Step-down chopper with DC motor load

A step-down (Type-A) chopper controls the speed of a separately excited dc motor by varying the average armature voltage $V_0 = kV_s$. The motor armature is an RLE load: resistance $R_a$, inductance $L_a$ and back EMF $E_b$.

Circuit

  +----CH----+-------------+
  |          |             |
  |          |            Ra
 Vs        FD ^            La
  |          |            (Eb)  motor
  |          |             |
  +----------+-------------+

Modes of operation

Mode 1: CH ON ($0 < t < T_{on}$). Source connects to the armature, $v_0 = V_s$:

$$V_s = R_a i_a + L_a\frac{di_a}{dt} + E_b$$

The current rises exponentially from $I_{min}$ to $I_{max}$; energy is stored in $L_a$.

Mode 2: CH OFF ($T_{on} < t < T$). The armature current freewheels through FD, $v_0 = 0$:

$$0 = R_a i_a + L_a\frac{di_a}{dt} + E_b$$

The current decays from $I_{max}$ to $I_{min}$. If $L_a$ is small or $k$ is low, it can reach zero before the next ON period (discontinuous conduction); then $v_0 = E_b$ for the remaining time.

 v0  Vs |____      ____
        |    |    |    |
      0 |____|____|____|____ t
 ia Imax|   /\        /\
        |  /  \      /  \
    Imin| /    \____/    \__ t
         Ton Toff

Steady-state relations (continuous conduction)

$$\begin{aligned} V_0 &= kV_s = E_b + I_a R_a \\ E_b &= K\phi\,\omega \\ \omega &= \frac{kV_s - I_a R_a}{K\phi} \end{aligned}$$

So speed is controlled by the duty ratio $k$: increasing $k$ increases $V_0$ and the speed. Torque $T = K\phi I_a$ depends on the load current.

Current limits:

$$I_{max} = \frac{V_s}{R_a}\cdot\frac{1-e^{-kT/T_a}}{1-e^{-T/T_a}} - \frac{E_b}{R_a}, \quad I_{min} = \frac{V_s}{R_a}\cdot\frac{e^{kT/T_a}-1}{e^{T/T_a}-1} - \frac{E_b}{R_a}$$

with $T_a = L_a/R_a$. Conduction is continuous if $I_{min} > 0$.

Features

• Operates in the first quadrant only (motoring, forward); braking needs a Class B or C chopper.
• High chopping frequency or an added series inductor reduces current ripple and torque pulsation.
• Smooth, efficient (no rheostat loss) control; widely used in battery vehicles, forklifts and traction.