Mutual Induction — Question 9

Question 9

Coupled RL circuits decay through two different magnetic modes

Two identical fixed coils each have inductance L and series resistance R, with $0<M<L$. There are no sources. Positive currents enter dotted terminals. The equations are $L\dot i_1+M\dot i_2+Ri_1=0$ and $M\dot i_1+L\dot i_2+Ri_2=0$.

1. Find decay times for equal and opposite current modes.

2. For initial currents i1=I0 and i2=0, derive both currents at later times.

Question 9 - Solution

Coupled RL circuits decay through two different magnetic modes

(a) Method and calculation

$$i_+=i_1+i_2,\quad i_-=i_1-i_2:\quad(L\pm M)\dot i_\pm+Ri_\pm=0.$$$$\boxed{\tau_+=(L+M)/R,\quad\tau_-=(L-M)/R}.$$

(b) Interpretation and further calculation

$$i_+(0)=i_-(0)=I_0\Rightarrow i_\pm=I_0e^{-t/\tau_\pm}.$$$$\boxed{i_1=\tfrac{I_0}{2}(e^{-t/\tau_+}+e^{-t/\tau_-}),\quad i_2=\tfrac{I_0}{2}(e^{-t/\tau_+}-e^{-t/\tau_-})}.$$

Physical check

The secondary initially rises in the positive direction as it opposes the falling primary flux; both currents ultimately dissipate away.