The Electrical-Mechanical Analogy — Question 9

Question 9

Mutual inductance acts like a coupling of inertial coordinates

Two identical lossless LC loops have positive mutual inductance $0<M<L$ and equations $L\ddot q_1+M\ddot q_2+q_1/C=0$ and $M\ddot q_1+L\ddot q_2+q_2/C=0$.

1. Find frequencies for equal-charge and opposite-charge modes.

2. Which mode is slower, and what happens as M approaches L from below?

Question 9 - Solution

Mutual inductance acts like a coupling of inertial coordinates

(a) Method and calculation

$$q_2=\pm q_1\Rightarrow(L\pm M)\ddot q_1+q_1/C=0.$$$$\boxed{\omega_+=1/\sqrt{C(L+M)},\quad\omega_-=1/\sqrt{C(L-M)}}.$$

(b) Interpretation and further calculation

$$M>0\Rightarrow L+M>L-M\Rightarrow\omega_+<\omega_-.$$$$\boxed{M\to L^-:\ \omega_+\to1/\sqrt{2LC},\quad\omega_-\to\infty}.$$

Physical check

The divergent ideal limit signals a vanishing modal inductance; real parasitic elements then matter.