The Electrical-Mechanical Analogy — Question 10

Question 10

A nonlinear capacitor corresponds to a hardening spring

A supplied lossless capacitor model has voltage $v_C(q)=q/C+\beta q^3$ with positive beta. In series with L it is analogous to a cubic restoring spring.

1. Find the capacitor energy by integrating voltage against charge.

2. For a small oscillation about a maintained bias charge qb, find the local natural frequency; assume the bias source supplies constant voltage.

Question 10 - Solution

A nonlinear capacitor corresponds to a hardening spring

(a) Method and calculation

$$U_C(q)=\int_0^qv_C(u)du=q^2/(2C)+\beta q^4/4.$$$$\boxed{L\ddot q+q/C+\beta q^3=0\quad\text{without bias}}.$$

(b) Interpretation and further calculation

$$q=q_b+\delta q,\quad v_C(q)-v_C(q_b)\simeq(1/C+3\beta q_b^2)\delta q.$$$$\boxed{\omega_{\mathrm{small}}^2=(1/C+3\beta q_b^2)/L}.$$

Physical check

The cubic term changes local stiffness; the constant-C amplitude independence no longer applies.