Objectives

The goal in the development of the mathematical model is to relate the voltage applied to the

armature to the velocity of the motor. Two balance equations can be developed by considering

the electrical and mechanical characteristics of the system.

Theory

A DC motor is a frequently used actuator in control systems. It converts electrical energy into

rotational mechanical energy. A common DC motor has several major parts. Such as armature,

commutator, brushes, axle, field magnet etc. A motor uses magnets to create motion. The

armature is an electromagnet, while the field magnet is a permanent magnet. Inside and electric

motor these attracting and repelling forces are used to create rotational motion. The axle holds

the armature and the commutator. An electromagnet is the base of an electric motor. The

armature is a set of electromagnets.

The armature is an electromagnet made by coiling thin wire around two or more poles of a metal

core. The armature has an axle and the commutator is attached to the axle. The flipping of the

electric field is accomplished by commutator and brushes. The brushes are just two pieces of

carbon that make contact with commutator.

Figure 2 – inside of a DC motor

The figure 2 shows the electric circuit of the armature and the free body diagram of the rotor.

The input of the system is assumed as voltage source (V) applied to the motor’s armature, while

the output is the rotational speed of the shaft dθ/dt. The rotor and shaft are assumed to be rigid.

Also assumed a viscous friction model, the friction torque is proportional to shaft angular

velocity.

Figure 4 – physical setup

Figure 2 – front, side, end-on view

of an armature Figure 3 – Commutator and Brushes

The physical parameters for the setup in figure 2 are,

1. J moment of inertia of the rotor (kg.m2)

2. b motor viscous friction constant (N.m.s)

3. Ke electromotive force constant (V/rad/sec)

4. Kt motor torque constant (Nm/A)

5. R electric resistance (Ω)

6. L electric inductance (H)

In general, the torque generated by a DC motor is proportional to the armature current and the

strength of the magnetic field. The magnetic field was assumed as constant. So the torque is only

proportional to only the armature current i by a constant factor Kt(electromotive force constant).

The bacl emf, e is proportional to angular velocity of the shaft by a constant factor Ke.

The Newton’s law and Kirchhoff’s law was applied to motor system to generate the equations

needed for modeling.

Methodology

The model in the figure 5 was created using simulink by adding components from the simulink

library. Next all the components were saved to a single subsystem block. Later the output (speed)

was observed for different inputs (voltages).

The physical parameters were set before simulation.

J = 0.01;

b = 0.1;

K = 0.01;

R = 1;

L = 0.5;

Figure 5 – simulink model

The model in figure 6 was created using simscape which is a extension to simulink. The blocks

in the simscape library which represent actual physical components were used to create the

model.

In order to simulate the response of this system the sensor blocks were added to the model to

simulate the measurement of various physical parameters and a voltage source to provide the

excitation to the motor. Next outputs were observed for various inputs.

Figure 5 – simscape model 01

Figure 6 – simscape model 02

Next a wheel and Axel Block was connected to the model in order to convert rotational motion

of motor shaft to Translational motion. Then the wheel was loaded with a Translational Spring.

The output was observed by varying the ‘spring rate’.

Figure 7 – simscape model extended

Results

Table 1 - Results for simulation of the simulink model

Speed(peak/graph)

Step Input 1V 0.0999

2V 0.1999

Ramp Input

Square Wave

Duty Cycle

20% 0.0486

50% 0.085

90% 0.0987

Repeating Sequence Stairs 0.3725

Table 2 - Results for simulation of the model in simscape without translational spring

Current(peak/graph) Speed(peak/graph)

Step Input 1V 0.98 9.05

2V 1.96 18.15

Ramp Input

Square Wave

Duty Cycle

20% 0.97

2.51

50% 0.98

5.61

90% 0.98

8.55

Repeating Sequence Stairs

5.06

30.74

Table 3 - Results for simulation of the model in simscape with translational spring

Current(peak/graph) Speed(peak/graph)

Step Input 1V 1.000

0.0126

2V 2.000

0.026

Ramp Input

4.435

Square Wave

Duty Cycle

20% 0.5605

0.0133

50% 0.8807

0.0896

90% 0.9908

0.0878

Repeating Sequence Stairs

3.868

0.1646

Discussion

The field magnet could be an electromagnet as well, but in most small motors it isn't in order to

save power.

Before creating the model in simulink the characteristic equations that are required to create the

model should be taken down. First of all, the equations based on physical laws should be written

down. Then the relationship between inputs and outputs should be derived. Next according to the

derived equations, the simulink model should be created.

Simscape provides an environment for modeling and simulating physical systems spanning

mechanical, electrical, hydraulic and other physical domains. It provides fundamental building

blocks from these domains such as electric motors, op amps, resistors, inductors etc. In order to

simulate and take an output to the scope in simulink, the simscape to simulink convertor should

be used.

The Translational Spring block represents an ideal mechanical linear spring, described with the

following equations:

F - Force transmitted through the spring

K - Spring rate

x - Relative displacement (spring deformation)

xinit - Spring initial displacement (initial deformation); the spring can be initially

compressed (xinit > 0) or stretched (xinit < 0)

xR,xC - Absolute displacements of terminals R and C, respectively

V - Relative velocity

T - Time

References

1] MATLAB tutorials. (2012). DC Motor Speed: Simulink Modeling.Available:

http://ctms.engin.umich.edu/CTMS/index.php?example=MotorSpeed§ion=SimulinkModeling.

Last accessed 25th Aug 2014.

2] Marshall Brain. (2013). How Electric Motors Work. Available:

http://electronics.howstuffworks.com/motor5.htm. Last accessed 25th Aug 2014.