MODULE I

PRINCIPLE OF
ELECTROMECHANICAL
ENERGY CONVERSION
1.0 INTRODUCTION

An electrical machine is an electromechanical device that converts either mechanical energy to

electrical energy or electrical energy to mechanical energy. The electromechanical energy conversion
occurs when there is a change in magnetic flux linking a coil, associated with mechanical motion.
When such a device converts mechanical energy (input) to electrical energy (output), it is called a
GENERATOR. When it converts electrical energy (input) to mechanical energy (output), it is called
a MOTOR. While all electromechanical energy converters have different structures, they all operate
using similar principles. Understanding these principles allows for the analysis and design of new
systems while also improving the existing ones.

The transformer is an electrical device similar to an electric machine that converts AC electrical
energy from one voltage level to another. The similarity between transformers, generators and
motors depending on the action of the magnetic field to accomplish the change in voltage level
necessitate them being studied together.

These three types of electric devices are part and parcel of modern daily life. Electric motors in our
home run refrigerators, freezers, blenders, air conditioners, fans and many similar appliances, while
generators are required to supply the power used by all these motors, transformers aid the process of
reducing energy loss between the point of electric power generation and the point of use.

1.1 ELECTRICAL ENERGY CONVERSION PROCESS

The conversion of energy from one form into another is known as the principle of conservation of
energy. According to the law, “energy can neither be created nor destroyed, it can only be converted
from one form to another”.

An electromechanical system is comprised of an electric system, mechanical system and a means by

which the electric and mechanical systems can interact. Interaction is possible through any and all
electromagnetic and electrostatic fields that are common to both systems, and energy is transferred
from one system to the other as a result of this interaction. Both electrostatic and electromagnetic
coupling fields may exist simultaneously and the electromechanical system may have any number of
electric and mechanical subsystems. A simplified form of an electromechanical system with one
electric subsystem, one mechanical subsystem, and one coupling field is illustrated in figure 1.1.
 ELECTRIC COUPLING MECHANICAL
SYSTEM FIELD SYSTEM
Figure 1.1: Block Diagram of a Simple Electromechanical System

It is known that irreversible energy converted to heat in such systems occurs due to

a) I 2 R losses in the windings of the electric system.

b) Friction and windage losses in the mechanical system.
c) Core losses in the coupling field.

An electromechanical system involves energy in four forms captured in the energy transfer equation
as

i. Motion Action
 Electrical   Mechanical   Stored Energy   Energy Losses 
         
 Energy Input   Energy Output   in Coupling Field   Due to Heat 

ii. Generating Action

 Mechanical   Electrical   Stored Energy   Energy Losses 
         
 Energy Input   Energy Output   in Coupling Field   Due to Heat 

Rewriting the energy balance equation (motoring convension) we have

 Electrical   Mechanical Friction and   Stored Energy Core 

  I 2R         
 Energy Input   Energy Output windage loss   in Coupling Field losses 

Now consider a differential time interval dt during which an increment of electrical energy dWe

flows into the system. If dW f is the energy supplied to the field and dWm the energy converted to

mechanical form then:

dWe  dWm  dW f (1.1)

it is natural for this equation to have positive signs for electrical and mechanical energy terms when
the device is monitoring. For generating mode, however, both terms assume negative signs. When a
system has no mechanical work done equation (1.1) is reduced to equation (1.2) indicating that the
electrical energy input is stored in the coupling field.
 dWe  dW f (1.2)

1.2 ANALYSIS OF MAGNETIC CIRCUITS

As previously stated, magnetic fields are the fundamentals by which energy is converted from one
form to another in motors, generators and transformers. Four basic principles describes how
magnetic fields are used in these devices:

1. A current-carrying conductor produces a magnetic field in the area around it (e.g. solenoid).
2. A time-varying magnetic field induces a voltage in a coil of wire if it passes through that coil.
(e.g. Transformer action).
3. A current-carrying conductor in the presence of a magnetic field has a force induced on it
(e.g. Motor Action).
4. A moving conductor in the presence of a magnetic field has voltage induced in it (e.g.
Generator Action).

This section will elaborate on the production of a magnetic field by a current-carrying conductor.
Later section of this course and indeed other courses explain the other three principles.

A simple electromagnetic system is shown in fig 1.2. This system consists of conductor wound N
times around a ferromagnetic member. The magnetic member contains an airgap of uniform length
between points a and b. A magnetic system (circuit) can be described as consisting of the magnetic
member and the air gap only.

The basic law governing the production of a magnetic field by a current–carrying conductor is
Ampere’s law. This states that the line integral of field intensity H about a closed path is equal to the
net current enclosed within this closed path of integration. That is,

 H .dl  i
n (1.3)

Wherein:

H = magnetic field intensity in Ampere-turns/ meter

dl = Differential length in meter

in = the net current enclosed in Ampere

 Applying Ampere’s law to the closed path depicted by the dashed line in fig 1.2. we have

a b
H
b i dl   H g dl  Ni
a
(1.4)

Figure 1.2: A simple magnetic cirtcuit

Where the path of integration is assumed to be in clockwise direction and the field intensity exists
only in the direction of the given path. The subscript i denotes the field intensity (Hi) in the
ferromagnetic member (iron or steel) and g denotes the field intensity (Hg) in the air gap. The right-
hand side of (1.4) represents the net current enclosed, since the coil cuts the path of integration N
times while carrying current i. Carrying out the line integration, Ampere’s law thus becomes:

H i li  H g l g  Ni (1.5)

Where li is the mean length of the magnetic material and l g is the length across the air gap. It was

earlier assumed that the magnetic circuit consist only of the ferromagnetic member and the air gap
and that the magnetic field intensity is always in the direction of the path of integration, so the
assumed direction of the magnetic field intensity is always in the direction of the magnetic field
intensity is valid except in the vicinity of the corners where the field intensity changes slowly rather
than abruptly. Even so, the mean path length approximation is widely used as an adequate means of
analyzing magnetic circuit of this form.

The magnetic field intensity H can be described as a measure of the “effort” that a current is putting
into the establishment of a magnetic field. This also causes a flux density B to be set up at every
point along the flux path. From basic physics, we know that for linear, isotropic magnetic materials
the flux density B is related to the field intensity by:

B  H (1.6)

Where:  is the permeability at the medium in Henry/Meter (H/m)

B is the flux density in wb/m2, also known as Tesla (T)

Equation 1.5 can be written in terms of flux density as

Bi Bg
li  l g  Ni (1.7)
i g

Now the total flux in a given area is given by:

   B.dA (1.8)
A

If we assume that the flux density is uniform over the cross-sectional area, then

 i  Bi Ai 
 (1.9)
 g  Bg Ag 

Where  i and  g are the total flux in the magnetic material and air gap respectively. Ai and Ag are

the associated cross-sectional area of the magnetic material and air gap respectively.

Also from physics, it is known that the streamlines of flux density B are closed; hence, the flux in the
air gap is equal to the flux in the core i.e.  i   g . If the air gap is small, Ai  Ag , and therefore

Bi  Bg . However, due to the phenomenon known as the “fringing effect”, the effective area of the

airgap is larger than that of the magnetic material. We shall keep this in mind and assume later for
our analysis that Ag  Ai .

Letting  i   g   and substituting into (1.7) we obtain:

li lg
   Ni (1.10)
 Ai  g Ag
What we have in equation (1.10) is an analogy to ohm’s law. The magneto motive force (mmf) Ni is
analogous to the voltage or electromotive force (emf), and the flux  is analogous to the current.
Recall also that resistance of a conductor is proportional to its length and inversely proportional to its
conductivity and cross-sectional area. Similarly, the term li / i Ai and l g /  g Ag are the reluctance of

the magnetic material and air gap, respectively. Generally, the permeability is expressed in terms of
relative permeability as

i  ri 0

 g   rg 0

Where 0 is the permeability of free space (4   4  10 7 wb or 4 107 H/m since wb/A and

ri and rg are the relative permeability of the magnetic material and air gap respectively. For all

practical purposes,  rg  1; however, ri may have value that range between 500 and 4000

depending upon the type of ferromagnetic material. If reluctance is denoted by  , so as to

distinguish it from resistance. Equation (1.10) can now be written as

Ni   i   g   (1.11)

li li
Where: i  and  g 
i Ai  g Ag

i is the reluctance of the iron and  g is the reluctance of the air gap.

The circuit shown in figure 1.3 shows the magnetic electric circuit analogy of the discussion thus far.
Figure 1.3: (a) A Simple Electric Circuit (b) Magnetic Circuit equivalent

In the magnetic circuit, mmf (Ni) forces flux  to flow through reluctances  , similarly, voltage V
forces current I to flow through resistances R.

Example: The magnetic circuit of figure 1.2 has dimensions: Ai  4  4cm 2 ,

l g  0.06cm, li  40cm; N  600 turns . Assuming r  6000 for iron. Find the exciting current for

Bi  1.2T and the corresponding flux.

Solution:

From equation 1.5 the ampere turns for the circuit are given by

Bi Bg Bi Bg
Ni  li  lg  li  lg
i g ri 0 0

Neglecting fringing

Ai  Ag Hence Bi  Bg

Bi  li 
Then i    lg 
0 N   ri 
 1.2  40 
i  0.06   102  1.06A
4  10  600  6000
7


  Bi Ai  1.2 16 104  19.2 10 4 wb

1.3 PROPERTIES OF FERROMAGNETIC MATERIALS

In free space and non-magnetic materials, the permeability 0 is constant so that B-H relationship is

linear. This however, is not the case with ferromagnetic materials used in electric machines, wherein
the B-H relationship is strictly nonlinear in two respects i.e. saturation and hysteresis. Hysteresis
non-linearity is a double valued B-H relationship exhibited in cyclic variation of H (i.e. exciting
current). This nonlinearity is usually ignored for magnetic circuit calculation and only important
when current wave form and power losses are to be accounted for. A typical normal B-H relationship
for ferromagnetic materials is shown in fig 1.4. it consists of an initial nonlinear zone, a middle
almost linear zone and a final saturation zone in which B progressively increases loss rapidly with A
compared to the linear zone. In deep saturation zone, the material behaves like free space.

Due to economic consideration, electric machines and transformer are designed such that the
magnetic material is slightly saturated (i.e. somewhat above the linear zone). In exact magnetic
circuit calculations, the nonlinear magnetization curve has to be used necessitating
graphical/numerical solutions.

Figure 1.3: Typical normal magnetization curve of ferromagnetic material

1.4 VOLTAGE, FLUX LINKAGE AND INDUCTANCE

By faraday’s law of induction, changing magnetic flux density creates an electric field (and voltage).
Written in integral form of Maxwell’s fourth equation as

  d  

 E.dt   dt  B.dA
c s
(1.12)

The integrated form of equation 1.12 for a coil of N turns is

d d
e  N  (1.13)
dt dt

Where  = “flux linkage” = N

The negative sign means that induced emf due to increase in  is in opposite direction to that of
positive direction of current in the coil is that direction which establishes positive flux and flux
linkages, equation 1.13 may be written as

d d
eN N
dt dt

On the other hand,

d d  di di
eN N . L (1.14)
dt di dt dt

d d
Where: L  N  (1.15)
di dt

Inductance relates flux linkage to current as


L (1.16)
i

In addition, inductance in terms of field qualities is given as

N 2 BA A N2
L N2   N 2 (1.17)
Hl l 
 l
Recall that reluctance;  
A

I
Defining    permanceof the magnetic circuit . It is the reciprocal of reluctance and analogous

to conductance in an electric circuit.

The inductance concept is easily applied to a magnetic circuit with two windings as show in figure
1.5. it is pertinent to note that the total flux is the sum of flux due ti i1 and that due to i2 given the
current directions as shown.

Figure 1.4: Magnetic circuit with two windings

Figure 1.5: Electric circuit analog of the magnetic system

For the sake of analysis, it is assumed that the permeability of the magnetic core is of infinite value,
this means that the reluctance of the core is zero. Hence, only the reluctance of the air gap is used for
analysis. i.e. i  0 , the new magnetic circuit is as shown below

Rg
Φ

N1i1 N2i2

Figure 1.6: Magnetic circuit with infinite iron core permeability

  A
flux:    N1i1  N 2i2   0 
 l g 

Assuming Ai  Ag  A

Flux linkage for coil 1:

 0 A   0 A 
1  N1  N12 i1    N1 N 2i2  
 l g   lg 

This can be written as

1  L11 i1  L12 i2

Where: L11 is “Self” inductances of coil 1

L12 is “Mutual” inductance between coils 1 and 2:

0 AN12 0 AN1 N 2
L11  and L12 
lg lg

Flux linkage for coil 2:

  0 A   0 A 
2  N 2   N1 N 2i1    N 2 i2 
2

 l g   l g 

Rewriting:

2  L21i1  L22i2

In matrix form

 1   L11 L12   i1 
  L L 22  i2 
 2   21

Note however, that not all the flux closes its path from the magnetic core but some portion closes its
path through air. This portion is called the leakage flux  l . The analysis done thus far is based on a

second component called the magnetizing flux. The total flux is given as

Figure 1.7: Leakage flux illustration in a magnetic circuit

  L  m

 l = Leakage flux

 m Magnetizing flux
The leakage flux is produced by current flowing in each of the windings and it links only the term of
that winding. The magnetizing flux on the other hand is produced by the current flowing in any of
the windings and links all the turn of both windings.

By deduction:

N i i1 Ni
Let  l1  and  l 2  2 2
 l1 l 2

N12 
l1  N1  l1  
Rl1 

N 22 
l 2  N 2  l 2 
Rl 2 

N12 
Ll1  
l 2 

N2
Ll 2  2 
l 2 

1.5 MAGNETIC STORED ENERGY

If the stored energy in a magnetic circuit in a time interval between t 1 and t2

t2
W   pdt …………………………………………………………..1.27
t1

Recall: p  ei

d d
Where e  N 
dt dt

t2
W   eidt
t1

2
  idt
1

For a linear magnetic circuit:


  Li  i 
L
 1 2 2
W 
2 L 1

2  12 

Similarly

2
W   id 
1

  Li; d   Ldi

i2 1
W  L  idi  L(i22  i12 )
i1 2

If i1  0, i2  i, and 1  0, 2  

The relationship between  and i is shown in figure

Figure 1.8: Illustration of Conergy in a non-linear system.

1.6 TOTAL CORE LOSS

When a magnetic material undergoes cyclic magnetization, two kinds of power losses occur in it.
These are:

1. Hysteresis loss
2. Eddy current loss