Copyright 20 13 Dorling Kindersley (India) Pvt. Ltd

14.1.3 
Maximum Value of Mutual Inductance and coupling coefficient
The maximum value of mutual inductance that a two-coil coupled system can have is the geometric 
mean of self-inductances of the two coils – i.e., L L
1 2
. This maximum value is realised when the flux 
produced by a current in one coil links totally with the other coil. This is impossible to achieve physically 
and can only be approached in practice. Winding both the coils over a common core made of some high 
permeability material like laminated iron makes M very close to (but less than) L L
1 2
in practice.
The ratio between the actual mutual inductance realised to the theoretical maximum value is defined 
as coupling coefficient and is symbolised by k. Coupling coefficient is a positive number between 0 
and 1. It is difficult to achieve anything more than few tenths for k-value in air-cored coils; however, 
coils wound on common core made of iron can have k-value very close to unity.
M
k L L
k
=
≤ ≤
1 2
0
1
H; 
14.2 
the tWo-WIndIng tranSforMer
A system of two coils with constant values of L
1
, L
2
and M with two pairs of terminals identified for 
application of excitation and/or measurement of response is called a two-winding linear transformer. 
Such a two-winding transformer will be referred to simply 
as a transformer in this section.
Fig. 14.2-1 shows a two-coil system with self-
inductance values of L
1
, L
2
and mutual inductance value 
of M. Also shown are two independent voltage sources 
driving the circuit at two pairs of terminals.
The mesh equations of the circuit are written assuming 
that the coils have zero winding resistance values.
L
di
dt
M
di
dt
v t
M
di
dt
L
di
dt
v t
1
1
2
1
1
2
2
2
−
=
−
+
=
( )
( )
The current i
2
leaves the dot in the second coil and therefore the mutually induced voltage in the 
first coil appears with negative polarity at the dot. Hence, the negative sign for mutually induced 
voltage in the first mesh equation. The current i
1
enters the dot of the first coil and therefore the 
mutually induced voltage in the second coil appears with positive polarity at the dot. When mesh 
Fig. 14.2-1 
A transformer with 
voltage source drive 
at both ports
M
+
–
+
–
i
1
i
2
L
1
L
2
v
2 
(
t
)
v
1 
(
t
)

---

14.8
Magnetically Coupled Circuits
equation is written by traversing the loop in clockwise direction the self-induced voltage and mutually 
induced voltages will enter with opposite signs. Hence, the negative sign for mutually induced voltage 
in the second mesh equation.
+
+
+
–
–
–
+
–
i
y
i
y
L
1
L
2
M
d
d
t
i
x
i
x
M
d
d
t
(a)
i
1
i
2
v
2 
(
t
)
v
1 
(
t
)
+
–
+
–
L
1
L
2
– 
M
M
– 
M
(b)
i
1
i
2
v
2 
(
t
)
v
1 
(
t
)
Fig. 14.2-2 
Two circuit models for a transformer (a) model using linear dependent sources 
(b) Conductive equivalent model
Refer to Fig. 14.2-2. The reader may easily verify that the circuit in Fig. 14.2-2 (a) and the circuit 
(b) have the same mesh equations as those of the circuit in Fig. 14.2-1. Hence, the coupled set of 
coils may be replaced by two decoupled coils and two linear dependent sources as in circuit model 
Fig. 14.2-2 (a) as far as the v–i behaviour at the terminals is concerned. This circuit model preserves 
the conductive decoupling – i.e., the galvanic isolation – that exists between the two sides of the 
circuit.
A coupled set of two coils can also be replaced by a T-shaped equivalent circuit comprising 
three pure decoupled inductors as in circuit model in Fig. 14.2-2 (a) as far as the v–i behaviour 
at the terminals is concerned. This circuit model hides the galvanic isolation that is present in the 
transformer. Therefore, it is called the conductive equivalent circuit of coupled coils. Note that one 
inductance (either L
1 
- 
M or L
2 
-
M) can be negative-valued for sufficiently large value of coupling 
coefficient. There is no negative inductance in the physical world. But then, the inductors that appear 
in the conductive equivalent circuit of coupled coils are not physical inductors – they are mathematical 
inductors that are arranged to result in same set of circuit equations as those of the coupled coils. 
Therefore, they can assume negative values – we do not have to construct them!
Note that a certain dot polarity was assumed for the equivalent models established above. The 
equivalent models for the second relative dot polarity are shown in Fig. 14.2-3.
L
1
L
2
M
(a)
L
1 
+ 
M
L
2 
+ 
M
–
M
(c)
+
+
–
–
i
x
i
x
i
y
i
y
L
1
M
d
d
t
(b)
L
2
M
d
d
t
Fig. 14.2-3 
Circuit models for coupled coils 
The circuit in Fig. 14.2-3 (b) shows the dependent source based circuit model for the two-coil 
system in (a) and the circuit in Fig. 14.2-3 (c) shows the conductive equivalent circuit of the two-coil 
system in Fig. 14.2-3 (a). Note the reversal of polarity in the case of dependent sources and change in 
inductance values in the case conductive equivalent circuit.
Differentiation in time is replaced by multiplication by j
w
in the phasor equivalent circuit for 
sinusoidal steady-state analysis in the equivalent circuit employing dependent sources. Winding 
resistance can be included as series resistors on both sides in both equivalent circuits.

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The Two-Winding Transformer 
14.9