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CHAPTER- 1 : FUNDAMENTALS OF MAGETIC CIRUITS
CHAPTER-1: FUNDAMENTAL OF MAGETIC CIRCUITS

PERSONAL REMARK :

Electrical machines is one of the core subject of the Electrical
Engineering.

Knowledge of electrical machines is very important because most
of the electrical energy (about 90%) is generated and consumed by
electrical machines.

Electrical machines are classified as Dynamic and static machines.

All the Rotating type machines are dynamic machines and
transformer is classified as of static machine.

All the dynamic machines operate in two modes-Generating and
motoring.

Generating Mode
Power (Electrical)
TG
Prime
Mover
Heat Power
Electric
Generator
TM,W
Losses
Power (Mechanical)

Motoring Mode
P electrical
TG W
Load
(mechanical)
Electric
Motor
TM
Losses
P mechanical

Transformer
Electric
power
Electric
power
MAGNETIC CIRCUITS

Coupling magnetic field exists in transformers and practically in all
the rotating electrical machinery. In low-power electrical machines,
magnetic field may be produced by permanent magnets. But in highpower electrical inachinery and transformers, coupling magnetic field
is produced by electric current.

Thus, an elementary knowledge of the analysis of magnetic circuits
is necessary for understanding the working of these devices.
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CHAPTER- 1 : FUNDAMENTALS OF MAGETIC CIRUITS

The object of this chapter is to develop some basic techniques for
PERSONAL REMARK :

the analysis of magnetic-field systems and to introduce the concept
of induced emf.
Introduction to Magnetic Circuits
The complete closed path followed by the lines of flux is called a
magnetic circuit. Some improtant terms relating to the study of
magnetic circuits are discussed here.
(I)
Magneto Motive Force (MMF)

In an electric circuit, the current is due to the presence of
electromotive force. By analogy, in a magnetic circuit, the magnetic
flux is due to the presence of a magnetomotive force.

The mmf is created by a current flowing through one or more turns.
This shows that mmf is equal to the product of current and the number
of turns in the coil.
Mmf = NI ampere–turns (or ATs).
Magnetic
flux path
a

b
I
Area of
cross section A
N
d
c
Figure 1: A simple magnetic circuit with N turns and current I
(II)
Magnetic Field Intensity

If the magnetic a circuit of figure 1 is homiogeneous and of uniform
cross-sectional orea. The magneto motive force per unit length of
magnetic circuit is termed as the magnetic field intensity its usual
symbol is H. If I  is the mean length of magnetic circuit abcd in
figure 1, then
H
IN

ampere-turns/metre
[or ATs/m]
(III) Permeability of Free Space

Suppose flux density at C, caused by magnetic field intensity H at x1
is B tesla and if C is one metre away from x1 then permeability of
free space µ0 is given by
µ0 =
B
H
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CHAPTER- 1 : FUNDAMENTALS OF MAGETIC CIRUITS
PERSONAL REMARK :
Magnetic
flux path
x
X

C
1m
Figure 2 : Pertaining to permeability of free space
Note that conductor x and point c are in vacume (free space), in air
or in any other non-magnetic medium.
For vacuum or non-magnetic materials
µ0 =
B
= 4 × 10–7 henries/m
H
(IV) Reluctance. The opposition offered by the magnetic circuit
to magnetic flux is called reluctance. Reluctance is analogous
to resistance in electric circuit. Just as resistance R =
reluctance R  is given by, R  =
.
,
a

AT/Wb
µ.A
where, l = length of the magnetic path, m
A = area of cross-section normal to flux path, m2
µ = µ0, µr = permeability (or absolute permeability) of the
magnetic material
µr = relative permeability of the magnetic material
µ0 = permeability of free space = 4 × 10–7 H/m
Since resistance , R =
l
, absolute permeability µ is analogous to
a
 p 
1
conductivity  

(V)
Permeance. Reciprocal of reluctance is called permeaence.

Permeance = Wb/AT
(VI) Magnetic flux density. magnetic flux is analogous to current
in a electric circuit.
As current I =
=
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emf
, flux  in a magnetic circuit is given by
resistance
mmf
Wb
Reluctance
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CHAPTER- 1 : FUNDAMENTALS OF MAGETIC CIRUITS
R =
=

and mmf = IN ; then magnetic flux is given by, 
µ0 µ r A
PERSONAL REMARK :

IN . µ 0 µ r . A
Wb

Magnetic flux density B is defined as the magnetic flux per unit
cross-sectional area of the core and its unit is Wb/m2 or Tesla (T).

B=
IN . µ0 µr
magnetic flux, 

core area, A

I
E
V/A or  we have R  =
AT/Wb

I
(VII) Calculation of ampere-turns. In any magnetic circuit, flux
Like R =
 is given by
Ampere turns required for the magnetic circuit,
I = × Reluctance

B
 H  µ 



φ

B



.   H
µA
A
µ
µ
=×
= Magnetic field intensity H in AT/m in that part of circuit × length 
of that part normal to flux path.
Comparison Between Magnetic and Electric Circuits

There are many similarities in magnetic and electric circuits.
The reader is usually more conversant with electric circuits. Thus, a
comparison of electric and magnetic circuiits in tabular form goes a
long way in the better understanding of magnetic circuits.
Table 1
Electric Circuit
Magnetic Circuit
Conductor length

I

Flux
core length
l
N
E
Current I
I
Core area, A
Conductor area A
A toroidal iron ring of length  ,
A toroidal copper ring of length core area A is excited by a coil of
 , cross-sectional area A is N turns carrying I amperes so that
connected to emf E so that flux is produced.
current I flows
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CHAPTER- 1 : FUNDAMENTALS OF MAGETIC CIRUITS
Similarities
PERSONAL REMARK :
1. Closed path for the electric
current is called an electric
circuit.
is called a magnetic circuit
2. Driving force is mmf I = IN ATs
2. Driving force is emf E, volts
3. Resistance, R =
p.l
V/A or 
A
3. Reluctance, R  =

+
+
p
R

A
E
–
driving force
5. Current, I =
resistnce
–
5. Magnetic flux,  

AT/Wb
µA
driving force
reluctance
I
Wb
R
6. Magnetic flux density, B =

T
A
(or Wb/m2)
7. Magnetic field intensity
7. Electric fleld intensity,
E
V/m
l
Also 
R 
T
or  
E
I=
A
R
I
6. Current density, J =
A

I
AT/Wb
µA
4. Equivalent circuit
4. Equivalent circuit
I
or

1. Closed path for the magnetic flux
IN
AT/m
l
H =
E IR I p . l
I

 .
 p.
l
l
l A
A
Also H =
IN I .Rl  l
1 
 


l
l
l
l µA µ A
or  . J V/m
8. Conductivity
path,  
of
current
l

or
H=
1
BAT/m
µ
8. Permeability of magnetic circuit,µ
so that B = µH Wb/m2 or T
So that , J = A/m2
Dissimilarities
1. The electric current actually
flows in an electric circuit. For
the existence of this current,
energy is drawn from the source
continuously. This energy gets
dissipated in resistance in the
form of heat.
2. Electrical insulator confine the
current to well defined paths.
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1. Strictly speaking, magnetic flux
does not flow in the magnetic
circuit. Energy in needed for
estabilishing the required flux.
Once the requisite flux is created,
no more energy is needed in
maintaining it.
2. There are no magnetic irsulators.
Even in the best known magnetic
insulator some flux can be
established.
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CHAPTER- 1 : FUNDAMENTALS OF MAGETIC CIRUITS
Ex.1 A mild steel ring has a mean diameter of 20 cm and a cross-
PERSONAL REMARK :

2
sectional area of 50 cm . For a relative permeability of 800,
calculate (a) reluctance of the ring and (b) current required
in 200 turn coil to produce a flux of 1m Wb in the ring.
Sol. (a) Reluctance =
 R =
(b)
l
. Here  = D = × 20 cm, A = 50cm2
µ0 . µr A
  20  10 –2
4  10 – 7  800  50  10 – 4
= 1.25 × 106 AT/Wb
mmf
flux.  = 1 × 10–3 =
1.25  106
 Mmf, I = 1 × 10–3 × 1.25 × 106 = 1250 ATs.
 Current in the coil, I =
I
1250

= 6.25 A
N
200

Series and Parallel Magnetic Circuits

Like electric circuits, a magnetic circuit may be made up of a series
circuit, a parallel circuit or a combination of series-parallel circuits.
Such magnetic circuits may be excited by one or more coils. Solution
of such magnetic-circuit configurations can be obtained by applying
Kirchoff's flux laws.
Area A2
Area A1
I
2
1
I

1
g .Area Ag
N
3

3
Area A3
Figure 3 : Composite magnetic circuit

Series Magnetic Circuit : Just like a series electric circuit, a series
magnetic circuit carries the same flux in each part of the circuit
configuration. Figure 3 shows a composite magnetic circuit consisting
of three different magnetic materials in series. There is also an air
gap as shown. The dimensions are indicated in figure 3. Since the
same flux  exists through each part of the magnetic circuit in
figure 3, the total reluctance is equal to the sum of four individual
circuit reluctances as under :
Total reluctance,
R =
g
3
1
2



µ 0 µ r1 A1
µ 0 µ r2 . A 2
µ 0 µ r3 . A 3
µ0 . A g
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CHAPTER- 1 : FUNDAMENTALS OF MAGETIC CIRUITS
Note that µr for air is one

PERSONAL REMARK :

Total mmf = flux × total reluctance of the series circuit

g 
3
1
2




=  
 µ 0 µ r1 A1 µ 0 µ r2 . A 2 µ 0 µ r3 . A 3 µ 0 . A g 
=
3
1
2



 g







A1 µ 0  µ r1 A 2 µ 0  µ r2 A 3 µ 0  µ r3 A g µ 0
=
Bg
B1
B2
B3
 1 
 2 
 3 
. g
µ0  µr1
µ0  µr2
µ0  µr3
µ0
= H1 .  1 + H2 .  2 + H3 .  3 + Hg .  g
Total mmf, IN = mmf in for [  1 +  2 +  3 +  g]
.... (i)
.... (ii)
Equation (i) may be re-written as under :
IN – H1 1 – H2 2 – H3  3 – Hg  g = 0
In general,  Mmf a closed magnetic circuit = 0
But this statement is similar to Kirchhoff's voltage law for a series
electric circuit. Therefore, Kurchhoff's mmf law for a series magnetic
circuit is as under
Kirchhoff's mmf law (KML) states that algebraic sum of mmfs
rises (or falls) taken in a specified direction in a closed magnetic
circuit is zero.
Parallel Magnetic Circuit : Consider a parallel magnetic circuit
shown in figure 4. Coil carrying a current I produces flux in path
EFAB. At point B. flux   gets divided into two paths,
 and  such that
    
or
 –  –  = 0
In general,
  = 0
The above statement is similar to Kirchhoff's current law. Therefore,
for a magnetic circuit also, total magnetic flux towards a junction is
equal to the total magnetic flux away from that junction. In other
words, algebraic sum of magnetic fluxes at a junction is zero, this is
called Kirchhoff's flux law (KFL).
I
A
1
N
F
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C
2
3
E
(a)
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CHAPTER- 1 : FUNDAMENTALS OF MAGETIC CIRUITS
R 1 
1 B
PERSONAL REMARK :
R1
3
2
T
R 2

I3
I1
I2
R 3
E
(b)
E
R2
R3
(c)
Figure 4.(a) Parallel magnetic circuit. Its equivalent
(b) magnetic circuit and (c) electric circuit.
From figure 4 (a) if reluctance of parth EFAB = R  1
reluctance of path BF = R  2 and reluctance of path ECDE = R  3,
then equivalent magnetic cirucit of figure 4(b) is obtained. From this
circuit, it is seen that total coil mmf, I = IN mmf required for
path [EFAB + BCDE]

IN = 1 Rl1 + 2 Rl2
Also, I = IN = mmf required for path [EFAB + BCDE]

IN = 1 Rl1 + 3 Rl3
Figure 4(c) gives the equivalent electric circuit.

Leakage Flux and Fringing

All practical magnetic circuitcs are so designed that most of the flux
produced by an exciting coil is confined to the desired magnetic path
of low reluctance. However, a small amount of flux does follow a
path through the surrounding air. Figure 5 shows a magnetic circuit
with a ferromagnetic core. The flux which returns by such path as a,
b, c is called leakage flux. It does not follow the intended path.
Therefore, leakage flux may be defined as that flux which does not
follow the intended path in a magnetic circuit. Leakage flux does
exist in all practical electromagnetic devices. Its effect on the analysis
of electrical machinery is carried out by replacing it by an equivalent
leakage reactance.
It is seen from Figure 5 that total flux thorugh the exciting winding =
useful flux + Ieakage flux. A term leakage factor gives a measure of
the leakage flux. It is defined as
leakage factor =
total flux handled by the exciting winding
useful flux
The value of leakage factor in electrical machinery is in the range
of 1.15 to 1.25.
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
As the magnetic flux lines cross the air gap, these bulge out as shown
PERSONAL REMARK :

in figure 5. This bulging out of the flux is called fringing. Larger the
air gap, more is the flux fringing. The effect of fringing flux is to
increase the effective cross cectional area of the air gap. As a result,
flux density in the air gap is not uniform and average flux density
gets reduced.
Useful flux
a
+
b
c
Fringing
flux
–
Leakage
flux
Ferromagnetic core
Figure 5: Fringing of the flux across air gap. Useful and leakage
fluxes are also shown

The effect of fringing can be taken into account empirically by adding
one gap length to each of the linear dimensions making up the gap
area. For example, if core dimensions are a and b and small gap
length is  g , then effective gap area taking fringing into consideration
is Ag = [(a + lg) × (h + lg)] > core area Ac = a . b. If core is circular
of radius r, then effective gap area Ag = [ (r + lg)2] > core area
Ac = r2.
B-H CURVE

A B-H curve, also called magnetisation curve or saturation curve, is
the plot of flux density B as the magnetic field intensity H is varied.

Figure 6. shows a typical B-H curve of a ferromagnetic material. It
has initial non-linear zone OA, zone from A to C is almost linear and
zone beyond C is called saturation zone. The flux density B in the
saturation zone increase less repidly with H as compared to its change
in the linear zone. As B-H curve is not a stratight line, relative
permeability r =
B
of a ferromagnetic material changes with
µ0 H
the flux density.

Therefore, during magnetic circuit calculations, the value of µr and
H should correspond to the flux density under consideration. In free
space or non-magnetic materials, µ0 is constant, therefore B-H
relationship is linear.
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CHAPTER- 1 : FUNDAMENTALS OF MAGETIC CIRUITS
PERSONAL REMARK :

B(T)
Saturation
zone
C
Linear zone
A
Non-linear zone
0
H(AT/m)
Figure 6 : Typical magnetization curve of a ferromagnetic material.
Ex. 1 For the manetic circuit shown in figure 7, l c = 50 cm,
dia = 2.85 cm, l g = 2 mm and N = 500. Air-gap flux
= 0.8 mWb.
(a)
Find the exciting current in the coil in case µr = 500. Neglect
magnetic leakage and fringing.
Repeat part (a) if flux fringing at air gap is taken into
account.
(b)
Core length
c = 50cm
I
 g=2mm
N
Area Ac = 2.85cm 
Figure 7 : Pertaining to Example 2.
Sol. (a)
Air-gap length, l g = 2 mm, µ r = 500, core area
Ac =

× 2.852 cm2

Gap reluctance, R  g =
Ig
µ0 A c

50  10 –3  4
4  10 – 7   (2.85) 2  10 – 4
= 24.9483 × 105 AT/Wb
Iron-core reluctance
Ic
50  10 –2  4

Ric =
µ 0 . µ r A c 4  10 – 7  500   (2.85) 2  10 – 4
= 12.474 ×105 AT/Wb
Total reluctance, Rl = Rlg + Rlc = 37.4223 × 105 AT/Wb
But
 Coil current I =
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mmf = flux × reluctance
 . Rl 0.8  10 – 3  37.4223  105

= 5.9876 A
N
500
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(b)
When flux fringing is taken into account, the effect gap
PERSONAL REMARK :

area Ag

=

 2.85

 0.2 = 1.6252 cm2
(d + 2g)2 = (r + g)2 =  

 2

Ig
Air-gap reluctance, Rlg = µ . A
0
g

2  10 –3
4  10 – 7    1.6252  10 – 4
= 19.185 × 105 AT/Wb
I=


[Rlc + Rlg]
N
Coil current, I =
0.8  10 –3
[12.474 + 19.185] × 105 = 5.0654A

A magnetic circuit of cast steel is arrange as shown in
Ex. 2
figure 8. Various dimensions are also indicated in the
figure. The exciting coil, with N = 1000 turns, sets up a flux of
1mWb in the central limb. Find the coil current if, for cast
steel,
Area=2×2cm
15cm
15cm
2mm
25
cm
I
2
2mm
I
N
I
Area=4×2cm
25
cm
2
15cm
Figure 8
(a)
µr = 
and (b) µr = 6000
Neglect fringing and leakage.
Sol. (a) When relative permeability µr =  , the reluctance offered by cast
steel is zero and therefore no mmf (= flux × reluctance) is required
by the steel. Reluctance is offered by the air gaps alone. Reluctance
R 1 offered by each of the air gaps in outer limbs is
R 1 =
2  10 –3
   10 – 7  4  10 – 4
= 3.98 × 106 AT/Wb
Reluctance R 2 offered by air gap in central limb is
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PERSONAL REMARK :
2  10 –3
R 2 =
  10 – 7  8  10 – 4

= 1.99 × 106 AT/Wb
The electrical analog of the magnetic circuit of figure 8 is shown in
figure 9. As two reluctances R  1 are in parallel, their resultant is
R1
and the circuit gets reduced to that shown in figure. For this
2
circuit, Kirchhoff's mmf law gives
Rl1 

=0
NI – flux  Rl 2 
2 

or
1000 × I = 1 × 10–3 [1.99 + 1.99] × 106

I=
1  10 –3 (3.98)  10 6
= 3.98 A

1mWb
1mWb
R 1
R 1
R 2
NI
R 1
R 1
2
NI
(b)
(b)
Figure 9 (a) and (b) : Electrical analog of magnetic circuit of figure 8
(b)
Relative permeability, µr = 6000. Here reluctance of cast steel
cannot be neglected. The electrical analog of the magnetic circuit of
figure 8 is now drawn in figure 10, where Rlc1 is the reluctance of
each of the outer two steel limbs and Rlc2 is the reluctance of the
central steel core. Reluctance of outer steel core (including the top
15 cm)
R c1 =
(.  0.15)
   10 – 7  6000  4  10 – 4
= 1.33 × 106 AT/Wb
For central steel core,
Rlc2 =
0.15
4  10
–7
 6000  8  10 – 4
= 0.25 × 105 AT/Wb
Resultant of outer reluctances
=
Rl1  Rl c1 (1.33  39.8)  105

= 20.565 × 105 AT/Wb

2
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CHAPTER- 1 : FUNDAMENTALS OF MAGETIC CIRUITS
PERSONAL REMARK :
R 2
R 1
1mWb
Rc1
NI
R 1
R 2
R 2
Rc2
Rc1
(b)

Rc1+R1
2
NI
(c)
Figure 10 : is redrawn in figure 10(c) for which Kirchhoff's
mmf law gives
Rl  Rl1 

Nl –  Rl c2  Rl 2  c1

2


IN = 1 × 10–3 [0.25 + 19.9 + 20.565] × 105

Exciting current, I =
  10 –3
[40.715 × 105] = 4.072 A
1000
Where e = emf induced, volts; N = number of turns in the coil
N  = flux linkages with the coil, Wb-turns
t = time, seconds
LENZ'S LAW
Changing magnetic flux linking a coil induces an emf in the coil.
When the coil circuit is closed a current begins to flow in the coil.
The direction of this induced emf, or induced current, is governed by
Lenz's law. According to this law; the induced current develops a
flux which always opposes the change responsible for inducing this
current.

In order to take into consideration the fact that the induced current
or induced emf opposes the change in flux, a negative sign is added


e=–
d
d
=–N
dt
dt
Lenz's law is the outcome of natural law : for every action, there is
an equal and opposite reaction. Lenz's law, in short, is effect opposes
the cause.
Change in flux linkages induced emf in a coil requires changes in
flux or flux linkages with respect to time. This change in the flux
linking a coil can be obtained by any of the five methods outlined
below :
(i)
Magnetic flux is constant and stationary, but coil rotates (or
conductor moves). Flux-cutting action occurs and emf so
induced is known as speed emf, motional emf or rotational
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emf*. This principle is used in all dc machines and in
PERSONAL REMARK :

synchronous machines with field winding on the stator.
(ii)
Coil is stationary, but dc electromagnet (or permanent magnet)
is rotated. Here also flux-cutting action generates speed emf.
This principle is used in large-sized synchronous generators.
(iii)
Coil is stationary, but the flux passing through the coil changes
with time. induced emf is called transformer emf.
(iv)
Reluctance of the magnetic circuit varies with rotor rotation.
As a consequence, flux variation occurs and emf is generated.
This principle is used in alternators.
SPEED EMF OR MOTIONAL EMF
Consider a single conductor of length l metre moving at right angles
to uniform magnetic field between N and S poles. The magnetic
field has uniform flux intensity B tesla. Conductor velocity u m/s is
perpendicular to its length l and flux density B. Let the conductor
travel a small distance dx in a time dt, from position 1 to position 2.
Therefore, area swept by the conductor in time dt = l. dx m2
N
N
N
B
2
v
B
v
v sin 

S
B
v
S
S
(b)
(c)
dx
Area
swept

1
(a)
2
Figure 11: Pertaining to the generation of speed emf in a conductor
Flux cut by conductor in time dt = [Uniform flux density]
× [area swept by conductor normal to field in time d(t)]
= B0. ldx = d Wb
As per Faraday's law, magnitude of induced emf e, is
e=
Since ,
d B dx
dx

 B .
dt
dt
dt
dx
= velocity of conductor v in m/s, the emf induced in one
dt
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conductor is given by
PERSONAL REMARK :

e = Blu
Now let the conductor move at an angle to the direction of magnetic
field B
Now component of velocity perpendicular to magnetic field is
u sin .
e = B  u sin 
In figure 11(c), conductor moves paralle to the flux lines, no
flux is cut and therefore induced emf is zero. In case  = 0 in
figure 1(c), e = 0,
Equation can also be expressed as
e
= B u sin 
l
or
E = u  B V/m
where E = electric field intensity in volts per metre. Here E is in a
direction normal to the plane containing u and B . Cross product of
u and B indicates the direction of E . For this purpose, stretch four
fingers of right hand along u , now move these fingers from u
towards B , then outstretched thumb gives the direction of E . In
figure 11, u and B are in x – y plane and E is along z-axis, normal
to the plane containing u and B . If this rule is applied to figure 12,
direction of generated emf would be into the paper which can be
denoted by cross.
TRANSFORMER EMF

The magnitude of transformer emf is given by e = – N
d
. The coil
dt
is stationary (or fixed), but flux is time-varying in nature.

Note that in motional or speed emf, motion or rotation is involved.
Therefore, speed emf is associated with energy conversion from
electrical to mechanical or mechanical to electrical. The transformer
emf does not involve any motion of the coil. Therefore, transformer
emf is not associated with energy conversion; it is, however, associated
with energy transfer only.
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PERSONAL REMARK :
z

E
90º
y
v

B
x
Figure 12 : Pertaining to the direction of electric field intensity E
GENERAL CASE OF INDUCED EMF
Suppose there is a relative movement between coil and the magnetic
flux and at the same time, flux is changing with time, Under these
conditions, both speed and transformer emfs would be induced in
the coil. Thus, total emf, e in the coil would now be given by
e = speed or rotational emf, er + transformer emf, et
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