Energy Calculations for
Mutually – Coupled Coils
ECE 201 Circuit Theory 1
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Energy Calculations continued
• To begin, assume that currents i1 and i2
are equal to zero
• Let current i1 increase to a value I1
• Compute the energy stored
W1
t1
I1
1 2
0 dw  0 v1i1dt  0 L1i1di1  2 L1I1
ECE 201 Circuit Theory 1
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Energy Calculations Continued
• Hold i1 constant and increase i2 to I2
• Voltage induced in L2 by I1 = 0
• Voltage induced in L1 by i2 = M12(di2/dt)
ECE 201 Circuit Theory 1
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Power Input to the pair of coils
di2
p  I1M 12
 i2 v2
dt
i2  I 2
W2
I2
 dw   I M
1
W1
0
I2
12
di2   L2i2 di2
0
1
W  W1  I1 I 2 M 12  L2 I 22
2
1
1
2
W  L1 I1  L2 I 22  I1 I 2 M 12
2
2
ECE 201 Circuit Theory 1
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If the procedure were reversed
1
1
2
2
W  L1 I1  L2 I 2  I1 I 2 M 21
2
2
1
1
2
2
W  L1 I1  L2 I 2  I1 I 2 M 12
2
2
When the medium is linear, M21 = M12, and
the total energy stored is the same!
ECE 201 Circuit Theory 1
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Instantaneously
1 2 1 2
w(t )  L1i1  L2i2  Mi1i2
2
2
This case had both currents entering a dotted
terminal. If one current enters a dotted terminal
and the other current leaves a dotted terminal,
the sign of the Mi1i2 term reverses.
1 2 1 2
w(t )  L1i1  L2i2  Mi1i2
2
2
ECE 201 Circuit Theory 1
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