Torque on a Current Loop — Question 9

Question 9

Current commutation produces a nonzero mean torque over a full revolution

A motor coil has peak torque tau0=NIAB. With fixed current direction its signed torque about the shaft is $\tau_0\sin\theta$. An ideal commutator reverses the current whenever needed so that the driving torque is $\tau_0|\sin\theta|$.

1. Find the angle-averaged torque with and without commutation.

2. Find the mechanical work per revolution with commutation and explain where that energy is supplied.

Question 9 - Solution

Current commutation produces a nonzero mean torque over a full revolution

(a) Method and calculation

$$\langle\tau\rangle_{\mathrm{fixed}}=\frac1{2\pi}\int_0^{2\pi}\tau_0\sin\theta\,d\theta=0.$$$$\langle\tau\rangle_{\mathrm{comm}}=\frac{\tau_0}{2\pi}\int_0^{2\pi}|\sin\theta|\,d\theta=\frac{2\tau_0}{\pi}.$$

(b) Interpretation and further calculation

$$W_{\mathrm{turn}}=\int_0^{2\pi}\tau_0|\sin\theta|\,d\theta=4\tau_0.$$$$W_{\mathrm{electrical}}\ge W_{\mathrm{turn}}\quad\text{with equality only when losses are absent}.$$The maintained-current source supplies this energy through the motor circuit.

Physical check

The average is over angle; it equals a time average only for uniform rotational speed.