Mutual Inductance
L1
I1
Lm
L1
+
L2
V1
-
I1
Lm
L1
L2
V1
dI1
dI
± Lm 2
dt
dt
dI
dI
V2 = ± Lm 1 + L2 2
dt
dt
© Robert York, 2006
I2
+
Lm
dI 2
dt
+
-
+
-
L1
Lm
dI1
dt
V2
-
L2
+
-
V1 = L1
L2
I2
+
dI
− Lm 2
dt
+
-
+
-
− Lm
dI1
dt
V2
-
• 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
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)
I1
I2
+
+
V1
V1 = jω L1 I1 + jω Lm I 2
V2
-
-
V2 = jω Lm I1 + jω L2 I 2
When there is no flux leakage, the mutual inductance is related to the primary and
secondary inductances as
Lm = L1 L2
For real transformers this can never be quite achieved, so we write
Lm = k L1 L2
© Robert York, 2006
where
0 < k <1
coefficient of coupling
Ideal Transformer
1:n
I1
If both coils share the same
flux, then Farady’s law gives:
I2
+
+
V1
V2
-
-
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
=
=
V2 N 2 n
I1
N2
⇒
=n
I2
N1
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:
I1
1:n
I2
+
+
V1
V2
-
-
© Robert York, 2006
ZL
V1 ( N1 / N 2 )V2 Z L
Z in = =
= 2
I1 ( N 2 / N1 ) I 2 n
Transformer Equivalent Circuit
Using the tee-equivalent for reciprocal networks, we find the following
equivalent circuit for mutual inductances or transformers
L1 − Lm
L2 − Lm
L1 ∝ N1
Lm
L2 ∝ N 2 = n 2 L1
Lm = k L1 L2 = knL1
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
© Robert York, 2006
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
L1 (1 − k 2 )
1:n
n=
2
k L1
L2
Lm
ideal
For good transformers, k is nearly 1, and this model also clearly shows why
real transformers do not work at DC
© Robert York, 2006