Feedback effect in DC motor speed control

Feedback (a tachogenerator measuring the speed) makes an armature-controlled DC motor respond faster and keeps its speed nearly constant despite load and parameter changes.

Open-loop model

Motor equations ($L_a$ neglected): $V_a = R_aI_a + K_b\omega$, $T = K_tI_a = J\dot\omega + B\omega + T_L$. Hence

$$\omega(s) = \frac{K_m}{\tau_m s+1}V_a(s) - \frac{K_L}{\tau_m s+1}T_L(s)$$ $$K_m = \frac{K_t}{R_aB + K_tK_b}, \quad K_L = \frac{R_a}{R_aB + K_tK_b}, \quad \tau_m = \frac{R_aJ}{R_aB + K_tK_b}$$

Steady-state speed drop on load: $\Delta\omega_{OL} = K_LT_L$. Any change in $R_a$, $K_t$ or $B$ changes the speed directly.

Closed-loop scheme

 ωr  +    e  +-----+ Va +-------+   ω
 --->(Σ)---->| KA  |--->| Motor |---+--->
      ^ -    +-----+    +-------+   |
      |      +-------+              |
      +------| Kg    |<-------------+
             | tacho |
             +-------+

With amplifier gain $K_A$ and tacho constant $K_g$, the loop gain is $K = K_AK_mK_g$:

$$\frac{\omega(s)}{\omega_r(s)} = \frac{K_AK_m}{\tau_m s + 1 + K}, \qquad \frac{\omega(s)}{T_L(s)} = \frac{-K_L}{\tau_m s + 1 + K}$$

Comparison

1. Faster response: the time constant is reduced by $(1+K)$.
2. Better speed regulation: the load-torque disturbance causes a speed drop $(1+K)$ times smaller.
3. Less sensitivity to changes in $R_a$ (heating), flux or friction.
4. Cost: lower overall gain (made up by $K_A$), a tachogenerator is needed, and with armature inductance included, very high $K$ can cause oscillation.

Example: with $K = 19$, a 100 rpm open-loop drop on load becomes only 5 rpm.