Armature-controlled DC motor transfer function and block diagram

In an armature-controlled DC motor the field current $i_f$ is kept constant (constant $v_f$) and the speed/position is controlled by changing the armature voltage $v_a$.

Assumptions and symbols

• $R_a, L_a$: armature resistance and inductance; $i_a$: armature current
• $e_b$: back emf; $K_b$: back-emf constant
• $T_m$: motor torque; $K_t$: torque constant
• $J, B$: inertia and viscous friction of motor plus load; $\theta$: shaft angle
• Flux $\phi \propto i_f$ is constant, so torque and back emf are linear.

Governing equations

1. Armature circuit (KVL):

$$v_a = R_a i_a + L_a \frac{di_a}{dt} + e_b$$

2. Back emf: $e_b = K_b \dfrac{d\theta}{dt}$
3. Torque: $T_m = K_t\, i_a$
4. Load (Newton's law): $T_m = J\dfrac{d^2\theta}{dt^2} + B\dfrac{d\theta}{dt}$

Laplace transform (zero initial conditions)

$$\begin{aligned} I_a(s) &= \frac{V_a(s) - E_b(s)}{R_a + sL_a} \\ T_m(s) &= K_t I_a(s) \\ \theta(s) &= \frac{T_m(s)}{s(Js + B)} \\ E_b(s) &= K_b\, s\,\theta(s) \end{aligned}$$

Block diagram

 Va +    ┌──────┐ Ia ┌──┐ Tm ┌─────┐ ω  ┌───┐ θ
 ──►(Σ)─►│  1   ├───►│Kt├───►│  1  ├─┬─►│1/s├─►
    ▲ -  │Ra+sLa│    └──┘    │Js+B │ │  └───┘
    │    └──────┘            └─────┘ │
    │ Eb        ┌────┐               │
    └───────────┤ Kb │◄──────────────┘
                └────┘

The back emf forms an internal negative feedback loop on speed $\omega = s\theta$.

Transfer function

Reducing the inner loop (forward path $G = \dfrac{K_t}{(R_a+sL_a)(Js+B)}$, feedback $K_b$):

$$\frac{\omega(s)}{V_a(s)} = \frac{K_t}{(R_a + sL_a)(Js + B) + K_t K_b}$$

Multiplying by the integrator $1/s$:

$$\frac{\theta(s)}{V_a(s)} = \frac{K_t}{s\left[(R_a + sL_a)(Js + B) + K_t K_b\right]}$$

If $L_a$ is negligible, this becomes $\dfrac{\theta(s)}{V_a(s)} = \dfrac{K_m}{s(1 + s\tau_m)}$ with $K_m = \dfrac{K_t}{R_aB + K_tK_b}$ and $\tau_m = \dfrac{R_aJ}{R_aB + K_tK_b}$, a type-1 second-order system.