---
title: "PID control of speed and torque"
subject: "Utilization of Electrical Energy"
chapter: 3
key: "semester_7/utilization_of_electrical_energy:pid-control-speed-torque-motor"
source: https://www.bctengineer.com/q/semester_7/utilization_of_electrical_energy/pid-control-speed-torque-motor
site: BCT Engineering
---

# PID control of speed and torque

Utilization of Electrical Energy · Chapter 3: Control of Electric Drive

- Asked in: 2071 Shrawan; 2070 Asar (asked 2 times)
- Video: [DC motor PID speed control](https://www.youtube.com/watch?v=HRaZLCBFVDE) — Curio Res
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/utilization_of_electrical_energy/pid-control-speed-torque-motor
- Whole chapter: https://www.bctengineer.com/md/semester_7/utilization_of_electrical_energy/chapter-3

## Question

With the help of mathematical expression and block diagrams discuss the PID control of speed and torque of electric motor?

## Short answer

PID output is u = Kp e + Ki integral(e) + Kd de/dt. Speed control uses cascade loops: an outer speed PID gives the armature current reference, and an inner current PI controls the converter, limiting current and giving zero steady-state speed error.

## Answer

A **PID controller** produces a control signal from the error $e(t)$ between reference and measured value using proportional, integral and derivative actions:

$$
u(t) = K_pe(t) + K_i\int_0^te(\tau)\,d\tau + K_d\frac{de(t)}{dt}
$$

$$
G_c(s) = \frac{U(s)}{E(s)} = K_p + \frac{K_i}{s} + K_ds
$$

| Action | Effect |
|---|---|
| Proportional $K_p$ | Fast response; reduces but does not remove steady-state error; too high causes oscillation |
| Integral $K_i$ | Removes steady-state error (exact speed under load); may slow response and cause overshoot |
| Derivative $K_d$ | Anticipates error, adds damping, reduces overshoot; amplifies noise |

#### PID speed and torque control of a DC motor (cascade control)

Motor equations (separately excited DC):

$$
V_a = R_aI_a + L_a\frac{dI_a}{dt} + K\omega, \qquad T = KI_a = J\frac{d\omega}{dt} + B\omega + T_L
$$

Torque is proportional to armature current, so **torque control = current control**.

```text
 w*->(+)->[Speed PID]-Ia*->(+)->[Current PI]
      ^-   (limit)          ^-       |
      |                     |   firing/duty
      |                     |        v
      |                     |   [Converter]
      |                     |        | Va
      |                     |   [DC motor]--> w
      |                     |     |  |
      |  current sensor <---+-----+  |
      +---- tacho/encoder <----------+
```

1. **Outer speed loop**: speed error $\omega^* - \omega$ goes to the speed PID (usually PI). Its output is the **current (torque) reference** $I_a^*$, limited to the maximum allowed current, so the motor accelerates at maximum safe torque.
2. **Inner current (torque) loop**: current error $I_a^* - I_a$ goes to the current PI controller, which sets the converter firing angle (rectifier) or duty cycle (chopper). This loop is fast; it quickly corrects supply-voltage changes and limits current during starting, overload and stall.
3. **Converter** supplies $V_a$ to the armature.
4. **Feedback**: tachogenerator/encoder for speed, CT/Hall sensor/shunt for current.

For an induction motor the same structure is used in vector (field-oriented) control: an outer speed PI produces a torque-current command $i_{qs}^*$, and inner current PI loops control the inverter.

#### Behaviour

- On a sudden load increase, speed drops; the speed PID raises $I_a^*$, torque increases and speed returns to the reference with zero steady-state error (integral action).
- During starting, the speed error is large, $I_a^*$ saturates at the limit, so the motor accelerates at constant maximum torque.
- Derivative action is used sparingly (filtered) because of noise in speed measurement.

**Advantages**: accurate speed, fast torque response, inherent current protection, simple tuning of each loop separately.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which action removes steady-state error?
   A. Proportional
   B. Integral
   C. Derivative
   D. None
   Answer: B. Integral action gives zero steady-state speed error. (Key point: Integral action)

2. Why is torque control equal to current control in DC motor?
   A. T = K Ia
   B. T proportional to speed
   C. T = Va
   D. T = Ra
   Answer: A. Torque is proportional to armature current. (Key point: Torque control)

3. In cascade control, the outer speed loop output is:
   A. Firing angle
   B. Current (torque) reference limited to max current
   C. Voltage
   D. Speed feedback
   Answer: B. The inner fast current loop follows this reference. (Key point: Cascade loops)

4. Why use derivative action sparingly?
   A. It amplifies measurement noise
   B. It removes damping
   C. It increases error
   D. It slows response
   Answer: A. Derivative amplifies noise although it adds damping. (Key point: Derivative action)

## Related in Utilization of Electrical Energy

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- [DC chopper: input power, speed, torque, speed range](https://www.bctengineer.com/question.md?key=semester_7%2Futilization_of_electrical_energy%3Achopper-400v-input-power-speed-torque) — Utilization of Electrical Energy, chapter 3, asked 2×
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- [Induction motor plugging: braking current and torque](https://www.bctengineer.com/question.md?key=semester_7%2Futilization_of_electrical_energy%3Ainduction-motor-plugging-braking-current) — Utilization of Electrical Energy, chapter 3
- [DC shunt motor speed at reduced voltage and starting resistance](https://www.bctengineer.com/question.md?key=semester_7%2Futilization_of_electrical_energy%3Adc-shunt-reduced-voltage-starting-resistance) — Utilization of Electrical Energy, chapter 3

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