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Mutual Inductance
L 1
L 1
© Robert York, 2006
L m
L m
dI1 V1 = L1 ± Lm
dt
dI2
dt
V =± L
dI
+ L
dt
dI
dt
L 2
L 2
1 2
2 m
2
I 1
+
V 1
-
I 1
+
V 1
-
L1
dI
2 Lm dt
−
L1
dI
2 Lm dt
+
-
+
-
+
-
+
-
L2
dI
1 Lm dt
L2
1 Lm dt
• Correct sign for mutual inductance found from Lenz’
law and dot convention
• Dot convention: current flowing into one dot will
induce current flow out of second dot
−
dI
+
V 2
-
+
V 2
-
I 2
I 2

Transformers
A transformer is just a special case where the mutual inductance is made as large as
possible by allowing both coils to share the same flux
This is usually achieved by winding them both on a common core of high permeability
material (soft iron or ferrite materials)
I 1
+
© Robert York, 2006
V 1 V 2
- -
+
I 2
V = jωLI + jωL I
1 1 1 m 2
V = jωL I + jωL I
2 m 1 2 2
When there is no flux leakage, the mutual inductance is related to the primary and
secondary inductances as
L = LL
m
1 2
For real transformers this can never be quite achieved, so we write
Lm k LL 1 2 where 0 k 1
= < < coefficient of coupling

I 1
Ideal Transformer
1:n
+ +
V 1 V 2
-
© Robert York, 2006
-
I 2
If both coils share the same
flux, then Farady’s law gives:
As the permeability of the core
increases, the relationship between
the primary and secondary
currents approaches a limiting
value set by the turns ratio:
V1 N1
1
= =
V N n
2 2
I1 N2
⇒ = n
I N
2 1
These two relationships define an ideal transformer. This is a fictitous element (note that µ→∞
implies infinite inductances so the impedance matrix is infinite) but a real transformer
approximates this behavior.
An idea transformer has the following useful property when one winding is terminated:
I 1
1:n
+ +
V 1
-
V 2
-
I 2
Z L
Z
in
V ( N / N ) V Z
I ( N / N ) I n
1 1 2 2
= = =
1 2 1 2
L
2

Transformer Equivalent Circuit
Using the tee-equivalent for reciprocal networks, we find the following
equivalent circuit for mutual inductances or transformers
L1Lm © Robert York, 2006
− L2−Lm Lm
L ∝ N
1 1
L ∝ N = n L
2
2 2 1
L = k LL = knL
m
1 2 1
If desired, this circuit can be cascaded with an ideal 1:1 transformer to
simulate the fact that a real transformer has electrically isolated ports

Alternative equivalent circuit
The following is also an identical equivalent that uses an ideal transformer to
explicitly incorporate the turns ratio and isolation between ports
© Robert York, 2006
L k
2
1 (1 ) −
2
kL1
1:n
ideal
For good transformers, k is nearly 1, and this model also clearly shows why
real transformers do not work at DC
n
=
L
L
2
m