Electric Machine-I (Course Code: EE 551 )

Chapter –1 ( Reviews on Magnetism )

Lecture No.2

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1.6 Hysteresis with ac excitation

Fig.1.14 (a) Electromagnet excited by ac voltage source

Fig.1.14 (b) Waveform of exciting current and Corresponding hysteresis loop

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The hysteresis power loss in the core is given by:
Wh =  Bm1.6 f Vol (Watts) (1.7)

Where, Bm = Peak value of magnetic flux density in the core (Wb/m2)

f = Frequency of the applied acvoltage (Hz)
Vol = Volume of iron core (m3)
ƞ = Steinmetz constant, whose value depends upon the grade of iron core.

1. 7 Magnetic circuit
The path followed by the magnetic flux is known as magnetic circuit. The path a-b-
c-d-a shown in Fig.1.15 is a magnetic circuit consisting of iron core and winding.

Fig.1.15 Magnetic circuit

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 Let I = Current through the exciting winding (Amp)
N = Number of turns in exciting winding
ϕ = Magnitude of magnetic flux (wb)
A = Cross-sectional area of the iron core (m2)
L = Mean length of the magnetic flux path (a-b-c-d-a)


Magnetic flux density in the core is given by: B=
A
N .I
And, magnetic field intensity in side the core is given by: H = L
B
For linear part of the magnetization curve: =
H
OR  = .N.I
N.I mmf
Or B = μ.H OR  = L = (ohm' s Law) (1.8)
A L Reluctance
. A

Where, N.I= mmf = magnetomotive force, which push the magnetic flux in the magnetic circuit.
L
Rel = = Reluctanc e of magnetic circuit
. A

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1.7.1 Series Magnetic Circuit:
Series magnetic circuit is such circuit, where the same magnetic flux passes through the all
sections of the magnetic circuit as shown in Fig.1.17.

Fig.1.17 Series magnetic circuit

The magnetic circuit has four different sections with different length and cross-
sectional area as follow:

Section-1 : L1 = ba + ai + ih, Area = A1, Permeability = μ1

Section-2 : L2 = bc + cd + ef + fg , Area = A2 Permeability = μ2
Section-3 : L3 = gh, Area = A3 Permeability = μ3
Section-4 : Lg = de, Area = Ag=A2 Permeability = μ0

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 The Total reluctance of the magnetic circuit = sum of the reluctances of all the sections.

L L L L
L 1 2 3 g
Thereefore, Total Rel =  = + + + (1.9)
.A . A . A . A . A
1 1 2 2 3 3 0 2
N .I
Hence Magnetic Flux in circuit ,  = (1.10)
. L . A

Note that the air gap offers very high reluctance. It reduces the magnetic flux in
the circuit. It is quite similar to addition of very high resistance in series with
low resistance in case of series electric circuit.

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1.7.2 Parallel Magnetic Circuit:
If the magnetic flux divides into two or more parallel paths, such magnetic circuit is
known as parallel magnetic circuit.

Fig.1.18(a) Parallel magnetic circuit Fig.1.18(b) Corresponding electric circuit

For electric circuit,

I = I1 + I2 If Rbcda = Rbefa, Then I1 = I2 Or I1 = I/2
Writing KVL for loop a-b-c-d: V = I×Rab + I1× Rbcda OR V = I×Rab + 0.5I × Rbcda
V
OR I =
(R + 0.5 R )
ab bcda
Similarly, for magnetic circuit:
ϕ = ϕ1 + ϕ2 If Reluctance of path-1 = Reluctance of path-2, then ϕ1 = ϕ2 = 0.5 ϕ
Therefore, N.I =   Rel(ab) +   Rel(bcda) OR N.I =   Rel(ab) + 0.5  Rel(bcda)
1
N .I
=
(Rel + 0.5 Rel )
(ab) (bcda)
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1.8 Electromagnetic Induction:
In 1831, English scientist Michael Faraday discovered the relationship between
magnetism and electricity. He observed the momentary induced current in a circuit,
when the magnetic flux linking with the circuit changes momentary with respect to
time. He had stated two laws known as Faraday’s Law Of Electromanetic Induction
as follows:

i) First law: “Whenever the magnetic flux-linkage in a conductor changes with respect
to time, an emf will induced on it”

ii) Second law: “The magnitude of emf induced is equal to the time rate of change of
magnetic flux-Linkage”

The magnetic flux-linkage could be changed in two different ways. Therefore, emf can be
produced in two different ways:
i) Statically induced emf
ii) Dynamically induced emf

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 1.8.1 Statically Induced emf:

No Relative motion between

conductor and magnetic flux

Fig.1.21 Illustration of statically induced emf

Let N = Number of turns in coil-B
ϕ = Magnetic flux passing through the N turns of conductive coil
Then the product N× I = Magnetic Flux-Linkage in coil-B

If the magnetic flux in the coil-B changes from ϕ 1 to ϕ2 in a small time interval from t1 to t2 ,
then according to second law given by Faraday’s law of electromagnetic induction,
magnitude of emf induced in a turn of coil-B is given by:
 - d
e (per turn) = 2 1 =
t −t dt
2 1
If there is ‘N’ number of turns in the coil-B, then total emf induced across the coil is
given by:
d
e =N (1.11)
dt
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The direction of statically induced emf and direction of current due to this emf can be
determined by Lenz’s law.

Lenz’s law statement: “Direction of induced current in the conductor will be such that the
magnetic field set up by the induced current opposes the cause by which the current was
induced.”

Example-1: Emf due to increasing current in coil-A

Coil-A Coil-B
Flux of Coil-A
S N N S
Flux of Coil-B
x + e = emf - y

Rv

+ - R
V

Fig.1.22 Illustration of Lenz’s law (case-I )

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Example-2: Emf due to decreasing current in coil-A

Coil-A Coil-B
Flux of Coil-A
S N S N
Flux of Coil-B
x - e = emf + y

Rv

+ - R
V

Fig.1.23 Illustration of Lenz’s law (case-II)

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1.8.2 Dynamically Induced emf:

v = Velocity
B

N S

I R
A
Induced
current
B= Magnetix Flux density There is relative motion between
conductor and magnetic flux
Fig.1.24 Illustration of dynamically induced emf

The distance moved ‘dx’ by the conductor in a small time

v interval of ‘dt’ is given by: dx = v×dt
Area swept by the conductor in a small time interval of ‘dt’
A is given by: A = L×dx = L× v×dt
Total flux swept by the conductor in a small time interval
of ‘dt’ is given by: d = B  A = B  L  v  dt
L v
dx
Hence, emf induced in the conductor is given by:
d B  L  v  dt
e = = = BLv
Fig.1.25 Detail of area dt dt
swept by the conductor Or e = BLv (1.12)
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The direction of dynamically induced emf and current in the conductor is
determined by Fleming’s Right Hand Rule. To use this rule, we shall use our right
hand as shown in Fig.1.26
Direction of v (thumb)

Direction of B
(fore finger)

Direction of induced
current (middle finger)

Fig.1.26. Illustration of Fleming’s right hand rule

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 (a) Conductor motion upward (b) Conductor motion downward
Fig.1.27 Direction of dynamically induced current

Fig.1.28 Dynamically induced emf with inclined direction of motion of conductor

If the direction of motion is inclined with the direction of magnetic flux density (B) as
shown in Fig.1.28, then the magnitude of emf induced is given by:
e = B.L.v.sin (1.13)
Where,  = Angle between direction of motion and direction flux density.
v. sin = Component of v in the direction perpendicular to direction of B
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1.9 Force developed on current carrying conductor:

When electric current is passed through a conductor lying in the magnetic field, a
force will develop on the conductor.

F = force
B

N S

A
B= Magnetix Flux density
current

Fig.1.29. Force developed on current carrying conductor

The magnitude of the force so developed is given by:

F = B.I.L (Newton) (1.14)
Where,
B = Magnetic flux density ( wb/m2)
I = Current passing through the conductor (Amp)
L = Length of the conductor lying within the magnetic field (m)

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The direction of force so developed is determined by Fleming’s Left Hand Rule. To
use this rule, we shall use our left hand as shown in Fig.1.30
Direction of F (thumb)

Direction of B
(fore finger)
Left Hand

Direction of current
(middle finger)

(a) (b)
Fig.1.31 Direction of force developed on the conductor

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