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Projects for StudentsPPrroojjeecctt 3377 Dynamic Model of a Permanent

Magnet DC Motor

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The stator consists of permanent magnets which create a magnetic field. The armature consists of an

electromagnet created by a coil wound around an iron core. The armature rotates due to the phenomenon of

attracting and opposing forces of the two magnetic fields. A magnetic field is generated by the armature by

sending an electrical current through the coil and the polarity is constantly changed by alternating the currentthrough the coil (also known as commutation) causing the armature to rotate.

The commutator is made up of two semicircular copper segments mounted on the shaft at the end of the rotor.

Each terminal of the rotor coil is connected to a copper segment. Stationary brushes ride on the copper segments

whereby the rotor coil is connected to a stationary dc voltage supply by a near frictionless contact.

Mathematical Model

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 mechanicalcharacteristics of the system.

Electrical Characteristics

The equivalent electrical circuit of a dc motor is illustrated in Fig. 6F.2. It can be represented by a voltage source

(Va) across the coil of the armature. The electrical equivalent of the armature coil can be described by an

inductance (La) in series with a resistance (Ra) in series with an induced voltage (Vc) which opposes the voltage

source. The induced voltage is generated by the rotation of the electrical coil through the fixed flux lines of thepermanent magnets. This voltage is often referred to as the back emf (electromotive force).

Fig. 6F.2 Electrical representation of a dc motor.

A differential equation for the equivalent circuit can be derived by using Kirchoff's voltage law around the

electrical loop. Kirchoff's voltage law states that the sum of all voltages around a loop must equal zero, or

According to Ohm's law, the voltage across the resistor can be represented as

where ia is the armature current. The voltage across the inductor is proportional to the change of current throughthe coil with respect to time and can be written as

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where La is the inductance of the armature coil. Finally, the back emf can be written as

where kv is the velocity constant determined by the flux density of the permanent magnets, the reluctance of the

iron core of the armature, and the number of turns of the armature winding. a is the rotational velocity of the

armature.

Substituting eqns. (6F.2), (6F.3), and (6F.4) into eqn. (6F.1) gives the following differential equation:

Mechanical Characteristics

Performing an energy balance on the system, the sum of the torques of the motor must equal zero. Therefore,

where Te is the electromagnetic torque, T' is the torque due to rotational acceleration of the rotor, T is the

torque produced from the velocity of the rotor, and TL is the torque of the mechanical load. The electromagnetictorque is proportional to the current through the armature winding and can be written as

where kt is the torque constant and like the velocity constant is dependent on the flux density of the fixed

magnets, the reluctance of the iron core, and the number of turns in the armature winding. T' can be written as

where J is the inertia of the rotor and the equivalent mechanical load. The torque associated with the velocity is

written as

where B is the damping coefficient associated with the mechanical rotational system of the machine.

Substituting eqns. (6F.7), (6F.8), and (6F.9) into eqn. (6F.6) gives the following differential equation:

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State Space Representation

The differential equations given in eqns. (6F.5) and (6F.10) for the armature current and the angular velocity can

be written as

which describe the dc motor system. Putting the differential equations into state space form gives

which is expressed symbolically as

where is the state vector, is the input vector, and is the output vector.

Transfer Function Block Diagram

A block diagram for the system can be developed from the differential equations given in eqns. (6F.11) and(6F.12). Taking the Laplace transform of each equation gives

If perturbations around some steady state value are considered, the initial conditions go to zero and all the

variables become some change around a reference state, and the equations can be expressed as follows:

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The above equations can then easily be put into block diagram form. The block diagram obtained from these

equations for a permanent magnet dc motor is shown in Fig. 6F.3.

Fig. 6F.3 Block diagram representation of eqns. (6F.19) and (6F.20).

The block diagram in Fig. 6F.3 can be simplified by making the assumption that the load torque is constant. Inthe case of a sun tracking servo system, the only load torque to be concerned with is the friction in the system,

which is relatively constant while the motor is moving. Since the change in TL is zero, it does not need to appear

in the block diagram. Also, if one only focuses on the angular velocity as the response of interest, the block

diagram becomes as shown in Fig. 6F.4.

This block diagram is then easily reduced by block diagram algebra to an overall transfer function. Several steps

in this process are shown in Fig. 6F.5, with the overall transfer function between the output angular velocity and

input applied voltage given within the last block in Fig. 6F.5.

Fig. 6F.4 Block diagram of the dc motor as modeled in this study.

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Fig. 6F.5 Overall transfer function for the dc motor.

Analysis and Conclusions

Step Response of a DC Motor

The block diagrams given in Figs. 6F.4 and 6F.5 were used as the basis for a simulation model in Matlab, where

a simple step response is computed to illustrate the dynamics of the dc motor. Two cases are simulated:

Case 1 uses the overall transfer function given in the last block of Fig. 6F.5

Case 2 uses the graphical interface within the Simulink toolbox to construct a block diagram similar to that given

in Fig. 6F.4 and automatically generate an equivalent state space model for use in Matlab

The Matlab file, MOTORTST.M, models these two cases. The block diagram of the motor as constructed within

the Simulink graphical interface for Case 2 is given in Fig. 6F.6. Note that this diagram is built on screen and, as

designed, it is essentially identical to the block diagram in Fig. 6F.4. The Simulink model was saved in

MOTORSL.MDL, which contains information about the individual blocks, the input/output connections to the

blocks, and a set of graphical commands for drawing the block diagram on screen.

Notice that the interaction between Matlab and the Simulink model is through a single command to extract the

state space linear model from the MOTORSL.MDL file. This is given in MOTORTST.M as

[A,B,C,D] = linmod('motorsl');

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Fig. 6F.6 Block diagram produced within Simulink from MOTORSL.MDL.

Once the state space matrices for a stationary linear system are known, the system simulation can proceed as

usual. Figure 6F.7 shows the step response of the dc motor using a set of nominal values for the motor

parameters (see MOTORTST.M). Notice that the response time is quite fast, taking only about 0.01 sec to reach

the new steady state rotor speed following a unit increase in applied voltage. Also note that the Case 1 (points)and Case 2 (solid line) simulations give identical results, indicating the equivalency of the block diagrams in

Figs. 6F.4-6F.6 and the ability of Matlab/Simulink to convert a graphical model to a state space computational

model.

Fig. 6F.7 Step response of a dc motor for the Case 1 and Case 2 simulations.

The Light Tracking Servo System (revisited)

As another illustration of the capability of the Simulink interface to Matlab, the Light Tracking Servo System

that was discussed in Case Study E was modeled in block diagram form within Simulink. The dc motor model

from above was simply placed as a subsystem of the Simulink sun tracker model as shown in Fig. 6F.8

(generated by SUNTRKSL.MDL). This Simulink file is then called within a new Matlab simulation model,

SUNTRK.M, to create the state space matrices for the sun tracker and to perform the same sensitivity analyses as

in the previous Case Study (see LTSERVO.M from the Light Tracking Servo SystemCase Study). The new

M/MDL-files (SUNTRK.M and SUNTRKSL.MDL) should be compared to our previous analyses of this system.

Note that since identical results are obtained, no simulation or sensitivity results are given here. The reader

should refer back to Case Study E for the results of the simulations

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Fig. 6F.8 Simulink model of Light Tracking Servo System.

Conclusions

This Case Study has developed the defining equations for a dc motor and performed some simple testing of themodel. It also illustrated the model-building capability of the Simulink toolbox and showed how to include a

complete system (the dc motor) as one component of a larger system (the light tracking servo system). This basic

procedure can be expanded to build very complex simulation models of realistic engineering systems. In practice,

each component model is tested separately to validate its performance, then groups of components are tested,

and so on, until the final system, including all the subsystem components, is complete. The complete model is

then verified and analyzed as a single comprehensive system.

This Case Study provides a simple example of this model-building procedure and it also illustrates how new

graphics-based software is making the analysis of real systems much more manageable. With tools like Matlab

and Simulink, the engineer can concentrate on the design and performance of the system, and not on the

underlying numerical methods and coding needed to perform the actual computations. Just a few years ago, 80%

of the engineer's work went into the implementation of methods for simulation and 20% into the analysis of the

simulation results. Today, this distribution of effort has been reversed and the increased emphasis on analysis haslead to better designs and a better overall understanding of the system performance.

Summary of Matlab Files for Case Study F

This table contains a summary of the Matlab files used as part of the example applications for this Case Study.

There are four M-files as follows:

File Names File Description

MOTORTST.M Main simulation file for testing the motor model.

MOTORSL.MDL Generated from the Simulink block diagram in Fig. 6F.6.

SUNTRK.MPerforms sensitivity analysis of sun tracker model usingSUNTRKSL.MDL to create the state space model. Similar in

construction to LTSERVO.M from Case Study E.

SUNTRKSL.MDL Generated from the Simulink block diagram in Fig. 6F.8.

Project 37

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