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PROBLEMS - chapter 1

NGUYEN Phuc
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PROBLEMS
This lecture note is based on the textbook # 1. Electric Machinery - A.E. Fitzgerald, Charles Kingsley,
Jr., Stephen D. Umans- 6th edition- Mc Graw Hill series in Electrical Engineering. Power and Energy
1.1 A magnetic circuit with a single air gap is shown in Fig.1.1. The core dimensions are:
Cross-sectional area Ac = 1.8x10−3 m2
Mean core length lc = 0.6 m
Gap length g = 2.3 x 10−3 m
N = 83 turns

Figure 1

Figure 1.1 Magnetic circuit.

Assume that the core is of innite permeability ( µ → ∞) and neglect the eects of fringing elds at the
air gap and leakage ux. (a) Calculate the reluctance of the core RC and that of the gap Rg . For a current
of i = 1.5 A, calculate (b) the total ux φ, (c) the ux linkages ) λ of the coil, and (d) the coil inductance L.
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1.2 Repeat Problem 1.1 for a nite core permeability of µ = 2500µ0 .

1.3 Consider the magnetic circuit of Fig.1.1 with the dimensions of Problem1.1. Assuming innite core
permeability, calculate (a) the number of turns required to achieve an inductance of 12 mH and (b) the
inductor current which will result in a core ux density of 1.0 T.
1.4 Repeat Problem 1.3 for a core permeability of µ = 1300µ0 .
1.5 The magnetic circuit of Problem 1.1 has a nonlinear core material whose permeability as a function
of Bm is given
 by 
µ = µo 1 + √ 3499
7.8
1+0.047(Bm )
where Bm is the material ux density.
a. Using MATLAB, plot a dc magnetization curve for this material ( Bm vs. Hm ) over the range 0 B ≤
2.2 T.
b. Find the current required to achieve a ux density of 2.2 T in the core.
c. Again, using MATLAB, plot the coil ux linkages as a function of coil current as the current is varied
from 0 to the value found in part (b).
1.6 The magnetic circuit of Fig.1.2 consists of a core and a moveable plunger of width lp , each of
permeability . The core has cross-sectional area Ac and mean length . The overlap area of the two air
gaps Ag is ahfunctioni of the plunger position x and can be assumed to vary as
x
Ag = Ac 1 − X0
You may neglect any fringing elds at the air gap and use approximations consistent with magnetic-circuit
analysis.
a. Assuming that µ → ∞, derive an expression for the magnetic ux density in the air gap Bg as a
function of the winding current I and as the

Figure 2

Figure 1.2 Magnetic circuit for Problem 1.6.

plunger position is varied ( 0 ≤ x ≤ 0.8X 0 ). What is the corresponding ux density in the core?
b. Repeat part (a) for a nite permeability µ.
1.7 The magnetic circuit of Fig.1.2 and Problem 1.6 has the following dimensions"

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AC = 8.2cm2 lC = 23cm
lp = 2.8cm g = 0.8 mm
X0 = 2.5 cm N = 430 turns
a. Assuming a constant permeability of µ = 2800µ0 , calculate the current required to achieve a ux
density of 1.3T in the air gap when the plunger is fully retracted (x =0).
b. Repeat the calculation of part (a) for the case in which the core and plunger are composed of a
nonlinear material
 whose permeability
 is given by
µ = µ0 1 + √ 1199
1+0.05Bm
8

where Bm is the magnetic ux density in the material.

c. For the nonlinear material of part (b), use MATLAB to plot the air-gap ux density as a function of
winding current for x = 0 and x = 0.5 X0 .
1.8 An inductor of the form of Fig.1.1 has dimensions:
Cross-sectional area AC = 3.6cm2
Mean core length lC = 15 cm
N - 75 turns
Assuming a core permeability of µ = 2100µ0 and neglecting the eects of leakage ux and fringing elds,
calculate the air-gap length required to achieve an inductance of 6.0 mH.
1.9 The magnetic circuit of Fig.1.3 consists of rings of magnetic material in a stack of height h. The rings
have inner radius Ri and outer radius Ro. Assume that the iron is of innite permeability ( µ → ∞) and
neglect the eects of magnetic leakage and fringing. For:
Ri =3.4 cm
R0 = 4.0 cm
h =2cm
g = 0.2 cm
calculate:
a. the mean core length lC and the core cross-sectional area AC .
b. the reluctance of the core RC and that of the gap Rg .
For N = 65 turns, calculate:
c. the inductance L.
d. current i required to operate at an air-gap ux density of Bg = 1.35T.
e. the corresponding ux linkages ***SORRY, THIS MEDIA TYPE IS NOT SUPPORTED.*** of the
coil.

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Figure 3

Figure 1.3 Magnetic circuit.

1.10 Repeat Problem 1.9 for a core permeability of ***SORRY, THIS MEDIA TYPE IS NOT SUP-
PORTED.*** .
1.11 Using MATLAB, plot the inductance of the inductor of Problem 1.9 as a function of relative core
permeability as the core permeability varies for ***SORRY, THIS MEDIA TYPE IS NOT SUPPORTED.***
= 100 to µx = 10000. (Hint: Plot the inductance versus the log of the relative permeability.) What is the
minimum relative core permeability required to insure that the inductance is within 5 percent of the value
calculated assuming that the core permeability is innite?
1.12 The inductor of Fig. 1.4 has a core of uniform circular cross-section of area AC , mean length lc and
relative permeability/Zr and an N-turn winding. Write an expression for the inductance L.
1.13 The inductor of Fig.1.27 has the following dimensions:
AC = 1.0 cm2
lC = 15 cm
g = 0.8 mm
N = 480 turns
Neglecting leakage and fringing and assuming µx = 1000, calculate the inductance.

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Figure 4

Figure 1.4 Inductor.

Figure 5

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Figure 1.5 Pot-core inductor .

1.14 The inductor of Problem 1.13 is to be operated from a 60-Hz voltage source. (a) Assuming negligible
coil resistance, calculate the rms inductor voltage corresponding to a peak core ux density of 1.5 T. (b)
Under this operating condition, calculate the rms current and the peak stored energy.
1.15 Consider the magnetic circuit of Fig. 1.5. This structure, known as a pot-core, is typically made in
two halves. The N-turn coil is wound on a cylindrical bobbin and can be easily inserted over the central post
of the core as the two halves are assembled. Because the air gap is internal to the core, provided the core is
not driven excessively into saturation, relatively little magnetic ux will "leak" from the core, making this
a particularly attractive conguration for a wide variety of applications, both for inductors such as that of
Fig. 1.27 and transformers.
Assume the core permeability to be µ = 2500µ0 and N = 200 turns. The following dimensions are
specied:
R1 = 1.5cm R2 = 4cm l = 2.5cm
h = 0.75cm g = 0.5mm
a. Find the value of R3 such that the ux density in the outer wall of the core is equal to that within
the central cylinder.
b. Although the ux density in the radial sections of the core (the sections of thickness h) actually
decreases with radius, assume that the ux density remains uniform. (i) Write an expression for the coil
inductance and (ii) evaluate it for the given dimensions.
c. The core is to be operated at a peak ux density of 0.8 T at a frequency of 60 Hz. Find (i) the
corresponding rms value of the voltage induced in the winding, (ii) the rms coil current, and (iii) the peak
stored energy.
d. Repeat part (c) for a frequency of 50 Hz.

Figure 6

Figure 1.6 Inductor.

1.16 A square voltage wave having a fundamental frequency of 60 Hz and equal positive and negative
half cycles of amplitude E is applied to a 1000-turn winding surrounding a closed iron core of 1.25 x 10-3
m2 cross section. Neglect both the winding resistance and any eects of leakage ux.

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a. Sketch the voltage, the winding ux linkage, and the core ux as a function of time.
b. Find the maximum permissible value of E if the maximum ux density is not to exceed 1.15 T.
1.17 An inductor is to be designed using a magnetic core of the form of that of Fig.1.6. The core is of
uniform cross-sectional area AC = 5.0cm2 and of mean length IC =25cm.
a. Calculate the air-gap length g and the number of turns N such that the inductance is 1.4 mH and so
that the inductor can operate at peak currents of 6 A without saturating. Assume that saturation occurs
when the peak ux density in the core exceeds 1.7 T and that, below saturation, the core has permeability
µ = 3200µ0 .
R h 2i
b. For an inductor current of 6 A, use Eq. Wd = B2µ dV to calculate (i) the magnetic stored energy
V
in the air gap and ( ii ) the magnetic stored energy in the core. Show that the total magnetic stored energy
is given by Eq. Td = i2 dLdθ(θ) .
2

1.18 Consider the inductor of Problem 1.17. Write a simple design program in the form of a MATLAB
script to calculate the number of turns and air-gap length as a function of the desired inductance. The script
should be written to request a value of inductance (in mH) from the user, with the output being the air-gap
length in mm and the number of turns.
The inductor is to be operated with a sinusoidal current at 60 Hz, and it must be designed such that the
peak core ux density will be equal to 1.7 T when the inductor current is equal to 4.5 A rms. Write your
script to reject any designs for which the gap length is out of the range of 0.05 mm to 5.0 mm or for which
the number of turns drops below 5.
Using your program nd (a) the minimum and (b) the maximum inductances (to the nearest mH) which
will satisfy the given constraints. For each of these values, nd the required air-gap length and the number
of turns as well as the rms voltage corresponding to the peak core ux.

Figure 7

Figure 1.7 Toroidal winding.

1.19 A proposed energy storage mechanism consists of an N-turn coil wound around a large nonmagnetic
( µ = µ0 ) toroidal form as shown in Fig. 1.7. As can be seen from the gure, the toroidal form has a circular
cross section of radius a and toroidal radius r, measured to the center of the cross section. The geometry of
this device is such that the magnetic eld can be considered to be zero everywhere outside the toroid. Under
the assumption that a r, the H eld inside the toroid can be considered to be directed around the toroid
and of uniform magnitude
Ni
H = 2πr

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For a coil with N = 1000 turns, r = 10 m, and a = 0.45 m:

a. Calculate the coil inductance L.
b. The coil is to be charged to a magnetic ux density of 1.75 T. Calculate the total stored magnetic
energy in the toms when this ux density is achieved.
c. If the coil is to be charged at a uniform rate (i.e., di/dt = constant), calculate the terminal voltage
required to achieve the required ux density in 30 sec. Assume the coil resistance to be negligible.
1.20 Figure 1.8 shows an inductor wound on a laminated iron core of rectangular cross section. Assume
that the permeability of the iron is innite. Neglect magnetic leakage and fringing in the two air gaps (total
gap length = g). The N-turn winding is insulated copper wire whose resistivity is ρ Ω.m. Assume that the
fraction fw of the winding space is available for copper; the rest of the space is used for insulation.

Figure 8

Figure 1.8 Iron-core inductor.

a. Calculate the cross-sectional area and volume of the copper in the winding space.
b. Write an expression for the ux density B in the inductor in terms of the current density Jcu in the
copper winding.
c. Write an expression for the copper current density Jcu in terms of the coil current I, the number of
turns N, and the coil geometry.
d. Derive an expression for the electric power dissipation in the coil in terms of the current density Jcu .
e. Derive an expression for the magnetic stored energy in the inductor in terms of the applied current
density Jcu .
f. From parts (d) and (e) derive an expression for the L/R time constant of the inductor. Note that this
expression is independent of the number of turns in the coil and does not change as the inductance and coil
resistance are changed by varying the number of turns.
1.21 The inductor of Fig. 1.8 has the following dimensions:
a = h = ω = 1.5cm b = 2cm g = 0.2cm
The winding factor (i.e., the fraction of the total winding area occupied by conductor) is fw = 0.55. The
resistivity of copper is 1.73x10−8 Ω.m. When the coil is operated with a constant dc applied voltage of 35 V,
the air-gap ux density is measured to be 1.4 T. Find the power dissipated in the coil, coil current, number

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of turns, coil resistance, inductance, time constant, and wire size to the nearest standard size. (Hint:Wire
size can be found fromhthe expression i
AWG = 36 − 4.312 1.267 A wire
×10 −8

where AWG is the wire size, expressed in terms of the American Wire Gauge, and Awire is the conductor
cross-sectional area measured in m2 .)
1.22 The magnetic circuit of Fig.1.9 has two windings and two air gaps. The core can be assumed to be
of innite permeability. The core dimensions are indicated in the gure.
a. Assuming coil 1 to be carrying a current I 1 and the current in coil 2 to be zero, calculate (i) the
magnetic ux density in each of the air gaps, (ii) the ux linkage of winding l, and (iii) the ux linkage of
winding 2.
b. Repeat part (a), assuming zero current in winding 1 and a current I2 in winding 2.
c. Repeat part (a), assuming the current in winding 1 to be I1 and the current in winding 2 to be I2 .
d. Find the self-inductances of windings 1 and 2 and the mutual inductance between the windings.

Figure 9

Figure 1.9 Magnetic circuit.

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Figure 10

Figure 1.10 Symmetric magnetic circuit.

1.23 The symmetric magnetic circuit of Fig.1.10 has three windings. Windings A and B each have N
turns and are wound on the two bottom legs of the core. The core dimensions are indicated in the gure.
a. Find the self-inductances of each of the windings.
b. Find the mutual inductances between the three pairs of windings.
c. Find the voltage induced in winding 1 by time-varying currents iA (t) and iB (t) in windings A and B.
Show that this voltage can be used to measure the imbalance between two sinusoidal currents of the same
frequency.
1.24 The reciprocating generator of Fig.1.11 has a movable plunger (position x) which is supported so
that it can slide in and out of the magnetic yoke while maintaining a constant air gap of length g on each
side adjacent to the yoke. Both the yoke and the plunger can be considered to be of innite permeability.
The motion of the plunger is constrained such that its position is limited to 0 ≤ x ≤ ω .

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Figure 11

Figure 1.11 Reciprocating generator.

There are two windings on this magnetic circuit. The rst has N1 turns and carries a constant dc current
I0 . The second, which has N2 turns, is open-circuited and can be connected to a load.
a. Neglecting any fringing eects, nd the mutual inductance between windings 1 and 2 as a function of
the plunger position x.
b. The plunger is driven by an external source so that its motion is given by
x (t) = ω(1+2sinωt)
where  ≤ 1 . Find an expression for the sinusoidal voltage which is generated as a result of this motion.
1.25 Figure 1.12 shows a conguration that can be used to measure the magnetic characteristics of
electrical steel. The material to be tested is cut or punched into circular laminations which are then stacked
(with interspersed insulation to avoid eddy-current formation). Two windings are wound over this stack of
laminations: the rst, with N1 turns, is used to excite a magnetic eld in the lamination stack; the second,
with N2 turns, is used to sense the resultant magnetic ux.
The accuracy of the results requires that the magnetic ux density be uniform within the laminations.
This can be accomplished if the lamination width t = R0 − R1 is much smaller than the lamination radius
and if the excitation winding is wound uniformly around the lamination stack. For the purposes of this
analysis, assume there are n laminations, each of thickness ***SORRY, THIS MEDIA TYPE IS NOT
SUPPORTED.*** . Also assume that winding 1 is excited by a current i1 = I0 sinω t.
a. Find the relationship between the magnetic eld intensity H in the laminations and current i1 in
winding 1.
b. Find the relationship between the voltage ve and the time rate of change of the ux density B in the
laminations.
c. Find the relationship between the voltage v0 = G v2 dt and the ux density.
R

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Figure 12

Figure 1.12 Conguration for measurement of magnetic properties of electrical steel.

In this problem, we have shown that the magnetic eld intensity H and the magnetic ux density B in
the laminations are proportional to the current i1 and the voltage V2 by known constants. Thus, B and H
in the magnetic steel can be measured directly, and the B-H characteristics as discussed in Sections 1.3 and
1.4 can be determined.
1.26 In order to test the properties of a sample of electrical steel, a set of laminations of the form of
Fig.1.12 have been stamped out of a sheet of the electrical steel of thickness 3.0 mm. The radii of the
laminations are Ri = 75 mm and R0 = 82 mm. They have been assembled in a stack of 10 laminations
(separated by appropriate insulation to eliminate eddy currents) for the purposes of testing the magnetic
properties at a frequency of 100 Hz.
a. The ux in the lamination stack will be excited from a variable-amplitude, 100-Hz voltage source
whose peak amplitude is 30 V (peak-to-peak). Calculate the number of turns N1 for the excitation winding
required to insure that the lamination stack can be excited up to a peak ux density of 2.0T.
b. With a secondary winding of N2 = 20 turns and an integrator gain G = 1000, the output of the
integrator is observed to be 7.0 V peak-to-peak. Calculate (i) the corresponding peak ux in the lamination
stack and (ii) the corresponding amplitude of the voltage applied to the excitation winding.

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Figure 13

Figure 1.13 Magnetic circuit.

1.27 The coils of the magnetic circuit shown in Fig. 1.13 are connected in series so that the mmf's of
paths A and B both tend to set up ux in the center leg C in the same direction. The coils are wound with
equal turns, N1 = N2 = 100.
The dimensions are:
Cross-section area of A and B legs = 7 cm2
Cross-section area of C legs = 14 cm2
Length of A path = 17 cm
Length of B path = 17 cm
Length of C path = 5.5 cm
Air gap = 0.4 cm
The material is M-5 grade, 0.012-in steel, with a stacking factor of 0.94. Neglect fringing and leakage.
a. How many amperes are required to produce a ux density of 1.2 T in the air gap?
b. Under the condition of part (a), how many joules of energy are stored in the magnetic eld in the air
gap?
c. Calculate the inductance.
1.28 The following table includes data for the top half of a symmetric 60-Hz hysteresis loop for a specimen
of magnetic steel:

Figure 14

Using MATLAB, (a) plot this data, (b) calculate the area of the hysteresis loop in joules, and (c) calculate
the corresponding 60-Hz core loss in Watts/kg. The density of M-5 steel is 7.65 g/ cm3 .

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Figure 15

Figure 1.14 Magnetic circuit for.

1.29 It is desired to achieve a time-varying magnetic ux density in the air gap of the magnetic circuit of
Fig.1.14 of the form
Bg = B0 + B1 sinωt
where B0 = 0.5 T and B1 = 0.25 T. The dc eld B0 is to be created by a neodimium-iron-boron magnet,
whereas the time-varying eld is to be created by a time-varying current.
For Ag = 6cm2 , g = 0.4 cm, and N = 200 turns, nd:
a. the magnet length d and the magnet area Am that will achieve the desired dc air-gap ux density and
minimize the magnet volume.
b. the minimum and maximum values of the time-varying current required to achieve the desired time-
varying air-gap ux density. Will this current vary sinusoidally in time?

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