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Mutual Inductance<br />
L 1<br />
L 1<br />
© Robert York, 2006<br />
L m<br />
L m<br />
dI1 V1 = L1 ± Lm<br />
dt<br />
dI2<br />
dt<br />
V =± L<br />
dI<br />
+ L<br />
dt<br />
dI<br />
dt<br />
L 2<br />
L 2<br />
1 2<br />
2 m<br />
2<br />
I 1<br />
+<br />
V 1<br />
-<br />
I 1<br />
+<br />
V 1<br />
-<br />
L1<br />
dI<br />
2 Lm dt<br />
−<br />
L1<br />
dI<br />
2 Lm dt<br />
+<br />
-<br />
+<br />
-<br />
+<br />
-<br />
+<br />
-<br />
L2<br />
dI<br />
1 Lm dt<br />
L2<br />
1 Lm dt<br />
• Correct sign for mutual inductance found from Lenz’<br />
law and dot convention<br />
• Dot convention: current flowing into one dot will<br />
induce current flow out of second dot<br />
−<br />
dI<br />
+<br />
V 2<br />
-<br />
+<br />
V 2<br />
-<br />
I 2<br />
I 2
Transformers<br />
A transformer is just a special case where the mutual inductance is made as large as<br />
possible by allowing both coils to share the same flux<br />
This is usually achieved by winding them both on a common core of high permeability<br />
material (soft iron or ferrite materials)<br />
I 1<br />
+<br />
© Robert York, 2006<br />
V 1 V 2<br />
- -<br />
+<br />
I 2<br />
V = jωLI + jωL I<br />
1 1 1 m 2<br />
V = jωL I + jωL I<br />
2 m 1 2 2<br />
When there is no flux leakage, the mutual inductance is related to the primary and<br />
secondary inductances as<br />
L = LL<br />
m<br />
1 2<br />
For real transformers this can never be quite achieved, so we write<br />
Lm k LL 1 2 where 0 k 1<br />
= < < coefficient of coupling
I 1<br />
Ideal Transformer<br />
1:n<br />
+ +<br />
V 1 V 2<br />
-<br />
© Robert York, 2006<br />
-<br />
I 2<br />
If both coils share the same<br />
flux, then Farady’s law gives:<br />
As the permeability of the core<br />
increases, the relationship between<br />
the primary and secondary<br />
currents approaches a limiting<br />
value set by the turns ratio:<br />
V1 N1<br />
1<br />
= =<br />
V N n<br />
2 2<br />
I1 N2<br />
⇒ = n<br />
I N<br />
2 1<br />
These two relationships define an ideal transformer. This is a fictitous element (note that µ→∞<br />
implies infinite inductances so the impedance matrix is infinite) but a real transformer<br />
approximates this behavior.<br />
An idea transformer has the following useful property when one winding is terminated:<br />
I 1<br />
1:n<br />
+ +<br />
V 1<br />
-<br />
V 2<br />
-<br />
I 2<br />
Z L<br />
Z<br />
in<br />
V ( N / N ) V Z<br />
I ( N / N ) I n<br />
1 1 2 2<br />
= = =<br />
1 2 1 2<br />
L<br />
2
Transformer Equivalent Circuit<br />
Using the tee-equivalent for reciprocal networks, we find the following<br />
equivalent circuit for mutual inductances or transformers<br />
L1Lm © Robert York, 2006<br />
− L2−Lm Lm<br />
L ∝ N<br />
1 1<br />
L ∝ N = n L<br />
2<br />
2 2 1<br />
L = k LL = knL<br />
m<br />
1 2 1<br />
If desired, this circuit can be cascaded with an ideal 1:1 transformer to<br />
simulate the fact that a real transformer has electrically isolated ports
Alternative equivalent circuit<br />
The following is also an identical equivalent that uses an ideal transformer to<br />
explicitly incorporate the turns ratio and isolation between ports<br />
© Robert York, 2006<br />
L k<br />
2<br />
1 (1 ) −<br />
2<br />
kL1<br />
1:n<br />
ideal<br />
For good transformers, k is nearly 1, and this model also clearly shows why<br />
real transformers do not work at DC<br />
n<br />
=<br />
L<br />
L<br />
2<br />
m