Energy Calculations for Mutually – Coupled Coils ECE 201 Circuit Theory 1 1 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 2 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 3 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 4 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 5 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 6