Linear and Ideal transformers notes

Chapter 6 Linear and Ideal Transformers


6.1 Mutual Inductance
 Figure 6.1.1 shows two
coils in air, wound on a i1 i2 = 0
former made from + +
iSRC1 v1 v21
nonmagnetic material.
– –
 Let the current i1 in coil 1
be time-varying, whereas
coil 2 is open circuited. A Figure 6.1.1 Coil 1 Coil 2

voltage
v 1 is induced in coil 1, in accordance with Faraday’s law:
dλ1 N 1 dφ 1e di 1
v 1= = =L1
dt dt dt (6.1.1)
where 1e is an effective flux of coil 1 associated with i1, which if it links all N1 turns

gives λ1 . Thus, λ1 =N 1 φ1e =L1 i1 . 1e accounts for the fact that in the case of cores of
low permeability, not all of the magnetic flux links all the turns of the coil (Figure
6.1.1).

 Let φ21 e be the fraction of the time-varying flux φ1 e that links coil 2, where φ21e is an
effective flux that if multiplied by N2 gives the flux linkage 21 in coil 2 due to i1. A

voltage
v 21 is induced in this coil in accordance with Faraday’s law:
dλ 21 N 2 dφ21 e di 1
v 21 = = =M 21
dt dt dt (6.1.2)

where λ21=N 2 φ21 e .

 The quantity
M 21 is defined as the flux linking coil 2 per unit current in coil 1. Thus:
λ 21 N 2 φ21 e
M 21= =
i1 i1 (6.1.3)

 If a time-varying current
i2 is applied to coil 2, with coil 1 open circuited, then

following the same argument, we have, analogous to Equations 6.1.1 to 6.1.3:
dλ2 N 2 dφ2e di 2
v 2= = =L2
dt dt dt (6.1.4)




6-1/23

, dλ12 N 1 dφ12 e di 2
v 12= = =M 12
dt dt dt (6.1.5)
λ 12 N 1 φ 12e
M 12= =
i2 i2 (6.1.6)

where M 12 is the flux linking coil 1 per unit current i2 in coil 2.
 We will show that M12 = M21 by determining the energy expended in establishing

steady currents
I1 and I 2 , starting from zero. It is convenient to assume that I1 and

I2 are established in two steps: i) i1 is first increased from zero to I1 with I 2 =0 ; and

ii) i 2 is then increased from zero to I2 with i 1=I 1 .

 For the sense of winding of coil 1 in Figure 6.1.1, the flux associated with i1 is

downward in this coil, according to the right-hand rule. While
i1 is increasing, the
di
v L1 1
induced voltage 1 = dt in coil 1 opposes the increase in i1 , in accordance with
Lenz’s law, by being a voltage drop across L1 in the direction of i1. This voltage is

concurrently a voltage rise across the current source, so that the total energy
w 11

delivered by the source is:
t di 1 I1 1
w 11=∫0 L1 i 1 dt=L1 ∫0 i 1 di 1 = L1 I 1
2
dt 2 (6.1.7)

 With i 1=I 1 , (Figure 6.1.2),
Flux
I1 i2 2
the voltage induced in coil 1 2
i2 + +
is that due to increasing iSRC1 v12 v2 iSRC2
and the total energy – –

supplied by the current 1' 2'

source iSRC2 in establishing Coil 1 Coil 2

1 2 Figure 6.1.2
I2 is 2 L2 I 2 , as in Equation

6.1.7.
di
i v 12=M 12 2
 As 2 increases it induces a voltage dt in coil 1. The sense of winding of

coil 2 and the direction of i 2 are such that the flux associated with i2 is also




6-2/23

, downward in coil 1. The effect of increasing
i2 is therefore the same as that of

increasing i1 , so that v 12 is of the same polarity as v 1 in Figure 6.1.1, and opposes
the current in coil 1. The current source iSRC1 has therefore to deliver additional

energy to maintain
I1 :

t t di 2 I2
w 12=∫0 v 12 I 1 dt=∫0 M 12 I 1 dt=M 12 I 1∫0 di 2 =M 12 I 1 I 2
dt (6.1.8)

 The total energy expended in establishing
I1 and I2 is:

1 1
w 1= L1 I 21 + L2 I 22 + M 12 I 1 I 2
2 2 (6.1.9)

 If
I1 and I2 are established in the reverse order, then following the same argument

as above, the total energy expended in establishing
I1 and I2 is:

1 1
w 2= L1 I 21 + L2 I 22 +M 21 I 1 I 2
2 2 (6.1.10)


w 1 = w 2 , because in a lossless, linear system, the total energy expended must

depend only on the final values of
I1 and I2 and not on the time course of i1 and i2 .

Otherwise, it would be possible, at least in principle, to extract energy from the
system at no energy cost, in violation of conservation of energy.
 It follows that:
M 12=M 21= M (6.1.11)
 M is the mutual inductance between the two coils and is a constant in linear
systems. In contrast, the individual inductances L1 and L2 are self-inductances.

Definition The mutual inductance of two magnetically-coupled coils is the
flux linkage in one coil per unit current in the other coil. It is independent of
which coil carries the current.
 If either the polarity of iSRC2, or the sense of winding of coil 2, is reversed in Figure

6.1.2, the flux due to i 2 becomes upward in coil 1. The polarity of v 12 is reversed
and becomes a voltage drop across the current source iSRC1. Energy is therefore
returned to the source and the sign of the energy term involving M becomes
negative in Equations 6.1.9 and 6.1.10. M, however, is always a positive quantity.

 Since I 1 and I2 are arbitrary values, they might just as well be replaced by



6-3/23