The Electrical-Mechanical Analogy — Question 8

Question 8

A negative restoring coefficient gives growth instead of oscillation

An active circuit model obeys $L\ddot q-q/C_*=0$, where L and Cstar are positive. Compare it with an inverted mechanical spring. Initially q=Q and i=0.

1. Find q(t) and its growth time scale.

2. Show that the formal conserved quadratic quantity is not positive definite.

Question 8 - Solution

A negative restoring coefficient gives growth instead of oscillation

(a) Method and calculation

$$\tau=\sqrt{LC_*},\quad\ddot q=q/\tau^2.$$$$\boxed{q(t)=Q\cosh(t/\tau),\quad i=(Q/\tau)\sinh(t/\tau)}.$$

(b) Interpretation and further calculation

$$\frac d{dt}\left(\tfrac12Li^2-\frac{q^2}{2C_*}\right)=i(L\dot i-q/C_*)=0.$$$$\boxed{H=\tfrac12Li^2-q^2/(2C_*)\ \text{has no lower bound}}.$$

Physical check

An active negative restoring element is not an ordinary passive capacitor; the sign change destroys stable oscillation.