---
title: "Dynamic model of armature-controlled DC motor"
subject: "Control Systems"
chapter: 2
key: "semester_3/control_systems:dc-motor-de-dynamic-model"
source: https://www.bctengineer.com/q/semester_3/control_systems/dc-motor-de-dynamic-model
site: BCT Engineering
---

# Dynamic model of armature-controlled DC motor

Control Systems · Chapter 2: Component Modeling

- Asked in: 2070 Chaitra (old course), 8 marks
- Video: [How to Model a DC Motor as a System of ODEs](https://www.youtube.com/watch?v=HCnMdfQgDmI) — DMD Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/dc-motor-de-dynamic-model
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-2

## Question

For armature controlled separately excited DC motor, identify the necessary differential equations governing its behaviour and hence derive the dynamic model of such motor.

## Short answer

The differential equations are: v_a = R_a i_a + L_a di_a/dt + K_b ω, T_m = K_t i_a, J dω/dt + B ω + T_L = T_m, dθ/dt = ω. The state-space model has states i_a, ω, θ and inputs v_a, T_L. The transfer function Ω(s)/V_a(s) = K_t / [(R_a + sL_a)(Js + B) + K_t K_b].

## Answer

In a separately excited DC motor under **armature control**, the field current $i_f$ is held constant, so the air-gap flux $\phi$ is constant, and the armature voltage $v_a$ is the control input.

#### Symbols
$R_a$, $L_a$: armature resistance, inductance; $i_a$: armature current; $e_b$: back emf; $T_m$: developed torque; $T_L$: load torque; $J$: total inertia; $B$: viscous friction; $\omega = \dot\theta$: speed.

#### Governing differential equations
1. **Armature circuit (KVL):**
$$
v_a = R_ai_a + L_a\frac{di_a}{dt} + e_b
$$
2. **Back emf** (proportional to flux and speed; flux constant):
$$
e_b = K\phi\,\omega = K_b\,\omega
$$
3. **Developed torque**:
$$
T_m = K\phi\,i_a = K_t\,i_a
$$
4. **Mechanical (Newton's law for rotation)**:
$$
T_m = J\frac{d\omega}{dt} + B\omega + T_L
$$
5. **Position**: $\dfrac{d\theta}{dt} = \omega$

In SI units $K_t = K_b = K$.

#### Dynamic model in state-space form
State variables $x_1 = i_a$, $x_2 = \omega$, $x_3 = \theta$; inputs $v_a$ and $T_L$:
$$
\begin{bmatrix}\dot i_a\\ \dot\omega\\ \dot\theta\end{bmatrix} =
\begin{bmatrix} -\frac{R_a}{L_a} & -\frac{K_b}{L_a} & 0 \\ \frac{K_t}{J} & -\frac{B}{J} & 0 \\ 0 & 1 & 0 \end{bmatrix}
\begin{bmatrix} i_a\\ \omega\\ \theta\end{bmatrix} +
\begin{bmatrix} \frac{1}{L_a} & 0 \\ 0 & -\frac{1}{J} \\ 0 & 0\end{bmatrix}
\begin{bmatrix} v_a \\ T_L\end{bmatrix}
$$
Output: $y = \omega$ (speed) or $y = \theta$ (position).

#### Transfer-function model (with $T_L = 0$)
Taking Laplace transforms:
$$
\begin{aligned}
I_a(s) &= \frac{V_a(s) - K_b\Omega(s)}{R_a + sL_a}, \qquad \Omega(s) = \frac{K_tI_a(s)}{Js + B}
\end{aligned}
$$
$$
\frac{\Omega(s)}{V_a(s)} = \frac{K_t}{(R_a + sL_a)(Js + B) + K_tK_b}, \qquad
\frac{\theta(s)}{V_a(s)} = \frac{K_t}{s\left[(R_a + sL_a)(Js + B) + K_tK_b\right]}
$$

#### Block diagram of the dynamic model
```text
 Va +    ┌───────┐ Ia ┌────┐ Tm +   ┌──────┐ ω  ┌───┐ θ
 ──►(Σ)─►│  1    ├───►│ Kt ├───►(Σ)►│  1   ├─┬─►│1/s├─►
    ▲ -  │Ra+sLa │    └────┘    ▲-  │Js+B  │ │  └───┘
    │    └───────┘              │   └──────┘ │
    │ Eb         ┌────┐        TL            │
    └────────────┤ Kb ├◄─────────────────────┘
                 └────┘
```

#### Remarks
- Electrical time constant $\tau_a = L_a/R_a$; mechanical time constant $\tau_m = J/B$. Usually $\tau_a \ll \tau_m$, so $L_a$ may be neglected, giving $\dfrac{\Omega}{V_a} = \dfrac{K_m}{1 + s\tau}$ with $K_m = \dfrac{K_t}{R_aB + K_tK_b}$, $\tau = \dfrac{R_aJ}{R_aB + K_tK_b}$.
- The back emf provides inherent speed feedback, which makes the motor more stable than a field-controlled motor.

## Check yourself

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

1. In armature control of a separately excited DC motor, what is kept constant?
   A. Armature voltage
   B. Field current, hence flux
   C. Armature current
   D. Load torque
   Answer: B. The field current is held constant so flux is constant, and the armature voltage v_a is the control input. (Key point: Armature control assumption)

2. Which pair of relations is correct for constant flux?
   A. e_b = K_b * i_a and T_m = K_t * omega
   B. e_b = K_b * omega and T_m = K_t * i_a
   C. e_b = K_b * theta and T_m = K_t * v_a
   D. e_b = R_a * i_a and T_m = J * omega
   Answer: B. Back emf is proportional to speed and developed torque is proportional to armature current. (Key point: Back emf and torque relations)

3. Which equation is the mechanical (Newton) equation of the motor?
   A. T_m = J dω/dt + B ω + T_L
   B. T_m = R_a i_a + L_a di_a/dt
   C. T_m = K_b ω
   D. v_a = J dω/dt + B ω
   Answer: A. Developed torque supplies inertia torque, viscous friction torque and the load torque. (Key point: Mechanical equation)

4. With T_L = 0, what is Ω(s)/V_a(s)?
   A. K_t / [(R_a + sL_a)(Js + B)]
   B. K_t / [(R_a + sL_a)(Js + B) + K_t K_b]
   C. K_b / [(R_a + sL_a)(Js + B) + K_t K_b]
   D. 1 / [s (R_a + sL_a)(Js + B)]
   Answer: B. The back emf feedback adds the K_t K_b term to the denominator of the forward path. (Key point: Transfer function)

## Related in Control Systems

- [Armature-controlled DC motor with gear and load](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Adc-motor-gear-load) — Control Systems, chapter 2
- [Armature-controlled DC motor transfer function and block diagram](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-armature-dc-motor) — Control Systems, chapter 2
- [State equations of DC motor circuit](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-dc-motor-armature-field) — Control Systems, chapter 8
- [Feedback effect in DC motor speed control](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afeedback-dc-motor-speed) — Control Systems, chapter 1
- [DC motor θm/Va via block diagram](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-dc-motor-bd-thetam-va) — Control Systems, chapter 2
- [DC motor with tachometer feedback and P-controller](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abd-dc-motor-tacho-p) — Control Systems, chapter 2

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