ELEC 242 Lab

Background

DC Motor: Torque vs. Speed

Consider a DC motor with armature resistance $R_a$ , torque constant $K_t$ , and back emf constant $K_e$ being driven by a voltage source $v_s$ .

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$J$ 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 $T=K_ti$ . Combining these along with Ohm's law, we get:
$\displaystyle T=K_ti=K_t\frac{v_s-E}{R_a}=K_t\frac{v_s-K_e\omega}{R_a}=\frac{K_t}{R_a}v_s-\frac{K_tK_e}{R_a}\omega=K_1v_s-K_2\omega$
where $\displaystyle K_1=\frac{K_t}{R_a}$ and $\displaystyle K_2=\frac{K_tK_E}{R_a}$

System Transfer Function

In terms of the angle of rotation of the drum $\theta$ , the equation describing the system is
$\displaystyle J'\ddot{\theta}+K_2\dot{\theta}+K_1K_pA\theta=K_1K_PA\theta_{des}+T_w$
where
$\bf A$ is the summing amplifier gain $\displaystyle \frac{v_{dif}}{v_{act}}=\frac{R_F}{47\rm k}$
$\bf J'$ is the equivalent moment of inertia seen at the shaft, including the effects of the load mass.
$\bf K_p$ is the gain of the potentiometer ( $\displaystyle v_\theta = K_p\theta$ ).
$\bf T_w$ is the equivalent torque caused by the weight of the load.
$\bf\theta_{des}$ is the desired angle of the shaft.
Making the substitutions $\frac{d}{dt}\rightarrow s$ and $x(t)\rightarrow X(s)$ we get
$\displaystyle J's^2\Theta(s)+K_2s\Theta(s)+K_1K_pA\Theta(s)=K_1K_PA\Theta_{des}(s)+T_w(s)$
Using superposition we define
$\displaystyle T_1(s)=\frac{\Theta(s)}{\Theta_{des}(s)}$ with $T_w=0$
and
$\displaystyle T_2(s)=\frac{\Theta(s)}{T_w(s)}$ with $\Theta_{des}=0$ .
Then
$\displaystyle \Theta(s)=T_1(s)\Theta_{des}(s)+T_2(s)T_w(s)$ .
Evaluating $T_1$ and $T_2$ we have
$\displaystyle T_1=\frac{K_1K_pA}{J's^2+K2s+K_1K_pA}=\frac{\frac{K_1K_pA}{J'}}{s^2+\frac{K_2}{J'}s+\frac{K_1K_pA}{J'}}$
and
$\displaystyle T_2=\frac{1}{J's^2+K2s+K_1K_pA}=\frac{\frac{1}{J'}}{s^2+\frac{K_2}{J'}s+\frac{K_1K_pA}{J'}}$

Some observations

  1. In the steady state, $T_1=1$ , i.e. there is no steady state gain error regardless of A. If $T_w\neq 0$ there will be a constant offset, but $\Delta\theta=\Delta\theta_{des}$ .
  2. $T_2$ falls off as $\frac{1}{A}$ , i.e. disturbance rejection increases with increasing gain.
  3. System dynamics depend on $A$ . The characteristic equation is $\displaystyle s^2+\frac{K_2}{J'}s+\frac{K_1K_pA}{J'}=0$ . This is our familiar second order system with $\displaystyle \alpha=\frac{K_2}{2J'}$ and $\displaystyle \omega_0=\sqrt{\frac{K_1K_pA}{J'}}$ . So as $A$ increases, the system eventually becomes underdamped and will oscillate.