ELEC 242 Lab

Interlude

DC Motor Dynamics

Consider a DC motor with armature resistance $R_a$ , torque constant $K_t$ , and back emf constant $K_e$ being driven by a source with output resistance $R_s$ :

\includegraphics[scale=0.650000]{ckt6.7.ps}
$J$ represents the combined moment of inertia of the motor armature and the load. Then we have $\displaystyle T=J\dot{\omega}$ . From the coupling equations we have $\displaystyle E=K_e\omega$ and $T=K_ti$ . Defining $R=R_s + R_a$ , Ohm's law gives $\displaystyle i=\frac{v_s - E}{R}=\frac{v_s}{R}-\frac{K_e\omega}{R}$ . But we also have $\displaystyle i=\frac{T}{K_t}=\frac{J\dot{\omega}}{K_t}$ . Combining these we get
$\displaystyle \frac{J\dot{\omega}}{K_t}+\frac{K_e\omega}{R}=\frac{v_s}{R}$
or
$\displaystyle J\dot{\omega} + \frac{K_tK_e}{R}\omega = \frac{K_t}{R}v_s$