# 1.7: DC Motor Model

- Page ID
- 17747

A DC motor (Figure 1.11) represents an an electro-mechanical system that draws electrical energy and converts it into mechanical energy. In an armature-controlled DC motor, the input is the armature voltage, \(V_

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[1]/span[1]/span, line 1, column 1
```

In order to develop a model of the DC motor, let \(i_

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[2]/span[1]/span, line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[2]/span[6]/span, line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[2]/span[8]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[2]/span[8]/span[2], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[2]/span[9]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[2]/span[9]/span[2], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[2]/span[10]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[2]/span[10]/span[2], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[2]/span[10]/span[3], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[2]/span[11]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[2]/span[11]/span[2], line 1, column 1
```

Figure 12: An armature-controlled DC motor.

In order to obtain a single input-output relation for the DC motor, we may solve the first euqation for \(i_a(s)\) and substitute in the second equation. Alternatively, we multiply the first equation by \(k_

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[2]/span, line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[4]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[4]/span[2], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[4]/span[3], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[4]/span[4], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[5]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[5]/span[2], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[5]/span[3], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[5]/span[4], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[10]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[10]/span[2], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[10]/span[3], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[10]/span[4], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[11]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[11]/span[2], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[12]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[12]/span[2], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[12]/span[3], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[4]/span[12]/span[4], line 1, column 1
```

**Example 1.11:** A DC motor model

We assume that the parameter values for a small DC motor are given as: \(R=1\Omega ,\; L=0.01H,\; J=0.01\; kgm^{2} ,\; b=0.1\; \frac{N-s}{rad} ,\; {\rm a}nd\; k_{t} =k_{b} =0.05\); then, the transfer function of the DC motor is obtained as: \[\frac{\omega (s)}{V_

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[6]/span[2]/span, line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[6]/span[3]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[6]/span[3]/span[2], line 1, column 1
```

Assuming a unit-step input, the output of the DC motor is given as: \[\omega \left(s\right)=\frac{500}{s\left(s+10.28\right)\left(s+99.72\right)}=\frac{0.488}{s}-\frac{0.544}{s+10.28}+\frac{0.056}{s+99.72}\] By applying the inverse Laplace transform, the time-domain output is given as (Figure 13a): \[\omega \left(t\right)=\left[0.488-0.544e^{-10.28t}+0.056e^{-99.72t}\right]u\left(t\right)\]

**Simplified DC motor Model.** The DC motor model developed above is a second-order ODE model with two unequal (electrical and mechanical) time constants, where \(\tau _

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[8]/span[1]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[8]/span[1]/span[2], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[8]/span[2]/span, line 1, column 1
```

A simplified model of the DC motor is obtained by ignoring the coil inductance (\(L\to 0\)), i.e., effectively ignoring the electrical time constant. The motor speed equation is modified as: \[R(Js+b)\omega (s)+k_

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[2]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[2]/span[2], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[2]/span[3], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[2]/span[4], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[3]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[3]/span[2], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[3]/span[3], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[3]/span[4], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[4]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[4]/span[2], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[4]/span[3], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[5]/span[1], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[5]/span[2], line 1, column 1
```

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[9]/span[5]/span[3], line 1, column 1
```

**Example 1.12:** A DC motor model (simplified)

Using the parameter values for a small DC motor (Example 1.11), its simplified transfer function model is obtained as: \[\frac{\omega (s)}{V_

```
Callstack:
at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.07:_DC_Motor_Model), /content/body/p[11]/span[1]/span, line 1, column 1
```

Figure 13: DC motor response to unit-step input: second-order motor model (left); first-order motor model (right).