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Circuits II
EE221
Unit 9
Instructor: Kevin D. Donohue
Mutual Inductance, Energy in Magnetically
Coupled Circuits, Analysis of Mutual
Inductance Circuits
Mutual Inductance
Coils with current flowing in them emit magnetic energy that
induce voltages in other coils in close proximity. The effects
of the magnetic link between these circuits is called mutual
inductance.
Show that current i1 in first coil induces voltage v2 in second
coil of the form:
di1
v2  M 21
dt
d12
where M 21  N 2
di1
11
12
Mutual Inductance Dot Convention
The geometry of the coil and flux paths are expressed in the
dot convention below. This notation is important for apply
circuit laws for analyses.
12
21
11
22
Dot Notation for KVL
If both currents are either entering or leaving the dot then the self
and mutual induced voltage will have the same sign.
M
I1
+
V1
L1
L2
-
V1  I1 L1s  I 2 Ms
I2
+
V2  I 2 L2 s  I1Ms
V2
-
If one current (loop direction) enters the dot and the other current
leaves the dot, then the self and mutual induced voltage will have
opposing signs.
M
I1
+
V1
-
+
L1
L2
V2
-
I2
V1  I1 L1s  I 2 Ms
V2  I 2 L2 s  I1Ms
Equivalent Circuits Example
Convert the Mutual Inductance components to an equivalent circuit
using current controlled voltage sources.
M
I1
+
V1
+
L1
L2
-
I2
V1  I1 L1s  I 2 Ms
V2  I 2 L2 s  I1Ms
V2
-
M
I1
+
V1
-
+
L1
L2
V2
-
I2
V1  I1 L1s  I 2 Ms
V2  I 2 L2 s  I1Ms
Examples
Find vo(t), given vi(t) = 2cos(4t) V.
1H
4
vi(t)
2H
1H
Show vo(t) = 0.4851cos(4t-14º) V
2
+
vi(t)
-
Examples
Equivalent Series Inductance
Circuits with Mutual Inductors