Ohm's Law — Question 2

Question 2

A cubic voltage law has distinct static and incremental resistance

A symmetric device at fixed temperature obeys $V(I)=R_0I+\gamma I^3$, with $R_0,\gamma>0$.

1. Find static resistance V/I and differential resistance dV/dI at $I=I_b\ne0$.

2. Find the first-order current change from a small voltage perturbation and quantify the error from using static resistance.

Question 2 - Solution

A cubic voltage law has distinct static and incremental resistance

(a) Method and calculation

$$R_s=V/I=R_0+\gamma I_b^2.$$$$r_d=dV/dI=R_0+3\gamma I_b^2.$$

(b) Interpretation and further calculation

$$\delta I\simeq\delta V/r_d=\frac{\delta V}{R_0+3\gamma I_b^2}.$$$$\frac{\delta I_{R_s}}{\delta I_{r_d}}=\frac{R_0+3\gamma I_b^2}{R_0+\gamma I_b^2}>1.$$

Physical check

Incremental response is governed by a tangent; a full operating ratio is a secant.