Mutual Induction — Question 10

Question 10

Sequential current ramps expose the mutual contribution to source work

Two fixed lossless coupled coils have positive M. Starting from zero currents, first raise i1 to I1 while holding i2=0. Then hold i1=I1 and raise i2 to I2. Both final currents enter dotted terminals.

1. Compute the work supplied by each current source over the complete sequence.

2. Compare total work with magnetic energy and explain how reversing the ramp order changes individual source work.

Question 10 - Solution

Sequential current ramps expose the mutual contribution to source work

(a) Method and calculation

$$W_1^{(a)}=\int_0^{I_1}L_1i_1di_1=\tfrac12L_1I_1^2,\quad W_2^{(a)}=0.$$$$W_1^{(b)}=\int_0^{I_2}MI_1di_2=MI_1I_2,\quad W_2^{(b)}=\tfrac12L_2I_2^2.$$

(b) Interpretation and further calculation

$$\boxed{W_1+W_2=\tfrac12L_1I_1^2+\tfrac12L_2I_2^2+MI_1I_2=U_f}.$$$$\boxed{W_1^{\mathrm{reversed}}=\tfrac12L_1I_1^2,\quad W_2^{\mathrm{reversed}}=\tfrac12L_2I_2^2+MI_1I_2}.$$

Physical check

The source already holding a nonzero current supplies the mutual term during the other ramp. Total work is independent of ramp order.