# Electrical Machine 1

Topics: Alternating current, Electromotive force, Transformer Pages: 5 (1320 words) Published: August 27, 2013
Electrical Machines I

Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao

2

Basic Principles
As mentioned earlier the transformer is a static device working on the principle of

Faraday’s law of induction. Faraday’s law states that a voltage appears across the terminals of an electric coil when the ﬂux linkages associated with the same changes. This emf is proportional to the rate of change of ﬂux linkages. Putting mathematically, e= dψ dt (1)

Where, e is the induced emf in volt and ψ is the ﬂux linkages in Weber turn. Fig. 1 shows a

Figure 1: Flux linkages of a coil

coil of N turns. All these N turns link ﬂux lines of φ Weber resulting in the Nφ ﬂux linkages. In such a case, ψ = Nφ and e=N dφ dt volt (3) (2)

The change in the ﬂux linkage can be brought about in a variety of ways • coil may be static and unmoving but the ﬂux linking the same may change with time. 3

Indian Institute of Technology Madras

Electrical Machines I

Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao

• ﬂux lines may be constant and not changing in time but the coil may move in space linking diﬀerent value of ﬂux with time. • both 1 and 2 above may take place. The ﬂux lines may change in time with coil moving in space. These three cases are now elaborated in sequence below, with the help of a coil with a simple geometry.

L B

X

-

+

Figure 2: Static coil

Fig. 2 shows a region of length L m, of uniform ﬂux density B Tesla, the ﬂux lines being normal to the plane of the paper. A loop of one turn links part of this ﬂux. The ﬂux φ linked by the turn is L ∗ B ∗ X Weber. Here X is the length of overlap in meters as shown in the ﬁgure. If now B does not change with time and the loop is unmoving then no emf is induced in the coil as the ﬂux linkages do not change. Such a condition does not yield any useful machine. On the other hand if the value of B varies with time a voltage is induced in the coil linking the same coil even if the coil does not move. The magnitude of B 4

Indian Institute of Technology Madras

Electrical Machines I

Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao

is assumed to be varying sinusoidally, and can be expressed as, B = Bm sin ωt (4)

where Bm is the peak amplitude of the ﬂux density. ω is the angular rate of change with time. Then, the instantaneous value of the ﬂux linkage is given by, ψ = Nφ = NLXBm sin ωt The instantaneous value of the induced emf is given by, e= dψ π = Nφm .ω cos ωt = Nφm .ω. sin(ωt + ) dt 2 (6) (5)

Here φm = Bm .L.X. The peak value of the induced emf is em = Nφm .ω and the rms value is given by E= Nφm .ω √ 2 volt. (7)

Further, this induced emf has a phase diﬀerence of π/2 radian with respect to the ﬂux linked by the turn. This emf is termed as ‘transformer’ emf and this principle is used in a transformer. Polarity of the emf is obtained by the application of Lenz’s law. Lenz’s law states that the reaction to the change in the ﬂux linkages would be such as to oppose the cause. The emf if permitted to drive a current would produce a counter mmf to oppose this changing ﬂux linkage. In the present case, presented in Fig. 2 the ﬂux linkages are assumed to be increasing. The polarity of the emf is as indicated. The loop also experiences a compressive force.

Fig. 2(b) shows the same example as above but with a small diﬀerence. The ﬂux density is held constant at B Tesla. The ﬂux linked by the coil at the current position is 5

Indian Institute of Technology Madras

Electrical Machines I

Prof. Krishna Vasudevan, Prof. G. Sridhara Rao, Prof. P. Sasidhara Rao

φ = B.L.X Weber. The conductor is moved with a velocity v = dx/dt normal to the ﬂux, cutting the ﬂux lines and changing the ﬂux linkages. The induced emf as per the application of Faraday’s law of induction is e = N.B.L.dx/dt = B.L.v volt.(Here N=1)

Please note,the actual ﬂux linked by the coil is immaterial. Only...

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