 4.3 Current per unit length along the load due to lOOOA
flowing through coils of different number of turns, tl,
at h = 40mm and the conductor radius is a = lomm.
 80

4.5

4.0

3.5

3.0
N
8
......
:3:
..\<:

2.5
0
"''
;>,
....+J
Ul
.:Q) 2.0
'Cl

":.
Q)

0
N=45

0.
'Cl
1.5
Q)

".:"
'Cl
H

1.0
'
. 30

5
15

Distance along the load, z, (m)

Fig. 4.4 Power density distribution along a nonmagnetic Load of

-8
2.8 X 10 nm resistivity due to the coils mentioned
in Fig. 4.3, the frequency is 50Hz.
 81

40_

35

30
~
,;
8
,; -
.r:
+J
- 25
h=35
0>
<:
....'"
55
....+J
<: 75
::1
20
...
'0."
+J
<:
......'"
::1 15
.,u

.,u'<:"
::1

H
10

0
0 .1 .2 .3 :'1 .5 .6 .7 .8 .9 1.0
Distance along the load, z, (m)

Fig. 4.5 current per unit length along the load due to lOOOA
flowing through 40 turns coils at different heights, h,
and the conductor radius is a = lOmm.
 82

3.5

2.5
~

N
E
'.>:
;3:

0' 2.0
"'>.'
....+J
00
<:
.gj
...Ql 1.5
~
0
a. h=35

"'u Ql

"'"<:
H
1.0
55
75

0.5

Distance along the load, z, (m)

Fig. 4.6 Power density distribution along a nonmagnetic load of

8
2.8 x 10- Qm resistivity due to the coils mentioned in
Fig. 4.5, the frequency is 50Hz.
 83

40_

35

a
-.... h~37.5

~ 30 a~7.5

.,;' h~4o
a~lo
'
...,
.<::

"'c:
....QJ
25
...,
....
c:
"
H
QJ
0. 20
...,
c:
QJ
H
H

"u
'tl
QJ
15
u
"c:
'tl
H

10

Distance along the load, z, (m)

Fig. 4.7 Current per unit length along the load due to lOOOA
flowing through 40 turns coils of different conductor
radii, a, at air gap of 30mm.
 84

3,5_

3.0

~
h=37.5
"'~ 2.5 a=7.5
:;::
.!<

h=40
Q
a=lO
p.
.
;:.,
.._,
2.0
....
Ul

"QJ
'0

":<
QJ

0
0. 1.5
'0
QJ
u
"'
'0
H
"
1.0

0.5

Distance along the load, z, (m)

Fig. 4.8 Power density distribution along a nonmagnetic load

8
of 2.8 x lo- nm resistivity due to the coils mentioned
in Fig. 4.7, the frequency is 50Hz.
 30
s
"..>:,;
,; - 25
...
.c
lJ'
<::
....'"
.......
<:: 20
..."'
'0."
...<::
......'" 15
"'
<.J

"''"
<.J

"'"'<::
H 10

Distance along the load, z, (m)

Fig. 4.9 Current per unit length along the load due to lOOOA
flowing through 40 turns coil of lOmm conductor
radius at h = 40mm.
 -H
p lLxlO

-0
lOxlO
6

-8
8xl

5 -8
"''-8 6xl0
3:
-"
0
0.

:>. 4 -8
....+Jrn 4xl0

"'"
'0
H

""
0
0. 3
'0 -8
Q) 2xl0
u
"
'0
H
"
2

0.,1~~~~~~~~~~~~~~~~~~~~~~~~~~~
0 .1 .2 .3 :'1 .5 .6 .7 .8 .9 1.0
Distance along the load, z, (m)

Fig. 4.10 Power density distribution along a nonmagnetic load of

-8
resistivity varies radially from 2 x 10 nm
-8
to 12 x .10 Qm, due to the coil mentioned in
Fig. 4.9, the frequency is 50 Hz.
 ll'i

5.0.

5.5.

5.0

'1.5

N
s 4.0
'-
:3:
"'
Cl 3.5
';.,"
....'-'Ul
3,0
'd"
QJ

...
QJ 10. 4xl0
:> 2.5
0
0.
'd
QJ
u -8 7.3xl0
'd
" 2.0 = 7. 3xl0
H "
4.8xl0
1.5

- - 2.7xl0 -8
1.0

05

Distance along the load, z, (m)

Fig. 4.11 Power density distribution along a nonmagneti€ load of

-8
resistivity varies axially from 2.7 x 10 Qm to
-8
10.4 x 10 Om due to the coil mentioned

in Fig. 4.9, the frequency is 50 llz.

 HH

4.4 Discussion and Conclusions

An equation has been derived for the distribution of the induced current

along the load length due to a current flowing in a nearby single

conductor. The adoption of the superposition princip~ls on this equation

resulted in the development of a new theory for the design of

induction heating work coils.

This theory does not assume a uniform magnetic field produced by

the coil and, also, it does take the conductor diameter and the spacing

between the turns into account.

The superposition theory was correlated with known equations[4.~

and the agreement was very good. This proves that the former is based

on reliable theoretical work and reasonable assumptions.

This theory was used to investigate the effect of the load non-

uniform physical properties and the influence of each parameter in

the power density induced in the load. The computer program neither

needed large memory nor required long running time.

The application of this theory showed that the uniformity of the power

density induced on the load is a function of the coil number of turns,

coil pitch and the air gap. The magnitude of the power density depends

largely on the number of turns, the air gap and, of course, the physical

properties of the load. The nonuniformity of the resistivity along

the load, due to the difference in the temperature, will result in

a nonuniform power density induced in the workpiece.

The application of the superposition theory to a number of cases and

it's ability to deal with different situations shows i t s flexibility

and encourages one to examine it practically.

 CHAPTER 5

THE SUPERPOSITION THEORY - PRACTICAL EXAMINATION

The next logical step in the quest for a flexible coil design method,

after having derived and verified the superposition theory

theoretically, is the experimental examination of this technique;

so that the practical problems can be established.

The objective of this chapter is to inspect the applicability of

the superposition theory at different conditions similar to those which

occur in induction heating applications. To achieve this it was

necessary to divide the work into two parts. The first is to test the

method when there is no ferromagnetic materials in the system which

would complicate the solution due to permeability effects. The

investigation concerning the effects of ferromagnetic materials was

then carried out in the second part of the investigation.

5.1 Preliminary Investigations

The derivation shown in Chapter 4 was for filamentary conductors

where the air gap and the spacings between the turns were large in

comparison with the conductor diameter. In induction heating these

distances are relatively small which can cause a non-uniform

distribution of current across the conductor.

 9]

The aim of this section is to establish the difference between the

theoretical case of filamentary conductors and the actual situation

where there is inter-conductor effect and also to examine the

superposition technique practically.

A series of experiments have been executed on an aluminium slab

subjected to magnetic fields induced from mains frequency current

carrying conductors. The first part of the experiments was to

study the case of single conductors, of different shapes and sizes,

above the aluminium slab; and the second was the examination of the

superposition method by employing a number of conductors instead of

a single one.

5.1.1 Single Conductor

The experimental rig of Fig. 5.1 was employed to study the influence

of a single conductor above an aluminium slab. The circuit

diagram is shown in Fig. 5.2.

Before starting the experiments, different precautions should be

considered. Since the experiments involve high current, care should

be taken to reduce the effect of the high magnetic field on the

different parts of the system. This has been done by shielding the

meters with mild steel sheets of 3.2 mm thickness. The current

carrying cables were positioned at the greatest possible distance

from the load.

Fig. 5.1 The Experimental Rig
 ,J
c:J..
current (10,000:10)
transformer
~

41 SV INDUCTIVELY
~
60A ~
~ HEATED
SOHz r-
~ IVORKPIECE
1
'---

variable high current

t
water cooled
voltage transformer transformer copper work coil.
max current
output SOOOA.

Fig. 5.2 The Circuit Diagram

.
 04

The conductor was placed over the centre of the load. If this was

not the case the induced current would concentrate on a point on

the load and not the point directly below the conductor. This

characteristic is due to the end effect. That is the concentration

of the magnetic field lines near the edge of the load.

The function of the wooden frame, Fig. 5.1, was to hold stable the

various conductors, parallel above the workpiece. It was able to hold

conductors of various sizes at different set heights.

Four conductors were used to carry the current above the slab. The

first three were circular water cooled copper conductors of lmm

wall thickness, BOOmm length and external diamaters of 6, 15 and 28mm.

These conductors will be referred to as "circular" conductors.

The fourth conductor was of 450mm length, made of solid rectangular

copper brazed to a rectangular water cooled copper tube, as illustrated

in Figs. 5.3 and 5.4. It will be referred to as "rectangular" conductor.

Those conductors were chosen because the induction heating work coil

is usually made of water cooled copper conductors, either of circular

or rectangular cross section. The rectangular conductor is used in

a mains frequency induction billet heating coil. The solid part is

greater than the penetration depth of copper at 50 Hz, which is

about lOmm.
 lmrn

-
•
~
' Bmrn
1 Omrn

_,
V (1)
24 mm.

water pa th / (8)
T
4mrn

[/ -~ (2)
sol id copper / 4mrn

(7)
- 1- (3)

4mrn
~Smm I
T 3rnm 1l
j (6) (5) (4) I
1---- lOmm _ _ . {

Fig_ 5_3 Rectangular conductor used in mains frequency

induction heating_

The number between brackets ( ) is the probe number_

 <)(,

Fig. 5.4 Rectangular Conductors

An aluminium slab of (500 x 300 x SO.S)mm dimensions was employed

as a workpiece. When operating at mains frequency the thickness of

this slab was greater than 4 times the skin depth. This load

satisfied two factors: firstly it was of nonmagnetic mabrial so that

there was no worry about the change in permeability at this early stage

of the practical work; and secondly the thickness of this slab was

more than 4 times the skin depth, hence the current beyond this depth

is very small. The current penetration into the load is

approximately
X
0
J = J e
X S

where J is the surface current density.

s
and J is the current density at depth x.
X

If x = 4o then J
X
0.018 J . i.e. J
S X
is small.

A high current should be fed through the conductor, if a measurable

eddy current is to be obtained. The high current necessary for the

experiments was supplied by a transformer capable of supplying SOOOA.

It's primary was connected to a 415 V, single phase regulating

transformer. Water cooled heavy duty cables, capable of carrying the

required current, connected the conductor across the secondary of the

transformer. A current transformer of lOOOO:lOA and digital multi-meter,

with current shunt were adopted to measure the conductor current.

 ~lfi

The surface current density induced on the load was measured by the

current density probe shown in Fig. 5.5. The theory, construction

and characteristics of filament type current density probes are

described by Burke and Alden [5.1], see also Appendix 5. A perspex

frame containing 2 brass pins was the type of probe suitable for

use on the surface of the aluminium slab. The distance between the

pins was 30.5mm and each one was connected to an insulated constantan

thin wire (O.lmm diameter), which was kept close to the surface. The

two wires were twisted together in the centre and the other ends were

connected across a voltmeter. By attaching this probe to the surface

of the load, the current density at that point could be measured.

Care was taken while constructing this probe as the constantan wire

must be extremely close to the surface of the load, but no contact

should occur in any part other than the two pins. A coaxial wire

connected the probe to the voltmeter so as to reduce pick-up.

The current density on the surface of the conductor was also measured to

assess the effect of the conductor current flow on the distribution

of the current density induced on the load. The probe was of 30mm

length, constructed from constantan wires. The two constantan wires

were laid along the conductor length and their ends were cleaned and

connected to the conductor. Seven probes positioned at 30° intervals

were constructed around the 28mm diameter circular conductor.

 I)')

Fig. 5.5 The probe for measuring the induced

current density on the slab.

 lOO

On the rectangular conductor; nine probes were used and they were

positioned as depicted in Fig. 5.3.

In order to obtain adequate resolution the probe must satisfy two

criteria: firstly, (J) and (p) must not change over the probe length;

secondly, the diameter of the filament must be small compared to

the skin depth. The first condition was satisfied as the load was

uniform and the current's direction was known and the second too was

satisfied by using probes of O.lmm diameter wire when the skin depth

was 12mm. The surface current density J is a function of the voltage

s
across the probe V, the resistivity p and the probe length ~ as:

V
(5.1)
p~

The experiments were carried out on a conductor which was placed at

a fixed distance h from the aluminium slab. A mains frequency current

of lOOOA was used.

The current density distribution on the 28mm circular conductor

and the rectangular conductor was measured by the attached probes

when there was aluminium slab beneath it and when there was no load.

The results are shown in Figs. 5.6 and 5.7. It can be seen that the

distribution was uniform on the circular conductor, this was because

the thickness of the conductor was much smaller than the penetration

depth. The current distribution on the rectangular conductor was non-

 Conductor

0.0

-::+----- - - .
0.0

1.o1--r-

Fig. 5.6 Normalised surface current density dis-

Fig. 5.7 Normalised surface current density dis-
tribution on circular conductor for no
tribution on the solid part of the rectanc;uLr
load and with load at h = 37mm._ The graph
is in polar coordinate. a = 14mm,
conductor. The distances from 0.0 show
the current density at the probe shown by
I = lOOOA, the arrow
J( Jli no load o with aluminium slab
at g = 23mm
uniform and the existence of the aluminium slab increased the

concentration of the current on the lower surface of the conductor.

This was due to the proximity effect on the conductor itself. That

is the variation of the current distribution of the conductor

because of the existence of the load. Ths calculation was based

on constant resistivity as the conductors were water cooled and

0
the temperature was constant at 20 C. The surface current density

was calculated from equation (5.1).

To deduce the distribution of the surface power density induced

on the load, a current density probe, Fig. 5.5, was attached to

the surface of the slab at different points along the z direction.

The z direction was perpendicular to the conductor, and the

probe was parallel to the conductor, that being the direction

of the induced current. The object of this exercise was to measure

the surface current density induced on the slab; hence the surface

power density distribution was determined by using equation (4.21).

The practical measurements of the current density induced on the ·

slab showed that the "theoretical" equation (4.9) of a filamentary

conductor cannot be applied practically.

h
I e (4.9)
A
1T 2
+ z

Many experiments have been carried out in order to find a more

suitable equation. It was found, empirically, that equation (4.9)

 ---- -----------

lu3

should be amended to the "practical" equation (5.2) below,

h
e (5.2)
h2 + a 2
z
e cosa

where a and B are functions of h , see Figs. 5.8 and 5.9, and
e
a is the angle between the line joining the conductor and point P

and the perpendicular line from the conductor to the slab,

see Fig. 4.l(a). It will be shown later on in the study that

equation (5.2) can be applied also to an aluminium cylinder of

radius equal to the thickness of the slab.

Graphs of the measured power density and those calculated from

the "theoretical" equation (4.9) and the "practical" equation (5. 2)

are shown in Figs. 5.10 -- 5.21 for the circular conductors.

Figs. 5.22-5.25, for the rectangular conductors, do not contain

curves for the theoretical equation as this equation was

derived for circular conductors only. The equivalent height of

the rectangular conductor was found to be equal to the air gap g.

The theoretical power density was calculated by employing equations

(4.9), (4.20) and (4.21). The calculation of the practical power

density was based on equations(5.2), (4.20) and (4.21). The measured

power density was determined from the measured surface current

density and equation (4.21). The calculation of the practical

11
power density was carried out on a program ALI" developed for a
 0·2

25 so 75 100 125 150

The effective height, he 1
(mm)
25 50 75 100 '125
The effective height, he 1
(mm)
f1g. 5.8The variation of the constant (a) with the
effective height lhel
Fig,S. 9 The variation of the constant !B) with
0
X

•
0
Rec tan gular conductor
Circular
""'""'"' ... ··~··}
Circular conductor of 15mm diameter
Circular conductor of 28 mm diameter
Above aluminium
slab 0
the effective hei·ght !he)

Rectangular conductor
Circular conductor of 6mm diameter
}

A Circular conductor of 6 mm diameter around the cylinder Above aluminium

• Circular conductor of 15mm diameter slab
0 Circular conductor of 28 mm diameter
A Circular conductor of 6mm diameter around the cylinder
 l {j.)

50

Equation (4.9)

Equation (5. 2) with

40
(l = 0.8 andS = 1.15

0 Measurements

a
.....
:;::
" ' '\
Cl
0.
- 30 .\
\
..,-
....."' \
{))

"
Q)
\
'0
\
:."
Q)

0
20
'\
"'
'0
\
Q)

" '\
"
'0
"
H
10

40 80 120
---
0 ---

160 260
Distance along the load, z, (mm)

Fig. 5.10 Surface power density distribution along the aluminium

slab due to 1000A flowing through circular conductor
of 6mm diameter at h = 80mm
 1 or)

90

80 Equation (4. 9)

Equation (5. 2) with

a = 0. 7 and S = 1. 1
0
70 Measurements

\
N \
fl
'
:;:
60 \
\
Q
0.
- \
>.
+J
- so
\
..... \
""'
Q)
'0
\
'"'
Q)
\
:. 40
0 \
"'
'0
Q)
\
u
'0 " \
"
H 30 \

20

10

40 120
--- -----160 200

Distance along the load, z, (mm)

Fig. 5.11 Surface power density distribution along the aluminium

slab due to lOOOA flowing through circular conductor
of 6mm diameter at h = 58mm
 -- -----------------
lo7

180

160
Equation (4.9)

Equation (5.2) with

a= 0.5 and B = 1.1
140 0 Measurements

\
120 \
\
\
\
lOO I
\
\
\
so \
\
\
60 \
I
\
\
40 \
\
\
\
20 \
''
' ...... ...,

40 80 120 160 200

Distance along the load, z, (mm)

Fig. 5.12 Surface power density distribution along the alumini tun
slab due to lOOOA flowing through circular conductor
of 6mm diameter a.t h c: 42mm
 500
\
\
Equation (4.9)
\
I Equation (5.2) with
I a = 0. 2 and B = o. 85
1
,3:E 0
N
Measurements
4 1
1
Q
0..
I
I
>,
.j..)
.....
Ul
I
c:
<))
'tJ

'~" 3
0
0.
'tJ
<))
I
u I
H
"c:
'tJ
I
I
I
\
I
I
I
I
I
I
lO I
\
\
50 \
\
''
40
.....
-- 80 120
Distance ulong the load, z, (mm)
160 200

Fig. 5.13 Surface power density distribution along the aluminium

slab due t.o 1000/\ f"lowinq through circulur conductor
of 6mrn dLuneter at h = 22mm
 j (I' 1

50

45

Equation (4.9)
40
Equation (5.2) with
a = 0. 9 and S = 1. 2
0
Me as uremen ts
35

30
" '\
\
25
\
\
\
\
20 \
\
\
15 \
\
\
\
10 '\
"\
~

0
--- ---
40 80 120 160 200
Distance along the load, z, (mm)

Fig. 5.14 Surface power density distribution along the aluminium

slab due to lOOOA flowing through circular conductor
of lSrrm1 diarnt.:ter at h = f37rnrn
 l1LJ

80

N
8
':;; Equation (4. 9)

70
0 Equation (5. 2) with
"'
>,
a = 0.8 and (3 = 1.2

....'"' 0
Measurements
"'
~
<lJ
'0 60
...
<lJ
~
0

"'
'0
<lJ
u
"
'0
~
H
so
'\
\
\
40 \
\
\
\
30 \
\
\
20
\
\
\
\
\.. ·o
10 '\

"
40 80 120
-- 160 200
Distance alonq the load, z, (mm)

Fig. 5.15 Surface power· density distribution along the q/.urniniura

slab due to 10001\ fJowinq t llrouqh ci.rcular cond1wtor

of l5mrn at ll 0"" 70r.;rn
 l ]_ 1

N 140
s Equation (4.9)
'3:
Equation ( 5. 2) with
0 a = 0.65 and B = l.l8
"'><' 0
Measurements
........
Ul
120

"
'0
(])

..,
(])

"
0

'0 "'
(])
lOO

u
'0 "'c \
H \
\
8
\
\
\
\
6 \
\
\
\
40

20

40 80 120 160 lGO

Distance along the load, z, (mm)

Fig. 5.16 Surface power density distribution along the aluminium

slab due to lOOOA flowing through circular conductor

of l5mm diameter at h = 52mm
 112

300

Equation (4. 9)

s
-..._ Equation (5.2) with
:?:
Cl = 0.28 and s
= 0.94

Q
250 0
Measurements
"'
>,
....+-'<Jl
"
OJ
'd
H
OJ

"'
0 200
"'
'd
OJ
0
u
'd
" I
H
"
I
150 I
I
I
I
I
lOO
\
\
\
\
\
\
5 \
\
\

"
40 80 120 lGO 200
Distance along the load, z, (mm)

Fig. 5.17 Surface power d<•nsi ty distribution along the aluminium

slab due to 1000/\ flowing through circular conductor
of lSmm diameter at h =: 30rnm
 113

Equation (~. 'l)

N
El Equation (5.2) with
'-
c a = 1.1 and ~ = 1.2
Q
.
0. o Measurements

;>,
.
....'Ul"'
~
(J)
'd
H
(J)
~
0
0.
'd
(J)
()
~
'd
~
H
1

---
40 80 120 160 200
Distance along the load, z, (mm)

Fig. 5.18 Surface power density distribution along the ~/uminium

slab due to lOOOA flowing through circular conductor
of 28mm diameter at h = 112mm
 lH

70

Equation (4.9)
60
Equation. (5.2) with
a = 0. 85 and S = l. 2

0 Measurements

so

40 " '\

\
\
\
30 \

\
\
\
20 \

\
\
\
10 ~
~

40
-------
80 120 160 200
Distance along the load, z, (mm)

Fig. 5.19 Surface power density distribution along the aluminium

slab due to lOOOA flowing through circular conductor

of 28mm diameter at h = 75mm
 I 1',

Equation (4.9)

120 Equation (5.2) with

a = o. 55 and B = 1.13
0 Measurements

lOO

:>,-
'"'
.....
Ul \
"Q)
\
"' so
>< \
~ \
0
0.
\
"'
Q)
{)
~ 60 \
" \
\
\
40
\
\
\
\
20

""
" """' -....

0 so 120 160 200

Distance along the load, z, (mm)

Fig. 5.20 Surface power density distribution along the aluminium

slab due to lOOOA flowing t.hrough circular conductor
of 28n~ diameter at h = 54mm
220

200

\ Equation (4.9)

180
\ Equation (5.2) with

\ ll = 0.35 and = l.O s

I 0 Measurements

160 I
I
I
140--
\
\
120
\
I
I
lOO
I
I
80
\
\
\
60
\
\
\
40 \
\
\
\
20 \.
'\.
" .......
40 80 120 160 200
Distance along the load, z' (mm)

Fig. 5.21 Surface power den~ity distribution along the aluminium

slab due to 1000/\ flowing through circular conductor
of ).1;1':1~;1 tli.amr•tc;r _,, h 'l7r;·:m
 l ' '
''

45

Equdtion (5.2) with

a = 0.89 and S = 1.2

40 0 Measurements

35
C•l
G
':;:
Cl
p. 30

;>,
,.,"-'
Ul

'0 ""' 25
"':<
Q)

0
0.
'0
20
u
::l
"'
'0
H
"
15

lO

40 80 120 160 200

Distance along the load, z, (mm)

Fig. 5.22 Surface power density distribution along the aluminium

slab due to lOOOA flowing through rectungular conductor
at g = 8'Jmm
 J _\ ~--)

140
Equation• (5.2) with
= 0.5 and s
= 1.2
e "
'- 120 0 Measurements
~

Cl
"'
><
+J lOO
·.-<
Ul

"OJ
'0

'OJ;."' so
0
0.
'0
OJ
u
"'
'0
60
"
H

40

20

0
0

40 80 120 160 200

Distance along the load, z, (mm)

Fig. 5.23 Surface power density distribution along the aluminium

slab due to lOOOA flowing through rectangular conductor
at g = 53mm
 -------·---------------------------------------------------------

1 'l\
' '

180
Equation (5.2) with
a= 0.35 and S = 1.07
0 Measurements
160

140

120

lOO

so

60

40

20

40 80 120 160 200

Distance along the load, z, (mm)

Fig. 5.24 Surface power density distribution along the aluminium

slab due to lOOOA flowing through rectangular conductor
at g = 40mm
 -----------------------

-lOO

360

Equation (5.2) with

320 a = 0. 2 and a
= 0. 9
0 Measurements

280

"' ~
240
....""UJ
-1-l
0

"
Q)
'r1
k 200
~
0

""
'r1
Q)
u
"'
'Cl 160
H"

120

so

.40

40 80 120 160 200

Distance along the load, z, (nun)

Fig. 5.25 Surface power density distribution along the aluminium

slab due to lOOOA flowing through rectangular conductor
at g :-: : 23rmn
 J:: I

hand calculator. In all cases the resistivity of the aluminium

-8
load was taken as 2.824 x 10 ~m which is corresponding to a

temperature of 20°C [s.2]. This was the temperature of the aluminium

slab during the experiments as the magnetic field was relatively

low and it did not heat the slab.

The power density distribution due to a single conductor, Figs. 5.10-

5.2l,showed that the "theoretical" equation (4.9) of a filamentary

conductor does not represent the practical situation accurately.

The difference between the measurements and the values of the power

density predicted by the theoretical equation (4.9) was up to

32%. Many experiments have been done to amend the theoretical

equation (4.9) to the "practical" equation (5.2). Although the

three correction factors were found experimentally, they can be

calculated theoretically. The factor coso is a function of the

distance z:

h
cos a ( 5. 3)
.{2 + z
2

The constants a and S are functions of the effective height h .

e
a is a linear function of h and it was found to be a straight line
e
represented by:

( 5. 4)
 i :::-:

The application of the computer routine (E02ACF) was found to

produce the equation of the other constant:

2 -3 2
8 = -(2.642 X 10-l) + (7.736 X l0- )h - (1.539 X 10 )h
e e

(5.5)

where h is in (mm). a and 8 are shown in Figs. 5.8 and 5.9

e
respectively. It should be mentioned here that in the case of an

aluminium cylinder of radius equal to the thickness of the aluminium

slab, the same constants can be applied. This will be shown later on

in the study. The practical equation (5.2) predicts the current

per unit length and can be applied within a range of lOO% to 10%

of the maximum induced power density, the error within this band

being less than 10%. The practical equation applies to both

conductors. The effective height of the circular conductor is:

h = ~2 a
2
e

It was found, experimentally, that the rectangular conductor can

be replaced by a filament on its lower surface i.e. h = g and a = 0.0.

The effect of the aluminium slab on the current distribution of

the circular conductor, within the range of the experiments, is

very small and can therefore be ignored. The current distribution

 U3

on the rectangular conductor has altered, due to the existence of the

aluminium slab. This is the reason for the increase in the

current density on the lower surface as shown in Fig. 5.7.

The surface current distribution of the conductor, which differs for

both conductors, see Figs. 5.6 and 5.7, does not affect the applicability

of equation (5.2), i.e. the induced power density distribution is

a function of the total conductor current. This case greatly

simplifies the application of this equation.

5.1.2 Number of Conductors

Before recommending the superposition theory as a coil design technique,

it must be verified experimentally. It was decided to examine this

theory in two stages. The first involved the conductor current

to be approximately uniform and the proximity effect between the

conductors to be at a minimum. The second stage is to examine the

superposition technique in a more realistic context, that is when

the current flows nonuniformally through the conductor, and there is

a proximity effect between the conductors.

The experimental rig and the circuit diagram were identical to those

of Figs. 5.1 and 5.2 respectively, but instead of using a single

conductor 10 conductors were introduced in the study.

 124

The practical applicability of the superposition theory with uniform

conductor current was tested by taking a single circular conductor

of 6~ diameter and situating it above the aluminium slab at

various air gaps (19-77)mm. A current of lOOOA flowed through this

conductor, and the induced voltage distribution on the load was

then measured. These voltages which are functions of the surface

current density, see equation (5.1), were fed into a computer

program "W-SC-FIT" to calculate the most accurate curve, representing

the induced voltage V(z) as a function of the distance (z) along

the load in an equation which takes the following form:

V (z) ( 5. 6)

where a , a , a , a and a are constants.

0 1 2 3 4

The above equation was then used to calculate the voltage from each

individual conductor at each point on the surface of the load, and

then they were summated to derive the predicted voltages (V) at

different points along the load. If the load was a slab then the

power density distribution on the load would be calculated

from equations (5.1) and (4.21). The equation suitable for the

cylindrical load will be dealt with in another chapter. The induced

power densities along the load have now been calculated and the results

can be presented in graphical form. The flow chart of this program

is shown below.
 / R e a d : Np , Z.,
~
vi' i=l, NP, Il,
r , N, d, £, p' F, 11 r' Rol, NSC
2 /
Call the subroutine E~2ACF to calculate A , ... ,A
1 5

I2
Calculate RA = - ' SLO, CON
Il

Set Z = zero

r
NO
~
"Z NP
YES

I
2 2 3 4
V= [A +A Z+A Z +A z +A Z +A Z ] V= ~SLO*Z) + coN] ~
1 2 3 3 4 5
*RA
*RA
I I
Store Z and V in arrays

7
NO
z >- £ z z + 0. 25

YES

7
1 2
NSC = 7
I I
Calculate p01-1er density
for slab (PD)
rCalculate power density
for cylinder (PD)
I
1 I
[ Store Z, PD in arrays

/ Plot V against z
Plot PD against Z /
STOP

11
The flow chart of the program ~1'-SC-FIT"
Then. five similar conductors were connected in parallel above the slab,

at identical air gaps to those which were applied for the single

conductor. The coil pitch (d) between these conductors was 60mm,

so that the proximity effect between them would be small. A total current

of 3000A flowed through them, and the induced voltage distribution

on the load was then measured. The current per conductor was 600A

which was less than that for the single conductor. This is because

it was not necessary to draw maximum current of 50001\ from the trans-

former as 3000A would produce measurable eddy current on the load.

Although many experiments have been done; only one graph is given,

Fig. 5.26, to prove the applicability of the theory by showing the

measured induced surface power density and that calculated from the

superposition theory. The differences between the two values remain

within an acceptable experimental error of 5%. The conductors were

of a relatively small diameter and the proximity effect between th~M

was small. In this case the current was distributed uniforrnlJ through

the conductors. However, further on in this study it will be seen

that this is not important criterion and that the superposition

method can be applied even when the conductors are in close proximity

to each other, and when the current flows nonuniform/J~·~. This Fig.,

also shows the nonuniformi:ty of the power density along the load.

This correlates with the theoretical results of Chapter 4.

To examine the superposition method under similar conditions to those

in induction heating applications; that is a nonuniform flow of

current through the conductors, groups of 5 and 10 rectangular

 220

200

180

160
~

"'
!:140
~
D120
-
a..
>-
~100
11)
c:
QJ
"0
... eo
QJ

~
0..60
"0
QJ
u
~40
-
c:

20

20 4() 80 100 14() 160 180 200

Distance along the load, z, (mm)

Fig.5.26 Power density distribution along the aluminium slab due

to 5 circular conductors at air gap of 39 mm, coil pitch
of 60 mm. and a current of 600A/conductor.

Prediction by superposition o Practical readings

 1.! J

40

36

;:;-
E 32
--
3:
...0 ~

28
..-
X
~

D 24
0..

.....>. 20
Vl
c
QJ
D
'-
QJ
16
)::
0
0..
"Cl 12
QJ
u 0
::J
"Cl
c
._, 8

0
0 80
100 120 140 160 180 200
Distance along the load ,z, (mm)

Fig-5.27 Power density distribution along the aluminium slab

due to 10 rectangular conductors at air gap of 36mm.
coil pitch of 11mm and a current of 300 A/conductor
Prediction by superposition o Practical readings.
conductors, Figs. 5.3 and 5.4, were connected in parallel ~bove the

aluminium slab. The air gap between the conductors and the load was

varied between 89-23mm. The coil pitch was between 62-llnun, that is

down to a lmm gap between the conductors. In the single conductor

experiments, a current of lOOOA flowed through the conductor, and

a total current of 3000A was used in those involving more than one

conductor.

In any of these experiments the difference between the measured

induced power density and that calculated by the superposition theory

did not exceed 8%. Only one graph is shown, Fig. 5.27,

because all the results prove one thing, that is the validity of

the superposition theory.

The applicability of the superposition theory was not affected by

the surface current distribution of the conductors which was measured

for an air gap of 36mm and coil pitch of llmm. The results, Figs. 5.28

and 5.29, are represented in graphs which match the pattern recorded

in Fig. 5.7," The surface current distribution is different on the

two conductors as a result of their position amongst other

conductors ..

5.1.3 Discussion of the Preliminary Investigations

The surface power density distribution on an aluminium slab induced

from single conductor was studied. It was found that the equation of

a filamcntary conductor (4.9) does not represent the actual situation

 0.0

l.O

Fig. 5.28 Normalised surface current density on the solid part

of a centre· rectangular conductor amongst 10 conductors
with llmm coil pitch and at 36mm air gap from aluminium slab

o.oj

The other
conductors are
in this side

l.O

Fig. 5.29 Normalised surface current density on the end

conductors of a group of 10 rectangular conductors with
llmm coi.l pitch and at JGmrn oJ.ir gap from aluminium slab
correctly. A better equation with three new factors was found

experimentally. Mathematical expressions were given to calculate

these factors.

The investigations on the power density induced on the aluminium

slab due to more than one conductor have shown that the superposition

method can be applied, regardless of the shape of the conductors or

the distribution of the current flowing through them. The proximity

effect on the conductors does not prevent the superposition method

being applied and nor does the positioning of the conductor alter

it 1 s effect on the load.

The experimental error was within 8% only and it is due to many

factors; among them are the accuracy of the instruments, and the

variation in the resistivity of the aluminium slab, due to the change

in the ambient temperature. This 8% error in the calculation of the

power density implies that the calculation of the voltage and the current

induced on the load will be lower than this margin of error as the

current is a function of the square root of the power.

The outcome of these experiments proved the theoretical work to be

correct, when the coil pitch is large the induced power on the load

resembles ripples and is not uniform. The magnitude of these ripples

will increase when one decreases the air gap, and when the coil pitch is

large they are noticeable.

This preliminary verification on a nonmagnetic load, which is

similar to magnetic load subjected to high magnetic field,

encourages one to examine the superposition method with ferromagnetic

materials and low magnetic field strength. This is to be discussed

in the next section.

5.2 Ferromagnetic Materials

Ferromagnetic workpieces are associated with applications within the

metal forming industry. These materials are very suitable to

induction heating due to their high permeability. Ferromagnetic

billets of different sizes are heated efficiently by induction prior

to forging, rolling and, to some· extent, extrusion [5.3].

Also, the employment of these materials as a flux guide to protect

the surrounding metal work from being heated by the stray flux is well

known. Packs of low-loss material, such as nickel-iron or silicon-steel,

are usually placed on the outside of the coil of a vessel heater or

metal melting furnace, so as to confine the magnetic field close to the

outside of the coil. This prevents the flux from spreading away from

the outside of the coil and linking with surrounding metallic objects.

The aim of this section is to assess the applicability of the

superposition method on applications whereby ferrornagnetic materials

exist in the system, such as off-the-bar forging and vessel heating. Tl1is

is to be done by dividing this section into twu parts. The first

part is to study the effect of single conductor on a fcrroma9nctic

load and to e:·:<"~mine the ~-:;upcrpo~;ition met:-lud on a wurl:pi(•Cc' ,)[ t.hi:-_; ;:i11d.
-------------------------------------,

The second part is to investigate the influence of ferromagnetic

laminations on the distribution of the power density induced on the

load and, also, to inspect the validity of the superposition method

with the existence of the laminations.

5.2.1 Ferromagnetic Workpiece

In some induction heating applications such as vessel heating, it

is required to heat the mild steel container to a temperature below

the Curie point. The main problem associated with these applications

is the variable permeability which should be taken into account when

designing the installations. Many expressions which attempt to

represent the variation of the permeability can be found in the

literature [5.4-5.6]. Among these expressions is the Frohlich

formula (5.7)

FA
- = -
FB + H
+ FC (5. 7)

This formula is used extensively in induction heating [5.7-5.9].

It is very accurate at low magnetic field and around the knee of the

magnetization curve [5.7] but it gives a saturation flux density of (FA+

FC.H} , while actual saturation curves never reach such a value.

As far as the Author is aware, there is no published work concerning

the design of a coil to heat ferromagnetic load subjected to low

magnetic field apart from the empirical formula of Thorn ton [s.lo J
 L.L

to estimate the coil number of turns. The difficulty in employing

this formula lies in the necessity to know the efficiency and the power

factor of the system beforehand.

The objective of the experimental work in this section is to assess

the possible use of the superposition met-Aod on applications containing

ferromagnetic workpieces subjected to low magnetic field. This is to

be done by investigating the distribution of the power dPn~-;i.ty on a

ferromagnetic workpiece subjected to low magnetic field produced by

single conductor and examining the superposition method when applied

to a load of variable permeability.

The experiments were carried out on an EN3 mild steel slab of

(500 x 300 x 25.4)mm dimensions which was taken as a workpiece instead

of the aluminium slab on the experimental rig shown in Fig. 5.1.

The electrical circuit was similar to that shown in Fig. 5.2.

First, it was necessary to investigate the current distribution

on the conductor above such a load, so that the effect of the conductor

current flow on the power density induced on the surface of the

workpiece can be established. The current distribution on the surfaces

of the 28mm diameter circular conductor and the rectangular conductor

were measured by the J-probes attached to them. This current

distribution was first measured when there was no load beneath the

conductor, then the same measurements were repeated with the mild steel

slab under the conductor. The results are shown in Figs. 5.30 u.nd 5.31
 0.~. -·1'-----------,..-+-

'

___,o_.-"oCl------ • l.O

Fig. 5.31 Normalized current density distribution on

the rectangular conductor
I
no load x mild steel at g
I

J~ Fig. 5.30 Normalized current density distribution

on the circular conductor

no load o mild steel slab at h 37mm

for the circular and rectangular conductors respectively.. The

difference between the two measurements at any probe on the

circular conductor did not exceed 4% when the height h was 37mm,and

less than this for a greater height. For the rectangular conductor,

this difference was within 8% only for a 22rnm air gap and less than

this, when the air gap was larger. As the change on the conductor

current distribution due to the mild steel slab was small, the influence

of this change on the power density induced on the load must be

insignificant.

The accurate calculation of the surface power density induced on the mild

steel slab necessitates taking the variation of the perme~ility

with the magnetic field strength into account. Frohlich formula

was employed in the calculation to represent this variation, as it has

the advantages of simplicity and the satisfactory representation of

the magnetization curve within the range of the experimental

magnetic field strength [5.4, 5.7]. This formula has been used in

much of the useful literature [5.7-5.9]. The constants of Frohlich

formula for an EN3 mild steel can be found in reference [5.8]. This

formula for this material takes the following form:

2.08 -6
+ (1.257 X 10 ) (5.8)
1380 + H

With the aid of a magnetic field strength probe, H-probe [5.11] the

magnetic field was measured on the surface of the workpiece.

 l37

The probe consisted of 3 turns of constantan wire wrapped around

a (lOO x 3)mm perspex sheet of 30mm width. The two ends of the

wire were twisted together and connected to the voltmeter by a

coaxial wire. By attaching this probe to the surface of the load,

the magnetic field strength can be derived from Faraday's law:

NAw!l H ( 5. 9)
0

where V The voltage across the H-probe (V)

H
N The number of turns of the probe (3 turns)

A The area of the probe (lOO X 3) (mm2)

w 2nf (rad/s)

H The magnetic field strength on the surface (A/m)

In addition, it was necessary to record the current density, by

the J-probe, on the surface of the slab induced by different conductors

at different heights above the slab.

The following method was taken to measure the power density on the

surface of the mild steel slab:

l. The surface magnetic field strength was calculated from the

measured voltage across the H-probe VH as:

H = (A/m) (5.10)
2. The corresponding permeability was calculated by substituting

the value of H from equation (5.10) into equation (5.8)

-8
3. The resistivity of the mild steel was taken as 16.78 x 10 Qm

[5.8]. The penetration depth at each point was calculated

from the values of the resistivity and local permeability as:

-r;;t;;
- 1ff\1 \1
o r
(5.11)

4. The surface current density was calculated from the measured

voltage across the J-probe VJ as:

J ~ (5.12)

5. As the skin depth was small in comparison with the thickness of

the slab, the surface power density was calculated by substituting

the values of p, J and 6 into the following equation

~ pJ2o
PD (5.13)
2

A program "Magnetic Material" was written to a hand calculator to cater

for these calculations.

The power density induced by the single conductor on the surface of

the mild steel slab is shown in Figs. 5.32-5,35. The involvement of

many factors in the calculations makes it difficult to find a simple

formula to represent the distribution of the power density induced on

the surface of the load. However, it was noticed that the maximum
 l 3')
420

N
s
'-
3:

0
. 240
p.

:>,
+'
·.-!
<Jl
c:Q)
'tJ
k
Q)
~ 180
0
0.
'tJ
Q)
u
~
'tJ
c:
H

60

40 80 120 160 200

Distance along the load, z, (mm)

Fig. 5.32 Surface power density distribution along the mild steel
slab due to lOOOA flowing through circular conductor of
Gmm diameter at different heights (h)
 .l(>O

320

2SO

N
8
'3 240

"'><- 200
"'
·rl
<Jl
<::
"'
'1j

...
:." 160
0
0.
'1j

"u
"<::
'1j

H
120

so

40

40 so 120 160 200

Distance along the load, z, (mm)

Fig. 5.33 Surface power density distribution along the mild steel
slab due to lOOOA flowing through circular conductor of
l5mm diameter at different heights (h)
 l·i l

320

280J h=37mm
N
a
'~
Q

"' 240
..,;..,'
..,
Ul

""'
'0
H 200
"'0
~

0.
'0
"'
lJ

"
'0 160
H
"

40 80 120 160 200

Distance along the load, z, (mm)

Fig. 5.34 Surface power density distribution along the mild steel
slab due to lOOOA flowing through circular conductor of
2Bmm diameter at different heights (h")
 .,,. '

160

140

120

El
"3:
~

lOO
Cl
0.

>,

'"'
.....
Ul
<: 80
QJ
'0

:.'0"'
QJ

0.
60
'0
QJ
u
"
'0
H
"
40

20

40 so 120 160 160

Distance along the load, z, (mm)

Fig. 5.35 Surface power density distribution along the mild steel
slab due to lOOOA flowing through rectangular conductor
at 4lmm air gap.
 i·i.'

power density induced on the surface of the load, i.e. beneath the

conductor, is a function of the effective height , Fig. 5.36, and takes

the following form:

K
PD = (5.14)
h
e

where K is a constant and in this case was found to be 9300.

The factors which affect the power induced on a nonmagnetic load have

the same effects for a ferromagnetic load; they are:

1. The induced power density decreases with the increase of both z and h.

2. A conductor of small diameter will concentrate the induced power

density beneath the conductor more than would a large diamater

conductor.

The next stage was to investigate the applicability of the superposition

method on the mild steel slab. This has been done by using 5 circular

conductors of 6MM dian:eter and 5 rectangular conductors respectively.

The voltages across the J-probe and the H-probe induced by a single

conductor were combined with the superposition principals to predict

the performance of more than one conductor. These calculations were

facilitated by the computer program "FITTINGl", which was developed

for the PRI~E 400 system. These values were then substituted into the

program "Magnetic Material" to predict the performance of a number

of conductors.
 144
450

400

350
"'~ •
~

Cl
p..
300
>,-
.j.J
.....
Ul

"
QJ
'd
250
'"
QJ

"
0
0.
'd
QJ
u 200
"
'd
....."
!3
.....s 150
"'"
::;:

lOO

50

20 40 60 80 lOO
The effective height, h , (mm)
e

Fig. 5.36 The maximum power density induced directly beneath

the conductor as a function of the effective height (h )
e
K
calculated from
h
e
x measurements
 ( STI\RT

/
/ Read N, X. , V. , Il, I2,
~ ~

NT, D, L

Call the subroutine E021\CF to

calculate 11 .•• 11
1 8

/ Write A , ... , AB
1 /
Calculate RA~ r 2 ;r ,
1
Slope of last part of curve = SLO
The consta.·nt of the straight line ~ CON

Distance X = zero
I

NO
~ YES

Calculate the voltage Calculate the voltage,from the!

from the polynomial equation of straight line
equation V ~ X*SLOP + CON
I I
X ~
X+O.Ol

/ Write X, VX
/
NO

0 YES

( STOP

The flow chart of the program ''FITTINGl"

 l·il,

The measured power density and that predicted by the superposition

method are shown in Figs. 5.37 and 5.38 for the circular conductors.

The agreement between the two values are very good and the

discrepancy did not exceed 6% for air gaps between 80mm and 22mm.

Before examining the superposition method with the rectangular

conductors; it was necessary to investigate the current distribution

on the conductors so that the effect of this distribution can be

assessed. The current density distribution on a rectangular

conductor in amongst others and the superposition method with

such conductors was investigated.

Despite the nonuniformity and dissimilar conductor current

distribution, Figs. 5.39 and 5.40, the superposition still applies

with very good accuracy, Fig. 5.41. The difference between the

measured and predicted power density was within 6% only. The

good agreement between the measured and predicted values for the

two kinds of conductors, circular and rectangular, proves that

the superposition technique can be applied with ferromagnetic

workpiece.
 j ·1 "i

700

•
600

500
•

400

300
0

200 0

lOO

40 80 160 200

Distance along the load, z (mm)

Fig. 5.37 Power density distribution along the mild steel slab
due to 5 circular conductors at air gap of 77mm, coil pitch
of 60mm, and a current of 600A/conductor.

Prediction by superposition 0 Practical readings

 1400

1200
0

N
s
"3: 1000 0 0
Q'
p. "
..,;.,
....Ul 800
0
<:
(])
'tJ

";.
(])

0
p. 600
'tJ
(])
lJ
~
'tJ
<:
H
400

200

40 80 160 200
Distance along the load, z, (mm)

Fig. 5.38 Power density distribution along the mild steel slab due to
5 circular conductors at air gap of l9mm, coil pitch of 60mm
and a current of 600A/conductor

Prediction by superposition 0 Practical readings

 l. 0 t--1.------

Fig. 5.39 Normalized current distribution on the rectangular

conductor when placed centrally among 5 conductors
with mild steel slab at g = 4lmm and coil pitch of 3lmm

The other conductors

are in this side

Fig. 5.40 Normalized current distribution on the end rectangular

conductor; the others being on one side only with mild
steel slab at g = 4lmm and coil pitch of 3lmm
 1600

" a

1400
"
•
1200
0
N
8
"-
3:

Cl 1000
0.

:>.
-1-'
·rl
Ul

"
Q)
'0
800
><
Q)

"
0

"'
'0
Q)
600
u
:J
'0
H
"
400

200

40 80 120 160 200

Distance along the load, z, (mm)

Fig. 5.41 Power density distribution along the mild steel slab
due to 5 rectangular conductors at air gap of 4lmm,
coil pitch of 3lmm and a current of 600A/conductor.

Prediction by superposition 0 Practical readings

 151

5.2.2 The Lamination Packs

In some induction heating applications, such as vessel heating,

lamination packs are installed around the coil. The main reason

for using them is to channel the flux in the region outside the

coil so that it does not link the surrounding metallic objects,

which will heat these objects and, also, might result in sparking.

The efficiency will be improved also by the employment of these

flux guides due to the reduction on the stray losses and the

magnetizing current.

The object of this section is to inspect the applicability

of the superposition method in a system containing laminations, before

this technique can be recommended as a coil design method to

installations involving the laminations. This is to be done

by studying the effect of the laminations on the power density

induced on the workpiece due to single conductor and number

of conductors.

The experiments were carried out on the same aluminium slab employed

in Section 5.1 and by using the same experimental rig as shown in

Fig. 5.1, with the addition of the laminations. The circuit

diagram is that of Fig. 5.2 and the configuration of the system

is shown in Fig. 5.42.

 ~lamination pack

hl

l conductor
'-
1
h g

slab

Fig. 5.42 The load, conductor and laminations

A pack of nickel-iron laminations was used in the experiments;

the pack contained 250 sheets of the following dimensions

45 x 280 x O.l5mm. It was supported by 2 stainless steel bolts.

The pack was then covered with a fibre glass tape to maintain a

close proximity between the laminations and to protect their

edges.

A circular conductor of 28mm diameter and a rectangular conductor

were used in order to investigate the laminations effect on a

single conductor. The air gap g between the aluminium slab

and the conductor was held constant at 40mm. The current in the

conductor was permanently fixed at lOOOA, The voltage induced

at different points on the load was measured by the current density

 - - - - - - - - - ------

probe and the current distribution on the conductors was also

recorded, so that the effect of this distribution could be assessed.

When investigating the effect of the laminations on the power

density induced on the load a pack of laminations was positioned

above the conductor at distance hl = 5, 11, 20 and 35mm respectively.

The pack was above the centre of the aluminium slab, where the

induced surface power density was measured. Figs. 5.43-5.50 show

these measurements together with the calculated values according

to equation (5.2). The differences between the two values did not

exceed 5%; this proves that the practical equation (5.2) can be

used even when there are laminations above the conductor. The

constant cc. is the same as in the case of no laminations, while the

other constant ~ inc·reases when decreasing the distance hl, see

Table 5.1. Decreasing the distance hl increases the power density

induced on the load i.e. increases the coupling.

hl ( nm) ~ for circular s for rectangular

conductor conductor

co(no 1.13 1.07

laminations)

35 1. 39 1.16
20 1.46 1.19
ll 1. 51 1. 25
5 1. 58 1.3

Table 5.1 The variation of S with hl

 154

160

Equation (5.2) with

a = 0.55 and B = 1.39
140
o Measurements

0
~ 120
~
Q
~

~
~
·M loo
w
~
w
~

~
w
~
0
~
so
~
w
u
~
~
~
H
60

40

20

40 so 120 160 200

Distance along the load, z, (mm)

Fig. 5.43 Surface power density distribution along the aluminium slab
due to lOOOA flowing through circular conductor of
2Smm at h = 54mm with lamination pack at hl = 35mm
 lSO

Equation (5.2) with

a= 0.55 and 8 = 1.46
160
o Measurements

s 140
'-
"'
0
p.

>· 120
+'
·rl
Ul
0:
"
'd
....
"0:. lOO

"'
'd
"u
"
'd
H
0: so

60

40

20

40 so 120 160 200

Distance along the load, z, (mm)

Fig. 5.44 Surface power density distribution along the aluminium slab
due to lOOOA flowing through circular conductor of
28mm diameter at h = 54mm with lamination pack at hl 20mm
 200

180
Equation (5.2) with
"= 0.55 and 8 = 1.51

0 Measurements
160
e
......
;::_ .,,
Cl
0.
140

...>o
·4
Ul
~

'd " 120

":."
0
0.
'd

"
u
lOO
"
'd
H
"
so

60

40

20

40 80 120 160 200

Distance along the load, z, (mm)

Fig. 5.45 Surface:! power density distribution along the aluminium slab
due to lOOOA flowing through circular conductor of
28mm ut h = 54mm 'Nith lamination pack at hl = llrnm
 1 ~.i 7

225
_____ Equation (5.2) with
a = 0. 55 and B = l. 58

200 0 Measurements
"''-G
e
0

"'..,
>.
175

.....
tl)
c
Q)
'0
150
H
Q)

"
0
0,
'0
Q)
u 125
"c
'0
H

lOO

75

50

25

40 80 120 160 200

Distance along the load, z, (mm)

Fig. 5.46 Surface power density distribution along the aluminium slab
due to lOOOA flowing through circular conductor of
28mm diameter at h = 54mm with lamination pack at hl 5mm
 l.SB

200

180 Equation ( 5. 2) with

a= 0.35 and B = 1.16

0 Measurements

160

s
'-
3:

Q'
140
p,

'
...."
+J

"'<::
QJ 120
'0
1-<
QJ
~
0
o,
'0 lOO
ru
lJ

"<::
'0
H

so

60

40

20
•

40 80 120 160 200

Distance along the load, z, (mm)

Fig. 5.47 Surface power density distribution along the aluminium slab
due to lOOOA flowing through rectangular conductor at g = 40mrn
with lamination pack at hl = 35rnm
 l~;'_}

200

180 Equation (5.2) with

C< ~ 0.35 and S ~ 1.19

0 Heasurements
160

140

120

loo

80

60

40

20

40 80 120 HiO 200

Distance along the loud, z, (mm)

Fig. 5.48 Surface power density distribution laong the aluminium slab
due to lOOOA flowing through rect<mgular. conductor at
g = 40mm with lamination pack at hl = 20mm
 Equation (5.2) with
a = 0.35 and S = 1.25

0 Measurements

125

lOO '

75

50

25

40 80 160 200

Distance along the load, z, (mm)

Fig. 5.49 Surface power density distribution along the aluminium slab
due to lOOOA flowing through rectangular conductor
at g = 40mm with lamination pack at hl = ll mm
 161

250

225
Equation (5.2) with
= 0. 35 and B = l. 31

200
"
0 Measurements

s
":;:
Q
175

"'><
.jJ
·.-<
Ul

"<ll
'd
150

.
k
:.
0
0.
'd
<ll
125
u
"
'd

"
H

lOO

75

so

25

40 so 120 160 200

Distance along the load, z, (mm)

Fig. 5.50 Surface power density distribution rtlong the aluminium slab
due to lOOOA flowin9 through rectangular conductor
at q = tJOmm with lamination pack ut hl == Smm
 hl (mm) The load surface power The load surface power
density induced from density induced from
circular conductor rectangular conductor
(W/m 2 ) (W/m2)

35 170 200

20 189 213

ll 203 233

5 223 256

Table 5.2 The power density induced beneath the conductor

Table 5.2 shows the values of the power density induced on the load

directly beneath the conductor for different values of hl.

The next stage was to investigate the current distribution on

the conductor itself in order to find out whether the change in the

distribution of the induced power density was due to a variation on

the conductor current distribution or not. The surface current

density distribution on the circular conductor was measured with

the lamination pack 5mm above it and the normalized readings

are shown in Fig. 5.51.

The same measurements have been taken on the rectangular conductor

for hl = 5mm. The readings are normalized with respect to the reading

at probe 5, see Fig. 5.3, and the results are shown in Fig. 5.52.
 l.O
+·

1.0

Fig. 5.52 Normalized current density distribution

on the rectangular conductor at
g = 40mm from an aluminium slab

no laminations * laminations at h:=~_,:

F1g. 5.51 Normalized current density distribution

on the circular conductor at h = 54mm

from an aluminium slab
no laminations * laminations at hl 5mm
Figs. 5. 51 and 5. 52 show that, within the range of the experiments,

the laminations do not have an important effect on the distribution

of the current on the conductor. This effect did not exceed 5%

for the circular conductor and 7% for the rectangular conductor.

As the change in the conductor current distribution was small; its

effect on the power induced on the load would also be small.

Hence the change in the power induced on the load was due mainly

to the presence of the laminations.

In induction heating applications, the laminations do not completely

surround the coil as most practical designs rely on a number of

packets of punchings uniformly distributed around the -coil. For

this research to be useful in practial applications; it is important

to know the effect of the laminations on the parts of the load

which are not exactly beneath them, as the power density induced

underneath the pack has already been investigated.

One lamination pack was placed above one edge of the load and the

induced voltage distribution was measured at both edges of the slab.

The distance hl was 5, 11, 20 and 35mm respectively. Figs. 5.53-5.56

illustrate the power density distribution for hl = 35mm and hl = 5mm

only. The other results are not shown because they lie between the

limits of these two cases. These curves show the power density

induced on two parts of the load, near the two edges, one is covered

with one pack of laminations and the other edge is left exposed.

__________________________....
 ll''
lUO

160

Under the laminations

140
0 On the other edge
N
8
'-
~
120
Cl
"'
>,
....+'
(I)

"
QJ
'0
lOO

"
QJ
3
0
0.
'0 so
QJ
()

"
'0
H
"
60

40 80 120 160 200

Distance along the load, z, (mm)

Fig. 5.53 Surface power density distribution along the aluminium slab
due to lOOOA flowing through circular conductor
of 28mm at h = 54mm with lamination pack at hl = 35mm above one
edge of the slab
 '-
l ()(;

225

200 Under the laminations

0 On the other edge

175

150
rl

"'c:
aJ
d
><
aJ
:. 125
0
"'
lOO

75

so
0

25
•

40 80
. 1 0
•
200
Distance along the load, z, (mm)

Fig. 5.54 Surface power density distribution along the aluminium slab
due to lOOOA flowing through circular conductor of
28mm diameter at h = 54mm with lamination pack at hl 5 mm
above one edge of the slab
 200

180

Under the laminations

0 On the other edge

160

"'......8
3:

Cl
. 140

"':;.,.
.._,
..... 120
{))

"
Q)
'0
...
Q)
:>
0

"'
'0
Q)
lOO
u
"
'0
H
"
80

60

40

20

40 80 120 160 200

Distance along the load, z, (mm)

Fig. 5.55 Surface power density distribution along the aluminium slab
due to lOOOA flowing through rectangular conductor at
g = 40mm with lamination pack at hl = 35mm above
one edge of the slab
 lbU

225

Under the laminations

200
0 On the other edge

e
17!0
'i!:
Q

"''
:>,
....'"' 150
"'<:
'd"
,
"";.
0
125
0.
'd
'"
u

H
"<:
'd
lOO

75

50

25

40 so 120 160 200

Distance along the load, z, (mm)

Fig. 5.56 Surface power density distribution along the aluminium slab
due to lOOOA flowing through rectangular conductor at
g = 40mm with lamination pack at hl = 5mm above
one edge of the slab
The power density induced on the first part is higher than that in

the second, at given points near the conductor. While the opposite

is true at a greater distance from the oonductor, it did not, however,

appear in the graphs because these values were small in comparison

with the maximum value. This point should be taken into account

in the applications where laminations were used, otherwise the calculated

induced power density distribution might not represent the actual

distribution.

Table 5.3 shows the power density induced on the load at those two

parts directly beneath the conductor. From this table it can be seen

that the power density induced on the other end varies only slightly

with hl and can thus be regarded as constant.

Circular Coaductor Rectangular Conductor

Distance PD under PD on the PD under PD on the

hl lamin. other part %Diff lamin. other par1 %Diff
(mm) W/m 2 W/m 2 W/m2 W/m2

35 170.34 153.45 9.92 199.02 165.25 16.97

20 183.17 157.97 13.76 219.08 179.11 18.25
11 202.87 153.07 24.55 234.53 180.0 23.25
5 223.57 15 7. 2l 29.7 242.93 181.95 25.1

Table 5.3 The power density on the load directly

beneath the conductor
 170

This phenomenon was investigated over all parts of the load in the

next experiment. The power density distribution induced on the load

at different distances from the laminations was investigated by

assuming the load to comprise of 6 parts, each part being of 50mm

width. The lamination pack was situated above the 5th part as

shown in Fig. 5.57. The induced voltage distribution was measured

on parts 5, 4, 3 and 2 at hl = 20mm and hl = Smm respectively.

The readings are not shown as the changes are small and cannot, therefore,

be shown in graph form. The shape of the curves would have been

as in Fig. 5.58, so that the smaller the distance hl the greater

the differences between the different parts.

The main aim of this section is to investigate the superposition

method when laminations e~ist in the system, so that this technique

can be used in applications such as vessel heating. When investigating

the effect of the laminations on the applicability of the superposition

r,1ethod it proved necessary to measure the voltage distribution, induced

on the load, when lOOOA flowed through the single rectangular

conductor, which was above the slab and below the lamination pack

when hl = 5mm. These readings were combined with the computer

program "W-SC-FIT" to predict the induced power density from 5

similar conductors which were parallel to each other and positioned

at 3lmm intervals. These 5 conductors operated under identical

conditions to those of the single conductor with the exception that

a current of 600A/conductor and not lOOOA was applied. The results

are shown in Fig. 5.59.

 l'il

~~~~~~~~~~-----lamination pack

conductor

I I I I 1
50mm
I I I I I

part
I r I I
I slab
I I I
I
( 1) I ( 2) I ( 3) I (4) I (5) I
(6)
I I
' '

Fig. 5.57 The lamination pack above part (5) of the load

(5)

( 4)

( 3)

( 3)
( 4)

(5)

Distance from the conductor

Fig. S.Se Power de>nsity induced on different parts of the load

 J 7:.·

1200

1100 ..
0

1000 (J

N
s
'-
:;: 900

Cl

"'>. 800
....
'-'
if)
c:Q)
'0 700
...
Q)
:.0
0.
600
'0
Q)
()

"'c:
'0
H 500

400

300

200

lOO

40 so 120 160 200

Distance along the load, z, (mm)

Fig. 5.59 Power density distribution along the aluminium slab

due to 5 rectangular conductors at air gap of 40mm,
coil pitch of 3lmm and a current of 600A/conductor with
lamination pack at hl = 5mm

prediction by superposition 0 practical readings

 173

The existence of the laminations did not affect the applicability

of the superposition method .. The discrepancy between the

measured power density and that predicted by the superposition

method is less than 7%. This small difference is an acceptable

experimental error and proves that this technique can be used in

the relevant applications.

5.2.3 Conclusions from the Nork on Ferromagnetic Materials

Although the numerical values and percentages are for the particular

cases under consideration; they do give indications about the

effects of the ferromagnetic materials and some conclusions can

be drawn.

The superposition method can be applied to a magnetic load and

to a system containing laminations. Also, the applicability of this

method is not affected by the current distribution on the conductor

which depends on the position of the conductor among other conductors.

The use of laminations increases the power density induced on the

load beneath the lamination pack. This power density is inversely

proportional to the distance between the laminations and the conductor.

The accurate calculation of the power density induced on a ferromagnetic

load requires the variation of the permeability to be taken into account.

5.3 Summary of Results and Suggestion

The surface current density distribution induced from a single

current carrying conductor on a nonmagnetic slab was investigated and

an equation governing this distribution was found. The equation differs

from that derived for a filamentary conductor by three factors. These

factors can be represented by mathematical expressions.

The superposition method can be employed with magnetic and nonmagnetic

loads. The applicability of this method was not affected by the

existence of the lamination packs. The permeability variation

must be taken into account when using this method with ferromagnetic

workpiece. This has been done by adopting a functional representation

to the magnetisation curve.

The superposition method can be applied to conductors of different

shapes irrespective of the current distribution through them.

Also, neither the positioning of the conductor, amongst others, alters

it's effect on the workpiece nor the proximity effect on the conductors

prevents this method being applied.

The practical verification of the superposition method raises a

question concerning the possible use of this method in induction

heating applications. From the experiments on aluminium slab , it

is evident that this technique can be used with nonmagnetic loads

as well as to magnetic loads above the Curie temperature.

Also, it is possible to apply this method to applications requiring

relatively low magnetic fields to be induced on a ferromagnetic

load such as in vessel heating. This has been shown by the

experimental work on the mild steel slab. Finally, the work with

lamination packs proves that the superposition method is suitable

to applications involving laminations such as metal melting and

vessel heating. Hence it is conceivable to employ the superposition

method on different applications.

The main reason for developing the superposition method was to

design a coil capable of producing a nonuniform power density

along the load. This method has not yet been practically proved

suitable to this task. Therefore, it is required to test the

flexibility of this method and it's ability to design such a coil.

Applications such as off-the-bar forging need nonuniform power density

along t!l"' lead. Part of the workpiece in this case will be hot and

the other part cold. This problem cannot be tackled in the laboratory

due to the high cost of building a heater capable of producing the high

power density required for the hot part of the workpiece.

It is reasonable to suggest the aluminium extrusion as the final

experimental examination to assess the flexibility of the superposition

method practically. This application necessitates nonuniform power

density to be induced on the load, and it can be modelled within

the laboratory facilities.

 CHAPTER 6

SUPERPOSITION AS A COIL DESIGN METHOD FOR

HEATING ALUMINIUM PRIOR TO EXTRUSION

The versatility and economic advantages of extrusion as a method

of forming al umini urn have caused a very wide growth in it~· s use

for different products. Superficially, the process appears to be

rather simple. It consists of heating a billet, inserting it into

the container in a hydraulic press and then applying sufficient

pressure to force it through a die.

Unfortunately, maintenance of consistently high quality in aluminium

alloy extrusions is difficult. The major problem is that aluminium

actually gains heat during extrusion. It is, therefore, desirable

to taper-heat the billet. If this is not the case, metallurgical

inconsistency in the end product might occur.

In direct extrusion of aluminium, a ram directly pushes a billet

through a stationary press container and selected die-shape. The

finally extruded section emerges onto the run-out as a correctly

shaped product. In this kind of extrusion it is prefPrred that the

extrusion speed of the ram and product temperature through the

die be constant throughout the extrusion cycle, thus avoiding

continual adjustment of the press. This isothermal effect may be

achieved by inducing into the heated billet a temperature at the

 177

front face; that part which enters the die first. In order to

appreciate the reason for this isothermal effect it must be explained

that additional heat will also be generated during the extrusion

operation:-

a) by heat energy, caused by the shear effect at the restricted

die opening,

b) through the generation of friction, since the hot aluminium

billet is not only forced into the die but also moves along

the stationary container whilst confined within it's bore.

In indirect extrusion the hot billet remains stationary in relation

to the press container, the rear end of the billet becomes progressively

cooler as the extrusion proceeds rather than the front end. This,

of course, is the exact reversal of the direct extrusion and care must

be taken to ensure the correct degree of reverse taper if optimum

results are to be achieved.

To demonstrate the usefulness of the design procedure developed

in the previous chapte~ it was decided to design an induction heating

coil capable of producing a linearly decreasing temperature taper.

This, of course, models the practical requirements met industrially

in the induction heating of aluminium prior to extrusion.

It was not possible to build a commercial size billet heater to test

the theory because of the capital cost and the limitations of the

laboratory power supplies. Megawatt power supplies and large amounts

of power factor correction capacitors are not uncommon in industrial

units. However, a laboratory scale model was built to demonstrate the

validity of the design procedure.

6.1 The Superposition Method with a Cy_lindrical l·lorkpiece

Before adopting the superposition method to design the required coil,

it is necessary to investigate the applicability of this technique

on a cylindrical load. To achieve this, a series of experiments

have been carried out to find the effect of single turn and multiturn

coil on an aluminium cylinder.

The circuit diagram is the same as that of Fig. 5.2. Instead of a

slab, an aluminium cylinder was used as the workpiece. This solid

cylinder of 50.8mm radius and 340mm length had it's surface machined

until smooth. As can be seen, the radius of this cylinder is equal

to the thickness of the aluminium slab which was used previously

in Chapter 5.

The experimental rig is shown in Fig. 6.1. The single turn and

multiturn coils have been constructed from a water cooled copper tube

of 6mm diameter. Three single turn coils of 70.8, 80.8 and lOO.Smm

inner radii were respectively surrounding the aluminium cylinder.

The single turn coil was around the middle of the cylinder which

was assumed to be the reference point. A constant current of 400A

was flowing through the coil. The voltages induced at different

distances along the cylinder were measured by the current density probes.
Fig. 6.1 The experimental•rig
The probes were made from thin constantan wires wrapped around the

cylinder and spaced at 20mm intervals.

The results of the single turn coils are shown in Figs. 6.2-6.4.

These graphs show the measured surface power density together with

those calculated by using the filamentary conductor theoretical

equation (4.9) and the practical equation (5.2). It can be seen

that the practical equation (5.2) which was developed for aluminium

slab can be applied with the same accuracy to an aluminium cylinder

of radius equal to the thickness of the slab.

The calculations of the surface power densities were carried out

9
by substituting the numerical values of the resistivity, 28.24 x 10- Qm,

and the measured surface current density into the equations given

in Appendix l.

The results of the single conductor show that the practical equation

which was developed for a semi-infinite aluminium slab can be applied

for an aluminium cylinder of radius equal to the thickness of the

slab. The discrepancy between the measured power density and that

calculated by the practical equation (5. 2) did not exceed 10% ~or

the range from maximum to 10% of the maximum power density. This

accuracy is sufficient for the practical applications. As the power is

a function of the square of the current, the error in calculating the

current or voltage distribution is lov;er than that of the power density.

 lBl

60
\
\

40
\
\ ~

\.
'\.
20

"' ...........
~

20 40 60 80 lOO

Distance along the load, z, (mm)

Fig. 6.2 Surface power density distribution along the aluminium cylinder
due to 400A flowing through single turn coil at air gap of 20mm
 Hl2

140
Equation (4. 9)

~ Equation (5.2) with

N 120 a = 0.34 and S = 1.05
E
'
3: o Measurements

C)
0...
- lOO
\
\
....>. \
\
VI
c: 80 \
QJ
C) \
\
'-
QJ \
3:
0 \
0... 60 \
"'Cl \
QJ
u
\
:::> \
"'Cl
c 40 \
~

20
"" '
''

20 40 60 80 100
Distance along the load, z , (m m)

Fig.(,.3 Power density distribution along the aluminium

cylinder due to 400A flowing in single conductor
at air gap of 30 mm.
 ji._)_)

64

56 Equation (4.9)

Equation (5.2) with

N Cl = 0. 56 and S = 1.17
E
~ 48 0 Measurements

0
-
0...
-
>.40

Vl
c
CJ
"'Cl
r...
32
CJ
3
0
c..
"'Cl
CJ
u 24
::::J
"'Cl

- c

16

-
20 40 60 0 10
Distance along the load, z, (mm l
Fig.6.4 Power density distribution along the aluminium
cylinder due to 400A flowing in single conductor
.at air gap of SO mm
 L, ,. ,

An ll turn coil of l00.8mm inner radius and uniform pitch of 7.lmm

has been used to verify the superposition method. The centre

of the coil was around the middle of the cylinder and this was the

reference point. The measured power density distribution induced

on the load is shown in Fig. 6.5 together with the predicted values

which were derived with the aid of the computer program "W-SC-FIT".

This graph shows that the superposition theory can be applied to

a cylindrical load and the difference between predicted and measured

power densities is within 3%. For large values of distance z; the

error is higher, this is due to the approximation made in

calculating the induced voltages at these distances. This is not

important practically because the power density induced at these

distances is relatively small in comparison with other values.

 5.5

5.

4.

·-
.r::; 4.
3
-""
~

0
0..

....>.
VI
c
Q)
0
'-
Q) 2.
:;.:
0
0..
"0 2.
Q)
u
:::>
"0

-c

1.

0
0
0 10 20 30 40 50 60 70 80 90 100 110 120
Distance along the load, z, (mm).

Fig.6.5 Power density distribution along the aluminium cylinder

due to 11 turns coil at air gap of 50 mm, coil pitch
of 7·1 mm and a current of 400A.

Pred1c t1 on by Superpos1 t ion o Practical Readings

6.2 The Temperature Distribution in a Solid Cylinder

Carslaw and Jaeger [6.1) gave the temperature distribution in a

long cylinder initially with uniform temperature when subjected to

a constant surface power density with no heat loss from the surface

to be:

2
2 -8 1 J [13 n (r/R))
Tr (PD)~2T+ r l
- - 2 I e
n o
} (6.1)
A 2R2 4
n=l

where Tr temperature rise at radius r after time t

=
At
1
2 (dimensionless) (6. 2)
yCR

thermal conductivity (W/m°K)

t time (s)

y
3
= density (kg/m )

c specific heat capacity (J/kg°K)

R = the radius (m)

J 0 (x) is Bessel's function of the first kind and zero order and the

Bn's are the positive roots of J (f3)

1
= 0.

At the beginning of the heating cycle, the increase in the temperature

of the surface is faster than that of the centre; this is the

transient period, which is represented by the summation term in

equation (6.1). Baker [6.2) showed that the the transient time
is equivalent to T = 0.25, at this point the summation term will be very

small and as such can be ignored. Once the transient time has

elapsed, the temperature of all parts of the cylinder will increase

at the same rate and equation (6.1) will be simplified to;

Tr (PD)I (2T + (6. 3)

The temperature at the surface (r = R) is:

Ts = (PD)I (2T + ~) (6.4)

and at the centre (r 0), the temperature is:

Tc = (PD)I (2.T - ~) (6. 5)

Hence the tempe~ature difference between the surface and centre is

R
Ts - Tc PD 2A (6.6)

By using equations (6.5) and (6.6), equation (6.3) can be

rewritten as

2
r
Tr = Tc + (Ts - Tc)
R2 (6. 7)

As was stated earlier, the above equations are for a cylinder with power

input to the surface which can be applied at high frequencies as

the penetration depth is very small when compared with the

load diameter. When a finite penetration depth 0 exists and there

is no heat loss the temperature distribution, with respect to

the temperature of the centre, can be calculated [6.3] by

 JUG

th~ following equation:

R
Tr - Tc = (PD) 2A (6. 8)

(6. 9)

Z(x) ~ berxber'x + beixbei'x (6.10)

(6.11)

If there· is a surface radiation, then the net power input will be

less than the total input power [6.4) and equation (6.8) will be:

PD
Tr - Tc (6.12)
p
n

where Pn is the net power density. The radiation loss can be

calculated from Stefan-Boltzmann's law:

-8 4 4
PR= 5.67 X 10 E(Ts - Tb ) (6.13)

where PR 2
the radiated power density (W/m )

2 4
£ : the emissivity coefficient (W/m K ) and

Ts, Tb are the surface and the boundary temperatures in°K.

The billet does not enter the die .directly after leaving the heater,

as there is a handling time or a soaking period which will reduce the

temperature difference between different parts of the workpiece.

Carslaw and Jaeger [6.1] showed that a cylinder with initial

temperature distribution Tr = f(r) and no heat input will have a

temperature distribution of:

2
R ro -a T Jo(ran/R)
=~
n
2 0f
~

Tr r 'f(r ')dr' + L e
2
R n=l J (a )
o n

o
JR r'f(r'p (a r'/R)dr')
o n
(6 .14)

where the a 's are the positive roots of J ' (aR) = J (aR) 0
n o 1

The final value of the cylinder's temperature is represented by

the first term of equation (6.14). The second term represents

the transient period.

From equation (6.7) f(r) is

2
r
f(r) Tc + (Ts - Tc) (6.15)
2
R

The temperatures of the centre Tc(t) and of the surface Ts ( t)

during the soaking period as functions of time are given by

[6. 2] as: 2
-AS t
ro
n
{.!:. e R2
Tc (t) Tc + (Ts-Tc) + 4 I } ( 6. 16)
S~Jo(Sn)
2
n=l

2
-AS t
ro
n
{.!:. l e R2
Ts ( t) Tc + (Ts-Tc) +- .'i } (6.1•7)
s~
2 4
ri=l
The quantities inside the parentheses are shown in Fig. 6.6.

If the workpiece is of length ~ and initially with axial

temperature distribution of f(z), then during the soaking period

•

this distribution can be represented [6.1] by the following

expression;
mrz
cos
9.
T -1 !9. f (z') dz' + 2
£ 0 £

t nnz•
I
0
f(z 1 )cos 9. dz' (6.18)

The first term is the steady-state temperature and the second term

is the transient. The transient die away with time constants T',

where
1 (6.19)
T' :::::
e

i.e. the temperature difference will drop to 37% of the initial

value after a time T' seconds.

The surface power density distribution required for the extruding

of an aluminium billet can be determined by considering the radial and

axial temperature distributions during the handling time, i.e. the

soaking period.
 l_ <) .J

At a normalized time t = 0.25, the billet will have an even radial

temperature, see Fig. 6.6. This time is equivalent to

16 seconds for an aluminium billet of 70mm radius at a temperature

0
of Soo c [6.4]. This time is far less than the usual handling time

which is around 40 seconds [6.4).

If the billet is initially with linear temperature distribution,

f(z) = Cz, the temperature difference between the two ends

T£(t) - T (t) as a function of time with respect to the initial

0

temperature difference T£- T is shown in Fig. 6.7 [6.4].

0

At the end of the handling time, the temperature difference between

the two ends will drop to 90% of the initial difference for a billet

of 1 metre length and to 58% of the initial difference if the length

was 300mm [6.4]. Hence, a billet leaving the heater with axial

and radial temperature differences, will enter the die with uniform

radial temperature and differential axial temperature.

As the temperature of any point is a linear function of the

surface power density on that particular part of the billet, see

equation (6.8), hence the temperature distribution of the billet

can be controlled by controlling the distribution of the surface

power density.
 l.O

o. 8 JTs(t) - TcJ
Ts - Tc

0.6

0.1

JTc(t) - TcJ
Ts - Tc
o. 2

0.05 0.1 0.15 0.2 0. 25

T

Fig. 6.6 The variation of the surface and centre temperatures,

Ts(t) and Tc(t) during the soaking period with respect
to the initial temperatures Ts and Tc

1.0

0.8
...,
0
E-< 0 0.6
E-<

...,
""
E-<
0.4
E-<""
0.2

0
0.5 l.O 1.5 2.0 2.5 3.0
t/t'

Fig. 6.7 The variation of the temperature difference between

the two ends during the soaking period as a function
of the initial difference
6.3 A Coil for Power Density Taper

A theory is pointless, unless it is of some practical use in the

solution of particular problems. The superposition method to be

employed here to design a coil capable of producing linear surface

power density distribution along an aluminium cylinder.

A computer program "W-FC-TEMP" was developed to calculate and

plot the power density distribution induced on a cylindrical load

from a nonuniform coil of unequal pitch Similar to the uniform

coil, the first stage in this program is to find the response of

a one turn coil in an equation of the form

(6.20)

It is then necessary to apply the superposition princip~ls to find

the total voltage induced at different points on the load. Then

using the required Bessel functions to calculate the power density

distribution and to plot it in graphical form.

The program, also, calculates the temperature distribution along

the length of the cylinder at the moment when it leaves the heater

i.e. at the start of the soaking period. The output of this part

is a three dimensionsgraph of Tr - Tc against the length and the

radius of the workpiece. The method of calculation in this program is

almost identical to that in the program "W-SC-FIT" for a cylindrical load

with the exception that different coil pitches inst<~ad of one value for

the pitch apply and the plotting of the temperature distribution. The

flow chu.rt of "W-FC-'l'EMP" i~.:; shown below.

 ~.-~;~~~/
·-··· ····--····· · - · - · · · · · - = · - - · - - - - · - - - - - - · ·····----··- -·-····-··
H_cu.U: Np, zi, Vi, .i::::l, NP, r , 1 ,
1 2
N, di, £. , p F, l1 r, Rol, NSC

Call the subroutine E 2ACF to calculate A , ... , A

1 5

Calculate RA ;

Set z zero

NO YES

2 2 3 4
V= [ A +A Z+A z +A z +A z +ASZ] V [(SLO*Z) + CON)
1 2 3 3 4
*RA
*RA

Store Z and V in arrays

NO
z z + 0. 25

Calculate the power density (PD)

Store Z and ~D in-array

I Calculate the temperature distribution

Plot V against z
Plot PD against z
Plot the temperature distribution

( STOP )

The flow chart of the program 11

W-FC-TEMP 11
The program was used to design a coil capable of producing

a linear surface power density on a workpiece. The work coil was

made from 6mm diameter water cooled copper tubing which meant

that the variable pitch coil could easily be manufactured with

the University's facilities. The workpiece was a 200mm long

aluminium cylindrical billet of radius 50.8mm and the air gap was

taken as 30mm which is in agreement with industrial practice [6.4].

The induced surface current density on the load due to a 400A

current flowing through a single turn coil was measured. These

values were then used to calculate the resultant surface current

density of a number of different work coil configurations using

the procedure outlined previously. The design of a work coil capable

of generating a linearly decreasing temperature taper on the surface

of the billet was then decided upon and details of the work coil

pitch are given in Table 6.1. The eleven turn work coil was con-

structed and the surface current and power densities it produced

were measured. The surface power densities induced in the load

2 2
varied from 3.2 kl~/m down to 0. 22 kW/m . The practical measurements
•
were compared with those required to produce the linearly decreasing

temperature taper and the results are shown in Fig. 6.8 correlation

to within an accuracy of 5% can be seen thus demonstrating the

validity of this novel method of work coil design.

 3. 5

3.0

N
s
'- 0
~
Cl 2.5
0.

.,....,
Ul
<:
"'
'd
2.0
"'"
~
0
0.
'd
u"'
~
'd
<:
H 1.5

1.0

0. 5

0
0 40 80 120 160 200
Distance along the load, z, (mm)

Fig. 6.8 Powt>r density distribution along the aluminium cylindt>r

due to 11 turns coil at air gap of 30mm, nonuniform coil
pitch and a current of 400A

Prediction by Superposition 0 Pr~ctical Reading

 i q 7

Pitch no. 1 2 3 4 5 6 7 8 9 10

mm 7 8 11 23 23 23 25 23 25 32

Table 6.1 The pitch of the coil

The expected temperature distribution in the workpiece exactly

after leaving the heater is shown in Fig. 6.9.

6.4 The Construction of a Versatile Coil

To demonstrate fully the usefulness of the superposition method,

a versatile coil capable of producing different patterns of surface

power density distributions along the aluminium cylinder has

been built. A 30 turn, 160mm inner diameter coil with uniform pitch

of l0.5mm was made of lOmm diameter copper tube. Each turn of

this coil had brazed to it a copper strip for electrical connection

so that neighbouring turns could be short circuited to produce a coil

with variable inter turn spacings, Fig. 6.10.

The surface power density distribution on the load produced by a

single turn of this coil is shown in Fig. 6.11.

11
The program W-FC-TEMP" was used to design a coil sui table to produce

a linear surface power density distribution along the aluminium

cylinder. The coil was of 12 turns with inter turn sp:1cinqs as

.: . se ;

'
-i
2.50
~ .00 -,I
- 2.00
l .50 ---,
1 .50
~
l .00
1 .. 00
'
.sa
.50
.ea -
I .00 CO
.1

ra,dius length
.7

.9

1. 1 1. 1. •

Fig. 6.9 Temperature distribution due to the coil mentioned in Table 6.1
 ~ 'J l ·'

\
''
I

The Versatile Coil

( Fig. 6.10
 120

lOO
"' El
"c
80
"'"'
:>.
+-'
·.-I
(})

"
<lJ 60
"'....
<lJ
:>
0
0.
'tl
<lJ 40
u
"
'tl
H
"
20

20 40 60 80 lOO 120 140

Distance along the load, z, (mm)

Fig. 6.11 Surface power density distribution along the

aluminium cylinder due to single turn of the
versatile coil with current of 400A .

•
shown in Table 6.2 with a coil current of 400A. The practical

measurements were compared with the predicted power density

distribution and the results are shown in Fig. 6.12.

Pitch no.! 1 2 3 4 5 6 7 8 9 lO 11

mm 10.5 10.5 10.5 10.5 21 10.5 32.5 10.5 32.5 10.5 32.5

Table 6.2 The inter turn spacings of the coil which produces
the surface power density distribution shown in
Fig. 6.12

The correlation between the calculated and measured values were very

good and the discrepancy did not exceed 10%. The predicted

curve is not very linear, this is dne to the fact that the

spacing between the turns can be varied by 10.5mm, which is the pitch

of the coil, or a multiple of this value only. The temperature

distribution of the workpiece exactly after leaving the heater,

would be that shown in Fig. 6.13.

6.5 A Simplified Coil Design Program

To demonstrate the simplicity of the superposition method as a

11
coil design technique; a nontu1iform coil design program Superposition 11

was written to be used with a programmable hand calculator.

As the calculator lacks u subroutine, which calculates the best

 5.5

5.0

0
4.5

0
4.0

0
N
s
...__
:;: 3.5
-"'
0
0

"' 3.0
0
....,:;.,
·.-<
Ul
<::
Q) 0
'0 2.5
...
Q)

"0..
0

'0 20
Q)
{)

"<::
'0
H
1,5 0

1.0 0

@.5

0 2{) 00 12(} 140 153 180 2()3 220

Distance along the load, z, (mm)

Fig. 6.12 Power density distribution along the aluminium cylinder due
to 12 turns versatile coil at air gap of 30mm,
nonuniform coil pitch and a current of 400A.

_____ prediction by superposition o practical reuding

,, . 32
~~--~ 4.00
3.50
'3.00
. ~, . 52 ---1

2.50
.:.90 :
:::::. r), ---.:
2.00
' _,V I
~
.50
• 00 I

1 . 00
.50
.00
• i .00
•1

.3

.5
radius
length
.7

.9

1 •1 1 •1

Fig. 6.13 Temperature distribution due to the coil mentioned· in Table 6.2
curve to fit the readings of the single conductor, the practical

equation (5.2) was taken to predict the surface magnetic field

strength, induced on the load, from a single turn around the

workpiece. The magnetic field strength induced from all turns are

to be calculated by the superposition principles in accordance with

the suggested coil pitches. The power density at any point is

a f unction of H 2 ; t h at 1s
.

PD ( 6. 21)

The value of the constant C should be given, see Appendix 1, as

the Bessel functions cannot be computed by the calculator. The

suggested coil pitches and the single turn equation parameters were

written independently of the program so that they can be varied

without affecting the program.

6.6 Conclusion

An equation for the distribution of the surface power density

on an aluminium cylinder induced from single turn coil has been

found. This equation matches the equation for aluminium slab of

thickness equal to the radius of the cylinder.

The practical investigations show that the superposition technique

can be applied to a cylindrical load. Applying this method to

a practical application i.e. the production of a linear power

density taper along the length of a billet, proved highly successful.

The computer program was relatively small, i.e. it did not need

a largo storage capacity or consume long computing time. A simplified

solution suitable for a hand calculator was also.given by applying

the superposition principles to the single turn equation.

The usefulness of the superposition technique has been fully

demonstrated by constructing a versatile coil capable of

producing different patterns of surface power density distribution

along the cylindrical workpiece.

 20J

CHAPTER 7

CONCLUSION AND SUGGESTIONS

FOR FURTHER RESEARCH

The results and their implications are summarised and possible

areas of further research are suggested.

7.1 Conclusion

A novel method of work coil design is described which facilitates the

design of induction heating work coils capable of inducing a nonuniform

power density along a workpiece.

A versatile coil capable of inducing different patterns of power density

distribution along the load has been constructed.

A coil producing a linear power density taper along an aluminium

cylindrical workpiece has been designed and constructed.

An equation was established for the distribution of the induced surface

current density along an aluminium slab and cylinder due to a

current in nearby conductor.

Three correction factors were found eXperimentally to amend the

theoretically derived equation of the surface current density

distribution along the load induced from a current in a nearby conductor.

Mathematical expressions were found for the three correction factors

of the equation of the surface current density distribution along

the load induced from a current in a nearby conductor.

The variation of the permeability has been taken into account in the

IDBasurement of the power density distribution induced on a

magnetic workpiece.

Different types of current density probes have been constructed

and used throughout the work.

Magnetic field strength probe has been constructed and used to measure

the magnetic field strength on the surface of a ferromagnetic workpiece.

A computer program has been developed to design a coil capable of inducing

a predetermined surface power density distribution along the load.

A simplified program has been written for a hand calculator to design

a coil.to produce a nonuniform power density along the load.

The axial and radial temperature distributionsin a cylindrical workpiece

inside a nonuniform coil have been predicted and plotted by the use

of the computer.

An equivalent circuit based on a coil-multilayer load of different

physical properties has been formulated to take the nonuniform physical

properties of the workpiece into account.

A computer program has been developed for the coil-multilayer load

equivalent circuit <:md tht~ effect of. the load nonuniforr:-1 phy~;ical

properties has been assessed.

A two dimensional finite difference computer program has been

developed to calculate the magnetic field and power distributions

in an inductively heated workpiece.

Computer programs have been developed to aid the understanding

of the equivalent circuit technique and to assess the effect

of different parameters.

A computer program has been developed to cater for the Bessel

functions required for different calculations.

The surface current density distribution on the coil conductor

was investigated and the effect of this distribution on the power

density induced on the load was assessed.

The investigations included the effect of the laminations on the

distribution of the power density induced on the load.

The equivalent circuit method has been explained and

different literature concerning this method·have been reviewed.

The use of the numerical techniques to solve induction heating

problems ha~ been reviewed and number of literature on this

subject have been discussed.

7.2 Possible Areas of Further Research

The major achievement of the present work is the development

of the superposition method as a design technique for a coil

capable of inducing a predetermined nonuniform power density

distribution along a load. This method was successfully

used to produce a linear power density taper along the

aluminium cylinder as required in the extrusion industry.

The applicability of the superposition method to mild

steel workpieces was also demonstrated. This technique could

therefore be used in applications such as off-the-bar forging

where a nonuniform power density is required, and ve.Mel-·~·:.::::~-.,_~_'~

heating where the design of the coil is still very empirical.

In off-the-bar forging, the heater has to deal with a workpiece

of differin<J physical properties along it's length as a result

of the temperature difference between the cold bar and the partly

heated end. Existing techniques allow the workpiece to cool

before reheating. This, of course, means a substantial loss

of energy and a very inefficient process. The requirement for

this application is to induce a nonuniform power density along a

workpiece having nonunifonn physical properties. It may be

possible to use superposition method to deal with these two

conditions. The theoretical applications on Chapter 4

showed that this technique can taKe the nonuniformity of the load's

resistivity into accotmt. The experimental work in Section 5.2.1

proved that the variation of the permeability can be dealt with

 • ~ _L l

by the superposi tion method. Finally, the production

of the linear power density taper along the aluminium cylinder,

confirmed the ability of this method to design a coil to

induce a nonuniform power density along the loud.

In certain chemical engineering operations it is necessary

to use induction to heat stainless steel or mild steel containers

into a low temperature, 200 to 300°C. The main problems associated

with vessel heating are that the assumption of a long solenoid

cannot be applied as the ratio of the vessel's diameter to it s

length is relatively large; and the variable permeability of the

mild steel vessel should be taken into account as the temperature

is well below the Curie point.

The design of the work coils for vessel heaters is a cumbersome

task, as the use of the existing coil design methods in this

application was not successful. Thorn ton [ 7 .l] presented

empirical formulae to aid on the design of the coils for

containers within a specific range, but he stated clearly that

these formulae cannot be generalised.

The superposition technique may be suitable as a design method

for this application, since this method does not assume the coil

as a solenoid and, also, the variation of the permeability can

be taken into account. Further research in these areas is

recommended.
 In the present economic climate abrupt changes in the price and

availability of fuels are commonplace, due to social and political

effects. Attention should be devoted more to the question of

energy conservation, and the newly developed multilayer coil is

a good example. Further work to demonstrate the applicability of

the superposition theories to the design of multilayer coils

is recommended.

The efficiency of the induction heating systems can, also,

be improved by the use of the laminations. The use of lamination

packs is commonplace in induction melting furnaces, but they are

rarely used with induction heating systems. Their employment

will improve the efficiency by decreasing the stray losses.

Further work to include the use of the laminations in the work

coil design procedures is recommended.

 l _·.

Appendix 1

Heating a Cylindrical \'orkpiece

by Induction

The distribution of the magnetic field in a cylindrical workpiece

can be determined by considering a cylinder of radius R,

resistivity P and permeability ~' when subjected to a surface

magnetic field HR along it's length (the symbol HR is used for the surface

magnetic field strength to be consistent with the radius). It is firstly

necessary to take a ring of unit length, radius r (r<R) and thickness dr.

If H is the magnetic strength and J the current density at radius x, x < r,

then the total flux within the ring is:

<j>r (Al.l)

The ernf induced in the ring is:

dE -~ (Al. 2)
at

a { r
i.e. dE
at
)l f 27TXH dx }
0

~
-21T~ t
0
aH
x - dx
at
(Al. 3)

The resistance of the ring is:

2nrp
R (Al. 4)
r dr
and the current is:

I ~
J dr (1\l.S)
r

dE
also I ~ (ill. G)
r R
r

-21TJ.I
0
tx aH dx
at
(1\l. 7)
211rp
dr

From equations (Al.S) and (Al.7)

arr
J
-J.I
pr fr X-
at
d
X (Al. 8)
0

Using Maxwell's equation and ignoring the displacement current

dH
J (1\l. 9)
dr

then, by equating (Al.B) and (Al.9) the following results:

r dH ~ ~ Jr x aH dx (Al.lO)
dr p
0
at

If H is a sinusoidal function then:

jwH (Al.ll)

From equations (Al.lO) and (Al.ll):

r dr ::;
dH H.
p
Jr x jwH dx (1\l. 12)
0
 r dH = junt Jr X H dx (Al.l3)
dr p
0

Differentiating (Al.l3) with respect to rand inserting the

limits yields:

2
d H dH
r +- = jWIJ Hr (Al.l4)
2 dr p
dr

let k
12 (Al.l5)
0

w~
i.e. k2 = - (Al. 16)
p

Equation (Al.l4) then becomes:

1 dH 2
+ --- j k H = 0 (Al.l7)
r dr

This is the diffusion equation, which governs the distribution

of the magnetic field strength H. The adoption of Bessel functions

facilitates the solution of this equation [Al.l). The magnetic field

strength at radit:1..'3 ....r., H , is:

r
H = CI (kr(]) + BK (krf:j) (Al.l8)
r 0 0

where C and B are constants

It is known [Al.2] that:

= 00 (Al.l9)
As the magnetic field strength in the centre of the cylinder

(r = 0) is not infinite, then B must be zero. This reduces the solution

to

(Al. 20)

i.e. H C (ber kr + j bei /q,r) (Al. 21)

r

The magnetic field strength at the surface (r = R) is H :

R

H c (ber kR + j bei kR) (Al.22)

R

H
R
i.e. c =
(Al.23)
ber kR + j bei kR

From equations (Al.2l) and (Al.22):

kr + j bci kr]
H H Lber (Al.24)
r R
ber kR + j bei kR

ber -
12 r 12
+ j bei l f r
0
i.e. H = HR
r
( (Al.25)
ber
12 R
- + j bei
12
-R
0 0

From equations (Al.9) and (Al.2l):

J =-Ck(ber'kr + j bei'kr) (Al. 26)

r
From equations (Al. 23) and (Al. 26):

=~[J k ber'kr + j bei'kr

J
r R J (hl. 27)
ber kR + j bei kR

i.e. J
12
=--
ber - 12 6
r + j be1.
''./2 r
6 (Al. 28)
r 6 HR 2 2
ber - R + j bei - R
6 6

The total flux lying inside a radius r is given by equation (Al.l).

Inserting equation (Al.21) into equation (Al.l) yields:

r
...
"'r
= 21f)J J C(ber k x + j bei k x)x dx (Al. 29)
0

The integration of the Bessel functions [Al.2] are

r
0
x ber x dx = x bei'x (Al. 30)

r
0
x bei x dx = -x ber'x (Al. 31)

Substitute equations (Al.30) and (Al.3l) into equation (Al.29)

to yield:

q, =
2 TI)JCr (bei'kr - j ber'kr)
(Al. 32)
r k
Substitute C from equation (A1.23) into equation (A1.32) to produce:

2n)lH r
R bei'kr - j ber'kr
[ ber kR + j bei kR
J (Al. 33)
<j>r k

The total flux in unit length of the cylinder is;

bei'kR - j ber'kR
(1\l. 34)
[ ber kR + j bei kR

which can be simplified to;

(1\1.35)

where

p
2 ber kR bei'kR - ber'kR bei kR
(1\l. 36)
kR 2 .2
ber kR + be1 kR

and

2 bei kR bei'kR + ber kR ber'kR (Al. 37)

Q
kR
b er 2kR + be1.2 kR

P and Q are as shown in Fig. Al.l.

The power dissipated in the ring is:

p (Al. 38)
r

~ (Jdr)2 2nrp (1\1.39)

dr
 ./ 1.)

lO
0·9
0·8
07
0·6
....~ 0·5
"'>
()I
04
'd
~ 0-3
"' 0·2
0·1
4 6 7 8
0 1 2 3
workpiece diameter to current depth ratio, 2r /ISw
w

Fig. Al.l P and Q for a solid cylinder

2 (Al. 40)
J 2nrp dr

The total power per unit length of the cylinder can be calculated

by substituting equation (Al.26) into equation (Al.40) and

integrating

JR2nlcl 2k 2 p 2
(ber' kr +
2
i bei' kr) r dr (Al. 41)
0

Using the following integration of Bessel functions [Al.2]

2 . 2
JX (her• x + be1' x)dx x(berx ber'x + beix bei'x) (Al. 42)
 .::•

Equation (Al.41) becomes;

2
Pw = 2n[c[ PkR(ber kR ber'kR + bei kR bei'kR) (Al.43)

2
Substitute for [c[ from equation (Al.23) and yield;

2 [ber kR ber'kR + bei kR bei'kRJ

P = 2npkRH (Al. 44)
2
W R ber kR + bei 2kR

By using the expression of Q; equation (Al.44) can be rewritten as:

(Al.45)

HR is the r.m.s. value of the magnetic field strength. If the peak

value HRM is to be adopted, then:

(Al. 46)

The power density is:

PD (Al. 4 7)

PD (Al. 48)

or (Al.49)
 221

Appendix 2

The Reluctances of the Equivalent Circuit

The magnetic circuit of a long coil surrounding a workpiece is

shown in Fig. A2.1.

The reluctance R of a flux path of length ~ and cross sectional

m

area A is given by:

R (A2.1)
m IJ IJ A
o r

According to equation (A2.1) above, the reluctance of the air

gap is:

t
c
R (A2. 2)
mg 2 2
IJ 'll(r -r )
0 c w

The reluctance of the cylindrical workpiece is

N I
c c
R (A2. 3)
mw ~w

where ~w represents the total flux within the workpiece,

it is given below and is calculated from Appendix 1.

= 1J or
1J H A (P-jQ)
w
(A2. 4)

The Workpiece cross sectional area is,

2
A 1fr (A2. 5)
w w
mrnf=NI R R R
c c me mg mw

and the magnetic field strength is

N I
c c
H = (A2. 6)
li-
e

Substitute equations (A2.4), (A2.5) and (A2.6) into equation (A2.3)

to yield;

li-
e
R (A2. 7)
mw
2
ll ll n (P-jQ)
o r w

The coil reluctance can only be calculated if the spacing between

the turns is ignored for the time being. The coil can then

be approximated to a semi-infinite slab, as it's curvature is

small when compared to the conductor thickness. The latter is

greater than the penetration depth [A2.1), the total flux per unit
~)
width being:- \)

(/\2. 8)
2
 { ,-· . .'

As the relative permeability of copper is unity, and the width

is the coil perimeter, the coil flux can therefore be defined as;

k (2nr ) (1-j) (A2. 9)

~c = r c

where k is a correction factor, allowing for the spacing between

r /•
) ~

the turns. Baker [A2.2]suggested that 1.5 ~ k ~ 1.0 with 1.15

' r
6)
being a typical value. Reichert [A2.3] on the other hand employed the

following expression for k

r

k
r
= 0.92
M c c
(1\2 .10)

where the conductor width is t .

c
The coil reluc.tance is

N I
c c
R
me
= (1\2.11)
<Pc

Substitute equations (A2.6) and (A2.9) into equation (1\2.11) so as

to yield;

£
c
R = -~~----;c--c:- (A2. 12)
mc ~
o
nk a r (1-j)
r c c

Fig. A2.l above applies when the return flux is ignored. In

induction heating the flux will return along an external path,

see Figs. 2.1 and 2.2. The length of the return flux path

t is a function of: the coil radius; the difference between the coil
r
and the workpiece lengths; and the workpiece penetration depth.
 .., ,_.,
'

This length was found experimentally [A2.4, A2.5] to be;

Jl Jl Jl + 2(0.45 + k)r (A2 .13)

r c w c

where k is: a correction factor; a function of the air gap;

the difference between the coil and workpiece lengths; and the

workpiece penetration depth. The value of k lies between 0.0

and 0.1 for the return flow through air. If the flux returns

through iron, the reluctance will be smaller and, therefore, the

range of k will lie between -0.22 and 0.1 [A2.6]. For a coil and

workpiece of the same length, equation (A2.13) reduces to

Jl = 2r (0.45 + k) (A2.14)
r c

Hence, according to equation (A2.1), the reluctance of the

return path will be:

2rc(0.45+k) 2(0.45+k)
R
mr
= 2
(A2.15)
11 11r 11 11r
0 c 0 c
 '\ 1:

Appendix 3

Illustration of Mutually Coupled Circuit Method

To illustrate the use of this method in determining the current

distribution in inductively heated workpiece consider the coil-billet

configuration shown in Fig. A3.1, [A3.1]. The coil current is known and

distributed uniformly along the coil length. The workpiece

is divided into n subconductors along current streamlines. The

subconductor sizes need to be small near the surface and relatively

large in the inside region. The coil is inductively coupled

to each subconductor, also each subconductor is coupled to

every other one. This will result in the equivalent circuit of

Fig. A3.2, where all inductances are mutually coupled to all others.

This circuit can be described by the following matrix equation,

(A3. 1)

where z is the n x n square matrix of workpiece impedances.

w

I is the column matrix of unknown workpiece currents

w

I is the coil current

c
Z is the colwrm matrix of mutual impedances between the
cw
coil and the workpiece subconductors
Fig. A3.l Coil-workpiece

.....--~---- - - -------....---....

• L
• n

1-----+---- - - _ _ __....._ __,

Fig. A3.2 Equivalent circuit for Fig. A3.l

where

for k 'I t (A3. 2)

(A3. 3)

Z -j"'L (A3. 4)
cwk ck

k l , 2 , 3 , ... ,n

1 l, 2, 3, ... , n

~k is the resistance of subconductor k, Lkk is the self inductance

of subconductor k, Lkt is the mutual inductance between

subconductors k and t.Equation (A3.1) is a set of complex linear

algebraic equations and may be solved for the workpiece subconductor

currents I .
w

The formulae required to calculate the self and mutual inductances

of circular turns of rectangular cross section and the mutual

inductance between such a turn and a solenoid are given by Graver [A3.21.

(A3. 5)

where r is the mean radius of the coil

c
Br b2 2 b2
c 1 7Tb 1 c
tn + 12 2 tn(l + + - - tn(l + - )
1.1 -)
c 3c 2 2 2
c 12b c
2 2
c c -1 b
+ - - tn ( 1 + £_) + I<E. b) tan (A3.6)
2 b2 3 c c
12b
 2 8r l 2 . 2
c c ~,
2 c
2 l.n(l + b 2) J (l + 2
96r c c
c·

b2 2 2
221 b b -l b
+ 3.45 2 + ---1.6112 + 3.2 2 tan c
c 60 c c

2 b2 . 4 2
c JJ c
- - - in(l + - ) + - - in(l + - ) } (A3. 7)
2 2 4 b2
lob c 2c

and b and c are the breadth and depth of the segment. in is the

natural logarithm.

The expression of the billet mutual inductances is;

(A3. 8)

where r and r are the radii of the two circles, d is the

1 2
distance between their centres, and

2 v'r r
1 2
k (A3. 9)

F and E are integrals to modulus k. The values of these integrals

are calculated using polynomial approximations.

The following formula gives the mutual inductance between a

solenoid and a coaxial current filament

 2
1-loTia 2 2
. a A n+l J
M /,2 2 [l + ~ c ( 2 2 2) x2n (1\3. 10)
2 A +h n=O n (A +h )

where
3
c0 8
(A3 .11)

(2 (n+2) -1) for n >- 1

cn = cn-2 2n
(1\3.12)

n 2
and x2n =
LK (- .!:__) n-p (A3.13)
p=O p A2

where K = 1.0
0

[2 (n+l-p)] [2 (n+l-p) +1] p (1\3.14)

K = K for p ± 1
p p-1 2
4p (p+l)

where a is the radius of filament, A is the radius of solenoid,

and h is the height of solenoid.

In order to adopt this expression for the mutual inductance between

a particular segment and the coil several steps are required.

First, the segment to be divided into two filaments a and b, then the

coil is to be divided in the plane of the segment into lengths h 1

and h . The mutual inductance between the whole solenoid and filament
2
a is then:

M (1\3.15)
a
wherG M is the mutual inductance between the filament a and the
1
upper portion of the solenoid, M is the mutual inductance between
2
a .J.nJ the lower portion of the solenoid. 11 and 11 arc the
1 2
lengths of upper portion and lower portion of the solenoid and

h is the total length.

This procedure to be followed also for filament b. The mutual

inductance between the solenoid and segment k is

~c (A3.16)
 -~ :J _l

Illustration of Finite Element Methods

To explain the variational formulations procedure in mathematical

terms, consider the following linear diffusion equation

1 a2~
l1 + a/ 1 = (M. 1)

and ~ is specified on the boundary.

The solution of equation (A4.1) can be obtained by solving the

equivalent variational problem. This consists of expressing

the diffusion equation in terms of an energy functional and

minimizing it. The required functional is given by Chari [A4.1] as:

J J { (~)
1
Fun =
2!1
R
ax
2 + c:t> 2 J.ds
,-2
+
jw
2p
JJ ~ .ds- J J J.~ds (M. 2)
R R

Chari has shown, through complicated mathematics, that the Euler

equation of the above functional is the linear diffusion equation (A4.1)

The field region, R, is divided into subregions of rectangular

or triangular shapes. <P defined in each subregion as the linear

interpolate of it's vertex values. Ftu1ctional minimization is

then achieved by substituting the value of~ in equation (A4.2)

and setting it's derivative to zero vlith respect to each of

the vertex potentials. This will lead, after some algebra,

to the final matrix equation representing functional minimization

to be solved for the required unknowns.

To illustrate the application of the weighted residual procedure

to induction heating problems, consider a long ferromagnetic

cylinder of arbitrary cross section subjected to a longitudinal

magnetic field tangential to the surface , Fig. A4.1 [A4.2J.

The value of this magnetic field is;

(A4. 3)

For ferromagnetic material, the permeability will be a function of H

and temperature. While the resistivity is a function of

temperature only.

The electromagnetic problem can be solved in terms of the axially

directed field strength:

H = H(x, y)k (A4. 4)

which must satisfy the following partial differential equation

within the cylinder·

 Fig. A4.l Cylinder subjected to longitudinal magnetic field

d <lB
+- (114. 5)
3y dt

subject to the boundary condition

H ~ H coswt (A4. 6)
0

on the conductor surface. In the magnetic state, the relation

between B and H is governed by

B ~(H)H (114. 7)

The finite clement formulation of the system, described by the

partial differential equation (l\4.5) and the boundary conditions

of equation (A4.6} can bP obtained by usinq a ,;zllerkin pre;cc:durc,

which is a weighted residual technique. The applicution of this

method, usually requires a specialist numerical analyst as it

is lengthy and complicated. However, the solution for this

problem is known and the resulting matrix equation for the

unknown values of H at interior nodes can be taken from

Lavers [A4.2]:

[ [s (p) J + jw[T] [!!]] {H) 0 (A4. 8)

where [s] and [T] are the finite element matrices and [!!} is

a diagonal matrix containing nodal permeabilities.

Equation (A4.8) is to be solved by a suitable method, so as

to determine the magnetic field and power distributions within

the billet.
 Appendix S

Current Density Probes

TlH!se probes were used to measure the current density on the surfacR

of a conductor [AS.l-AS.s]. The probe consisted of two thin

wires laid along the surface of the conductor.

The entire length of the wires, with the exception of the two

extreme ends, was insulated electrically from the conductor~

The two ends were electrically connected to the conductor.

The other two ends of the wires were twisted together and connected

to a voltmeter. The probe was aligned with the direction of

the current in the conductor. The voltage across the probe was

equal to the resistance drop in the element of the conductor

imnediately adjacent to the wire plus the induced voltage in

the probe, i.e.

V V + V. (AS .l)
d ~

where V The voltage across the probe

Vd The resistance drop in the element of the conductor

adjacent to the probe

V. The voltage induced in the probe

~

Since vd ptJ (AS. 2)

V. Nd~ (AS. 3)
l dt

and f ~
B/1 (AS.~)
 where p The conductor resistivity

£ The probe length

J The current density on the surface of the conductor

N The number of turns

~ The flux in the probe-conductor circuit

B Magnetic flux in the probe-conductor circuit

A The area of the probe-conductor circuit

Substitute equations (A5.2), (A5.3) and (AS.4) into

equationr (AS.l), yields

d
V = p£J + dt (BA) (AS. 5)

The area A is very small as the probe was close to the conductor.

Equation (AS.5) can be written as

V = p£J (AS. 6)

For the measurements of the induced surface current density

on a cylinder, the thin wire was wrapped tightly around the billet

so that the emf induced on the surface of the cylinder was equal

to that on the probe. As the leads of the probe wer~ twisted together

then the voltmeter measured only the vol tagc on the surface of the

billet, which was

V = E£ (AS. 7)
 ......

where E The electric field intensity, which is

E = pJ (A5.8)

and ~ = 21TR (AS. 9)

substitute equations (A5.8) and (A5.9) into equation (A5.7) to yield,

V 21fRpJ

where R was the billet radius.

 REFERENCES

1.1 IIOBSON, L.: "Guide to induction heating equipment", (BNCE, London)

1984.

1. 2 BAKER, R.M.: "Heating of non-magnetic electric conductors by

magnetic induction- longitudinal flux", AIEE Trans, 1944,

Vol.64, pp. 273-278.

1.3 BAKER, R.M.: "Design and calculation of induction heating coils",

AIF.E Trans, 1957, Vol.76, Pt.II, pp. 31-40.

1.4 REICHERT, K.. : "The calculation of careless furnaces with

electrically conducting crucibles:•, Electrotechnik, 1965,

Vol.49, No.6, pp. 376-397.

1.5 VAUGHAN, J.T. and WILLIAMSON, J.W.: "Design of induction-heating

coils for cylindrical non-magnetic loads", AIEE Trans, 1945,

Vol.64, pp. 587-592.

1.6 VAUGHAN, J.T. and WILLIAMSON, J.lv.: "Design of inducrion-heating

coils for cylindrical magnetic loads .. , l\IEE Trans, l91JG,

Vol.65, pp. 887-892.

1. 7 REICHERT, K.: "A numerical method for calculating induction

heating installations .. , Electrowarme Int., 1968,

Vol.2G, No.4, pp. 113-123.

1.8 DENERDASH, N.A .. , MOHAJ11NED, O.A., NEHL, T.W. and MILLER, R.H.:

1
'Solution of eddy current problems using three dimensional

finite element complex magnetic vector potential .. , IEEE Trans,

1982, Vol.PAS-101, No.ll, pp. 4222-4229.

1.9 KOLBE, E. and REISS, W.: "Distribution in space of current

density in induction heated bodies under consideration of the

temperature field", Elcctrowarmc, 1967, Vol.25_, pp. 243-250.

2.1 See reference 1.3

2.2 See reference 1.4

?..3 See reference 1.5

2.4 See reference 1.6

2.5 AL-SHAIKHLI, A.K.M.: 11

Methods of induction billet heating

work coil design", M.Sc. Dissertation, Electronic and

Electrical Engineering Department, University of Technology,

Loughborough, 1981.

2.6 See reference 1.9

2. 7 DUDLEY, R.E. and BURKE·, P.E.: "The prediction of current

distribution in induction heating installations", IEEE Trans,

1972, Vol.IA-8, pp. 565-571.

2.8 GROVER, F.W.: "Inductance calculations-working formulas and tables

(Von Nostrand, New York) 1946.

2.9 LAVERS, J.D.: Private conmnllli.cations.

2.10 SILVESTER, P .. P. and Rl\FlNEJl\D, P.: "Curvilinear finite elcr:1ents for

2-dimensional so.turable !'lo.gnctic fields", IEEE Trans, 1974,

Vol.PAS-93, pp. 1861-1870.

2.11 COGGON, J.H.: "Electromagnetic and electrical modelling by

the finite element method", Geophysics, 1971, Vol. 36, pp. 132-155.

2.12 DONEA, J., GIULIANI, S. and PHILIPPE, A.: "Finite elements in

the solution of electrimagnetic induction problems", International

Journal for Numerical Methods in Engineering, 1974, Vol.8,

pp. 359-367.

2.13 CHARI, M.V.K.: 11

Finite-elcment solution of the addy-current

problem in magnetic structurr!S 11 , IEEE Trans, 1973, Vol.Pl\S-93,

No.l, pp. 62-72.

2.14 See reference 1.8

2.15 NEMCYI'O, K. and TABUCHI, M.: "Thermal analysis of induction

heating by the fini tc element using a computer", lOth Conqrcss

of the International Union for Electroheat, Stockholm, 18-22 June

1984.

2.16 LAVERS, J. D.: "Finite element solution of non linear two

dimensional TE-Mode eddy-current problems", IEEE Trans, 1983,

Vol.MAG-19, pp. 2201-2203.

2.17 SABONNADIERE, J .C.: "Methods interactives de calcul des systems

de chauffage par induction", lOth Congress of Electroheat, Stockholm,

18-22 June, 1984.

2.18 HOLMDAHL, G. and SUNDBERG, Y.: "The calculation of induction

heating by the use of computers", paper 633, Vth Int. Congress

on Electroheat, Wiesbaden, BRD, 30 Sept-5 Oct, 1963.

2.19 See reference 1.7

2.20 GIBSON, R.C.: "BIEDDY, A computer program for calculating the

induction and other heating of long rectangular and other

regularly shaped slabs and billets", ECRC/MM20, 1974.

3.1 See reference 1.3

3.2 NOSOVA, L.N.: "Tables of Thompson functions and their

first derivatives", (Pergamon Press, Basus), 1961.

3. 3 DAVIES, J. and SI MP SON, P. : "Induction heating handbook"

(McGraw-Hill, Maidenhead), 1979.

3.4 See reference 2.20

3.5 Sec reference 1.4

 .....
.-, ''

4.1 See reference· 3.3

4.2 PEEK, F.W.: "Dielectric phenomena in high-voltage

engineering", (t·1cGraw-Hill, New York), 1929.

4.3 KRAUS, J.D.: "Electromagnetics", (McGraw-Hill, New York), 1953.

4. 4 HAYT, W.H.: "Engineering electromagnetics", (McGraw-Hill, New

York), 1967.

4. 5 CALLAGHAN·, E.E. and MASLEN, S.H.: "The magnetic field of a

finite solenoid", National Aeronautics and Space Administration,

Technical Note D-465.

5.1 BURKE, P.E. and ALDEN, R.T.H.: "Current density probes",

IEEE Trans, 1969, Vol.PAS-88, No.2, pp. 181-185.

5.2 See reference 3.3

5.3 See reference 1.1

5.4 FISGiBR, J. and MOSER, H.: "Die nachbildung von

rnagnetisieru.I1gskurven durch einfache algcbraiche oder

transzendente funktionen", Archiv Electrotechnik, 1956,

Vo1.42, pp. L86-299.

5. 5 WIDGER, G.F.T.: .. Representation of magnetisation curves

over extensive range by rational-fraction approximations·•,

IEE Proc., 1969, Vol.ll6, No.1, pp. 156-160.

5.6 BRIANSKY, D.: "Single analytical formula for the entire B-H

curve", Int. J. Eng. Educ., 1967, Vol.5, pp. 199-201.

5. 7 LIM, K.K. and HAMMOND, P.: 11

Universal loss chart for the

calculation of eddy-current losses in thick steel plates",

lEE Proc., 1970, Vol,_l..!:.:?_, No.4, p[>. B'i7-8G4.

5.8 See reference 2.20

5.9 G!l3SON, H..C.: "SLEDDY, a comput.er program for calculating

the induction and other heating of metal slabs and long

cylindrical billets", ECRC/MM16, 1973.

5.10 THORNTON, C.A.M.: "Resistance heating of mild steel

containers at power frequencies .. , IEE Proc., 1952, Vol.99,

Pt. II, pp. 85-93.

5.11 FAM, w.z.: "Measurements of losses in saturated solid

magnetic cores", IEEE Trans, 1971, Vol.MAG-7, No.l, pp. 198-201.

6.1 Ci\RSLAW, H.S. and JAEGER, J.G.: "Conduction of heat in

solids", (Oxford University Press- Oxford), 1959.

6. 2 BAKER, R.M.: "Classical heat flow problems applied to

induction billet heating", AIEE Trans., 1958, Vol.77,

pp. 106-112 0

6. 3 DREYFUS, L. A. : "High frequency t:ea ting and temperature

distribution in surface hardening of steel 11 , (l~oyal Swedish

Academy of Engineering Sciences- Stockholm), 1952.

6.4 See reference 3.3

7.1 See reference 5.10

Al.l WARREN, A.: "Mathematics applied to electrical Engineering",

(Chapman and Hall Ltd, London), 1958.

Al.2 DWIQJT, H.B.: ''Tables of integrals and other mathematical

data", (Macmillan, New York), 1961.

i\2.1 Sec reference 3.3

i\2.2 St•e reference l. 3

 A2.3 See reference 1.4

1\::! . -1 SCHOENBACI-IER, K.: "Zur bcrechnung von induktionsofcn",

ETZ-1\, 1952, Vol. 73, pp. 736-738.

A2.5 SCHOENBACHER, K.: "Induktivitiit einlagiger zyli.nderluftspulen",

Electrotechnik, 1949, Bd.3, Nr.lO, pp. 327-329.

1\2.6 SIEGERT, H.: "Inductive heating .. , Techn. Rundschau, l9hl,

Nos.l3 and 38.

A3.1 See reference 2. 7

A3.2 See reference 2.8

1\4.1 See reference 2.13

A4.2 See reference 2.16

1\5.1 DANNATT, C. and REOFEARN, S.W.: "The efficient utilization of

conductor material in busbar sections" 1 tvorld ?ower 1 Dec.l930,

yol.XIV, No.LXXXIV, pp. 492-496.

1\5.2 DALEY, J. L.:

11
Current distribution in a rectangular conductor .. ,

AIEE Trans., 1939, Vol. 58, pp. 687-690.

A5.3 01\VIES, E.J.: "An experimental and theoretical study_of

eddy-current couplings and brakes", IEEE Trans., Aug.l963,

Vol.82, pp. 401-419.

A5.4 See reference 5.1

1\5.5
BOWOEN, A.L. and 01\VIES, E.J.: "Analytic separation of the

factors contributing to the 0ddy-current loss in

magnetically nonlinear steel", IEE Proc., Sept.l983,

V__()_!_:_l30, Pt.B, No.S, pp. 364-373.

 The Listing of the Programs and the Publicat.ions

i. The Listing of the Progr<Jms:

SHORT

LONG

EQUIV

BESSEL

slab

W-P-PO\vER

PROXIMITY

W-SC-FIT

FITTINGl

\v-FC-TEt1P

ALl

Magnetic Material

Supe rposi tion

ii. Publications:

1. 11
Improvements in the design of induction billet heaters .. ,

19th UPEC, University of Dundee, April 1984.

2. "Methods of analysis for induction heating work coils",

20th UPEC, Huddersfield Polytechnic, April 1985.

J. "Illustrating electromagnetics using an industrial process .. ,

IJEEE, 1986, Vol.23, No.l. (to be published)

SLIST SHORT
REAL PS,V,F,DW,ROWW,URW ALC,DC,ROWC,URC,AKR,AKE,P,Q
REAL BER,BEI,KER,KEI,BERD,BEID,KERD,KEID
COMPLEX THETA,THY,KO,IO,KOD,IOD
READC5,*lPS,V,F,DW,ROWW,URW,ALC,DC,ROWC,URC,AKR,AKE
PI =3 .14159265
PIS=9.8696
X=10000000.0
DDW=SQRTC<ROWW*Xl/C4.*PIS*F*URWll
DDC=SQRTCCROWC*Xl/C4.*PIS*F*URCll
BA=DW/C1.414214*DDWl
C CALCULATION OF BESSEL FUNCTIONS
IF<BA.GT.8lGO TO 5001
A1=BERCBAl
A2=BEI<BA)
A3=BERDCBAl
A4 =BEID <BA l
GO TO 5002
5001 A1=REAL<IOCBAll
A2=AIMAGCIOCBAll
A3=REALCIODCBAl l
A4=AIMAGCIODCBAll
5002 CONTINUE

DEN= <<A 1 l * <A 1 l l + CCA2 l * CA2 l l

B1 = CCA1 l* CA4 l l- <CA2 >* CA3 l l
B2 = <<A 1 l * CA3 l l + CCA2 l * CA4 l l
C1=B1/DEN
C2=B2/DEN
P=2.0*C1/BA
Q=2.0*C2/BA
AK=C8.*PIS*Fl/CX*ALCl
AW=PI*CDWl**2/4.
AC=PI*CDCl**2/4.
AG=AC-AW
PC=PHDC
RW=URW*AW*Q
RC=0.5*AKR*PC*DDC
XC=RC
XW=URW*AW*P
RE=AKE*1.8/PC
XE=ALC/RE
X1=AG+XW+XC
RW1=CRW*CXEl**2l/CRW**2+CX1+XEl**2l
XO=XE* CRW**2+X1 **2+XE*X1 l/ CRW**2+ CX1 +XE >**2 l
RO=RC+RW1
Z=SQRTCR0**2+X0**2l
 PF=RO/Z
WRITEC1 ,5JPF
5 FORMATC1H0,4HPF =,F6.4J
EFF=CRW1/ROl*100.0
WRITEC1 ,6JEFF
6 FORMATC1H0,13HEFFICIENICY =,F7.4J
VA=PS/PF
ZO=Z*AK
WRITE C1 , 10 JZO
10 FORMATC1H0,4HZO =,F12.10J
EC=SQRTCVA*ZOJ
WRITE C1 , 9 JEC
9 FORMATC1H0,4HEC =,F9.5J
AIN=SQRTCVA/ZOJ
WRITE C1 ,13 lAIN
13 FORMATC1H0,5HAIN =,F12.3J
TN=V/EC
WRITE C1, 7 JTN
7 FORMATC1H0,4HTN =,F7.2J
AI=AIN/TN
WRITE ( 1 , 8 JAI
8 FORMATC1H0,9HCURRENT =,F10.4J
STOP

END
$INSERT BESSEL

OK,
SLIST LONG
REAL PS,V,F,DV,ROVV,URV,ALC,DC,ROVC,URC,AKR,P,Q
REAL BER,BEI,KER,KEI,BERD,BEID,KERD,KEID
COMPLEX THETA,THY,KO,IO,KOD,IOD
READ<5,*lPS,V,F,DV,ROVV,URV,ALC,DC,ROVC,URC,AKR
DDV'503.3*SQRT<ROVV/(F*URVll
DDC=503.3*SQRT<ROVC/(F*URCll
X=1000000.0
BA=DV/(1.414214*DDVl
C CALCULATION OF BESSEL FUNCTIONS
IF<BA.GT.8lGO TO 5001
A1=BER<BAl
A2=BEI <BA l
A3=BERD<BAl
A4=BEID<BAl
GO TO 5002
5001 A1=REAL<IO<BAll
A2=AIMAG(IO<BAll
A3=REAL<IOD<BAl l
A4=AIMAG<IOD<BAll
5002 CONTINUE
DEN= ( (A 1 l * (A 1 l l + C<A2 l * CA2 l l

B1 = CCA 1 l * CA4 l l- ( ( A2 >* <A3 l l

82 = ( <A 1 l * <A3 l l + C<A2 l * <A4 l l
C1=B1/DEN
C2=B2/DEN
P=2.0*C1/BA
Q=2.0*C2/BA
SP=<7.896*Fl/CALC*Xl
AV=0.785*<DV**2l
AC=0.785*<DC**2l
AG=AC-AV
PC=3.14159265*DC
AP=URV*AV*Q
AA=0.5*AKR*PC*DDC
TB=AA
TF=URV*AV*P

TA=AG
EFF=<AP/CAA+APll*100.0
VRITE<1 ,13lEFF
13 FORMATC1H0,13HEFFICIENICY =,F6.3l
Z=SQRT<<<AP+AAl**2l+<CTA+TF+TBl**2ll
PF=<AP+AAl/Z
VRITE<1 ,15lPF
15 FORMATC1H0,4HPF =,F6.4l
VA=PS/PF
TR=Z*SP
E=SQRTCVA*TRl
AIN=SQRT<VA/TRl
TN=<V/El
WRITE C1 , 1 8 l TN
18 FORMAT<1H0,4HTN =,F7.2l
AI=AIN/TN
WRITE C1 , 20 lA I
20 FORMAT<1H0,9HCURRENT =,F10.4l
STOP
END
$INSERT BESSEL
OK,
SLIST EQUIV
c *********************************************************************
c COIL OF CONCENTRIC CYLINDERS
c THIS PROGRAMME APPROXIMATES A BILLET AS N CONCENTRIC CYLINDERS
c ********************************************************************~
c
REAL*4 L,K1 ,K,N1 ,I1
INTEGER S,P
DIMENSION XPTC100l,EFFC100l,W1 C100l,I1 C100l,Q1 (100),
*RESC100l,URC100l
WRITEC1,800l
800 ·FORMATC//'H0\1 MANY CONCENTRIC CYLINDERS ARE REQUIRED'//)
READC1 ,*lN
WRITE ( 1 , 15 l
15 FORMAT(//' ENTER 1 IF RANGE OF VALUES REQD'/' ENTER 2 IF ONE •
*'VALUE REQD'//l
READ ( 1 , * l S
IFCS.EQ.1 l\IRITEC1 ,16)
16 FORMATC//' WHICH PARAMETER DO YOU WISH TO VARY'/'ENTER 1 FOR F'
*I' 2 FOR N1'/' 3 FOR L'l' 4 FOR RB'/
*' 5 FOR RC'//)
IFCS.EQ.1 lREADC1 ,*lM

\/RITE (1, 1 l
1 FORMAT(//' **** NCOIL ****'//)
C ***** INPUT RESISTIVITY PROFILE *****
c
WRITE ( 1 ,803)
803 FORMATC//'ENTER RESISTIVITY OF INNER CYLINDER,RES1 '//)
READ ( 1 , * l RES ( 1 l
WRITEC1,5000l
5000 FORMATC//'ENTER RESISTIVITY OF OUTER CYLINDER,RES2'//l
READ ( 1 , * JRES C2 l
WRITEC1,776l
776 FORMAT(//'ENTER INITIAL PERMEABILITY,URCOJ '//)
READC1,*lURO
WRITEC1 ,777)
777 FORMATC//'ENTER FIRST CONSTANT OF PERMEABILITY,AF '//)
READ(1 ,*lAAF
WRITE ( 1 , 77 8 l
778 FORMATC//'ENTER SECOND CONSTANT OF PERMEABILITY,BF '//)

READ(1 ,*lBBF
WRITEC1,779l
779 FORMATC//'ENTER THIRD CONSTANT OF PERMEABILITY,CF '//)
READC1 ,*lCCF
\/RITE ( 1 , 2 l
2 FORMATC//'ENTER VOLTAGE,U1'//l
READC1,*JU1
IFCM.EQ.1 JGO TO 100
WRITE ( 1 , 3 l
3 FORMAT(//'ENTER FREQUENCY,F'//l
READC1 ,*>F
IFCM.EQ.2lGO TO 103
100 WRITEC1,4l
4 FORMATC//'ENTER POWER,N1'//l
READC1 •* lN1
IFCM.EQ.3lGO TO 105
103 WRITEC1,7l
7 FORMATC//'ENTER LENGTH,L'//l
READC1 ,*lL
IFCM.EQ.4lGO TO 106
105 WRITEC1,9l
9 FORMATC//'ENTER BILLET RADIUS,RB'//)
READC1 ,*lRB
106 WRITEC1,10l
10 FORMATC//'ENTER COIL RESISTIVITY,RESC'//l
READ C1 , * l RESC
WRITE ( 1 , 1 2 l
12 FORMATC//'ENTER K1'//)
READC1,*lK1
IFCM.EQ.5lGO TO 107
WRITE C1 , 1 3 l
13 FORMATC//'ENTER COIL RADIUS,RC'//l

READC1 ,*>RC
107 WRITEC1,14l
14 FORMATC//'ENTER K'//)
READC1,*>K
I F CS • E Q • 1 l WR I TE <1 , 1 7 l
17 FORMAT(//' HOW MANY DATA POINTS ARE THERE ,LESS THAN 100 ONLY'//)
IF CS.EQ.1 lREAD<1 •* lNN
IF <S • EQ. 1 l WRITE <1 , 1 8 l
18 FORMAT(//' ENTER PARAMETER VALUES,LESS THAN 100 ONLY'//)
IF< S. EQ. 1 l READ< 1 , * l <XPT CI l , I= 1 , NN l
IF<S.EQ.1 >GO TO 19
CALL SUBCU1 ,N,RES,UR,F,N1 ,L,RB,RESC,K1 ,RC,K,EFFC1 l,W1 <1 l
*, I 1 <1 l , Q 1 ( 1 l , URO , AAF , BBF, CCF l
GO TO 21
19 CONTINUE
DO 20 P=1,NN
IF<M.EQ.1 >CALL SUBCU1 ,N,RES,UR,XPTCPl,N1 ,L,RB,RESC,K1 ,RC
*,K,EFF<Pl,W1 CPl,I1 CPl,Q1 CPl,URO,AAF,BBF,CCFl
IF CM.EQ.2 !CALL SUB<U1 ,N,RES,UR,F ,XPTCPl ,L,RB,RESC,K1 ,RC
*,K,EFF <P>, W1 CPl, !1 CPl ,Q1 CPl ,URO,AAF ,BBF ,CCF)
IF<M.EQ.3lCALL SUBCU1 ,N,RES,UR,F,N1 ,XPTC

Pl ,RESC,K1 ,RC,K,
*EFF CPl ,W1 CPl, !1 <Pl ,Q1 (Pl ,URO,AAF ,BBF ,CCF)
IF<M.EQ.4lCALL SUBCU1 ,N,RES,UR,F,N1 ,L,XPTCPl,RESC,K1 ,RC
*,K,EFF<Pl,W1 CPl,I1 <Pl,Q1 <Pl,URO,AAF,BBF,CCFl
IFCM.EQ.5lCALL SUBCU1 ,N,RES,UR,F,N1 ,L,RB,RESC,K1 ,XPTCPl
*,K,EFF<Pl,W1 CPl,I1 CPl,Q1 C
20 CONTINUE
C **** RESULTS ****
c
YR I TE ( 1 , 2 2 l
22 FORMAT!//' ****RESULTS****'//)
c
21 IF cS . EQ . 1 l GO TO 2 3
YR I TE ( 1 , 2 4 l
24 FORMAT!//' EFF Y1 I1 '
* Q1 '//)
YRITEC1 ,25lEFFC1 l,Y1 (1 l,I1 (1 l,Q1 (1 l
25 FORMATC5E14.5l
GO TO 500
23 YR I TE ( 1 , 26 l
26 FORMAT!//' XPTC!l EFF Y1 I1
*' Q1 '//)
DO 27 I= 1 , NN
\IRITEC1 ,28lXPTCil,EFFCil,Y1 (!l,I1 Cll,Q1 c I l
28 FORMATC6E13.5l
27 CONTINUE
500 CALL EXIT
END
c
SUBROUTINE SUBCU1 ,N,RES,UR,F,N1,

L,RB,RESC,K1 ,RC,K,EFF,
*W'1 ,!1 ,Q1 ,URO,AAF,BBF,CCFl
c
REAL*4 L,K1 ,K,N1 ,KERD,KEID,KER,KEI ,KK,I1
COMPLEX I OT, I ODT, KOT, KODT, RF, Z! 1 0 l , C ( 1 0 l , H ( 1 0 l , JD ( 1 0 l , A,
*ZAA,ZZ,IO,KO,IOD,KOD,AFC10,10l
INTEGER P,N,M,T
DIMENSION IOTC10,10l,IODTC10,10l,KOTC10,10l,
*RESC10l,URC10l,RRC10l,XC10,10l,AC20,20l
*, KODT ( 1 0, 1 0 l , RC 1 0 l , SD ( 1 0 l , YT1 ( 1 0, 1 0 l , YT2 ( 1 0, 1 0 l
* , YT3 C1 0 , 1 0 l , YT4 ( 1 0, 1 0 l , YT5 C1 0 , 1 0 l , YT6 ( 1 0, 1 0 l ,
* YT7 ( 1 0, 1 0 l , YT8 C1 0, 1 0 l , XD ( 1 0 l , Y ( 1 0 l
c
D=RB/N
R ( 1 l =D
IFCN.EQ.1 lGO TO 802
DO 802 I=2,N
RC! l=RCI-1 l+D
802 CONTINUE
PIE=3.1415926
U0=4*PIE*1E-7
Y=2*PIE*F
PC=0.92

AM=PIE*CRC**2-RB**2l
DDRR=CRESC2l-RESC1 ll/CN-1 l
DO 5001 I=2,N
RESCI l=RESCI-1 l+DDRR
5001 CONTINUE
C ***** CALC. OF SKIN DEPTHS *****
c
TT
E=O
UR ( 1 l = URO
DO 1500 JP=1 ,N
1 51 0 DO 807 I=1 ,N
IF CTTE.EQ.1 lSDCil=SQRTC2.0*RESCil/CW*UO*URCilll
IF CTTE .NE.1 lSDC I l=SQRTC2.0*RESC I l/ CW*UO*URCJPl l l
807 CONTINUE
SDC=SQRTC2.0*RESC/CK1*W*UOll
c ***** CALC. OF XDCil *****
c
DO 808 I=1 ,N
XDCil=-SQRTC2.0l/SDCil
808 CONTINUE
RFCR=PC*RESC*RC*2.0*PIE/CK1*L*SDCl
RFCI=RFCR
RF=CMPLXCRFCR,RFCil
XR=W*UO*PIE*RC/C2.0*(0.45+Kll
XO=W*UO*AM/L
DO 806 I=1 ,N
RRCI l=-RESCI l*2.0*PIE*RCI l/L
806 CONTINUE
c **

*** CALC. OF XCI,Jl ****

c
XC1,1 l=SQRTC2.0l*CRC1 l/SDC1 l l
IF (N. EQ. 1 l GO TO 811
DO 810 I=2,N
XC I , I l = SQRT ( 2 • 0 ) * CR ( I l I SD ( I l l
XCCI-1 l,Il=SQRTC2.0l*CRCI-1 l/SDCill
810 CONTINUE
C ***** CALC. OF YCil *****
c
c ****************************************************************
M= CN-1 l
T=N-2
MM=2*N
MN= C2*N-1 l
NM=C2*N-2l
c ****************************************************************
DO 811 I= 1 , M
YCI l=RESCI l/RESCI+1 l
811 CONTINUE
c ********************************************************************
c
CALC. OF BESSELL FUNCTIONS
c *********************************************************************
c
I F (X ( 1 , 1 l • GT • 8 l GO TO 1 0 1
YT1 ( 1 ,1 l =BER CX ( 1 ,1 l l
YT2 ( 1 , 1 l =BE I (XC 1 , 1 l l
YT3 ( 1 , 1 l =BERD (XC 1 , 1 l l
YT4 ( 1 , 1 l =BE ID CXC 1 , 1 l l
YT5 ( 1 , 1 l = KER (X ( 1 , 1 l l
 ----- --:---- - -------------------------

YT6 ( 1 , 1 l = KE I (X ( 1 , 1 l l
YT7 (1, 1 l=KERDCXC1, 1 l l _
YT8 ( 1 , 1 l = KE ID (X ( 1 , 1 l l
IOTC1 ,1 l=CMPLXCYT1 (1 ,1 l,YT1 (1 ,1 ll
I ODT ( 1 , 1 l = CMPLX ( YT3 ( 1 , 1 l , YT4 ( 1 , 1 l l
KOT ( 1 , 1 l = CMPLX ( YT5 ( 1 , 1 l , YT6 ( 1 , 1 l l
KODT ( 1 , 1 l = CMPLX ( YT7 ( 1 , 1 l, YT8 ( 1 , 1 l l
GO TO 100
1 01 I OT ( 1 , 1 l = I 0 (X ( 1 , 1 l l
I ODT ( 1 , 1 l = I OD (X ( 1 , 1 l l
KOT ( 1 , 1 l =KO (X ( 1 , 1 l l
KODT ( 1 , 1 l = KOD (X ( 1 , 1 l l
1 00 CONTINUE
IF CN.EQ.1 JGO TO 620
DO 102 I=2,N
IFCXCI,Il.GT.8lGO TO 103
YT1 CI,Il=BERCXCI,Ill
YT2 ( I , I l =BE I (X (I , I l l
YT3 ( I , I l = BERD (X ( I , I l l
YT4 ( I , I l =BE ID (X (I , I l l
YT5 ( I , I l =KER (X ( I , I l l
YT6 (I , I l = KE I (X ( I , I l l
YT7 (I , I l =

KERD (X <I , I l l
YT8 ( I , I l = KE ID< X ( I , I l l
I OT ( I , I l = CMPLX ( YT1 ( I , I l , YT2 ( I , I l l
I ODT ( I , I l = CMPLX ( YT3 (I , I l , YT4 (I , I l l
KOTII,Il=CMPLXIYT51I,Il,YT61I,Ill
KODT<I,Il=CMPLX<YT7CI,Il,YT81I,Ill
GO TO 104
10 3 I OT ( I , I l =I 0 (X ( I , I l l
I ODT ( I , I l = I OD (X ( I , I l l
KOTII,Il=KOIXII,Ill
KODT (I , I l = KOD (X (I , I l l
104 CONTINUE
IF (X ( I -1 , I l • GT. 8 l GO TO 1 05
YT1 ( ( I -1 l , I l = BER (X< ( I -1 l , I l l
YT2 ( ( I -1 l , I l =BE I (X ( (I -1 l , I l l
YT3 ( ( I -1 l , I l = BERD (X ( ( I -1 l , I l l
YT4 ( ( I -1 l , I J =BE ID (X ( (I -1 l , I J l
YTS ( <I -1 J , I J = KER (X ( ( I -1 J , I J l
YT6 ( <I -1 J , I J = KE I (X ( (I -1 J , I ) )
YT711I-1 l,Il=KERDCXIII-1 J,Ill
YT8!(I-1 l,IJ=KEIDCXIII-1 l,IJJ
I OT ( (I -1 l , I J = CMPLX <YT1 < <I -1 l , I J , Y

T2 ( (I -1 J , I J l
I ODT ( <I -1 l , I l =CMPLX ( YT3 ( (I -1 J , I J , YT4 ( <I -1 J , I l J
KOT ( (I -1 l , I J = CMPLX ( YT5 ( (I -1 l , I J, YT6 ( (I -1 J , I l l
KODT ( (I -1 l , I l =CMPLX ( YT7 ( (I -1 J , I l , YT8 ( (I -1 J , I l l
GO TO 102
1 05 I OT ( ( I -1 l , I J =I 0 <X ( ( I -1 J , I l J
IODT«I-1 l,Il=IOD<X«I-1 l,Ill
 KOT ( CI -1 l , I l =KO (X ( CI -1 l , I l l
KODT ( (I -1 l , I l = KOD (X ( (I -1 l , I l l
102 CONTINUE
c *****************************************************************
C SOLUTION OF N SIMULTANEOUS EQUATIONS
c *****************************************************************
c
c ***************************************************************
C SETTING MATRIX OF COEFFICIENTS
c
C **** SET ARRAY TO ZERO ****
c
DO 900 I=1 ,MM
DO 901 J=1 ,MN
ACJ,Il=O
901 CONTINUE
900 CONTINUE
A ( 1 , 1 l =I OT ( 1 , 1 l
A ( 1 , 2 l = - I OT ( 1 , 2 l
A ( 1 , 3 l =- KOT ( 1 , 2 l
AC2 ,1 l=IODTC1 ,1 l*YC1 l
A ( 2 , 2 l =-I ODT ( 1 , 2 l
A C 2 , 3 l =- KODT ( 1 , 2 l
A

CMN,NMl=IOTCN,Nl
ACMN,MNl=KOTCN,Nl
ACMN,MMl=1
IFCN.LE.2lGO TO 902
DO 902 I=2,M
J=2*I-1
JI=J-1
A ( J , J I l =I OT ( I , I l
A(J, CJI+1 l l=KOTCI ,I l
A ( J , ( J I + 2 l l =-I OTC I , CI + 1 l l
ACJ, (JI+3ll=-KOTCI, CI+1 ll
A ( CJ + 1 l , J I l = I 0 DT ( I , I l * Y ( I l
A ( ( J + 1 ) , ( J I + 1 ) ) = KODT ( I , I l * Y ( I l
A ( ( J + 1 l , ( J I +2) l =-I ODT (I , (I+ 1 l l
AC (J+1 l, CJI+3) )=-KODTCI, CI+1 l l
902 CONTINUE
DO 700 I=1 ,MN
DO 61 2 J = 1 , MN
C ***** EVALUATE PIVOT COEFFICIENT *****
c

IFCJ.EQ.IlGO TO 612
AF ( J , I l =A ( J , I l I A ( I , I )
DO 61 3 J J = 1 , MM
C *****EVALUATE LINE (J-I) *****
c
ACJ,JJl=ACJ,JJl-AFCJ,Il*ACI,JJl
613 CONTINUE
612 CONTINUE
DO 61 ~ J • 1 , MN
IFCJ.EQ.IlGO TO 614
ACJ,Il•O
c ***** THIS IS TO REDUCE ROUND-OFF ERROR ****
614 CONTINUE
700 CONTINUE
.C ***** CALC. OF CONSTANTS CCil *****
c
DO 620 I •1 ,MN
CC I l • A CI , MM l I A (I , I l
620 CONTINUE
IFCN.NE.1 lGO TO 750
c ***************************************************************
c FOR N•1 THERE IS ONLY ONE CONSTANT GIVEN BY:-
CC1 l•1 .O/IOTC1 ,1 l
JDC1 l• C-SQRTC2.0l*IODTC1, 1 l*CC1 l l/SDC1 l
GO TO 210
c ***************************************************************
c
c ****************************************************************
c
CALC. OF JDCil AND HCil
c ****************************************************************
c
750 HC1 l•CC1 l*IOTC1 ,1 l
JDC1 l• C-SQRTC2.0l*IODTC1 ,1 l*CC1 l l/SDC1 l
DO 200 I•2,M
J•2*I-1
HCI l: C ( J -1 l* I OT ( I , I l +CC J l *KOT CI , I l
JD CI l • XD CI l * CC CJ -1 l *I ODT CI , I l +CC J l * KODT < I , I l l
200 CONTINUE
JDCN l •XDCN l *CC C2*N-2 l*IODTCN ,N l +C C2*N-1 l*KODTCN ,Nl l
URCJP+1 l•C CAAF/CBBF+HCJPlll+CCFl/UO
1500 CONTINUE
IF CTTE.EQ.1 lGO TO 1520
TTE• 1
GO TO 1510
1520 CONTINUE
c ****************************************************************
c CALC. OF IMPEDANCES ZCil
c ****************************************************************
c
DO 210 I • 1 , M

21 0 CONTINUE
ZCNl•RRCNl*JDCNl
AA•AIMAGCZCNll
B•REALCZCNl l
KK•XR/CXR+XO+AAl
ZB•KK*B
ZA•CXO+AAl*KK
ZAA•CMPLXCZB,ZAl
ZZ•RF+ZAA
WRITEC1,1755lZZ
1755 FORMATC//'ZZ•',
2E14.5l
ZBR=REAL(ZZl
ZAI=AIMAG(ZZl
ZM=CABS<ZZl
C ZM IS MAGNITUDE OF FURNACE IMPEDANCE ZZ
c
C ZAI=IMAGINARY PART OF ZZ
C ZBR=REAL PART OF ZZ
c
V1=(U1/ZMJ•<SQRT(ZBR/N1 ll
I1=U1/((V1••2l•ZMl
Q1=(I1••2l•<V1••2l•ZAI
EFF=100•KK•BI<RFCR+KK•Bl
RETURN
END
c
$1 NSERT BESSEL
c
COMPLEX FUNCTION DIV<Y,Xl
COMPLEX Y,X
REAL MODA,MODB,MODX
c
C ****CALC. OF COMPLEX DIVISION *

***
C **** CALC. OF Y/X ****
c
A=REAL<Xl
B=AIMAG<Xl
MODA=ABS<Al
MODB=ABS(Bl
Z= 1 . 0
IF<MODA.GT.MODBJGO TO 500
IF<MODB.LT.0.1 JGO TO 501
502 IF(MODB.LT.1 lGO TO 510
X=X/10
MODB=MODB/10
Z=Z•10
IF<MODB.GT.1 JGO TO 502
GO TO 510
501 IF<MODB.LT.0.1 lX=X•10
Z=Z/10
MODB=MODB•10
IF<MODB.LT.0.1 lGO TO 501
GO TO 510
500 IF<MODA.LT.0.1 lGO TO 503
 ------~------------

504 IFIMODA.LT.1 lGO TO 510

X=X/10
Z=Z*10
MODA=MODA/10
IFIMODA.GT.1 lGO TO 504
GO TO 510
503 IFIMODA.LT.0.1lX=X*10
Z=Z/10
MODA=MODA*IO
IFIMODA.LT.O.I lGO TO 503
510 Y=Y/Z
MODX=CABSIXl
Y=Y/IMODX**2l
Y=Y*CONJGIXl
DIV=Y
RETURN
END
c
OK,
SLIST BESSEL
REAL*4 FUNCTION BERCXJ
X4= CX/8. >**4
Z=1-64*X4
X8=X4*X4
Z=Z+113.77777774*X8·
X12=X8*X4
Z=Z-32.36345652*X12
X16=X12*X4
Z=Z+2.64191397*X16
X20=X16*X4
Z=Z-0.08349609*X20
X24=X20*X4
Z=Z+0.00122552*X24
X28=X24*X4
Z=Z-0.00000901*X28
BER=Z

END
c
c
REAL*4 FUNCTION BEICXJ
X2= CX/8. >**2
X4=X2*X2
X6=X2*X4
Z=16*X2-113.77777774*X6
X10=X6*X4
Z=Z+72.81777742*X10
X14=X10*X4
Z=Z-10.56765779*X14
X18=X14*X4
Z=Z+0.52185615*X18
X22=X18*X4
Z=Z-0.01103667*X22
X26=X22*X4
Z=Z+0.00011346*X26
BEI=Z
. RETURN
END
c
c

REAL*4 FUNCTION KERCXJ

Y=X/2.
Z=-ALOGCYl*BERCXl+0.25*3.1415926*BEICXl
X2=CX/8.l**2
X4=X2*X2
Z=Z-0.57721566-59.05819744*X4
X8=X4*X4
Z=Z+171.36272133*X8
X12=X8*X4
Z=Z-60.60977451*X12
X16=X12*X4
Z=Z+5.65539121*X16
X20=X16*X4
 Z=Z-0.19636347*X20
X24=X20*X4
Z=Z+0.00309699*X24
X28=X24*X4
Z=Z-0.00002458*X28
KER=Z
RETURN
END
c
c
REAL*4 FUNCTION KEI!Xl
Y=X/2.
X2=!X/8.l**2
X4=X2*X2
Z=-ALOG!Yl*BEI!Xl-0.25*3.1415926*BER!Xl
Z=Z+6.76454936*X2
X6=X4*X2
Z=Z-142.91827687*X6
X10=X6*X4

Z=Z+124.23569650*X10
X14=X10*X4
Z=Z-21 .30060904*X14
X18=X14*X4
Z=Z+1 .1'l509064*X18
X22=X18*X4
Z=Z-0.02695875*X22
X26=X22*X4
Z=Z+0.00029532*X26
KEI=Z
RETURN
END
c
c
REAL*4 FUNCTION BERD!Xl
X2=!X/8.l**2

X4=X2*X2
X6=X4*X2
X10=X4*X6
X14=X10*X4
X18=X14*X4
X22=X18*X4
X26=X22*X4
Z=-4*X2+14.22222222*X6
Z=Z-6.06814810*X10+0.66047849*X14
Z=Z-0.02609253*X18+0.00045957*X22

Z=Z-0.00000394*X26
BERD=Z*X
RETURN
END
c
c
REAL*4 FUNCTION BEID!Xl
X2=!X/8.l**2
 X4=X2*X2
X8=X4*X4
X12=X8*X4
X16=X12*X4
X20=X16*X4
X24=X20*X4
Z=0.5-10.66666666*X4
Z=Z+11.37777772*X8-2.31167514*X12
Z=Z+0.14677204*X16-0.00379386*X20
Z=Z+0.00004609*X24
BEID=Z*X
RETURN
END
c
c
REAL*4 FUNCTION KERDCXl
Y=X/2.
X2=CX/8.l**2
X4=X2*X2

X6=X2*X4
X10=X6*X4
X14=X10*X4
X18=X14*X4
X22=X18*X4
X26=X22*X4
Z=-ALOGCYl*BERDCXl-BERCXl/X+0.25*3.1415926*BEIDCXl
ZZ=-3.69113734*X2+21 .42034017*X6
ZZ=ZZ-11 .36433272*X10+1 .41384780*X14
ZZ=ZZ-0.06136358*X18+0.00116137*X22
ZZ=ZZ-0.00001075*X26
ZZ=ZZ*X
KERD=Z+ZZ
RETURN
END
c
c
REAL*4 FUNCTION KEIDCXl

Y=X/2.
X2= CX/8. >**2
X4=X2*X2
X8=X4*X4
X12=X4*X8
X16=X12*X4
X20=X16*X4
X24=X20*X4
Z= -ALOGCYl*BEIDCXl-BEICXl/X-0.25*3.1415926*BERDCXl
ZZ=0.21139217-13.39858846*X4
ZZ=ZZ+19.41182758*X8-4.65950823*X12
 - ZZ=ZZ+0.33049424*X16-0.00926707*X20
ZZ=ZZ+0.00011997*X24
ZZ=ZZ*X
KEID=Z+ZZ
RETURN
END
c
c
COMPLEX FUNCTION THETACXl
REAL*4 K,J
Y=8./X
Y2=Y*Y
Y3=Y2*Y
Y4=Y3*Y
Y5=Y4*Y
Y6=Y5*Y

K=0.0110486*Y
K=K-0.0000906*Y3
K=K-0.0000252*Y4
K=K-0.0000034*Y5
K=K+0.0000006*Y6
J=-0.0110485*Y-0.3926991
J=J-0.0000901*Y3-0.0009765*Y2
J=J+0.0000051*Y5+0.0000019*Y6
THETA=CMPLXCK,Jl
RETURN
END
c
c
COMPLEX FUNCTION THYCXl
REAL*4 K,J

Y2=Y*Y
Y3=Y2*Y
Y4=Y3*Y
Y5=Y4*Y
Y6=Y5*Y
K=0.7071068-0.0625001*Y
K=K-0.0013813*Y2+0.0000005*Y3
K=K+0.0000346*Y4+0.0000117*Y5
K=K+0.0000016*Y6
J=0.7071068-0.0000001*Y
J=J+0.0013811*Y2+0.0002452*Y3
J=J+0.0000338*Y4-0.0000024*Y5
J=J-0.0000032*Y6
THY=CMPLXCK,Jl
RETURN
END
c
c
 COMPLEX FUNCTION KO!Xl
COMPLEX CN,Z2;F,K,THETA
Y=SQRT!3.1415926/!2*Xll
W=SQRT( 1./2. l
Z=W
CN=CMPLX ( -W, -Z l
K=THETA!-Xl
Z2=CN*X+THETA!-Xl
F=CEXP !Z2 l
KO=Y*F
RETURN
END
c
c
COMPLEX FUNCTION IO!Xl
COMPLEX CN,ZX,F,FY,B,D,DF,THETA,KO
REAL M,N
Y=1/(SQRT<2*3.1415926*Xll
W=SQRT!1./2. l

CN=CMPLX!W,Zl
ZX=CN*X+THETA!Xl
F=CEXP!ZXl
FY=F*Y
B=KO!Xl
M=AIMAG!Bl
N=REAL!Bl
D=CMPLX!-M,Nl
DF=D/3.1415926
IO=FY+DF
RETURN
END
c
c
COMPLEX FUNCTION KOD!Xl
COMPLEX CN,ZX,F,FY,FX,THETA,THY
Y=SQRT!3.1415926/!2*Xll
 W=1/SQRT<2.0l
Z=W
CN= CMPLX ( -W, -Z l
ZX=CN*X+THETA<-Xl
F=CEXP<ZXl
FY=F*Y
FX= -FY*THY <-X l
KOD=FX
RETURN
END
c
c
COMPLEX FUNCTION IOD<Xl
COMPLEX CN,ZX,F,FY,FX,B,D,DX,THETA,THY,KOD
REAL*4 M,N

Y=SQRT(1/(2*3.1415926*Xll
W=SQRT<1./2. l
Z=W
CN=CMPLX<W,Zl
ZX=CN*X+THETA<Xl
F=CEXP<ZXl
FY=F*Y
FX=FY*THY<Xl
B=KOD<Xl
M=AIMAG(Bl
N=REAL<Bl
D= CMPLX C-M, N l
DX=D/3.1415926
IOD=FX+DX
RETURN
END
OK,
pr slab.fortran

slab.fortran 03/26/85 1716.9 gmt Tue

c THE SLAB PROGRAM- this has been written by Ali Al-Shaikhli on his Ph.D
c project.It calculates the magnetic field ,power
c density and power distribution in a
c non magnetic rectangular slab.It also calculates the
c total power induced in the slab.

print' (a,/l' ,'This program carries out finite difference analysis'

print• (a,/l' ,'on a non magnetic rectangular slab.'
print' (a,/l' ,•It calculates the magnetic,power density and power
!distribution in the slab.'
print• (a ,I l •, • It also calculates the total power induced in the
1 non magnetic slab.'
print• (a,/l' ,•To run the slab program ,type slab and press return•

print• (a,/l' ,'You will be asked to input values for HAG(the surface
1 magnetic field strength'

1 magnetic field strength'

print' (a,/l' ,•and FREQ(the supply frequency).'
print'(a,/l','You will also be asked to input a value for q(7l.If
1 q<7 l is set to o.o•
print• (a,ll' ,'then you will get a detailed output including arrays
1 ROXA,ROX,ROY and MU.'
print' (a,/l' ,•To use the program for your own example you will hav
1e to do some simple editing•
print• (a,/)' ,'This is outlined in the EDDY report.•
print• (a,/l' ,'Also included in the report is a section which tells
1 you how to get a print-•
print• (a,/l' ,'out of the results at the computer centre.'

c PARAMETER STATEMENTS SETTING UP THE MESH GEOMETRY

c the number of horizontal meshes in the quarter section of the slab

parameter(mx=9 l

c the number of vertical meshes in the quarter section of the slab

parameter(my=10l

the number of refractory meshes

parameter(mr=4l

c the total number of mesh nodes

parameter(m=(mx+1 l*(my+1 l+mr+1 l

c the total number of mesh elements in the quater section

 --- --- --------------·-- ---- -

slab

parameterCn=mx*myl

c mac will be used to dimension array ac which is used in

c subroutine gauss

parameterCmac=mX**2l

c nbc should be greater than mx*mX*(my-1 l for the subroutine gauss to be

c succesfull

parameter(nbc=1000l

c mq will be used to dimension general array q

parameterCmq=15l

c DECLARATION OF REAL AND COMPLEX ARRAYS

real hxCmxl,hyCmyl,muCnl,rotab,roxaCnl
real rox

Cnl,royCnl,muo,pgCml,bCm),q(mql
real pCml,mgCnl,hag,freq,max
complex hCnl,acCmacl,bcCnbcl,ca,cb,cc

c pringo is a logical variable which determines the amount of output

c the user will get

logical pringo

c SETTING UP THE GEOMETRY OF THE SLAB

c setting up the horizontal mesh spacings

data hx/.075,.07,.039,.025,.018,.009,.006,.005,.003/

c setting up the vertical mesh spacings

data hy/.01 ,.009,.007,.005,.004,.003,.003,.003,.003,.003/

c slbthx is the length of the cross-sectional face of the slab

c which is being used for the finite difference analysis

data sl

bthx/.5/

c slbthy is the width of the cross-sectional face of the slab

c which is being used for the finite difference analysis

data slbthy/.1/

c the resistivity of non magnetic aluminium

data rotab/2.824e-8/

c HAG is the surface magnetic field strengthCA/ml

c FREQ is
 the supply rrequency CHzl

print*,'INPUT VALUES OF HAG AND FREQ'

read*,hag,rreq

print*,'INPUT VALUE OF QC7l'

read *,qC7l

c ir q(7l is 0.0 user will get a fuller output,otherwise the

c user will only get the power density,power generated ,and
c the magnetic field distributions in addition to the total power
c induced in the slab

pringo=(q(7l.eq.O.Ol

c THE FOLLOWING SECTION CHECKS FOR MESH CONSISTENCY

c checks if one or more of the horizontal meshes is less than 1e-10.

c ir so the user is informed And the program stops

100 xa=O.O
do1001i=1,mx
xa=xa

+hx Ci l
i r ( hx ( i ) .1 t. 1 e -1 0) then
print*,'HORIZONTAL MESHCesl IS/ARE TOO SMALLCless than 1e-10 ml'
stop
endir
1001 continue

c check ir twice the sum or the horizontal mesh spacings is consistent

c with the dimension slbthx

if(absCxa+xa-slbthxl.gt.slbthx*1e-5lthen
print*,'TWICE THE SUM OF HORIZONTAL MESHES IS NOT CONSISTENT WITH
1 DIMENSION SLBTHX'
stop
endir

c check if one or more of the vertical meshes is smaller than 1e-10

xa=O.O
do 10015 i=1 ,my
xa=xa+hyCil
ir Chy ci l .1 t .1 e-1 o lthen
print*,'VERTIC

AL MESHCesl IS/ARE TOO SMALL Cless than 1e-10 ml'

stop
endir
10015 continue

c check if twice the sum of the vertical mesh spacings is consistent

c with the dimension slbthy

irCabsCxa+xa-slbthyl.gt.slbthy*1e-5lthen
pri
nt*,'twice the sum of vertical meshes not consistent with
1 dimension slbthy'
,print*,'TWICE THE SUM OF VERTICAL MESHES NOT CONSISTENT
1 WITH DIMENSION SLBTHY'
stop
end if

c CALLING SUBROUTINE SETUP TO USE THE FINITE DIFFERENCE EQUATIONS

c DEFINED AND SETUP THE MATRIX

call setup(pg,b,h,hag,hx,hy,rox,roy,mu,ac,bc,
1rotab,roxa,mg,pringo,max,
2freq,q,mq,m,mx,my,mac,nbc,n,p)
stop
end

c ****************START OF SUBROUTINE SETUP ***********

subroutine setup(pg,b,h,hag,hx,hy,rox,roy,mu,ac,bc,
1rotab,roxa,mg,pringo,max,
2freq,q,mq,m,mx,my,mac,nbc,n,p)

declaration of real and complex arrays

real hx(mxl,hy(myl,mu(n),rotab,roxa(nl
real rox(nl,royCnl,muo,pg(mJ,b(mJ,q(ml
real p(ml,mg(nl,hag,freq,max
complex h(nl,ac(macl,bc(nbcl,ca,cb,cc

logical pringo

10 format(10f7.2,6xl
15 format(9f8.0,2xl
20 format(9f8.0,xl

mxpi =mx+1
mslab=mxpi*my+mxpi

c checking for any array bound irregularities in subroutine setup

if(m.gt.mslab.
1 and.mac.ge.mx*mxl
2 goto 101

print*,'

THERE ARE ARRAY BOUND IRREGULARITIES IN SUBROUTINE SETUP

1 PLEASE CHECK'

print• (4(a,i6/l l',

1 • mx=' ,mx
2, •rny=',my
3, 'm=' ,m
4, 'mac =' , mac
100 return
•1 01 mxmi=mx-1

c the following sets muo to be equal to 4pi*10-7

xa=8*atan(1 .0)
omega=xa*freq
muo=xa*2e-7

c THE FOLLOWING SECTION CALCULATES THE MESH AREA*RESISTIVITY

C FOR EACH MESH ELEMENT AND ALSO SETS THE PERMEABILITY OF EACH
c MESH TO MUO

do 104 j=1 ,my

i a = mx * Cj -1 l
do 103 i=1 ,mx
ia=ia+1
muCial=muo
roxa<ial=rotab*hx<il*hyCjl
103 continue
104 continue

ifC.not.pringolgoto 1051
print*,' TH

E MESH AREA X RESISTIVITY FOR EACH MESH ELEMENT•

print• C4e16.7l', Croxa<j l,j=1 ,nl

c THE FOLLOWING SECTION CALCULATES THE RESISTANCE VECTORS ROX AND

c ROY AT EACH NODE

1051 xa=2/omega
do 1 055 j =my, 1 , -1
ia=j*mX
xb=xa/(hyCj l*hy(j))
do 1054 i=mx,1,-1
xc=xa/ Chx ( i l*hx Ci l)
if Cia .eq.1 )then
royCial=xb*roxCial
else
roy Cia l=xb* Croxa Cia l+roxa ( ia-1))
end i f
if ( i .eq.1 lroy Cia l =xb*roxa Cia l
if Cj .eq.1 l goto 1052
roxCial=xc*CroxaCial+roxaCia-mxll
goto 1053
1052 roxCial=xc*roxaCial
1053 ia=ia

-1
1054 continue
1055 continue

ifC.not.pringolgoto 1084
print*,'THE RESISTANCE VECTORS ROX AT EACH MODE (array roxl'
print • C4 e 1 6 . 7 l ' , Crox Cj l , j = 1 , n l
print*,'THE RESISTANCE VECTORS ROY AT EACH NO
DE (array ray l •
print• (4e16.7l', (roy(j J ,j•1 ,nl

c THE FOLLOWING SECTION SETS UP THE RIGHT HAND SIDE

c OF THE MATRIX EQUATION IN THE ARRAY H

1084 j•n-mx
do 1 087 i • 1 , j
h(il•O.O
1087 continue
do 1088 i•mx,j,mx
h(i l•rox<i l*hag
1088 continue
j : j +1
do 1089 i•j ,n
h ( i J=roy ( i )*hag
1089 continue
h(nl=h(nl+rox(nl*hag

do 1092 j•1 ,my

do 1091 i•1 ,mx
mu(j*mx-mx+il•mu(j*mx-mx+il*hx(il*hy(jl
1091 continue
1092 continue
do 1094 j•my,2,-1
ia•j*mX
i b• ia-mx+2
do 1093 i•ia,ib,-1
mu(i J•mu<i J+mu(i-1 l+mu(i-mxl+mu(i-mx-1 l
1093 continue
mu<ib-1 l•mu(ib-1 l+mu<ib-1-mxl
1094 continue
do 1095 i•mx,2,-1
mu ( i l • mu ( i J +mu ( i -1 l
1095 continue

if(.not.pringolgoto 2000
print*,'THE NODE ARRAY MU'
print • ( 4e 1 6. 7 l • , (mu (j l , j • 1 , n l

c THE SUBROUTINE GAUSS IS NOW CAL

LED UPON TO SOLVE THE FINITE

c DIFFERENCE MATRIX USING THE METHOD OF GAUSSIAN ELIMINATION

2000 call gauss(rox,roy,mu,h,n,ac,mac,bc,nbc,mx,myl

c THE MAGNETIC FIELD STRENGTHS OBTAINED BY SUBROUTINE GAUSS WILL

c NOW BE USED TO CALCULATE

c al the power at each node

 I
c b) the power density over each mesh element
c c) the total power in the slab from the power generated at
c each node

121 ic=my-1
do 130 j=1 ,ic
ia=j*mx
ib=ia-mx
do125i=1,mxmi
ca=h(ia+i+1 l-hUb+i l
cb=h(ia+i l-h<ib+i+1 l
cc=<ca+cbl/hy(j)
c b = ( c a- c b l I hx ( i l
rox<ib+il=roxa<ib+il*(cc*conjg(ccl+cb*conjg(cbll*0.03125
125 continue
ca=hag-h<ia l
cb=h<ia+mxl-hag
cc=(ca+cbl/hy(j)
cb=(ca-cbl/hx<mxl
rox<ial=roxa<ial*(Cc*conjg(ccl+cb*conjg(cbll*0.03125
·130 continue
ia=mx*my-rnx

do 140 i=1,mxmi
ca=hag-h<ia+il
cb=hag-h<ia+i+1 l
cc=(ca+cbl/hy(myl
cb= <ca-cbl/hx<i l
rox<ia+il=roxa(ia+i l*(CC*Conjg(ccl+cb*conjg(cbll*0.03125
140 continue
cc=<hag-h(mx*myll/hy<myl
cb=(hag-hCmx*myll/hx<mxl
rox(mx*myl=roxa<mx*mYl*(Cc*conjg(ccl+Cb*conjg(cbll*0.03125

c the following section calculates the power generated at each

c node of the quarter section slab mesh

pg<1 l=rox(1 l
pg(mxpil=rox<mxl
pg(mxpi*my+1 l=rox <mx*my-mxmi l
pg<mslabl=rox<mx*myl
do 150 i=2,mx
pg(i l=rox<i-1 l+rox(i l
pg<mslab-mxpi+i l=rox Cmx*my-mx+i-1 l+rox<mx*my-mx+i l
150 continue

do 160 j=2,my
ia=mxpi*j-mxpi
ib=mx*j-mx
pg<ia+1 l=rox<ib+1 l+rox(ib+1-mxl
pg(ia+mxpil=rox<ib+mxl+rox(ibl
do 155 i=2,mx
pg ( i a+ i l =ro x ( i b + i l +ro x ( i b +i -1 l +ro x ( i b + i - mx l +rox ( i b + i - mx- 1
155 continue
 ------ -----~-~---------------------------------------------

160 continue

c setting ~he power at the refractory nodes to zero

i a =msl ab+1
do 161 i=ia,m
pg(i )=0.0
161 continue

c the following section calculates the power density within each mesh
c element of the quarter section slab

ia=O
ib=O
do 1 64 j =1 , my
do 162 i=1 ,mx
b<ia+i )=4/hy<jl/hx<i l*rox<ib+i l
162 continue

c setting the power densities at the surface to zero

ia=ia+mxpi
ib=ib+mx
b(ial=O.O
164 continue

ia=mslab-mx
do 165 i=ia,mslab
b(i )=0.0
165 continue

print• <a,/l' ,'THE POYER DENSITY<Yim3l IN EACH ELEMENTOF THE MESH'

print 15,<b<jl,j=1,9l
print 15,(b(j),j=11,19l
print 15, (b (j l ,j=21 ,29 l
print 15,<b<jl,j=31,39l
print 15,<b<jl,j=41,49l
print 15,<b<jl,j=51,59l
print 15,<b<jl,j=61,69l

print 1 5, <b <j l , j =71 , 79 l

print 15, <b<jl,j=81,89l
print 15, <b<jl,j=91,99l

c adding the powers generated at each node and then multiplying by

c four to give the total power in the slab

xa=O.O
do 1665 i=1 ,mslab
xa=xa+pg<il
1665 continue
xa=4*xa

print' <l,a,f15.2,/l' ,'TOTAL POYER<Yiml IN THE SLAB=' ,xa

print• (a,/l' ,'POYER GENERATED <Yl AT EACH NODE

--------~------------------------------------

OF THE MESH'
print 10,(pgCjl,j=1,10l
print 10, CpgCjl,j=11,20l
print 10,CpgCjl,j=21,30l
print _1 0 , ( pg ( j ) , j = 31 , ll 0 l
print 10, (pgCj l ,j=ll1 ,50)
print 10, Cpg(j l ,j=51 ,60)
print 10, CpgCj l ,j=61 ,70l
print 10, CpgCj l ,j=71 ,BOl
print 10, Cpg(j l ,j=B1 ,90)
print 1 0 , ( pg ( j l , j = 91 , 1 00 l
print 10,(pg(jl,j=101,110l

c expressing the power densities in each mesh as a percentage of the

c largest mesh power density

c the largest mesh power density being denoted by max

c finding the largest mesh power density

max=b(1 l
do 166 k=1,m
if(b(kl.gt.maxlthen
max=b(

c expressing the power densities as a percentage of the largest

c mesh power density mxa

do 167 k=1 ,m
pCkl=(b(kl/maxl*100
167 continue

print' (/,a,/l' ,'THE POWER DENSITY IN EACH ELEMENT OF THE MESH

1 EXPRESSED AS A PERCENTAGE OF THE LARGEST MESH POWER DENSITY'

print 10 , ( p ( j l , j =1 , 9 l
print 10 , ( p ( j l , j =1 1 , 19 )
print 1 0, ( p ( j l , j = 21 , 2 9 l
print 1 0, ( p ( j l , j = 31 , 3 9 )
print 10, (p(j l ,j=ll1 ,49l
print 10, (p(j l ,j=51 ,59l
print 10,(p(jl,j=61,69l
print 10,(p(jl,j=71,79l
print 10,(p(jl ,j=B1,89l
print 10,(pCjl,j=91,99l

print• U,a,f15.2,/l' ,'THE LARGEST MESH POWER DENSITYCw/m3l=' ,max

if(.not.pringolgoto 169
print*,'MAGNETIC FIELD STRENGTHS WRITTEN AS COMPLEX NUMBERS'
print' ( lld 1 6 • 6 l ' , ( h ( j l , j = 1 , n l

c mg is the array of the modulus of the complex magnetic field

c strengths
169 ia'=O
do 170 i=1 ,n
mg ( i l =cabs ( h ( i l l
170 continue

print• Ca,/l' ,•MAGNETIC FIELD STRENGTHS AT EACH NODE OF THE MESH'

print 2 0, ( mg (j l , j = 1 , 9 l
print 20, ( mg ( j l , j = 1 0, 1 8 l
print 20,(mg(jl,j=19,27l
print 20,CmgCjl,j=28,36l
print 20,Cmg!jl,j=37,45l
print 20,Cmg(jl,j=46,54l
print 20, <mg(j l,j=55,63l
print 20,Cmg!jl,j=64,72l
print 20, <mg(j l,j=73,81 l
print 20,Cmg(jl,j=82,90l

print• C/,a,f10.2,/l' ,'THE SURFACE MAGENTIC FIELD STRENGTH=' ,hag

return
end

c **********START OF SUBROUTINE GAUSS T

0 SOLVE THE MATRIX *****

subroutine gauss(rx,ry,mu,h,nxy,a,na,b,nb,mx,myl
complex hCnxyl,a(nal,b(nbl,ca
real rx<nxyl,ryCnxyl,muCnxyl

logical ntgo

mxmi=mx-1
mxmx=mX*ffiX
mxy=mx*mY

c checks if there are any array bound errors in subroutine gauss

if(na.ge.mxmx.
1 and.nxy.ge.mxy.
2 and.mx.gt.2.
3 and.my.gt.2. lgoto 100

print' (a,/l' ,'THERE ARE ARRAY BOUND ERRORS IN SUBROUTINE GAUSS'

print• (6(a,i6/ll',
1 • mx = '

,mx
2, 'my=' , my
3,'mxy=' ,mxy
4, 'nxy=' , nxy
5, 'mxrnx=', mxmx
6,'na=' ,na
return

100 mxpi=mx+1
mxt=2*mx
mxmxmm=mxmx-mx

c i
f ntgo is true then the subroutine gauss will not give the
c right solution because the array b is not large enough

ntgo=nb.lt.mxmx*<my-1 >
i f <ntgo )then
print*,'NBC IN THE MAIN PROGRAM IS NOT LARGE ENOUGH FOR THE MATRIX
1 TO BE SOLVED'
print*,'nbc=' ,nb
stop
end if
kb=mx
if <ntgol kb=O
ka=-kb

c setting up the elements in array a

a(1 l=cmplx<rx(1 l+ry(1) ,mu(1))

a<2>=-rx<1 >
do 200 i=3,mxmx
a <i l=O.O
200 continue
ia=mx-2
do 240 i=1·, ia

ib=i*mxpi
a<ib+1 )=cmplx<rx<il+rx<i+1 )+ry<i+1 l,mu<i+1 ))
a(ib+2l=-rx<i+1)
a<ibl=-rx<il
240 continue
a Ci b +mx +1 ) = - r x Cmx -1
a(ib+mx+2)=cmplxCrx<mx-1 l+rx<mxl+ry(mx),mu(mx))
ie=mxy-mxt+1

c start of the gaussian elimination process

do 600 i=1,ie,mx
ia=1
ka=ka+kb
ca=1/a(1 >
do 410 j=2,mx
b(ka+j-1 >=ca*a(j)
410 continue
b(ka+mxl=-ca*ry<i>
h<i l=hCi )*ea
ib=i
do 430 J=mxpi,mxmx,mx
do 420 k=1,

mxmi
a (k+j-mx-1 l=a Ck+j )-a Cj l*b(ka+k)
420 continue
a (j-1 l=-a (j l*b<ka+mx >
ib=ib+1
hCibl=hCibl-aCjl*h<i>
430 continue
 do 440 k=1,mx
aCmxmxmm+kl=ryCil*b(ka+kl
c print*,Ch(jl,j=1,90l
440 continue
a Cmxmxl=a Cmxmxl +cmplx Cry ( i l+rx Ci +mx )+ry ( i +mx) ,mu Ci +mx))
h ( i + mx ) = h Ci +mx ) +r y ( i ) * h ( i )
ib=i+1
ic=i +mxmi
do 550 ia=ib,ic
ka=ka+kb
ca=1 /a (1 l
506 do 510 j=2,mx
bCka+j-1 l=ca*a Cj)
510 continue
bCka+mxl=-ca*ryCial
hCial=hCiahca
id=ia.
do 530 j=mxpi,mxmx,mx
do 520 k= 1 ,mxmi
a Ck+j-mx-1 l=a Ck+j )-a (j l*b (ka+k)
520 continue
aCj-1 l=-a <J l*bCka+mx)
id

= i d+1
hCidl=hCidl-aCjl*hCial
530 continue
do 540 k=1 ,mx
a(mxmxmm+kl=ryCial*bCka+kl
540 continue
a(mxmxmml=aCmxmxmml-rxCia+mx-1
a(mxmx-1 l=a(mxmx-1 l-rx(ia+mx-1
a Cmxmx l=a (mxmx l+cmplx Cry ( i a) +rx ( ia+mx-1 ) +rx ( ia+mx l +ry Cia+mx)
1 ,mu Cia+mxl)
545 hCmx+ial=hCmx+ial+ryCial*hCial
550 continue
600 continue
ja=O
i=2
ic=mxpi
id=mxy-1
ie=mxy-mxmi
do 640 ia=ie,id
ca=1 /a Cja+i-1)
do 610 j=i,mx
aCj+jal=ca*a(j+jal
610 continue
h Cia l =ca*h ( ia

ib=ia
do 630 j=ic,mxmx,mx
do 620 k=i,mx
a Ck+j-!+1 l=a Ck+j-i+1 l-a (j l*a (ja+k l
620 continue
ib=ib+1
hCibl=hCibl-a Cj l*hCia l
630 continue
ja=ja+mx
 h(mxyl=h(mxyl/a(mxmx)
i =mx
ja=mxmx-mxt
do 700 ia=id,ie,-1
do 690 j=1 ,mx
h(ia )=h( ia )-h(ia-i +1 +j )*a (ja+j)
690 continue
i = i -1
ja=ja-mx
700 continue
id=mxy-mx
810 do 850 ia=id,1 ,-1
do 830 j=1 ,mx
h(ial=h(ial-h(ia+j)*b(ka+j)
830 continue
ka=ka-kb
850 continue
return
end

r 17:38 0.878 76 level 7

SLIST 11-P-POIIER
DIMENSION AC6l,HC6l,ALC6l,FC6l,ROII1 C6l,URC6l,AICC6l,AAPC999l
1 , N ( 6 l , LH ( 6 l , B ( 6 l , C ( 6 l, D( 6 l , AF ( 6 l , DD ( 6 l, AXZ ( 999 l , AAJ ( 999 l , R0\12 (6 l
COMMON XZC6,999l,AJ(6,999l,APC6,999l
READC5,*lLM, CACI l ,HCI l ,ALCI l ,F CI l ,R0\11 Cl l ,R0\12 Cl l ,URCI l,
1 AI C ( I l , N ( I l , LH ( I l , I= 1 , LM l
DO 1000 LS=1,LM
BCLS l=AICCLS l*HCLSl* CSQRTC1 .0-CACLSl/HCLSl >**2 l l/3.14159265
CCLSl=HCLSl**2-ACLSl**2
ROII=ROII1CLSl
DR= CROII2CLSl-ROII1 CLSl l/700.0
C IIRITEC1 ,20lDD
C20 FORMATC1H ,4HDD =,F15.6l
C IIRITEC1 ,4lA
C4 FORMATC1H ,3HA =,F15.9l
C IIRITEC1,8lH
CB FORMATC1H ,3HH =,F15.9l
C IIRITEC1,11lAL
C11 FORMATC1H ,8HLENGTH =,F15.9l

C IIRITEC1,12lN
C12 FORMATC1H ,3HN =,I8l
C IIR I TE ( 1 , 61 l F
C61 FORMAT!1H ,11HFREQUENCY =,F15.4l
C IIRITE!1,90lAIC
C90 FORMAT!1H ,14HCOIL CURRENT =,F9.3,4X,'AMP' l
C IIRITE!1,62lROII
C62 FORMATC1H ,5HROII =,E15.7,4X,'OHM-M' l
C IIRITE(1,63lUR
C63 FORMATC1H ,4HUR =,F15.3l
C IIRITE!1,9l
C9 FORMAT!1H ,2X,'Z' ,13X,'CURRENT DENSITY' ,10X,'POIIER DENSITY')
C IIRITEC1,7l
C7 FORMAT!1H ,1X,•---• ,12X,•---------------• ,10X,•-------------• l
IND=O
D!LSl=AL!LSl/(N!LSl-1 l
AF!LSl=AL!LSl/700.0

Z=O.O
5 IND=IND+1
XZCLS, INDl =Z
SUM=O.O
NL=NCLSl
DO 10 K=1 ,NL
I=K-1
SUM1=1 ./(C(LSl+Z**2+2.0*Z*D!LSl*I+DCLSl*DCLSl*I*Il
SUM=SUM+SUM1
10 CONTINUE
 PI=3.14159265
U0=4.0*PI/10000000.0
AXA=UR!LSl*FlLSl
XAX=PhUO*AXA
DD!LSl=!ROW/XAXl**0.5
AJ!LS,INDl=B!LSl*SUM
AP!LS,INDl=ROW*AJ!LS,INDl*AJ!LS,INDl/!DD!LSll
DIS=ABS!Zl
C WRITE!1,6lDIS,AJ!LS,INDl,AP!LS,INDl
C6 FORMAT!1H ,F6.3,5X,F15.6,14X,E10.4l
ROW=ROW+DR
Z=Z-AF!LSl
E=Z+AL!LSl
XZ!LS,INDl=ABS!XZ!LS,INDll
IF !E.GE.O.OlGO TO 5
1000 CONTINUE

AL1=0.0
YB=O.O
YET=-9999.0
YEV=-9999.0
DO 1500 MM=1,LM
DO 2000 J=1,IND
IF!YET.LT.AP!MM,JllYET=AP!MM,Jl
IF!YEV.LT.AJ!MM,JllYEV=AJ!MM,Jl
IF!AL1 .LT.AL!MMllALI=AL<MMl
2000 CONTINUE
1500 CONTINUE
IF!LH!LSl.EQ.3lCALL C1051N
IF!LH!LSl.EQ.1 lCALL TREND
' IF!LH!LSl.EQ.2lCALL 55600
CALL DEVPAP!210.0,297.0,1 l
CALL MOVT02(0.0,0.0l
CALL LINT02!0.0,297.0l
CALL MOVT02!210.0,0.0l
CALL LINT02!210.0,297.0l
CALL WINDOW!2l
CALL ERRMAX!100l
CALL AXIPOS!I ,40.0,90.0,150.0,1 l
CALL AXIPOS!1 ,40.0,90.0,180.0,21

CALL AXISCA!1 ,9,0.0,AL1 ,1 l

CALL AXISCA!1 ,9,0.0,YEV,2l
CALL AXIDRA!2,1 ,1 l
CALL AXIDRA (-2, -1 ,2 l
CALL MOVT02!25.0,285.0l
CALL CHAHOL('A !A/ml*.' l
CALL MOVT02(190.0,78.0J
CALL CHAHOL (' Z (m h. ' l
CALL.MOVT02!40.0,70.0J
CALL CHAHOL!'Fig.4 • • CURRENT PER UNIT LENGTH ALONG THE LOAD*·' J
 CALL MOVT02C40.0,54.0l
CALL CHAHOLC'N= Turns d= mm h= mm *. • )
CALL MOVT02C40.0,46.0l
CALL CHAHOLC'a= mm I= Amp *. • l
DO 1700 II =1 , LM
DO 1750 I= 1 , I NO
AXZ CI l =XZ CI I , I l
AAJ (I l =AJ CI I , I l
1750 CONTINUE
CALL GRACURCAXZ,AAJ,INDl
1700 CONTINUE
CALL PICCLE
CALL DEVEND
IFCLHCLSl.EQ.3lCALL C1051N
IFCLHCLSJ.EQ.2JCALL S5600
IFCLHCLSl.EQ.1 lCALL TREND

CALL DEVPAPC210.0,297.0,1 l
CALL MOVT02CO.O,O.Ol
CALL LINT02C0.0,297.0l
CALL MOVT02C210.0,0.0l
CALL LINT02C210.0,297.0l
CALL WINDOWC2l
CALL ERRMAXC100l
CALL AXIPOSC1 ,40.0,90.0,150.0,1)
CALL AXIPOSC1 ,40.0,90.0,180.0,2)
CALL AXI SCA C1 , 9, 0. 0, AL 1 , 1 l
CALL AXISCAC1 ,9,0.0,YET,2l
CALL AXIDRAC2,1 ,1)
CALL AXIDRAC-2,-1 ,2l
CALL MOVT02C25.0,285.0l
CALL CHAHOLC'PDCW/SQ ml*.' l
'CALL MOVT02C190.0,78.0l
CALL CHAHOLC'ZCml*.' l
CALL MOVT02C40.0,70.0l
CALL CHAHOLC'Fig.4 • . POWER DENSITY DISTRIBUTION ALONG THE LOAD *·' l
CALL MOVT02C95.0,62.0l

CALL CHAHOLC'NONMAGNETIC LOAD*.' l

CALL MOVT02C40.0,54.0l
CALL CHAHOLC'INITIAL RESISTIVITY = Ohm. m *. • )
CALL MOVT02C40.0,46.0l
CALL CHAHOLC'FINAL RESISTIVITY = Ohm. m*. • l
CALL MOVT02C40.0,38.0l
CALL CHAHOLC'RELATIVE PERMEABILITY=*·' l
 CALL MOVT02(40.0,30.0J
CALL CHAHOL<'N= Turns d= mm h= mm *. • J
CALL MOVT02(40.0,22.0l
CALL CHAHOL('a= mm I= Amp F= Hz*. • l
DO 1800 II=1 ,LM
DO 1 850 I= 1 , I NO
AXZ<Il=XZ<II,Il
AAP <I l =AP <I I , I l
1850 CONTINUE
CALL GRACUR<AXZ,AAP,INDJ
1800 CONTINUE
CALL PICCLE
IF<LH<LSJ.NE.2JCALL DEVEND
STOP
END
OK,
SLIST PROXIMITY
READ!5,*lA,H,AL,F,ROW,UR,AIC,PW
N=2
B=AI C*H* ( SQRT <1 • 0- !A/H>**2 l l /3.141 59265
C=H**2-A**2
DD=503.3*SQRT!ROW/!F*URll
C WRITE!1 ,20lDD
C20 FORMAT!1H ,4HDD =,F15.12l
WRITE!1 ,4lA
4 FORMAT!1H ,17HINDUCTOR RADIUS =,F5.4,4X,'m' l
WRITE< 1 , 8 l H
8 FORMAT!1H ,9HAIR GAP =,F5.4,4X,'m' l
WR I TE <1 , 1 1 l AL
11 FORMAT<1H ,8HLENGTH =,F6.3,4X,•m• l
WRITE<1,44lPW
44 FORMAT!1H ,16HPOWER REQUIRED =,F15.3,4X,'Watts/meter• l
WRITE< 1 ,61 lF
61 FORMAT!1H ,11HFREQUENCY =,F15.1 ,4X,'Hz' l

WRITE<1 ,66lAIC
66 FORMAT!1H ,14HCOIL CURRENT =,F15.3,4X,'Amp' l
WRITE!1,62lROW
62 FORMAT!1H ,18HLOAD RESISTIVITY =,E10.4,4X,'Ohm.m• l
WRITE(1 ,63lUR
63 FORMAT!1H ,14HPERMEABILITY =,F10.1 l
AF=AL/100
16 ASS=O.O
D=AL/ <N-1 l
Z=O.O
5 SUM=O.O
DO 10 K=1 ,N
I=K-1
SUM1=1 ./(C+Z**2+2.0*Z*D*I+D*D*I*Il
SUM=SUM+SUM1
10 CONTINUE
AJ=B*SUM

AP=ROW*AJ*AJ/!2.0*DDl
AAP=AP*AF
AAP=ASS+AAP
Z=Z-AF
E=Z+AL
IF <E.GE.O.OlGO TO 5
IF <AAP-PWl28,25,25
28 N=N+1
GO TO 16
25 . WRITE!1,22lN
22 FORMAT!1H ,17HNUMBER OF TURNS =,I10,4X,'Turns• l
WRITE< 1 , 7 lAAP
7 FORMAT<1H ,22HPOWER PER UNIT WIDTH =,F15.3,4X,'Watts/meter• l
ER=100.0*<AAP-PWl/PW
WRITE!1 ,53lER
53 FORMAT!1H ,12HDIFFERENCE =,F5.2,'%' l
WRITE!1 ,77l
77 FORMAT!1H ,•---------------------------------------------------• l
STOP
END
SLIST \1-SC-FIT
C THE SYMBOLS ARE
c N NUMBER OF POINTS
c X DISTANCE FROM SINGLE TURN CMMl
c Y VOLTAGES FROM SINGLE TURN CUVl
c All CURRENT FOR SINGLE TURN CAMPl
c AI2 CURRENT FOR THE COIL CAMPl
c NT NUMBER OF TURNS
c D COIL PITCHE CMMl
c AL LOAD LENGTH CMMl
c R0\1 LOAD RESISTIVITY COHM.Ml
c ROL LOAD RADIUS OR SLAB'S PROBE LENGTH <MMl
c UR LOAD RELATIVE PERMEABILITY
c F FREQUENCY CHzl
c LH CHOICE OF THE DEVICE,TREND=1 ,S5600=2,C1051N=3
c NSC CHOICE OF SLAB=1 ,CYLINDER=2
c
c
REAL BER,BEI,KER,KEI,BERD,BEID,KERD,KEID
COMPLEX THETA,THY,KO,IO,KOD,IOD
DIMENSION AZC999l,AVC999l,PDC999l,AJC999l
1,SAJC999l,HRMC999l

REAL*8 XC50l,Y(50l,AC50l,REF
INTEGER N,NT
READ CS,* lN, <X< I l, Y< I l, I= 1 ,N l ,AI 1 ,AI2 ,NT ,D,AL ,R0\1 ,ROL ,F, UR,NSC,LH
CALL E02ACFCX,Y,N,A,5,REFl
C \IR I TE ( 1 , 3 0 l <A ( I l , I =1 , 5 l
C30 FORMATC1H ,5X,E12.4l
SLO= CYCN)-Y(N-1 l l/CXCNl-XCN-1 l l
CON=Y<Nl-CSLO*X<Nll
CL=D* <NT-1 l
R=-<CL/2.0l
C \IR I TE C1 , 9 l
C9 FORMATC1H ,'DISTANCE' ,7X,'VOLTAGE' l
C \IRITEC1,90l
C90 FORMATC1H ,•--------• ,?X,' •l
ML=1
1000 SUM=O.O
RA=AI2/AI1
DO 200 J=1,NT
I=J-1
ZA=R+CI*Dl
Z=ABSCZAl

IFCZ.GT.XCNllGO TO 300
SUMO=AC1 l+CAC2l*Zl+CAC3l*CZ**2ll+CAC4l*(Z**3ll+CAC5l*<Z**4ll
SAMO=O.O
SUM1=SUMO+SAMO
GO TO 400
300 SUM1=CSLO*Zl+CON
400 IFCSUM1 .LT.O.OlSUM1=0.0
SUM= ( SUM+SUM1 l
200 CONTINUE
SUM2=SUM*RA
 ZZ=R+ CCL/2 .0 J
AZCMLl=ZZ
AV CML J =SUM2
ML=ML+1
R=R+0.25
IFCZZ.LE.ALJGO TO 1000
K=ML-1
DO 500 I=1 ,K
AVCIJ=AVCIJ/1000000.0
500 CONTINUE
IFCNSC.EQ.1JROL=ROL/1000.0
IFCNSC.EQ.2JR=ROL
PIY=3.141592654
U0=4.0*PIY/10000000.0
U=UO*UR
11=2.0*PIY*F
R=R/1000.0
PER=2.0*R*PIY
RL=ROII*PER
D2=C2.0*RO\Il/CII*Ul
D1=SQRTCD2l

BA=R*SQRTC2.0l/D1
C CALCULATION OF BESSEL FUNCTIONS
IFCBA.GT.8lGO TO 5001
A1=BERCBAl
A2=BEICBAl
A3=BERDCBAl
A4=BEIDCBAl
GO TO 5002
5001 A1=REALCIOCBAJl
A2=AIMAGCIOCBAll
A3=REALCIODCBAJl
A4=AIMAGCIODCBAlJ
5002 CONTINUE
DEN=CCA1 l*CA1 ll+CCA2l*CA2ll
B1 = CCA3 >* CA3 J J + CCA4 l * CA4 J l
B2 = CCA 1 l * CA3 J l + CCA2 l * CA4 J l
C1=B1/DEN
C2=B2/DEN

IFCNSC.EQ.2JGO TO 550
DP=D1/C2.0*ROII*ROL*ROLJ
DO 510 I= 1 , K
PDCIJ=DP*AVCII*AVCIJ
510 CONTINUE
GO TO 999
550 CONTINUE
CPD=U*PIY*F*R*C2/2.0
 DO 11 0 I= 1 , K
AJ<Il=AV<Il/RL
SAJ<I l=AJ(I l*AJ(I l
HRM<Il=D2*SAJ<Il/C1
PD!Il=CPD*HRM!Il
11 0 CONTINUE
999 CONTINUE
c WRITE ( 1 , 1001 l
C1001 FORMAT<1H ,2X,'DISTANCE' ,7X,'VOLTAGE' ,6X,'POWER DENSITY' l
c WRITE <1 , 1 1 0 0 l
C1100 FORMAT<IH ,2X,'--------' ,7X,'-------• ,6X,•-------------' l
c DO 120 I=l ,K
c WRITE <1 , 1 30 l AZ <I l , AV< I l , PD <I l
C130 FORMAT<1H ,3X,F6.1 ,5X,E12.5,5X,E12.5l
C120 CONTINUE

YB=O.O
YET=-9999.0
YEV=-9999.0
DO 2000 J=I,K
IF!YET.LT.PD<JllYET=PD(Jl
IF<YEV.LT.AV<JllYEV=AV(Jl
2000 CONTINUE
YET= I .1 *YET
YEV=1 .I*YEV
IF<LH.EQ.3lCALL C1051N
IF<LH.EQ.1 lCALL TREND
IF<LH.EQ.2lCALL S5600
CALL DEVPAP<210.0,297.0,1 l
CALL MOVT02<0.0,0.0l
CALL LINT02!0.0,297.0l

CALL MOVT02!210.0,0.0l
CALL LINT02!210.0,297.0l
CALL WINDOW<2l
CALL ERRMAX!IOOl
CALL AXIPOS(I ,40.0,90.0,150.0,1 l
CALL AXIPOS<I ,40.0,90.0,180.0,2)
CALL AXISCA(I ,9,0.0,AL,1 l
CALL AXISCA<I ,9,0.0,YEV,2l
CALL AXIDRA!2,1,1 l
CALL AXIDRA<-2,-1,2)
CALL MOVT02(25.0,285.0l
CALL CHAHOL<'V <VOLTl*.' l
CALL MOVT02(182.0,78.0l
CALL CHAHOL('Z(mml*.' l

CALL GRACUR<AZ,AV,Kl
CALL MOVT02!50.0,70.0l
CALL CHAHOL!'SURFACE VOLTAGE DISTRIBUTION ALONG THE LOAD*·' l
CALL MOVT02!95.0,60.0l
IF<NSC.EQ.1 lCALL CHAHOL<'ALUMINUM SLAB*.' l
IF<NSC.EQ.2lCALL CHAHOL('ALUMINUM CYLINDER*.' l
CALL MOVT02(40.0,50.0l
CALL CHAHOL('RESISTIVITY=2.8E-8 Ohm.m *·' l
CALL MOVT02(140.0,50.0l
CALL CHAHOLC'RELATIVE PERMEABILITY=!*.' l
CALL MOVT02C40.0,40.0l
CALL CHAHOLC'N= Turns d= mm g= mm *·' l
CALL MOVT02C113.0,30.0l
CALL LINT02C125.0,30.0l
CALL MOVT02C126.0,30.0l
CALL CHAHOLC'PREDICTION BY SUPERPOSITION*.' l
CALL MOVT02C145.0,40.0l
CALL CHAHOLC'o PRACTICAL READINGS*.' l
CALL MOVT02C40.0,30.0l
CALL CHAHOL('I= Amp F= Hz*.'l
CALL PICCLE
CALL DEVEND
IFCLH.EQ.3JCALL C1051N

IFCLH.EQ.2lCALL S5600
IFCLH.EQ.1 >CALL TREND
CALL DEVPAPC210.0,297.0,1 l
CALL MOVT02CO.O,O.Ol
CALL LINT02C0.0,297.0l
CALL MOVT02C210.0,0.0l
CALL LINT02C210.0,297.0l
CALL WINDOWC2l
CALL ERRMAXC100l
CALL AXIPOSC1 ,40.0,90.0,150.0,1 l
CALL AXIPOSC1,40.0,90.0,180.0,2l
CALL AXISCAC1,9,0.0,AL,1 l
CALL AXISCAC1,9,0.0,YET,2l
CALL AXIDRAC2,1 ,1 l

CALL AXIDRAC-2,-1,2)
CALL MOVT02C25.0,285.0l
CALL CHAHOLC'PDCW/SQ ml*.' l
CALL MOVT02C190.0,78.0l
CALL CHAHOLC'ZCmml*.' l
CALL GRACURCAZ,PD,Kl
CALL MOVT02C50.0,70.0l
CALL CHAHOLC'POWER DENSITY DISTRIBUTION ALONG THE LOAD *·' l
CALL MOVT02C95.0,60.0l
IFCNSC.EQ.1 >CALL CHAHOLC'ALUMINUM SLAB*.' l
 IFCNSC.EQ.2lCALL CHAHOLC'ALUMINUM CYLINDER*.' l
CALL MOVT02Cqo.0,50.0l
CALL CHAHOLC'RESISTIVITY=2.8E-8 Ohm.m *·' l
CALL MOVT02(1q0.0,50.0l
CALL CHAHOLC'RELATIVE PERMEABILITY=1*.' l

CALL MOVT02C113.0,30.0l
CALL LINT02C125.0,30.0l
CALL MOVT02C126.0,30.0l
CALL CHAHOLC'PREOICTION BY SUPERPOSITION*.' l
CALL MOVT02C1q5.0,qO.Ol
CALL CHAHOLC'o PRACTICAL READINGS*.' l
CALL MOVT02cqo.o,qO.Ol
CALL CHAHOLC'N= Turns d= mm g= mm *·' l
CALL MOVT02Cq0.0,30.0l
CALL CHAHOLC'I= Amp F= Hz*.'l
CALL PICCLE
IFCLH.NE.2lCALL DEVENO
STOP
END
$INSERT BESSEL
OK,
SLIST FITTING1
REAL*B XC50l,YC50l,AC50l,REF,D,AI1 ,AI2,AL
INTEGER N,NT
READ C5 , * l N, CXC I l , Y CI l , I =1 , N) , AI 1 , AI 2 , NT, D, AL
CALL E02ACFCX,Y,N,A,5,REFl
WRITE C1 , 30 l CA CI l , I= 1 , 5 l
30 FORMATC1H ,5X,E12.4l
SLO= CYCNl-YCN-1 l l/CXCNl-XCN-1) l
CON=YCNl-CSLO*XCNll
CL=D*CNT-1 l
R=-CCL/2.0l
WRITE C1 , 9 l
9 FORMATC1H ,'DISTANCE• ,?X,'VOLTAGE' l
WRITEC1 ,90l
90 FORMATC1H ,•--------• ,?X,' 'l
1000 SUM=O.O
RA=AI2/AI1
DO 200 J=1 ,NT
I =J -1

Z=ABSCZAl
IFCZ.GT.XCNllGO TO 300
S UMO =A C1 l + CA C2 l * Z l + CA C3 l * CZ* * 2 l l + CA C4 l * CZ**3 l l + CA C5 l * CZ* *4 l l
SAMO=O.O
SUM1 =SUMO+SAMO
GO TO 400
300 SUM1=CSLO*Zl+CON
400 IFCSUM1 .LT.O.OlSUM1=0.0
SUM= CSUM+SUM1 l
200 CONTINUE
SUM2=SUM*RA
ZZ=R+CCL/2.0)
WRITEC1 ,6lZZ,SUM2
6 FORMATC1H ,F6.1,10X,E15.7l
R=R+10.0
IFCZZ.LE.ALlGO TO 1000
CALL EXIT
STOP
END
OK,
SLIST \1-FC-TEMP
c NON UNIFORM COIL
c
c ------------------
c
C THE SYMBOLS ARE
C CD CONDUCTOR DIAMETER CMMl
C AG AIR GAP CMMl
C N NUMBER OF POINTS
C X DISTANCE FROM SINGLE TURN (MMl
C Y VOLTAGES FROM SINGLE TURN CUVl
C AI1 CURRENT FOR SINGLE TURN CAMP>
C AI2 CURRENT FOR THE COIL <AMPl

c NT NUMBER OF TURNS
c D<I> COIL PITCHES <MMl
c AL LOAD LENGTH <MMl
c R0\1 LOAD RESISTIVITY <OHM.Ml
c R LOAD RADIUS
c UR LOAD RELATIVE PERMEABILITY
c F FREQUENCY <Hzl
c THC THERMAL CONUCTIVITY CW/M Kl
c ALL NUMBER OF CYLINDERS
c SSS NUMBER OF DISCS
c
c
REAL BER,BEI,KER,KEI,BERD,BEID,KERD,KEID
COMPLEX THETA,THY,KO,IO,KOD,IOD
DIMENSION Z<120l,AVC120l,PD<120l,D<50l,AXC120l,AJ<120l
1 , SAJ ( 1 20 l , HRM ( 120 l , AR ( 1 20 l , HR C1 20 l , SRR ( 120 l , BR ( 120 l , ARM ( 120 l
2 , BI ( 1 20 l , SBR ( 1 20 l , SB I ( 1 20 l , XHR ( 1 2 0 l , XHRZ ( 1 20 l , SXZ ( 1 20 l , I TIT ( 80 l

REAL*B X(9l,YC9l,A(9l,REF,D
1 ,AX,AZ,V,V1 ,RA
REAL T
COMMON T ( 11 , 11 l , WX ( 242 l
DIMENSION \IXLC4l,WXHC4l,WYLC4l,WYHC4l
DATA WXLC1 l,WXL(2l,WXLC3l,\IXLC4l/20.0,140.0,20.0,140.0/
DATA WXH ( 1 l , WXH ( 2 l, \IXH ( 3 l , \IXH ( 4 l /1 40.0, 260.0, 1 40.0, 260.0/
DATA WYLC1 l,WYLC2l,WYLC3l,\IYLC4l/110.0,110.0,20.0,20.0/
DATA WYH ( 1 l , WYH ( 2 l , WYH ( 3 l , \IYH ( 4 l /200.0,200.0, 11 0. 0, 11 0. 0/
READC5,*lCD,AG,N, CXCI l, Y(l l, 1=1 ,Nl ,All ,AI2,NT,AL, (DC I l, 1=1 ,NTl
1 ,RO\I,R,UR,F,THC,ALL,SSS
WRITE(1 ,3000l
3000 FORMATC1H ,34X,'W-PDC• l
WRITE<1 ,3001 l
3001 FORMATC1H ,34X,'*****' l
WRITEC1 ,3002 l
3002 FORMAT<1H ,//22X,'COIL AND LOAD SPECIFICATIONS' l
WRITEC1 ,3003 l
3003 FORMATC1H ,21X,•----------------------------' l
WRITE<1 ,6001 lCD,R
6001 FORMATC1H ,5X,'CONDUCTOR DIAMETER= • ,F6.1 ,1X,'MM',
1 20X, 'LOAD RADIUS = ' , F4. 1 , 1 X, • MM' l
 WRITE Cl ,60021AG,ROW
6002 FORMATIIH ,16X,'AIR GAP = ',F6.1,1X,'MM'
1,7X,'LOAD RESISTIVITY= ',E9.4,1X,'OHM.M' I
WRITE Cl ,60031NT,UR
6003 FORMATIIH ,BX,'NUMBER OF TURNS= • ,I5,1X,'TURNS'
1,15X,'LOAD PERMEABILITY = ',F6.01
WRITE Cl ,60041AI2,F
6004 FORMATIIH ,11X,'COIL CURRENT= ',F6.0,1X,'AMP' ,20X,'FREQUENCY =
1,F6.0,1X, 'Hz' I
WRITEI1 ,46501THC
4650 FORMATI1H ,3X,'THERMAL CONDUCTIVITY= • ,F5.1 ,IX,'W/M K' I
WRITEI1 ,6005 I
6005 FORMATI1H ,//30X,'THE RESULTS' I
WRITEI1,60061
6006 FORMATI1H ,29X,•-----------',/3Xl
CALL E02ACFIX,Y,N,A,5,REFl
WRITEI1,10l
10 FORMATIIH ,14X,'THE CONSTANTS OF THE SINGLE TURN EQUATION' l
WRITEI1,77l
77 FORMATIIH ,14X,'-----------------------------------------')
WRITE I 1 , 3 0 lA I 1 l , A I 2 l , A I 3 l , A I 4 I , A I 5 l

30 FORMATIIH ,51E10.3,4Xll
SLO= IYINl-YIN-1 l l/ IXINl-XIN-1 l l
CON=YINI-ISLO*XINll
RA=AI2/AI1
S=AL/SSS
AK=AL/S
K=AK+I
PIY=3.141592654
U0=4.0*PIY/10000000.0
U=UO*UR
W=2.0*PIY*F
R=R/1000.0
PER=2.0*R*PIY
RL=ROW*PER
D2=12.0*ROYl/IW*Ul

D1=SQRTID2l
BA=R*SQRTI2.0l/D1
C CALCULATION OF BESSEL FUNCTIONS
IFIBA.GT.BlGO TO 5001
AI=BERIBAl
A2=BEIIBAl
A3=BERDIBAl
A4=BEIDCBAl
GO TO 5002
5001 AI=REALCIOIBAll
A2=AIMAGIIOIBAll

A3=REALIIODIBAll
A4=AIMAGIIODIBAll
5002 CONTINUE
DEN= I IAI >* IAI l l+ I IA2 >* IA2 l l
B1 = I I A3 >* I A3 l l + I I A4 l * I A4 l l
82 = I I A1 l * I A3 l l + I I A2 l * I A4 l l
CI=BI/DEN
C2=B2/DEN
 AX C1 J =DC 1 J
DO 80 I=2,NT
AXCI l=AXCI-1 l+DCI J
80 CONTINUE
WRITEC1 ,9000)
9000 FORMATC1H ,//28X,'THE COIL PITCHES' J
WRITEC1 ,9001 J
9001 FORMATC1H ,27X,'----------------' J
WRITE C1 , 40 J
40 FORMATC1H ,20X,'DISTANCE',10X,•THE COIL PITCHES' J
DO 20 I= 1 , NT
WRITE C1 , 6 0 J AX CI J , I , D CI J
60 FORMATC1H ,20X,F6.1,14X,2HDC,I2,3HJ =,F6.1J
20 CONTINUE

ZC1 l=O.O
DO 70 I=2,K
ZCI l=ZCI-1 l+S
70 CONTINUE
DO 90 I=1 ,K
V1=0.0
DO 1 00 I I = 1 , NT
ABC=ZCIJ-AXCIIJ
AZ=ABSCABCJ
IFCAZ-XCNJJ200,200,300
200 V=AC1 l+CAC2l*AZJ+CAC3l*AZ**2l+CAC4l*AZ**3l
*+CAC5l*AZ**4l
GO TO 400
300 V=CSLO*AZJ+CON
400 IFCV.LT.O.OJV=O.O
V1=V1+V
100 CONTINUE

AVCIJ=V1*RA/1000000.0
90 CONTINUE
CPD=U*PIY*F*R*C2/2.0
DO 110 I=1 ,K
AJCIJ=AVCIJ/RL
SAJCI l=AJCI hAJCI l
HRMCIJ=D2*SAJCIJ/C1
PDCIJ=CPD*HRMCIJ
110 CONTINUE
c WRITEC1 ,9003J
C9003 FORMATC1H ,//23X,•VOLTAGE AND POWER DENSITY' J
c WRITEC1 ,9004J
C9004 FORMAT ( 1 H , 22X, • ------------------------- • J
c WRITEC1 ,1000)
C1000 FORMATC1H ,2X,'DISTANCE' ,7X,'VOLTAGE' ,6X,'POWER DENSITY' J
c
C1100 FORMAT<1H ,2X,•--------• ,7X,'-------' ,6X,•-------------' l
c DO 120 I=1 ,K
c WRITE ( 1 , 13 0 l Z <I l , AV( I l , PD <I l
C130 FORMAT<1H ,3X,F6.1 ,5X,E12.5,5X,E12.5l
C120 CONTINUE
YB=O.O
YET=-9.999 .0
YEV=-9999.0
DO 2000 J=1 ,K
IF<YET.LT.PD<JllYET=PD(J)
IF<YEV.LT.AV<JllYEV=AV(Jl
2000 CONTINUE
LLL=ALL+1
DR=R/ALL
H1=R/(2.0*THCl
H2=BA/R
. H3=B2*BA
AR <1 l =0. 0

HR<1l=O.O
SRR <1 l =0. 0
DO 4100 I=2,LLL
AR <I l =AR <I - 1 l +OR
HR <I l =H2 * AR <I l
SRR <I l = <AR <I h AR <I l l I ( R* R l
41 00 CONTINUE
C CALCULATIONS OF BESSEL FUNCTIONS
DO 4200 I=1 ,LLL
IF<HR<Il.GT.8lGO TO 4250
BR<Il=BER<HR<Ill
BI ( I l =BE I <HR <I l l
GO TO 4200
4250 BR!Il=REAL<IO<HR!Illl
BI <I l=AIMAG<IO<HR<I l l l
4200 CONTINUE
DO 4400 I=1,LLL
SBR<I l=BR<I l*BR<I l
SBI <I l=BI (I l*BI <I l

XHRZ<Il=XHR<Il/H3
SXZ<Il=H1*<SRR<Il-XHRZ<Ill
4400 CONTINUE
DO 4050 I=1 ,LLL
ARM<Il=1000.0*AR<Il
4050 CONTINUE
c WRITE!1,4000l
C4000 FORMAT!1H ,//23X,'TEMPERATURE DISTRIBUTION' l
c WRITE<1 ,4350)
C4350 FORMAT<1H ,22X, '--------------------~---' l
DO 4500 J=1 ,K

DO 4600 L=1 ,LLL

T<J,Ll=SXZ<Ll*PD<Jl
C WRITE(1 ,4450lJ,L,T(J,Ll
C4450 FORMAT<1H ,22X,'T(' ,I3,' ,• ,I3,' l =' ,E12.5l
4600 CONTINUE
4500 CONTINUE
PRINT 44
44 FORMAT<'WRITE TITLE OF PLOT' l
 READ<1 ,49JITIT
49 FORMAT<80A1 l
PRINT 42
42 FORMAT('TYPE 1=S5600 , 2=C1051N' l
READ<1,*lKP
NY=2*K*LLL
ANX=K
ANY=LLL
63 PRINT 45
45 FORMAT<'ENTER IVIEY' l
READ( 1 •* l IVIEY
IF<KP.EQ.1 lCALL S5600
IF<KP.EQ.2lCALL C1051N

CALL MOVT02<20.0,15.0l
CALL CHAARR<ITIT,80,1l
CALL DATE
CALL ISOFRA<Ol
IF<IVIEY.EQ.4lGO TO 85
CALL I SOPRJ ( K, 1 • 0, ANX, LLL, 1 • 0, ANY, T, I VI EY, NY, YX l
89 CALL PICCLE
PRINT 82
82 FORMAT('1=PLOT AGAIN, 2=STOP' l
READ ( 1 , * l I PLOT
IF<IPLOT.EQ.1 lGO TO 63
IF<IPLOT.EQ.2lGO TO 99
85 DO 11 I= 1 , 4
IVIEY=I-1
CALL YIND02(YXL<I>,YXHCil,YYLCil,YYHCill
11 CALL ISOPRJ<K,1.0,ANX,LLL,1.0,ANY,T,IVIEY,NY,YXl

GO TO 89
99 CALL DEVEND
STOP
END
SUBROUTINE DATE
INTEGER ADATE<8l
CALL DATE$A<ADATEl
CALL MOVT02(240.0,200.0l
CALL CHAARR<ADATE,8,2l
RETURN
END
$INSERT BESSEL
OK,
 ALl
. ' i>l
Lnl;;_;..
';' \. !"
·. :,_ ·
' " ' .'1/'Y ._, 1· A.r:.r-1. AL-::-::h.J.ikhli UHll_
'('L: ;, ,:,1,.~;:,! '... \ I ' _ j);\ I ur.~
~ ;· ,i ;_;,: ·•- t·:-.,,.:;~_:\\1:-.L\TlU:-:
~-'1~<0~~~-k~~~~~~~~-\;lcU~'<~~---r~~~~~--~~~~~~-~[)~A~f~l-~~~
: LOC COD~: K!.Y (..<li\1Mt'Nr~ l_OC CODE. KE.Y COMMf:·r\tfs--
ADR KODE lAST£ flt.r,U~H.IUJNGCN Af.JH KODE TASTE tiCMERKUNGEN
ADR CODF TOUCHF ('OMMENlAHlF<; ADR CODE TOUCHE COMMENTAtRF''•
1 __ u _____ LRN _ ... __ __ 5 000 0 3 I
1 76 2ND Lbi 5 65 X
---~ " ll A 5 03 3
3 42 STO 5 .• 93
--:;-boo cil 1 z 5 02 2
•, ~-- 91 R/S 6 04 4
····---(J .. 4-3
RCL 6 Ol 1
-;.boo' Ill 4 6 06 6
!! 33- x2 6 52 EE
0 85 + 6 05 5
---1()' -- 4 3 RCL 6 94 +I-
1: bo o 0 1 6 95
-- y; 33 x2 6 33 x2
lJ
1--,C -·
95 6 65 X
1·1 34 ;;: 6 02 2
-- 1;, 65 X 7 93
le 43 RCL 7 02 2
l?- 00( - ·o 6 a 7 07 7
,-- -·· 17 06 6
RCL 7 ·\ 0~ 5
--it 4 h 7 .• 02 2
X X 7 .• 08 8
RCL a/cosa 7 06 6
0 1 7 52 EE
X
2 7 .. 04 4
'2 .. - ·-· -
= z a/cosa 8 <: 94 +I-
+ + 8 _95
RCL 8 41 R/S
0 4 LRN
. x2
B trhe values
30
-----
75 STO of the con j-,
31 43 RCL 0 2 stants and'
~__3 CX)c 5 0 5 I current to
33 33 .)(2 STO be stored
34 95 h 2 -a 2 + a (2\ 3
1-~---'-j--=:.--'--- ·- - .. __ -=co"-=-saooi'l here,outsi f>
~-~~-5 l/X 1/ :-- - - ... h the progra
36 42 STO STO so that it
·· · 2 az 2
~!: 00--Cf, 0 7
fl-/ (he eo sa (2\ 4 does not
3S 43 RCL a affect the
_:__]~ 00 c 0 4 • STO program wh f'
__ 51_0 33 x2 (2\ 5 the change
4 1 75 a
4 2 43
--
. - ---
RCL ··> STO
43 00 0 0 5 . 0 6
-44 -33 x2
------- -----~.
.

4; 95
4 r; 34 rx
_3 ;~ _65_ ... X
__40 43_ HCL 2
2 z
4 9 00 0· 0 7 l(h +a-- f-_iL___ _l_ _ ___[ _ _ _ _ _ _ _ _ _
__5 ·~
65 __ ex coscr
X .
"'"·''"''/•. "···~.''I,, .
I"""' '"'•''"'•1/',

5 1 43 RCL 62 [,'1] t:J1 7? ··,ru rt:'J !-1:1 :c.ro C

5 ;• 00 0 0 2 r~ 63 m 0 73 kCl 0 B4 u c
5 :; 65 X X
1;.1 m r..r:: 9? '"'" ,. ...!" !
~'.:1 1, ~~~~.~
f--------.. ---------·· - - -· ------"
5 43 I RCL L__ _ _ _ _ _ _ _ _ll ___~_ __ .J__ _ _ _ _ '---------'-L____:r_,_~~~:~_l N", '-~"_"'~:-:._• s __ j
 ' ~ . ~ • -. ' • .I

KOll!T0!(\1
:::~;:_::.!:~\~;_·~\~- c r, --~-.J\: K.l·1.. Al-Shaikhli . ·---- ____ i_;;_~-i ~-,v
~ROGHAMt.~EUH DA ll 1-'EClLLE P!~ i'Etie;E.. ~;-,1.'-</-.'fl(L'~
LQC
ADR
-C.ODL,
KOlJE
V,LY
TASTE
COMMtN1~
BE.MERKUNGEN
LOC
AOR
CODE.
KOOE
I<EY
TAST(
COfi.H·/.LNl~;
ULM( ~<KUNt;;( N
LOC
AOk
ICOOE
KOOE
I<,[Y
TASTE
COMMl.Ni1~'
HEMF..RIO:UNGEN
ADR CODF TOUCHE COMMENTA1RE5 ADR CODE TOUCHE lOMMfNTA!RFS r~DR CODE TOUCHE COMMENTAIRE
tJ _____ __ LRN ___ -------·----- ___ _5 -~- __ 43... ___ RCL __ 11 _76 2ND Lbl
1 _76. _2ND__ Lbl . v-. 5 ,. 02 0 27 11. 15 E PD
2 _1~- ]li!? .. l<. __,1_ 5 i 95 11 43 RCL
3 42 STO _ _?_H _42_ _STO 11 01 0 14
4 oo o).i 9i s •; 01__ o 12 ... ~ 11 91 R/S
s 91 -R/s..... ..... 6ll os 5

C+ ~+~- :~~~-fr;
LRN
1
:V ::. -.. ~ ~ ···-~··· ~ AF The consta ts
~- ··42· -STO- . - ····H·-- ... 6 :< 89 2ND 7f
~
9 ~0 --~-:t_
__
--- ----- - ·--·- ..... 6 4
-65
-· _____ X
.
.
STO
29
of Frohlic
formula an
~- __91_ R/S . . 6S 43 RCL BF the resist vil
1-1...2_ _Q2. ...... 2. .. ..6/ 01 0 ..... 12 STO of the mat rL
12 43. --·-'--·-· --· · - - - - _6..!_ _95_ r---"=·· 28 to. be stor
13 08 ____8_ _ - · - - - - 6 835 _J/X ...... CF outside. th
e-l-.:_~Q.l ... ____ 1___ ___ ·-- 1-6~ _65_ ... X STO program
~r-93.. c__....1._____ -····-·· ... ... _..]_n . 43__ _RCL 27
16 04 4 ---···--··---- 1-_21_. oLQ __26 _____ _ . ··-
. p
17 91.. . 7 -- - - · - - · - - - _73. _9.5.... ......E-----··· STD
1e o7__ _ 7____ --·---------- 1--1__:;_ __3..4__ _ lx 26.
19 03 3 - ·--·-········ ---· f..-?.~. -~~- -~-0 ...
:_l

•...
20 65 X 7 5 01 0 l3 0
-z 1 4j __:R~E. ::~.~~--== -·'[(, 4~- .. RCL · ·.·
22 0 01 P.).... f....-··-···--·· .. 7 /01.0. 11 . ··. ..
2 3 95 _18_ ..]3_ _ x 2 ......

;: 42 STO ~~03-~f--JL...--- ···-···

1--::..::~r-,1____95:~. - - L@_ _ __ ) f _ _ _B_<l _43..": - ..RCL ....... .
26 03 f..-.3__ -· ~'- ()_2_g ___ 2_~ - . .. . . . .. ..

27 00 . -~-- _8 Z l-f;_5 ____ X... _ .........

28 93 ---·-···. ···--···-··-- ..J1.3_ -~3._ .. RCL .

....1.?. _QS . 5 . 8 ' OL 0 13

_ _3_9 _?2_: EE 8 55.
_ _1_1_ __()} 3 8 ' 02 2
3 2 65
J..--.:.~ --······
X ..B~ 95 =
~-3_~~- _ . ~C:J:,. . B _ e 42. STO
3 4 o2 o 26 1---·----·---·- __.s '' loLQ --~L ___ _ PD
3 5 95 __ .::_ ___ - - · · - - _ __g_(J_ 36.....2ND• .LM . ....

3 6 35 ~L){ _ -· . --·-- ·- 1-221--J,L ____A·- .... H. .. . ·.

3 7 65 X J.-.9-~ 43_ . RCL

3 8 43 RCL _9_: pl
3 0 10.
3 9 00 _0 =_jij~i. . ..... ···- ..-. . 9.·1 _9_1_ _ R/S
1-4 0 95 __ .=..____ ----·-··-- _ 9 5 7(3__ 2~. Lbl.
4 1 42 STO 9 6 12 ___ L ..... .J. ---
4 2 Oleo U ........__ ,I_______ 9 7 43 RCL
4 3 4:3 ---- RCL -=~TRi_() -11
4 p: -,:; . ·- .. . -· .... ...
1-:....:: __1_--"-1'- ....1¥1 ..... ·--- .·-·- __9. ~ .91.... R/S.....•...
1--~! ..§_;>_ -· + _ . ·- --· 10 o_ ...76.. 2ND Lb1
4 6 43 RCL .lO..' .13.. C.
4 7 b2 0 ::::.:::JJJ. .:::: . . . ·-=-·::::=:-..::: .l.O.~ .43.... RCL ..... ·.

4
4 9
a ;; ~~i- ------·-·- i6-~b!io
··-··-··-- ... ·-·-·---· R/S
.. J2
·.

5 0 65 -~- _ . ·--···-···--- ;LQ.' _7§._ ..2.ND.Lb1 MERG(O COOtS

KO ... eo"'A liO .. S-KOOES
TOlK.HES COMI\I .. ([S

5 1 43 RCL 10 r, 14 D
-·-·-----------
5 2 2 0 29 10 ; 43 RCL
5 3
5:,
95
85
10 rn 0 13
TLXAS INSIIHJMLNTS
+ 1o 91 R/S

(
 ,
A.K.f·L Al-Sho.lkhl.i DJ\IL
(l.'"• : r_!M

1\L v cnt::\r-.~t N 1 s
Ti\SH: BCMEl~KVNGEN
10UCHf: C0MM[NTAI RE_
i---'.'~1---- __I,!lN___ .. -~--~- -~5-~ 7r SilR J,l _'!~
_ : .J.§~- __l_!!!>_Lb1 5 i 25 CLR 11 33
____! _],1__ --~~!'_____ 5 43 ll::L ~~i1 34
_ _2__ 42_ ~-!:i'rQ S 1 Xl 0I ~ 1 11 42 STD
_4_ oo_ o_ ___ P_l _5 "~ 75 ~ _ -: 11 D1 <5 15
---~ _9_1 -~ R(S _ 6 '' 43 RCL ~~ ~--~~ ~- 11 7l SilR
'• 43 RCL 6 D2 <1 ~ 21 11 25 CLR
=+,oD_ _o___---__~0x-_ _• 4 __ ___ _6 2 95 11 43 RCL
, 33 _ ~ 2 ~~~ _§:':~_)3 x2 11 DD 01
_2~ _ls_ -~-~- ____ 6 ·'~ 34 rx . ---~- . _______ u 75
e-J-r:>_ 4}_ RCL _ ~ _ 6 42 . STO 12 43 RCL
__ ;I-1_ ~--D ~ p2 5 ~~ . 6 ' Dl c 5 ~15 12 D2 D) 26
~~~ 3_3__ ---~)(__ 6 / __71_ . __$BR _ _____ __ ___ 12 95
J~ 95_______=:__~- __ 6 >< 25 CLR 12 33 x2 .

14 42 STQ_ 6 9 43 RCL __ _ _ 12. 34 /X

15 00-C O=jj 0. 7 ' DD Cl ~P 1
1 12 42 STD
Hi 34 /X --------------- _7_~ _25____- _____ ----------~-- 12 D1 C5 15
17 42 STD __ ------~ ~-?' _!13_____ )<~L_ ~-- -~~~- --~-- 12' 71 SBR
18 oo < 7 0 7 _ _________7__3_ D2 __ c~ ___2L__ _ ___ _ 12. 25 CLR
j.__:1::9+.::6:.:5~_ _.:X-:___ f--________ __2_:!_ _2!?_ ____"; _ _ ---~--------- 12 43 RCL
;~ 1~( -2 -ici --- -- -}: · 2
;~ ~-- )x~----~--·- 13 00 0
u· 75
0 1 ..

~~ 5?'5= -~--~~ :_ 7 42 STD -~ _ 13 43 RCL

,~ *i~-~C:j--- ----~--- 7 ,, Qlci5 __ _1.s_ . ______ _ 13 o2 o7 27
7 9 71 f----§B_lj_ -~--- 13. -~~95 =
25 55 ___.:;::____ ~-- -----~----- _ 1-8__ ~0 __ 2!? ~-~CLR. __ _ __ __ ____ _ n 33~ ·-x2
26 89 2ND 7f ~---------~-· __8 _l__ ___1} __ RCL____ ---~----- _ 34 13 rx
1-"2:.:_7_j_e_9=._5-1- ------~~-- ~-~-~--~ . !3 __~ 00 Cl_. >Ll._~- -------~-- 42 13~STD
28 42 ST9_______ ---~--- f-8-~'' __75 ---- --- ..... -~----- --~- 13 ' 01 D 15
~!l_ .Q_QI -~ J!L 8 _ _84 25 ... RCL .. 13 71 S!lR
_;JOL_()Q __ p_ --~-- 8 ·. 02 c 3 23 14 25 CLR
__21_ ~- __ STO _ 8 95 14 43 RCL
32 1 0 14 8 33 · x2 14- 00 D 0 1
_3_:3_ _'1_3_ -- - ~I, - --~ - - 8 '' ... 34 I _ /X: 14 75
34 o_ QJ ~_l_ ____ ------- - - ~ 8 .':. 42 STD 14 43 RCL
~---:3:.:::5~~7;:_5-+---=-~--- ~- ~-~--~-~--- -~--~--- --~---?.~ Q1_(§=i.s:~ ~~ __ _ __ __ _ 14' D2 D 28
36 43 RCL 9171 ·- ---··- SBR·-··-- -- -- __ ____ --------------- _14 < 45 =
~-----;.::..~-,::::__.J_~--- -- --------- .,

37 1 0 19 25 CLR I··· 14, 33 x2

- - --- -- 9 2
3B 95
3~!33-
-·----~~--------

=
-:xr- 9 :! 43 RCL 14 34 rx
4o 34 ~ - -~rx--~- -- --- ~9 ~'~ DO c 9l 1 14 42 STD
9 c 75 ~ 15 D1 0 15
~ 42 ----sTo- - 9 G 25 RCL 15. 7l SilR
42 1 Q< 15 -- f.--- - ---- - _9 _, o2~<:ij -24___
------~-~---~
15· 25 CLR
-~- - f--- ---
_j~ ;2!._ _I'BR____ - 9 [; 95 15 ' 43 RCL
~ 25 CLR 9 ~- 33 15 . 00 0 .)!\ 1
45 43 RCL 100 34 155 75
46 oo<
4775---~~--
i~l'fl - 1D1 42 STD 15 25 RCL
----H·-~- -----
_102 01 Op 15 15 D2 0 29
48 43 RCL 1o:• 71 SBR 15 .. 95
c
ti~ 02 p--2o___ _
10-: 25
~ ~

CLR 15 33
!---=-::_ ---- ----------~ -- -~
50 95
--'-CJ--~-- -- 2 ~- ~
-= lQ'• .43 RCL KOMI;IINA T tON~·K()tl[!o

i-2_1_ _l)___ _:_J( --~- 1DL DD D .01 6211!J C'J 72jsTO; £!I 83 :c,.ro; C)
~_;;? ~ 34~ ~-- rx 1o~ 75 63 [!!! EJ
64 m c
73:Hc>: EJ
74·soM m
841!::!11C]
92 INY_ ;'.&1
3
5 42
T LX-~-~~;:;-;~,
STD 10 43
s' ell c s 1s
--~~~~~~_L_ _ __u_~~~~
1D o2 D,, 25
RCL
,; I
19771r-A•S Instruments
 2 1__ { <;); .i.. , i· \}~-:.-..l
1'!-\1 d
.;;·· 1\,K,l'-L !d-ShaiLllli ! ~;, i l !~()lJFFOH~l
[;/\1 \.lflj

KE.Y cur-.!IME.NlS
1 AS T E BCMEHKUNGEN
TOUCHE C'.OMMENTAIRE
Hi~:- .3L _7x __ ___ 2J : 2 2 7 ___n;v LR
16 --1
r---- 42 STO '' LRN
'1_(): Q1 0 15
'16:l 71 ___SBR (! S_ Constant -
-;
16 ~ 25 CLR STO
16 ,; 43 RCL ,, -
~2
lG -- ,ol o:l 14 I Current
16,- 33 x2 :.: STO
16" 65 X :1
-~ 3
16 u 43 RCL _,
h _Height_
17<j 01 Qj 10 STO
.!:._7 ' 95 ~ 4
17l 42 STO _a Cond, Radi
17 3 01 0 11 ____STO __
.. --·------- ---------~---

:!} 4_ 14 3_ RCL __ _ ~ 5
,_,
1-2.:" 01 0 14 _a Constant ---
17 6 91 ____R/_'§_____ ____H _ __ STO
1
~?_ _76_- 2l'lJ?_~b1 - ----"-
2
~ 6- ------ - ------
£_B__1?_ ---~--- PD 3
------1 - -
pp T]1__e _c_OJ15t<!J
17 9 43 RCL 4 STO of the PD
__:_:c -- --'--
18 0 01 0 11 ---- c 11
181 91 R/S f· Zl he distanc
;
T82 7& 2ND Lbl The starti STO of the
·----- -----·-
~_?_5 CLR of the 19 turns from
118 4 43 R,CL sul:u;~_t:i,rl_"_ 22 t!'_.,__z er2 ___
1185 00 () -·-a-;4~- 0 _STQ____ point .C'l'l _
~_8 6 1--_3 3_ 1--- _X 20----- the load
··-··--.
-

18 7 85 + Z3
-----·---·-----
1188 43 RCL STO
189101 o, 15
x2
I -
4 21
19;;1· 33 ' Z4
-

I "

I
~· I
0
-- I - -: --
-

---
_,

200 55 '-•
201 25·· RCL --

---1------ - --------- ---

2.0.? oo__ o
2Q.2 _22.. =

--

- -----

•uuCoH.•, (.0MI\INFE~
 IHJ>MOvt:Ht:NTS IN 11ft: DESIGN Of' INOU<...TION BlLI.Y.T
llf!';\TERS.

A KM Al-Shaikhli and L Hobson

Louqhborou')h Uni~rsity of Technol09Y• UK.

INTRODUCTION t • Time (s)

X • Depth in the slab -•aured fro. t~e
Induction heating ot' metals is well established surface (m)
throughout the ~tal fo~ing industry for heating z The distance along the load Masured
metals prior to fotqi.ng, rollin9 and e~etrusion. from a point directly beneath the
Althouqh the unit cost of electricity is high com- conductor (RI)
pared with fossil fuels, induction heating techni-
ques offer: fi\Any advantages including high efficiency, •
y
n
• Poaitive roots of J Cel • 0
Denaity (k~ m-3) 1
high power densities, and improved working environ-
ment. These process advantages have ensured that 6 .(-L
• WfLI
Skin depth (m)
induction heati.n9 has achieved a substantial .arket
share of the heatin9 installations within certain • • Resistivity (0 at
Permeability (Hnl-11
parts of the aetal for~ing industry. Other areas a" • Temperature (°K)
however have so tar re~~ained virtu.~.lly untouched,
especially those requiring a n.on-unifor. surface 3 THE SUPf!RPOSITION METHOD
power density along the length of the workpiece or
those requiring ~~&ny different workpieces diameters The temperature of a load is a function of the sur-
to be heated durin<; the workin9 day. face power density. By controlling the current
density distribution on the load the temperature dis-
In industrial practice the desio;n of induction billet tribution alon9 its length can be controlled. To
heaters is lar<;ely based on the equivalent circuit achieve this aim, the surface current density distri-
method devised by Baker [1,2] and developed by bution in the load produced by a single conductor was
Reichert {ll and Vauqhan and Williamson [4,5). The investigated and it vas found that the distribution
method assumes a W'lifor.~~ magnetic field stren9th could be repreaenf.ed by the following equation~
along the length of the workpiece and .any empirical
factors are incorporated to take into account such 2 3 n
J(:r.) • ao + al:r. + a2z: + a3:r. + • •• • + anz: {1)
things as a shortness of coil and spacing between
turns. Numerical solutions usin9 finite difference If there are a number of siMilar conductors then the
and finite element techniques are in existence [6,7J surface current density at any point on the load can
but they also assume a uniform . .gnetic field be found by applying superposition principles. Con-
strength and hence a surface power density along the sider Fig. 1 which repreaents N identical conductors
length of the workpiece. Even though they are at parallel to each other at unequal distances frotll
best two dimensional solutions they require specia- each other but at a specific air gap from a metallic
list knowledge and powerful computational facilities slab. If a current 1 flows through each conductor
not norma!ly available to U.K. induction heating then the surface current density (Jp) at point P is
manufacturers. 9iven by equation (2).

In order to deal realistically with froblems involv- Jp (21

inq a non•unifor. surface power density alonq the
lenqth of a workpiece, the surface current density where Jp , Jp •••••. , Jp are the surface current
2
distribution produced by a sinqle conductor has densititAs prOduced by th~ individual conductors
been investigated. The superposition principle to 1, 2. N at point P and they are given by:
find the power density produced by a number of
2 n
conductors has been ~rified and a work coil design
to produce a particular temperature distribution in
Jpl •o • alzl + a2z:l + •••• + anz:l (Jl
n n
a billet has been carried out. Investigations have
also included the effect of conductor size, operat-
Jp2 • •o + al z:2 + 4 2%2 • + anz2 (<)

inq frequency, gqnetic per.eability and ahape of

workpiece, and current distribution within the work
Jp. •o • .alz:N • a2z:N ' • •• '
n N
n (5)

conductor itse'lt. The same equations can apply i.f it is required to

produce a specific current density Jp at point P by
changing the distances z , z , ••• zN.
1 2
a 0 , a , · • ·, an • COnstants to be foW'Id for each By using this method, the induced surface current
1 density and hence the power density and the tetnpera-
case
b • Thickness of slab (m) ture distributions along the load can be controlled.
c Specific heat (J K-l kq-l)
f • Frequency (Hz) It should be mentioned here that the relation bet-
Jo(x) Beasel function of the first kind ween the current in the conductor and the induced
and zero order surface current density in the load is a linear
J (zl The aurface current density on relation and it should be taken into account i! the
the load (M- 2 J current used in multi-turn coil is different from
A radius in the cylinder (m) that of single conductor.
'• The radius of the cylinder (m)
4 THE EXPERIMENTAL WORK achieved by inducing into the heated billet a tempe-
rature at the front face, that part which enters the
verify the aupeqx)sition method practically, a
'IY.o die first. In order to appreciate the reason for
series of testa wre carried on an aluminium slab of this isothermal effect it must be explained that addi-
50.8 11111 thickness and a cylinder of 50.8 mm radius, tional heat will also be generated during the extru-
In all the experiments a single current carrying sion operation:-
conductor was placed above the slab or arowtd the
cylinder at a specific air gap. The surface current a) By heat energy • caused by the shear effect at
density distribution induced in the load was tneasw:ed the restricted die opening,
by a current denaity probe !&I. Using the results
of the single turn coil the performance of a multi- b) Through the generation of friction, since the
turn coil can be predicted based on the calculation hot aluminium billet is not only forced into the
of the cu:rrent density by the superposition method die but also mows alonq the stationary container
and the power density based on the equations given whilst confined within its bore.
in the Appendix.
In indirect extrusion the hot billet remains stationary
Two types of water cooled conductors were used in in relation to the press container, the rear end of
the experiments with the slab. The first type was the billet becomes progressively cooler as the extru-
a circular conductor of 6 ... diameter, in these sion proceeds rather than t:he front end. This, of
conductors the current flows wtifor111ly. The second course, is the exact reversal of the direct extrusion
type was a rectanqular conductor, Fiq. 2, which are and care must be taken to ensure the correct deqree
used in uins frequency induction heatin9 installa- of reverse taper if optimum results are to be achieved.
tions. The current does not flow uniformly in this
kind of conductor. ~ demonstrate the usefulness of the design procedure
developed in the previous sections it. vas decided to
1:1 the experiments on the cylindrical load only design an induction heatin coil capable of producing
water cooled circular conductors of 6 mm diameter a linearly decreasing te~~~perature taper. This, of
were used. course, models the practical requirements met
industrially in the induction heating of aluminium
All the experimental work was carried out using a prior to extrusion.
mains frequency supply.
It was not possible to build a commercial size billet
f'ig. 3 shows the surface current density distribution heater to test the theory because of the capital cost
alonq the length of the slab due to current of 600 A and the limitations of the laboratory power supplies.
flowing throuqh a single circular conductor with a Megawatt power supplies and large amounts of power
)9 mm air gap. The correspondinq power density factor correction capacitors are not uncommon in
distribution induced from 5 conductors carrying industrial units. However, a laboratory scale model
600 A/conductor at 60 mm coil pitch is shown in was built to demonstrate the validity of the design
f'iq. 4. The agreement between the practical read- procedure. The work coil vas made from 6 an diameter
ings and the calculation values is 5\ i.e. within water cooled copper tubin9 which tneant that the
the range of experimental error. variable pitch coil could easily be .anufactured with
the University'• facilities. The workpiece vas a
The surface current density induced in the slab due 200 mn long aluminium cylindrical billet of radius
to sinqle rectanqular conductor carrying 600 A with 50.8 11'111 and the air gap vas taken as 30 111111 which is
an air qap of 36 mm is shown in Fig. 5. The super- in agreement with industrial practice llo(.
position theorem vas verified by us1ng 10 conductors
very close to each other, coil pitch of 11 mm, The induced surface current density o.n the load due
carrying a current of 600 A/conductor as shown in to a 600 A current flowing through a single turn
Fig. 6. The disc-repancy in this·ca.se-is·lesS than coil WAs· rneasurea··and· the re"SUlts. -~-·in Fiq. 9.
9\. These values were then used to calculate the resul-
tant surface current density of a number of different
Figs. 7 and e show the results obtained from the work coil configurations using the procedure out-
experiments using a cylindrical load. The current lined previously. The design of a work coil capable
in the single turn and the 11 turns coil was 600 A of generating a linearly decreasing temperature
and the air gap was so mm. The coil pitch of the taper on the surface of the billet vas then decided
11 turns coil was 7.1 mm. The difference between upon and details of the work coil pitch are given in
practical readings and calculated values is less table 1. The eleven turn work coil was constructed
than 3\ for power densities down to 5\ of the maxi- and the surface current and power densities it
mum value. At power densities below this the produced were measured. The surface PQwer densities
accuracy vas not as good but because of the magnitude induced in the load varied from 7 kw;m 2 down to
of the discrepancies vas small this vas neglected. o.S kW/m2. The practical measurements were compared
with those required to produce the linearly decreas-
S PPODUCTION OF A PCMER OENSI'IY TAPER ing temperature taper and the results are shown in
Fig. 10 correlation to vithin an accuracy of 5\ can
There are applications in induction heating, such be seen thus demonstrating the validity of this
as off-the-bar forging or the extrusion of aluminium novel method of work coil design.
which require a non-uniform power density to be
produced along the bar or billet.

Jn direct extrusion of aluminium, a ram directly

pushes a billet through a stationary press container
Pitch No. 1 2 l 3 4 5
• 7 0 • 10

and selected die-shape. The finally extruded sec-

tion emerges onto the rW1-out as a correctly shaped
product. In this kind of extrusion i t is preferred
~ 7 oIn 23 23 23 25 23 25 32

that the extrusion speed of the ram and product ' '
temperature through the die be constant throuqhout Table 1: The pitches of the coil
the extrusion cycle, thus avoiding continual adjust-
~nt of the press, This isothermal effect may be
• CONCLUSION 10 DAVIES, E J 'Induction Heating Handbook'
and (HcGraw-Hill, 1979)
The practical investiqations show that the super- SIMPSON, P1
position method can be applied on loads of different
ahapea. The non-uniform flow of current through APPENDIX
the conductor. did not affect the applicability of
this .ethod within the ranqe of the teat carried The Relationship Between Surface Current and Powec
out. Densities and the Temperature of the Load
Applying this .-thod to a practical application, In induction heating, the heat flows almost in the
i.e. the production of a linear power density taper radial direction only because of the high power
along the length of a billet, proved highly success- u.ually uaed in this kind of heatin9. The one
ful. dimension solution to the problea of heat flow in a
slab and cylinder heated by induction is shown be1dw.
The computer proqr. . h relatively BIMll, i.e. it
does not need a larqe atoraqe capacity or conatne a) Slab heated fro. one side
long computing time. This ..ethod is therefore suit-
able for desk top computers which are cheap and The temperature {a) at a depth {x) in a slab of total
readily avialable within even the smallest c~ny. thickness {b) heated on ono side at a constant power
density {PO) is 9iven by (9j.
Although the results are shown for tests on auaina
frequency and non-magnetic loads, further work now
being carried out to assess the effect of frequency, a • 1 4 <
proxilllity of laainations packs and non•uniform - l- 2 L
• n•l
current distribution in conductor. Work also vas
carried out on magnetic material to assess the
effect of the permeability and the possible use of (lA.)
this method in the oft-the-bar forging applications.

REFERENCES

1 BAKER, R M: 'Heating of Non-magnetic Electric The firse term is the final temperature of the slab.
Conductors by Magnetic Induction - The S\mll!l&tion term is the transient temperature rise
Longitudinal Flux 1 , AIEE Tra.ns, and can be ignored for valued of t 31- 0.25. The tra.n-
1944, ~· pp 273-279. aient period represents a small part of the heatinq
2 BAKER, R M: 'Design and calculation of Induction :r~~i!:r.~=~~~e 1 ~ ::r~::~==s~t~ 7 ;~c;~ds foe
Heating Coils', AIEE Trans, 1957, Equation {i~) then s~plifies to
,!! (pt II), pp 31-40.

3 REICHERT, K: 'The calculation of Coreless a • ~

k T + ~
2k [u - xfb)
2
- !.3 J (2A.)
FUrnaces with Electrically Conduct-
ing Crucibles', Electrotechnik,
1965,!! (6), pp 376-397. ·· a • 2b k {2t + [ {l - x/b) 2 - 'l }PO
3' (3A)

• VA.UGHAN, J T
and
'Design of Induction Heating
Coils for cylindrical Non- i.e. the temperature at any depth (x} is a function
WILLIA.MSON, J W; .aqnetic Loads 1 , AIEE Trans, of the surface power density which is in turn a
1945, !!· pp 587-592. function of the surface current density llo) and
it is given by equation (4A.} below
' VAUGHA.N, J T
and
'Design of Induction Heating
Coils foLCylindrical Magnetic - pD •
2 •
.J 2
WILLIA.MSON, J W: Loads', AIEE Trans, 1946, 65,
pp 887-892 0 -
This shows that the temperature at any depth is a
function of the surface current density •
• REICHERT, K: 'A Nwnerical Hethod for calculating
Induction Heating Installations', b) cylinder of radius R
Electrowarme International, 26 (4),
1968, pp 113-123, Englhh Tri'iisla- The temperature (a) at a radius (r) in a cylinder
tion by the Electricity Council, of radius (R) heated by induction la 9iven by [9]:
overseas Activities, Translation
Service.
7 DEHERDASH, N A, 'Solution of Eddy CUrrent
a • (PO)~ {2t + r ,-
2R
2

HOHA.."lMED, 0 A, Problems Using Three Dimensional

NEHL, T W and Finite Element COmplex Magnetic
MILLER, R H: Vector Potentional', IEEE Trans, (SA)
1982 1 PAS-101, (11),
pp422~
kt
• BURI<E,
and
PE 'Current Density Probes', IEEE
Trans, 1969, ~ (2),
where t • --2
ycR
ALOEN, R T H: pp 181-185.
as in the case of slab the summation term can be
9 CARS~, H S 'Cbnduction of Heat in Solids', i9nored when t • 0.25 and equation (SA) can then
and (Oxford University Press, 1959). be simplified to
JAEGER, H C:
 2
r --)
o•Po!!.(2t+ - 1
(611)
k 2R2 4

henci the temperature at any radius is a function of

the surface power density which is, as in slab, a
function of the surface current density [10]

1
[ber (•) ber (•) + bet (•) bei 1 (•) J
(7A)
bn
12
(•) + bep (•)
Where .. lz R

'
As in the case of slab the temperature at any radius
is a function of the surface current density.

...
I

I 2 l
[ -a••-
N

. . <!>
~ <!> 0 0
:
<!>0 -
-1•...-!
0
0
0 0 : ~
24
:--- z, ____, 0
0
0
0
0

•• _., ''
... _____.., ,./
0
I
0
-- water po.
0
~
,/
sol.i4 copper

?;b/?///.neto.Hic slo.b 777 7 7 7 7 7

Fig 1. Number of ilkntko.l conductors ot o. specific air~

frM a ~~~eto.llic load .

fig. 2. Recto.h9Jlo.r conductor used in ~~~Gins

frequency induction hi!Qting.
 210

•
"'
160

~140
:!
~120

"
eo 120 160 • 200
Disto.nce otonq t~ load. z. (!ne)

Fig 4. Powr density distribution along the alu~niniulll slo.b due

fig, 3. Current density distribution along the aluninit.n
to 5 circular conductors at air gap of 39 mm, coil pitch
slab dut to 600A ftOoting in single circular
conductor at air 90-P of 39 11111. of 60 ••· and a current of 600A/conduc:tor.

- - - - Prediction by superposition a Practical t~~:~dings,

40

. -;;
'i
l6

)2

;."
" s .... "
'"'
~

;;
"c
~
"
~ ~
c
c ' •
c
.!l
20

j 16
':l
)
~
~
~
] ':l 12
~c

'
40 . 110
Oistunu along the toad. z, lmml
,., 200 '' 20 c.o 60 60 100 120 140
OiStanu along the load ,z, {«wnl "' 110 200

fig 6 Power density distribution along the alllllliniulfl slob

Fig, S. Current density disfr1buf1on along the: alu111iniu11
due to 10 rtda~lar conductors at air gap rll611ft.
slob due to 600A rtoving in single reclo.f91~'
conductor at air qop of ]6 111111, coil pitch of 11111111 and a currtnt of 600 AtcondJctor'
Prtdic lion by superposition o Practical rtcdinoJ.
 130

120
SI
110
"'
e 41
-. ""
~
•-.' " ... "
"" "
..;
~
J6
D
" .,
d'
~

·;;
))
."
~

0
~

'E "24 0
~

~ so
.
i ~

..
V 21
'Z
'Z 13 ~
•
~

~
~
~

~
30
12
9 ,.
6
10

..
3

" Distance along the load, z, (IMII

160
0
0 10 20 D .. "' .....
Oido.nu olonc;J
120

Fig Power density distribution along the aluminium cylinder

Fiq. 7. Current density distribution along the aluminiull'l
cylinder due to UJJA flowing in single circular due to n turns coil at air gap of SO mm, <Oil pitch
conduchrat air gap of SO 111rn.
of 7-11'111'1 and a current of 600A

Prediction by Superposition o Practical Readings

"
,
~

"e
70

-.. "
" "
a·
~

~ so
~
0
k

~
• 54 •"
~
14842 ~

'Z 30
~
~, ]6
~
<
v30
20
~ 24
~
.= 18
12 10

.. .,
Distance o.long
160 OL-~-c.~~~~~~~..-~~
o 20 "Distance
" eo 1 m u ~ ~ ~
along the loo.d, z ,fmm).
fig. 9. Current density distribution o.bng the alu11iniue Fig 10. Power densitr distribution along the olucniniu111 cylinder
cylinder due to 600A tlowinq in single con!llctor
at air gap of l01t1111. dut to 1t turns coil at air gap of Jhun, non uniforM
coil pitch and o current of 600A.

---Prediction by Superpositions. o Practical Reo.dings

 METHODS or ANALYSIS fOR INDUCTION HEATING WORX COILS

A.K.M.Al-Shaikhli and L. Hobson

o.partment of Electronic and Electrical Enqineerinq,
touqhborouqh University of Technoloqy,
Louqhborouqh, Leicestershire, LEll 3TU,
Enqland.

is derived usinq the reluctance of each of the

maqnetic circuits and Ampere's law (3,4).
Industrial dadqners of induction heatJ.nv work colla
depend larqely on relatively atmpla equivalent The work coil voltaqe, Y0 , is related to the total
circuit techniques supplemented J:Jy empirical data flux J:Jy Farady'a lav and hence an expression far
acCUIII.ulated over many years. on the other hand the total circuit ilapedance, z, can be obtaine4. An
acad.mtea and other reHarch vorkua have developed. electrical equivalent circuit is therefor• derived
hiqbly sophisticated numerical tachniquaa to. solve and shown in Fiq.2 where
particular problem.a. Thb paper explaina the
principlaa behind each fora of work coil deaiqn and
aaaesaaa the ranqa of their applicability. .
2co.4S+kl
Suqqaatione are made to improve the exietlnq work (r2 - r2)
coil deaiqn techniques and J:Jrief details are viven • •
of a nev -athod of daaiqn aore read1ly applied to
certain industrial applications of major present
•.
day interest, i.e. aluminium extrusion and ofl-the- 2 • .2
ZV • W PO W •C r V (Q+jP) • RV+jXV
l>ar forving. -.;;-
l. INTRODUCTION
zc • Wp
0
W N! k~ ac re (l+j) • R +jX
0 0
Induction heating has a number of inherent advantages
over ita fuel fired competitive proceases1 including
••
where N0 ia the number of turns. r 0 , rv, le and f.w
fast heatinq rates, precise temperature control, are the radii and lengths of the coil and the
auitabi11ty for incorporation in fully automated workpiece respectively. 0 and P are dimensionless
proceaae•, small atandby losses, and qood vorld.nq constant., 40 1a the coil akin depth and Jc end ~
environment vith little extraneou• haatlnq and a are empirical factors.
low noise level. These process advantages have
ensured that inductiOil heatinq' has achieved a The equivalent circuit can be solved by circuit
substantial 111Al'ket share of the heat1nq installations analysis to qive work coil turns, current carrying
within certain parts of the metal forminq industry capacity, efficiency and power factor.
(1). other areas~ however, have ao far remained
virtually untouched, especially thoM requirinCJ a It should be: 11.entioned here that other fonu~ of the
non-uniform surface power density alonq the lanqth equivalent 'circuit may J:Je used by choollling ditlerebt:·
of the workpiece (2). positions for Xr in the circuit (3,5,6).

Indu•trial designers of induction heatinCJ work-coils The effect of the different parameters in the
depend largely on relatively simple equivalent equivalent circuit depends on the case under con-
circuit techniques supplemented J:Jy empirical data sideration. To asseslthe influence of the external
acCUIDUl&ted over many years. on the other hand reactance Xr and the empirical factors k and krr
academics and other research workers have developed three computer proqrama were developed J:Jaeed on
highly sophisticated numerical techniques to solve three different equivalent circuits. A specific
particular problesu. Exiatinq methods of work-c()il vork co!l/no~etic billet configuration vas
design almost universally assume that tha load ia asataned. and calculations were carried out for
subjected to a uniform maqnetic field. frequencies of SO Bz: and 3 kHz. The required
number of turns obtained from the three circuits is
I t 1a very difticult, therefore. to adapt theae vithin 8\ difference from each other for the SO Bz
methods to solve the probl&IU of non-unifoX'III. power frequency, this difference increases to J:Jecome 10\
density distribution alonq a workpleoe. when the frequency 1s 3 kHz. Varying k J:Jetween
-o.22 and O.l chanqed. the required number of turns
This paper explains the principles bahi ~ each form by up to 10\. The maximuaa effect of chanqifl9 kr
of work-coil desiqn and assesses the ra.nqe .of their fr0111 1.0 to 1.5 vas 20\ difference in the required
applicability. SUqqestions are made to improve the numJ:Jttr ot turns.
existinq work coil deaiqn techniques and details
are viven of a nev Mthod of desiqn mora readily The equivalent circuit Mthod suffers, "nudnly, from
a.pp11ad. when non-uniform surface power denalty three tmperfections.
distriJ:Jutions are requ11'114-:
(1) The need of empirical data. The values of these
2. EQUIVALENT CIRCUIT METHOD .factors are not necessarily known for every
application.
The basic a .. embly of induction billet heater
consists of a vater cooled coil aurroundinq a met- (2) The aa~tion of uniform material properties
allic workpiace. A relatively larqe air qap between· which is not· correct aa the resistivity and
the coil and the workpiece 18 required ao that the permeability of the workpiece chanqe with the
billet can IIIOYe: freely through the heater and, in teatperature end magnetic field strength. Thia deficit
addition, thermal inwlation ia provided to rad.ue. can partially be overCODe by considering the work-
the radiated heat losses froaa the billet to the piece aa consisting or a number of concentric
coil. 'nle maqnetic flux produced J:Jy the work coil cylinders as it will be seen in the next section.
has alternative pa.rellel paths through either the
workpiece, the coil or the air qap as shown in Cl) The assumption of uniform magnetic field along
Fiq.l. the length of the workpiece hinders this ~~ethod
from being applied to applications which require
An expression for the total ampere-turn requirement nonunifom power densities in the load. This
probleaa can be aolved by. aployinq tha nlaU.vely determine the induced current.a, power distribution
new auperp0.1tion met.hod--(2).·- dd mechanical forces produced by any tvo dJ.Junaion-
al, linear an4 axiaymmetric induction Mating prob-
2.1. Improvement on the Equivalent Circuit Method leaa. The limitationa of the ~Nthod are that u.q-
netio aatarials cannot be included in the problem
One of the Umitationa of the equivalent circuit 9ecmeb:'y and that the number of ele~Nnts into which
~~~t~tho4 la that it doea not tAke the nonuniforaitY the conductors can be divided is limited by the aize
of the vorkpiece into account. This can ba overcome of the mat.J:ix which . .y be solved. This number la
by a ..Wilinq the vorkpiece to conaht of ~e than of the order of 300 unknovna tor the IBM 3033 cOCD-
one concentric cylinder, each with different putar (10) which haa a memory capability of approx-
physical properties. imately one MCJabyte. Difficult!. . alae arise when
dealing with curved surfaces and vhen the penetra-
riq. 3 shova a typical billet ~atinq application tion c!epth la Maall in comparison vith the lCHid
in which the coil of ra4.ius r._ .urrounda a bUlet of dblendcns. The nwnber of unknown• can partly be
radius rv• The billet la ahoWn divided into n reduced by finely .ubdividing only the reqion of
concentric cylinders. The impedance of the ccmpodte goreatest interest, typically bewteen a diatance
load can ba datemined by UIPloyinq the complex 0.1 to 0.33 timea the akin depth G (LO) fra~ the
PQyntinq' a vector (4) to deriw an expre .. ion for .urface and using Larger aubdiviaicns elaewhere.
the tmp.dance, Zx, on radius x within a cylinder 'l'hia, of courn, requires more complicated software.

3.3. Finite Eleml!lnt Method

The finite element . . thod haa bec:cme an important

and practical numerical analysia tool. It has
then aolvinq Maxvell' a ~ationa on the boundaries found application in elmoat all areas of en;ineering
batvaan the concentric cylinders. and appli.S uthematica. •

A computer proqram vaa developed for a work coil/ In this Mthod, the field reqion t.a subdivided into
multicyllndar billet. 'l'vO caHs v.re considered. a finite number of discrete sized aubreqicns or
The first is, a load of 8 concentric cylinders with finite elements. '1"he unknovn quantitiea at each
ditfarant. physical properti... The HCond is tor a element are presented by suitable interpolation
unifon. load with physical propertiea equal to the tunctiona that contain the node values of each
average properties of th.a first can. The differ- element.
ence between the required number of turns for the
two loada vaa found to ba 23\. 'l'bia shOVII the L'ral the . .thematical point of viav the finite
importance of the load's nonunUormity in the element Mthod ia based on integral formulations.
calculations. By vay of caaparUon the older finite difference
methoda are usually based on 41ffarent1al formula-
3. NUMERIOL METHODS OF WOIUC COIL DESIGN tions. Finite element inteqral fcra11,1lations are
obtained by tvo different procedure•• Variational
Three distinct nuaerical approaches have been fcrllllll&tions (11, 12) and weiqhted reaidual formu-
applied to induction Matinq problems a finite lationa (13). Both of these techniques generate
c!itferenca, finite element and mutually coupled th8 final assembly of algebraic equation• that must
circuit aethods. In the finite difference and be .alnd fozo the unknown nodal par&~~etera.
finite element methods the vorkpiece is divided into
subraqions and the relevant non-linear partial It JaUat ba borne in mind that when employing the
differential ~ationa are replaced by a aet of finite el-.nt methods. to. 110lve induction heatinq
algebraic ~ations to ba solved in each .ubreqion prcbleiiUI, one muat (10}
by m.ans of iteratiw procedures. The llll.ltually
coupled circuit Mthod (7) divides the .ystea into (i) Include an •exterior :element• to represent the
a numba:r of aubconductors INtually coupled with reqion outside the con.
each othar and applias ICirchcff'• Lava to obtain a
aya~ of linear equations deacribinq the prcblell (U) Use a aparse matrix aclution routine.
which in turn are aol v.d by procedure a siailar to
those uaed in otlwr nu.rical •thoi•. The Typically, induction heating problem& involving the
application of trulae numerical methods to induction coraputation of the electrcmaqnetic and thermal
he&tlft9' problems will now be discussed in c!etail. fields at. 500 nodes requires approximately o. 75
lllltCJabyt&s et computer memory (10) even vhe.n usinq
3.1. MutualLY Coupled Circuit Method a sparse matrix solution routine. Problems involv-
ift9' 500. to 2QCX) unknovns can be run at a reasonable
Thia method vas developed to solva problems in coilt ·on an IBM 3033 which compares very favourably
induction heatinq and melting applications by with the 111\ltually coupled circuit method IIW!ntioned
Kolbe an4 Rei .. (7) and later developed by Duc!ley previously.
and Burk•· (8). '1'he reqion to be 1nvest1qated U
subdivided into sobconductors and using Jcnovn 3.3. Finite Differences
expressions (9) the reaistance and self inductance
of each subccnductor, toqether with tlw JaUtual 'l'hi• vas the firat numerical technique applied to
inductances between each subconductor are determined. induction heating problems (14) and ia still in
The expre aaions are usually preaentad in the fcna of use today. The technique provides a relatively
an impedance matrix and the application~of Xirchoff'• atraiqhtforvard means of formulation and aolvinq
LAva had to a system of linear equation• with com- two dimenlional problems.
plex coefficients of the form
'nie bada of the finite difference method is the
rzJ (I) rvJ conv.rsion of the 9overninq differential equations
of the problem into a aet of algebraic eqU.ations by
where [z] ia the aquare matrix compriainq all of the use of the Taylor aeriea approximations, The
the coil and billet self and .utual impedances, equations are solved at every point on a qrid
[I] is the column matrix of unknown coil and billet constructed. over the required space domain. The
segment currents and [v] is the column matrix ot mathematical operations are performed not upon
driving voltaqes. continuous function· but rather in terms of
equation• about one c!iacrete point. By thia
The mutually coupled circuit method can be u..d to approach operation& such as c!ift'erentiation and
1nte9rat1ora . .Y b4l reduced to silllpla ariU..tio bave also included the effect of conductor size,
torm. and can then be conveniently .alved ua1nq operatinq frequsncy, aaqnetic permeability and
di9ital computers. shape of workpiece, and current distribution within
the work conductor itself,
A proqru has been written to predict the maqnatic
field and power density distributions in a
haaocJeneou.e noru~~&qnetic rectanqul.az: wor1cp1eca
subjected to a unlfo~ .urface maqnatio field 1. BOBSON,L. Guide to Induction Heating Equipment,
atreJ\9tho BNCE, London, 1984, 66 paqes.

Tf\e ~-;ts:;·-iteid'' ard power- densttr- 4tiib:UiQtl.On: 2. AL-SBAIKBLI,A,X,K. and UOBSON,L. ·~provementa
vi thin an aluainl\111 voxkplece Of croea-Hction 1n the Desian of Induction Billet Beaters•, 19th
500 - by 150 • auhjecUd to a .urface UtJU;tio. UPEC, university of Dundee, April, 1984.
tidd atzenvth of LoS Arl at a frequency.of ~so.. n
are eon t.D F19•• 4 Ul4 5 reepectlnly. . . . .. 3. BAD!R,Jl.M. "Desiqn and calculation of Induction
Heat.f.n; Coib". AIEB 'l'rans, 76 1 Pt II, pp.3l-40,
The prQ91'. . v&a relatiwly simple to· blpl...nt an4 ~(arch, 1957.
qava aatiefactory re.ulte when applied to a at=pla
two dimensional problUio 4. REXCHERT,J:., "The calculation of coreless
Furnace• vith Electrically Conductinq crucibles•,
4. CONCLUSIONS AND RECOMMENDATIONS Electrotechnik, 49, 6, pp.376-397, February, 1964.

An accurate knovledc)a of the power density and 5. VJUJGBAN',J.T. an4 WILLIMSON,J.W. "Dedqn of
temperature dbtr1hut1on 1n an inductively heated tnductton Heating Coils for cylindrical Nonmaqnetic
load la of qraat importance. Analytical .xpreaeiona Lo&da", AIEB Trana, 64, pp.587-589, August, 1945.
are available for simple one ~atonal qeomatriaa
auch ae a cylinder or a dab (15) 'but for the 6. VAUGBAK,.:r ,T, and WILLIAMSON,.:r .w. "Desiqn of
solution of even the .tm:pl.. t two 4.1mauilonal Induction Beatinq Coiia for Cylindrical Maqnetic
g~trlea numerical technique. are requu.d. Loads", AIEl!: Trane, 65, pp.SB7-892, 1946.

ID induatrlal practice U. 4esi<pL of induction 7. J:DLBE,E. and Rehs,W. "Distribution in Space

billet heatera !a l.u9ely baeed on the equivalent of current Density in Xnduction Heated Bodiea Under
circuit metho4 vhich u~N~Dea a unifom .aqnetic Constderatton of the Temperature Field",Electrovarme,
field atrenqth alonq the length of tb vorkpiece 25, 7, pp.243-250, July, 1967.
and many empirical factor• are inco~rate4 to take
into account euch thinqa u a ahortneaa of coil and B. DUDLEY,R.F, and BUJUCE,P.E. "The Prediction of
spacinq between turn•. Numerical aolutions uainq CUrrent Distribution in Induction Beatinq Installa-
finite dift'erence, tinit.e element, and mutually tions•, IEEE 'l'rans, IA-8, 5, pp.565-571,
coupled circuit technique• are in existence but September/october, 1972.
they also aaiiUIDe a unlfcma magnetic field atrenqth
and hence a eurface pover density along the lenqth
of the work piece. EVen though they are at &..t':
two dtmenaional aolutiona they ~ire epeaialiat
knovledqe and powerful compUtational !acilitiea
not nonaally available to 0.11:.. induction Matinq 10. LAVERS,J.D. Private COIIIDUtlications.
manufacturera.
11. COGGON,J .B. "Electrau.gnetic and Electrical
Induction billet heating le a vell eatablbhed Mo4eU.n~ by the Finite Element Method", Geophydca,
industrial process for preheatinq prior to !orqinq, 36, 1, pp.l32-1SS, February, 1971.
rollinq and extrusion. There are areas, however,
in vhich iDduction heating' haa not achieved the 12. NEMOTO,J:., an4 TMOCBI,M. "Thermal Analysis of
market share that perhaps one 11liqht axpect e.q. Induction Beating by the Finite Element Usinq a
off-the-bar !orqinq or the extrusion of ali.DIIintum. COatputer", lOth Conqre .. of the International
In both applications the ability to create a non- Union for Electroheat, Stockholm, 18-22 June, 1984.
uniform eurface power density alonq the billet or
bar is reqUired, Existinq methods of work coil 13, LAVERS,J.D. "Finite Element Solution of Non-
design which are based on the asewaption of an Linear Two Dimensional TE - Mode Eddy current
infinitely lonq solenoid and/or the use of a·~qx:eat Problems•, MAG-19, 5, pp.220l-2203,
amount of empirical data are not eadly adapted to September,
the de*iqn of non-unifora surface power densities.
Further invutiqations to develop a new technique 14. REICHERT,K. "A Nu=erical Method !or Calculating
for work coil desiqn is reqUired which can be Induction Beatinq lnstallationa", Electrowarme Int.
adapted to solve probl&IIUI requirinq a non-uniform 26, 4, pp.l13-123, 1968.
surface power density. In order to deal rea.listic-
ally vith problems involving a non-uniform .urface 15. DAVIES,J. and SIHPSON,P. Induction Beating
power d.ndty along the length of a vorkpiece, the Handbook, KcGrav-Bill, Maidenhead, 1979, 426 paqes.
surface current density dhtribution produced by a
sinqle conductor has been invt~stiqated. The euper- 16. BOBSON,L. and AL-SHAIKHLI,A,J:..M. "Illuatratinq
podtion principle to t'ind the power density Electromaqnetics Using an Industrial Procesa•,
produced by a number of conductors has been !:!!!!• 23, 1, January, 1986, (to be published).
veri!!~ and a work coil design to produce a
particular temperature distribution in a billet
has been carried out. (2, 16). Investigations
 WORK PIECE

AIR GAP

ODD DD D COIL

Fig. 1 'l'ba flux paths in induction heating sy•tem

•
•.
,r--1'----.\

V •.
• X .
• z}·
w

••

Fig.2 The ~valent electrical circuit

1
·--·--·-
 1~1

0·6

0·6

0·4

0·2

0
1 3 s 6 7 8
Fig. 4 Variation of .aqnetic field distribution (&a a fraction of the
surface fidd) with depth y.

POWER (W)

140

120

100

80

60

40

Fig. 5 Power distribution down the centre line of the cross-sectional face
 ILLUSTRATING ELECTROMAGNETIC$ USING AN INDUSTRIAL PROCESS

Dr.L.Hobson and Mr.A.K.M.Al-Shaikhli

Department of Electronic and Electrical Engineering,

Loughborough University of Technology,

Loughborough, Leicestershire, LEll 3TU

ABSTRACT

The authors have attempted to illustrate basic electromagnetic

phenomena using readily available laboratory equipment and

unsophisticated current density measuring probes. A series of

experiments are described explaining induction heating theory and

relating it to the requirements of the industrial process of aluminium

extrusion.
 ILLUSTRATING ELECTROMAGNETICS USING AN INDUSTRIAL PROCESS

Dr.L.IIobson and Mr.A.K.M.Al-Shaikhli

Department of Electronic and Electrical Engineering,

Loughborough University of Technology,

Loughborough, Leicestershire, LEll 3TU

SYMBOLS

a = The radius of the conductor (m)

A Constant found from practical measurement (dimensionless)

B " " " " " "

f = Frequency (Hz)

h = Perpendicular distance between centre of conductor or

filament and surface of the slab or cylinder (m)

2
h
e
= ih 2 - a = Effective height (m)

H = Magnetic field strength (Am-l)

H = Magnetic field strength at point P (Am-l)

p
H = Radial component of magnetic field strength (Am-l)
r
H = Axial component of magnetic field strength (Am-l)
z
I = Current (A)
2
J = The surface current density (Am- )
-2
J = The surface current density at point P (Am )
p
i = The probe length = The perimeter of the cylinder (m)

V = The voltage across the probe terminals (V)

2
wp = The power density induced on point P (Wm- )

z = The distance of point P from plane of filament or conductor (m)

6
I P
nf~
= Skin depth (m)

p Resistivity (0m)

~ Permeability (llm-l)

(i) L.llobson
1. INTRODUCTION

Electromagnetic field theory is often taught in an extremely

abstract manner which does not appeal to the vast majority of

students. Laboratory experiments designed to illustrate the funda-

mental concepts of electromagnetics are also often contrived to avoid

practical measurement problems and over simplified to such an extent

that their industrial relevance is not obvious. In an attempt to

reverse this situation the authors have suggested a series of

laboratory experiments which illustrate basic electromagnetic

phenomena and relate it to a particular industrial process.

When a current carrying conductor passes in close proximity to a

metallic structure then eddy currents are induced in the metal which

can give rise to heating effects or even excessive mechanical forces

within the structure. The proximity effect is a fundamental feature

of electromagnetics and is put to use industrially in the induction

heating of metallic objects prior to rolling, forging or extrusion.

Induction billet heaters are made with many turns of water cooled

copper tubing and the design of these coils requires knowledge of the

induced currents from not one conductor but many in close proximity to

each other and to the workpiece(l). One of the most common

industrial processes using induction billet heating is the heating of

aluminium prior to extrusion. Aluminium workpieces heated prior to

(2)
extrusion require a temperature taper and hence a surface power

density taper along their length. The exact requirements depend on

the exact extrusion process being used.

The use of a long filament close to a semi-infinite slab of metal

would be the easiest example which could be analysed theoretically.

-1-
L.Hobson
However this situation is far from those seen in practice. The

interaction of a finite sized conductor close to a circular billet is

of greater industrial relevance but theoretical analysis of this

situation is not easy and graphical results with experimentally

verified correction factors are preferred.

The series of experiments put forward consists of an

investigation of the interaction of a single current carrying

conductor and a metallic workpiece; the illustration of the validity

of the method of superposition, i.e. the prediction of the total

surface current density induced by a number of conductors in close

proximity; and finally the design of a simple induction billet heater

which can produce a non-uniform current density distribution often

required in the industrial process of induction heating of aluminium

prior to extrusion.

2. SINGLE CONDUCTOR

Fig. 1 shows a filament carrying current I at a distance h from

the surface of a semi-infinite metal slab. Given that the slab is a

good conductor the magnetic field strength at any point P at distance

z along the surface of the slab can be calculated by assuming an

imaginary filament within the slab at a distance h from the surface

and carrying current -I. Both the conductor and the image will

produce a magnetic field strength H, according to Ampere's law, equal

to

I
H ( 1)

-2-
L.Hobson
 The axial components Hz of the conductor's magnetic field strength and

its image will aid each other, while the radial components 11 will
r
cancel each other. The resultant field on point P will therefore
be:-

11 2 11 (2)
p z

but 11 h
11 cos8 11
.{2 + z 2
~ ~

z (3)

Hence

11 (4)
p

If the filament were replaced by a circular conductor of finite

radius a, then the magnetic field at P is modified to the following

expression()),_

2
11 ~
I - a
p (5)

The surface current density on the semi-infinite slab in a

direction opposite to that of the conductor current also given by( 4 ):

J ~ 11 12 -1
(Am )
p p 0 (6)

and the surface power density is:-

w (7)
p

-3-
L.llobson
 If the metallic workpicce is a long cylinder then it can, to a

first approximation, be modelled by the semi-infinite slab

configuration providing the radius of the cylinder is many times the

4
electromagnetic skin depth within the material ( ).

A laboratory experiment has been designed in which a single

circular turn of 6rnrn diameter water cooled copper tubing was wrapped

around an aluminium cylindrical workpiece of SO.Brnrn radius. With air

gaps between the coil and workpiece of 30rnrn and SOrnrn, and a coil

current of 600 A at 50 Hz, the current density at lOrnrn intervals along

the surface of the billet was measured. The measurements were made

using surface current density probes. A number of surface current

density probes were positioned along the length of the billet, Fig. 2.

Each probe consisted of a single turn of constantan wire electrically

insulated from the billet. The output of the probe was measured by a

high impedance voltmeter shielded from the high magnetic field. The

voltage reading was directly related to the current density within the
. (5) (6)
surface of the workp1ece , as:-

V
J = (8)
p~

Fig. 3 shows the results obtained and whilst the shape of the curves

measured was consistent with equation (6) a more accurate expression

for the surface current density is given by

J
p
= (9)

where A and B are constants found from practical measurement.

-4-
L.Hobson
 They are functions of the effective height h , A is a straight
e
line represented by

A = 10-2 h ( 10)
e

The application of the closest fit computer routine was found to

produce the equation of the other constant

8 = (-(2.642 X 10-l) + (7,36 X 10- 5 )h - (1.539 X 10-g)h 3 (11)

e e

The variation of A and B with h is shown on Fig. 4. For simplicity

e
and within the accuracy levels required the use of graphical results

given in Fig. 3 are preferred.

A schematic diagram of the electrical circuit used is shown in

Fig. 5. A 415 v, SO Hz variable voltage transformer was connected to

the input of a 415/5 V transformer which gave an output current

capability to the work coil in excess of 2000 A. A current

transformer on the high current output was used to measure the work

coil current in the majority of experiments.

3, MULTIPLE CONDUCTORS

Induction billet heaters have, of course, many turns and the

second part of the proposed laboratory experiment investigated the

interaction of multiple conductors round a cylindrical aluminium work-

piece. The results verify the principle of superposition associated

with the surface current density distribution along the length of the

billet.

The same aluminium billet used in the first experiment was

-5-
L.Hobson
surrounded by an 11 turn work coil made from 6rnrn diameter water cooled

copper tubing. The air gap used was SOrnrn and the coil pitch was

constant at 7.lrnrn between each turn. The surface current density

distribution along the billet was measured and the results correlated

with those predicted from theory. The summation of the effects of

all the conductors could be done using the rather complicated

expression of equation (9). To simplify the procedure it is

suggested that the resultant surface current density at any point P be

calculated using the closest fit curve of the practically measured

values of Fig. 3. Correlating the results in this manner gives an

error of less than 5% as shown in Fig. 6.

4. TEMPERATURE TAPER
(1)
Research work at Loughborough has produced a novel computer

aided technique for the design of induction billet heating work coils

which can give any particular non-uniform surface power density or

temperature taper required.

To demonstrate the usefulness of the design procedure it was

decided to design an induction heating coil capable of producing a

linearly decreasing temperature taper. This, of course, models the

practical requirements met industrially in the induction heating of

aluminium prior to extrusion.

A computer program was developed to calculate the power density

at any point P on the surface of the workpiece when heated by multiple

turn work coil of arbitrary pitch. The values of the surface current

density of 30mm air gap single turn coil, Fig. 3, were used to

calculate the resultant surface current density of a number of

-6-
L.Hobson
different work coil configurations using the procedure outlined

previously.

A work coil was constructed again using 6mm diameter copper

tubing but with variable coil pitch as detailed in Table 1. A work

coil with the spacings given in Table 1 should produce a linear temp-
. 'b ut1on
erature d 1str1 . (l) usua 11 y assoc1a
. t e d w1t
. h a 1 um1n1um
. . .
extrus1on.

In the third part of the laboratory exercise the practical measure-

ments of surface current density for the coil should be correlated

with the theoretical predictions. The results are shown in Fig. 7.

Pitch 1 2 3 4 5 6 7 8 9 10

Spacing (mm) 7 8 11 23 23 23 25 25 25 32

Table 1: Coil pitches to give linear temperature taper.

It was not possible to build a commercial size billet heater to

test the theory because of the capital cost and the limitations of the

laboratory power supplies. Megawatt power supplies and large amounts

of power factor correction capacitors are not uncommon in industrial

units. However, the laboratory scale model does demonstrate the

validity of the design procedure.

The type of laboratory experiment has been used to illustrate the

basic electromagnetic phenomena associated with induction heating and

the essentials of taper heating associated specifically with induction

heating of aluminium prior to extrusion. It has been used with final

year undergraduates undertaking an electroheat option and to technical

-7- L.Hobson
staff trainees of the Electricity Supply Industry involved in

specialist short courses on Industrial Applications of Electricity(?).

5. CONCLUSIONS

The authors have attempted to illustrate the basic electro-

magnetic phenomena of the proximity effect using readily available

laboratory equipment and unsophisticated current density measuring

probes. The temptation to use unrealistic workpiece/coil config-

urations has been avoided and by the use of graphical superposition

techniques complex mathematics have been eliminated. Throughout the

series of experiments the work has been related to induction heating

theory and finally the requirements of the highly relevant industrial

process of aluminium extrusion have been explained.

-8- L.Hobson
 REFERENCES

. 1. AL-SHAIKHLI,A.K.M. and HOBSON,L. 'Improvements in the design of

induction billet heaters', 19th UPEC, University of Dundee,

April 1984

2. HOBSON,L. 'Guide to induction heating equipment', (BNCE: London)

1984

3. BAKER,R.M. 'Heating of nonmagnetic electric conductors by

magnetic induction-longitudinal flux', AIEE Trans., 1944, Vol.63,

pp.273-278

4. DAVIES,E.J. and SIMPSON,P. 'Induction Heating Handbook',

(McGraw-Hill: Maidenhead), 1979

5. BURKE,P.E. and ALDEN,R.T.H. 'Current density probes', IEEE

Trans., 1969, PAS-88, (2), pp.l81-185

6. KARGAHI,M.R. and HOBSON,L. 'Medium frequency metal melting using

conducting crucibles', 19th UPEC, University of Dundee, April

1984

7. HOBSON,L. 'Short courses on the industrial applications of

electricity', Int. J. Elect. Eng. Education, April 1985, ~. (2)

-9-
L.Hobson
LIST OF FIGURES

l.(a) Conductor near a semi-infinite slab.

(b) The magnetic field strength on point P.

2. The probes along the length of the billet.

3. Current density distribution along the aluminium cylinder due to

600 A flowing in single conductor at air gaps of 30rnm and 50rnm.

4. The variation of the constants A and B with the effective height

5. The electrical circuit diagram.

6. Current density distribution along the aluminium cylinder due to

11 turn coil at air gap of 50mm, coil pitch of 7.lmm and

current of 600 A.

7. Current density distribution along the aluminium cylinder due to

11 turn coil of nonuniform coil pitch designed to produce a

linearly decreasing power density.

B. Power density distribution along the aluminium cylinder due to

11 turn coil at air gap of 30rnm, nonuniform coil pitch and a

current of 600 A.

UI/PBl/3.12.84

-10-
L.Hobson
 z

7 7 7 /stab 7 7
I
C'!Ymage

Ia l

Ibl
 8

EQUATION 9

EQUATION
AIR GAP OF
7 30 mm
0 MEASUREMENT)

EQUATION 7 )
AIR GAP OF
)
So mm
0 MEASUREMENT)
6

30 mm AIR GAP

SO nun AIR GAP

"' El
I

11'10
~
~

....
4
"' '\. . \
\
~ \ \ D
.., \\. \
~ 3
H

"'z~ \
"'z \
~ \
D
u
2
~
\.
'\.
1 ' -..;::

·- -~----

40 so 120 160 200

DISTANCE ALONG THE LOAD, Z (nun)

L.Hobson
 current (10,000: 10)
transformer
r----'"VY'I"'-------r- b INDUCTIVELY
415V
HEATED ALUMINIUM
60A
CYLINDRICAL
50Hz WORK PIECE.

variable high current

f
water cooled
voltage transformer transformer copper work coi I .
max current
output SOOOA.
 so
SUPERPOSITION

0 MEASUREMENTS
"'<:
I
40
~

"'o
..-<

..,X 30
-
!::H
[(I
z
1:!
I'<
z
20
~
D
u

10

0
0 40 80 120 160 200

DISTANCE ALONG THE LOAD, Z (mm)

L.Hobson
 40

SUPERPOSITION
35
o MEASUREMENTS

30

N 25
I
Ei
.0:
~

IIlo
.....
..,X 20
. 0
~
....
"'~z
E-<
15
z
~
p
u

10

0
0 40 80 120 160 200

DISTANCE ALONG THE LOAD, Z (mm)

L.Hobson
 8

SUPERPOSITION

6 0 MEASUREMENTS

IS 5
8

0
0 40 80 120 160 200

DISTANCE ALONG THE LOAD, Z (mm)

L.Hobson