Other Single-Loop Circuits — Question 7

Question 7

Motor back emf links electrical current to mechanical speed

A motor has resistance r and back emf $E_b=k\omega$, opposing an applied ideal source E. A series resistor R is also present. Assume a quasisteady loop with fixed parameters.

1. Find current as a function of angular speed and its stall value.

2. Find the speed for zero current and interpret speeds above it using the sign of source power.

Question 7 - Solution

Motor back emf links electrical current to mechanical speed

(a) Method and calculation

$$E-k\omega-I(R+r)=0\Rightarrow I(\omega)=\frac{E-k\omega}{R+r}.$$$$I_{\mathrm{stall}}=E/(R+r)\quad(\omega=0).$$

(b) Interpretation and further calculation

$$I=0\Rightarrow\omega_0=E/k.$$$$\omega>\omega_0:\ I<0,\quad P_{\mathrm{source,delivered}}=EI<0.$$An externally driven motor then returns electrical energy to an absorbing source.

Physical check

Regeneration requires mechanical input and a source capable of accepting power; the resistance still dissipates I squared times R.