analysis of PMDC motor DEI - EEWeb...

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1. Introduction

ADRDE

Aerial Delivery Research & Development Establishment (ADRDE), a pioneer establishment of

ministry of Defense, Government of India engaged in design & development of Flexible

Aerodynamics Decelerators commonly known as ‘parachute’ and the allied technologies.

ECSG Group

ECSG i.e. Electronic Control System Group is related to various field of research like

instrumentation and telemetry. It works for the designing of different on board and remote units

required for data acquisition and its analysis. This group plays an important role in the designing

of any parachute related operations.

Project Overview

The objective of the project “Permanent Magnet DC Motor Analysis” is to analyze and study the

behaviour of Permanent Magnet Direct Current (PM DC) Motor if one or other parameters of the

PM DC motors are altered and to study their response under different excitations and load

conditions, by modelling and simulating the Permanent Magnet DC Motor under the conditions

decided in a computer controlled environment.

This task is accomplished using mathematical software tool MATLAB®

and SIMULINK®

.

2. Permanent Magnet DC motors

The direct current (DC) motor is one of the

into mechanical power. Its origin can be traced to the disc type machines conceived and tested

by Michael Faraday, the experimenter who formulated the first principles of electromagnetism.

The ease with which the DC motors lends

Compatibility with thyristor systems and transistor amplifiers, plus better performance due to the

availability of new improved materials on magnets and brushes and epoxi

interest in DC machines.

PMDC motors have found their applications in the robotic systems requiring small actuators that

can easily be tested and formulated.

Basic Theory

The stator magnetic field is generated by permanent magnets;

structure. The stator magnetic flux remains essentially constant at all levels of armature current

and, therefore, the speed torque curve of the PM motor is linear over an extended range (see fig

1).

Fig. 2.1. Permanent Magnet DC motor

Permanent Magnet DC motors

The direct current (DC) motor is one of the first machines devised to convert electrical power



The ease with which the DC motors lends it to speed control has long been recognized.


availability of new improved materials on magnets and brushes and epoxies, has revitalized the


can easily be tested and formulated.

The stator magnetic field is generated by permanent magnets; no power is used in the field



Fig. 2.2. Variation of current and speed with Torque

2

first machines devised to convert electrical power



to speed control has long been recognized.


es, has revitalized the


no power is used in the field



Fig. 2.2. Variation of current and speed with Torque

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With modern ceramic magnets, the stalled torque will tend to be more linear than for a

comparable wound field motor.

The linear speed torque curve is a result of use of permanent magnets, since in the case of these

motors, the armature reaction flux stays orthogonal to the permanent magnet flux, since the

permeability of ceramic magnetic material is very low (almost equal to that of air). In addition

the high coercive force of the magnetic materials resists an change in flux whenever the armature

field enters, thus the result is the linear torque characteristics.

Advantages of the PM DC motors:

1. The electrical power need not be supplied to generate the stator magnetic flux. And hence

reducing the power losses. Thus, the Pm motor thus simplifies power supply requirements,

while at the same time it requires less cooling.

2. They had linear torque-speed characteristics.

3. High stall (accelerating) torque.

4. A small frame and lighter motor for a given output power. This is because their radial

dimension is typically one forth that of the wound field for a given air gap.

3. Motor Equations and Transfer Function

In PM DC the permanent magnets produce a constant field

electromagnetic torque, Tem is given by;

Where, kT is the torque constant of the motor. In the armature circuit a back emf,

by the rotation of the armature conductors at speed wm in the presence

by;

When a controllable voltage Vi

determined by the Vi, the back-

winding inductance La, as;

The interaction of Tem (electromagnetic torque generated) with the load torque,

Where, J and B are total equivalent inertia and damping, of the motor load combination and

is the total working torque of the load.

The equation (3) and (4) are termed as the electrical and the dynamic

Motor Equations and Transfer Function

In PM DC the permanent magnets produce a constant field flux; Φf. in steady state the

is given by;

…(1)

is the torque constant of the motor. In the armature circuit a back emf,

by the rotation of the armature conductors at speed wm in the presence of field flux, and in given

…(2)

is applied to the armature terminals to establish

-emf Ea, the armature winding resistance Ra, and the armature

…(3)

(electromagnetic torque generated) with the load torque, T

…(4)

Where, J and B are total equivalent inertia and damping, of the motor load combination and

is the total working torque of the load.

Fig. 3.1. DC Motor Equivalent Circuit

The equation (3) and (4) are termed as the electrical and the dynamic equations respectively.

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. in steady state the

is the torque constant of the motor. In the armature circuit a back emf, Ea is generated

of field flux, and in given

is applied to the armature terminals to establish Ia. Thus Ia is

, and the armature

TL(t) is given by;

Where, J and B are total equivalent inertia and damping, of the motor load combination and TL(t)

Fig. 3.1. DC Motor Equivalent Circuit

equations respectively.

Motor Transfer Function

Equation (1), (3) and (4) provide a model for the motor which describes a relation between its

variables. Now for the motor to be used as a component in a system, it is described in a transfer

function between motor voltage and its velocity.

For this purpose we can assume that

Laplace transforms to the motor equations, we obtain;

Fig.3.2. Block diagram representation of the Motor and Load

Solving these equations, we obtain;

As to gain better insight into the dc motor behavior, the friction term, which is usually small, will

be neglected by setting B = 0 in equation (8). Now considering the equation (8); we define

Motor Transfer Function



between motor voltage and its velocity.

For this purpose we can assume that TL equals zero for the time being, and then apply the

Laplace transforms to the motor equations, we obtain;

… (5)

… (6)

… (7)

. Block diagram representation of the Motor and Load

Solving these equations, we obtain;

… (8)


n equation (8). Now considering the equation (8); we define

… (9)

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equals zero for the time being, and then apply the

… (8)


n equation (8). Now considering the equation (8); we define

… (9)

Using τe and τm in the expression for the G(s) yields

Since in general, , it is a reasonable approximation to replace s by

foregoing expression. Therefore

The physical significance of the mechanical and the electrical time constants is that they tell how

quickly the angular speed of the motor or the armature current of the motor builds up in response

to step input to the motor terminals respectively.

It is important to note that to remain the system linear, the following assumptions are taken that

the during the current build up the rotor speed is kept constant and while the angular speed build

up the te is assumed to be negligible as then the armature current can change instantaneously,

when a step input is applied at the input of the motor terminals.

Fig 3.3. Electrical time constant te; speed wm is assumed

to be constant.

… (10)

in the expression for the G(s) yields

… (11)

, it is a reasonable approximation to replace s by

… (12)


angular speed of the motor or the armature current of the motor builds up in response

to step input to the motor terminals respectively.


ent build up the rotor speed is kept constant and while the angular speed build


when a step input is applied at the input of the motor terminals.

Electrical time constant te; speed wm is assumed Fig 3.4. Mechanical time constant tm; load torque is

assumed to be constant.

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in the


angular speed of the motor or the armature current of the motor builds up in response


ent build up the rotor speed is kept constant and while the angular speed build


Fig 3.4. Mechanical time constant tm; load torque is

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4. Open Loop Analysis of the PM DC Motor

The open loop analysis is done using MATLAB®

and SIMULINK®

.

a. Open loop analysis using MATLAB® .m-file

With the aid of the PM DC motor parameters provided, the open loop analysis us done by

considering the stability factors and making the necessary plots for this analysis. This includes

the step response and the bode plot diagram.

For this purpose, separate m files were created for the constants, file to change these constants

and the main files.

constants.m

% start of code

% motor parameters

L = 0.1200 % henry

R = 7 % ohms

Kt = 0.0141 % Nm/A

J = 1.6000e-006 % kgms2

Ke = 0.0141 % Nw/Watt

B = 6.0400e-006 % Nm-s/rad

% end of code

The above file generated constants.m stores the relevant constants describing a motor. The

file described below change_data.m is for any change in the values by the user at any later

moment to be used in the calculation of response of a motor different than defined by the motor

parameters provided in the above file constants.m.

change_data.m

% any change in data

display = input('do you want to change the values(y/n): ', 's');

if strcmp(display,'y')

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L = input('enter the value of the inductance,L: ')

R = input('enter the value of the resistance,R: ')

Kt = input('enter the value of constant Kt: ')

J = input('enter the value of J: ')

Ke = input('enter the value of Ke: ')

end

% end of code

This file open_loop.m analyzes the open loop response of the motor with the constants supplied by the file

constants.m or by the user through change_data.m. The step response is showed with a series of plots completely

describing the state of the system.

open_loop.m

% start of code

% include constants.m

constants

% includes change_data

change_data

% electrical transfer function

Z = tf([1],[L R]);

% dynamic transfer function

I = tf([1],[J B]);

% closed loop transfer function

T = feedback(Z*Kt*I, Ke)

% bandwidth frequency of the motor

fs = sqrt((R*B+Ke*Kt)/L/J)

% system response to step input

figure (1), step (T)

grid on

% frequency response

figure (2), bode (T)

grid on

% end of code

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The output of the analysis is as follows:

Output of the code

L = 0.1200 % henry

R = 7 % ohms

Kt = 0.0141 % Nm/A

J = 1.6000e-006 % kgms2

Ke = 0.0141 % Nw/Watt

B = 6.0400e-006 % Nm-s/rad

do you want to change the values(y/n): n % providing the choice to change

% the system parameters

Transfer function: % overall system transfer

% function

0.0141

----------------------------------------

1.92e-007 s^2 + 1.192e-005 s + 0.0002411

fb = 35.4355 % system’s break frequency

% in Hertz

Here, the fb is the break frequency calculated for the system, as this defines the bandwidth of

the system, as this will limit the system’s response to that particular frequency only and after that

the system will start getting immune to the input signal applied at the frequencies above the

break frequency. The plots are shown below;

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Fig. 4.1. The Open Loop Step Response

Fig. 4.2. Frequency Response

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b. A Comparative Analysis

The responses here obtained are very limited in their expressions in the sense that they do not

provide the comparative results between the responses if any parameter is changed.

For accomplishment of the fore mentioned task, separate .m files were created, each having

one varying parameter i.e. Resistance, R; Inductance, L; Inertia of the load and the motor system,

J and the damping of the load and the motor system, B.

The results are based on the fact that each of the above mentioned parameters are changed one

by one and the response of the motor and the behavior of the natural frequency and the damping

ratio of the whole system are plotted are against them.

The .m files created for the comparative analysis of the system by varying the system

parameters are presented in the appendix 1 for reference.

Variation in System Parameter:

a. Resistance, R

Fig.4.a.1

Important Observations:

Step Response of the system

1. As the resistance of the system is increased

the overall response of the system to the step

input that is the output angular velocity of the

system decreases.

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Fig.4.a.2

Frequency response of the system

1. The break frequency slightly increases when

the value of the resistance is increased.

Fig.4.a.3

Current Response

1. The steady state values and the peak values of

the current decreases drastically as the

system’s resistance is changed (increased),

upon the application of step input.

Fig.4.a.4

Current’s frequency response

1. The magnitude of the current in the in

the steady state upon the application of

a sinusoidal input decreases as the

resistance is increased, and so the

break frequency.

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Fig.4.a.5

Damping ratio vs. resistance characteristics

1. Upon increasing the resistance of the

system the damping ratio defined for

the system also increases, i.e. the

system slow down will be faster as the

system’s resistance is increased.

2. Also notice that the system response at

higher resistive values is highly over

damped response, which may not be

suitable for certain applications.

Fig.4.a.6

Break frequency vs. resistance characteristics

1. It is evident from the accompanied

graph that the break frequency

increases as the system resistance is

increased, though the change is small.

The following series of characteristics describes the behavior of the necessary parameters of a

system such as rise time, tr; peak time, tp; maximum overshoot, Mp and the settling time for the

step response of the system. It is important to note that some of the quantities do not need to be

calculated in certain responses like maximum overshoot doesn’t have any significance under the

over damped or critical response of a system. The important observations are summarized below

the characteristics.

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Fig.4.a.7

Upon increasing the resistance of the system generally the following is noticed:

1. The rise time decreases.

2. Peak time decreases.

3. Maximum overshoot decreases.

4. Settling time decreases.

The increase in the resistance can cause the system response to become more and more over

damped, thus, decreasing the efficiency of the system. Thus, resistance parameter should be

appropriately chosen while designing a system, keeping system response in mind.

b. Inductance, L

The variation of inductance causes many changes to the system response as the most important

was the increase in the settling time of the system. Many other changes that occurred due to the

variation in the inductance value of the armature are described corresponding to the graphs

below.

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Fig.4.b.1

Step Response

1. As it is evident from the graph

accompanied the angular speed over

shoots its steady state value upon

increasing the inductance. It shows

that the motor acceleration is

increased as the armature inductance

is increased. The system’s settling

time and rise time also increases as

the inductance is increased.

Fig.4.b.2

Frequency Response

1. On increasing the armature

inductance the magnitude of the

output response remains the same but

the bandwidth of the system decrease

slightly.

Fig.4.b.3

Response of the Current

1. Increasing armature inductance has

its greatest effects on the current

response when the system is excited

to a step response. As shown, the

current builds more slowly and settles

late.

2. The rise time is also increased.

3. The maximum over shoot decreases

as the inductance increases.

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Fig.4.b.4

Frequency Response of Current

1. The bode plot indicates that on

increasing the armature inductance

the system bandwidth in the

changes to the current increases.

Fig.4.b.5

Damping Ratio vs. Armature Inductance

Characteristics

1. Increasing armature inductance

decreases the damping ratio of the

system. That is the system tends to

be more oscillatory as the value of

the inductance tends to increase.

2. But since, the value of ζ doesn’t

tend to zero the system remains

under damped.

Fig.4.b.6

Natural Frequency vs. Armature

Inductance Characteristics

1. The same decreasing trend is

observed with the natural

frequency of the system.

2. Therefore it is not a good choice

for a motor to have high

inductance being utilized in a

servo system.

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c. Inertia of the Motor and the Load System, J

Fig.4.c.1

Step Response

1. As per the accompanied response

of the system it is evident that the

high inertia system tends to

decrease the system output.

2. They decrease the total settling

time.

3. Thus, they over all effect the

system efficiency.

Fig.4.c.2

Frequency Response

1. As the system gains inertia, the

frequency response of the system

shows that its bandwidth decreases.

2. And thus, making it unsuitable for

high frequency servo applications.

Fig.4.c.3


1. The current is also greatly

affected by the system’s high

inertia as both the peak to peak

value and the steady state value

tends to decrease.

2. Thus, reducing the overall output

of the system.

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Fig.4.c.4


1. Current bandwidth is affected as

it decreases with the increase in

the inertia of the system.

Fig.4.c.5

Natural Frequency vs. Armature

Inductance Characteristics

1. Increasing inertia greatly

increases the damping ratio

taking it over the ζ = 1 value thus

making it highly over damped

system.

Fig.4.c.6

Damping Ratio vs. Armature Inductance

Characteristics

1. It too has adverse effects on the

system bandwidths making it highly

unresponsive to the high frequency

servo systems.

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The following plots show the behavior discussed above for the system parameters, to the step

response. They are briefly summarized below:

Fig.4.c.7

1. Rise time reduces with the increase in the inertia of the system.

2. Peak time reduces nonlinearly with the increase in the system inertia.

3. As the system remains under damped there is no significance of calculating the rise time

of such a system.

4. Settling time again increases with the increase in the system inertia.

Thus, system inertia ideally is suitable for only some extent in order to provide torque to the

system and setting up optimum bandwidth of the system, otherwise high inertial systems are

inefficient.

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d. Damping Coefficient of the Motor-Load System, B

Fig.4.d.1

Step Response

1. Damping affects the steady state

step response of the system by

decreasing it, if we increase the

system damping.

Fig.4.d.2

Frequency Response

1. Frequency response doesn’t get usually

affected as the change in the magnitude

and bandwidth are very small.

2. Notice the undistinguishable graphs.

Fig.4.d.3


1. The overall steady state value of the

current decreases.

2. Doesn’t affect the peak value of the

current.

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Fig.4.d.4


1. System magnitude of the current

increases as the damping increases

but the change is negligibly small.

Fig.4.d.5

Natural Frequency vs. Damping

Coefficient Characteristic

1. Damping pushes a system to more

under damped state though the

change is small.

Fig.4.d.6

Damping Ratio vs. Damping Coefficient

Characteristics

1. The damping increases the system

bandwidth.

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The following plots show the behavior discussed above for the system parameters, to the step

response. They are briefly summarized below:

Fig.4.d.7

1. The rise time of the system decreases with the increase in the damping coefficient of the

system, though the change is negligibly small.

2. The peak time too decreases with a increase in the damping.

3. As the system remains under damped there is no significance of calculating the rise time of

such a system.

4. The settling time decreases with the increase in the damping.

The damping thus, for the above conditions for negligibly affecting the system response is

generally neglected for an ideal system.

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5. Open Loop Analysis of PMDC Motor using

SIMULINK®

a. Without Load

Open loop analysis of the DC motor using SIMULINK tools is shown below. The arrangement

of the motor system is shown in the figure (1). Here, the PM DC Motor is provided with the step

input and various parameters of importance such as armature current, Ia; angular speed, ωm at the

output shaft are plotted as a function of time.

The SIMULINK(R)

provide as various tools to accomplish the same task by different means and

ways thus, greatly simplifying the design process and speeding up the things as we go up in the

design process, for example, designing the motor block separately greatly enhances the

debugging process of the system and also decreases the complexity of the system.

Fig.5.1. DC motor model analysis using SIMULINK®

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Simplified model of the same system with DC MOTOR sub block.

Fig.5.2. Simplified DC Motor Model for Analysis

Fig.5.3. Internals of the DC MOTOR sub block

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Fig.5.4. Response showing the angle and the angular speed of the system as the function of time.

The step response carried over here, represents the same facts that were obtained in the section 4.

a. here in the simulation the step is applied at the t = 1 sec and output is obtained at the scope.

The first graph of the step input applied at t = 1 sec. the other represents the increase in the

degrees of rotation with the application of the voltage to the motor.

The last graph of scope output is representing the angular speed of the system. The angular speed

is settled to a constant value over the time and attains a steady state value.

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b. with Load

The open loop analysis is done here with the inclusion of the ramp load torque at 0.5 seconds

having a uniform slope of 0.085 Nm/s. the switch, switches the load from zero value to the the

torque at 0.5 secs.

The connections of the block are shown below:

Fig.5.5. Open Loop Response With Load

Fig.5.6. A Simplifed Model Of The Same

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After the setup is completed the model is simulated with the simulation time being equal to 1

seconds. The variation of armature current, Ia; load torque TL and the angular speed ωm are

plotted as a function of time.

Fig.5.7. Response of the Various Signals as the Function of Time

As it is evident from the graphs plotted in the figure 3. As the load torque started to increase at

0.5 sec the armature current starts to increase while the angular speed starts decreasing. The

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motor has a specified limit for the load torque known as its rated torque beyond which the motor

cannot work efficiently.

The above phenomenon is further backed by the following two characteristics between Current

vs. Torque (fig. 4) and Angular Speed vs. Torque (fig. 5).

Fig.5.8. Current (y axis) vs. Torque (x axis) variation

Fig.5.9. Angular Speed (y axis) vs. Torque (x axis) variation

6. Open Loop Analysis of the PM

Input

DC Motor’s Four Quadrant Operation

PWM Generation with Bipolar Voltage Switching

For controlling motor in either of

between the two polarities of the voltages. In order to do that a triangular wave is passed through

a dead band generator or a comparator which provides voltages by comparing the incoming

triangular signal to its different bands and allocating fixed signals or voltages accordingly.

Such bipolar voltage switching helps us to control the motor under the four quadrants of

operation. As presented by the fig. 1 the four operations are motoring in the forwar

backward direction, and braking in forward and backward directions respectively.

Fig.6.1. Four Quadrant Operation of DC Motor

To accomplish our task of generating such bipolar voltage PWM in the MATLAB

an input as sine wave and took an inverse of this wave. Thus we got an increasing function with

an interval of 3 units, between –

signum function which has an dead band of zero.

Analysis of the PMDC Motor with PWM

DC Motor’s Four Quadrant Operation:

PWM Generation with Bipolar Voltage Switching

For controlling motor in either of the direction continuous voltage switching is necessary



ignal to its different bands and allocating fixed signals or voltages accordingly.


operation. As presented by the fig. 1 the four operations are motoring in the forwar


Fig.6.1. Four Quadrant Operation of DC Motor

To accomplish our task of generating such bipolar voltage PWM in the MATLAB


–pi/2 to pi/2. This signal was then provided as an input to the

signum function which has an dead band of zero.

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with PWM

the direction continuous voltage switching is necessary



ignal to its different bands and allocating fixed signals or voltages accordingly.


operation. As presented by the fig. 1 the four operations are motoring in the forward and the


To accomplish our task of generating such bipolar voltage PWM in the MATLAB®

we provided


. This signal was then provided as an input to the

30

The signum function compared the values according to the constant provided as the duty cycle

parameter between -8 to 8. The signum function thus provided output varying between -1 to 1

units.

Any suitable amplitude could be set for the final output.

Using the above idea, .m file and SIMULINK®

models were generated and there response

was analyzed.

a. Analysis using .m file

The pwm.m file creates bipolar PWM with user defined duty cycle and amplitude. It also

calculates the minimum frequency of the PWM wave for the smooth operation of a particular DC

motor.

pwm.m

% start of code % include constants constants % include change_data change_data % bandwidth frequency of the motor fb = Ke*Kt/2/pi/J/R % pwm amplifier frequency limits calculation fs_min = 10*fb % generation of pwm t = (-20:.0001:20); f = sin(t); plot (t,f) hold on f1 = asin(f); f2 = 5*asin(f); n = input('enter variable between (-8,8) to set the duty cycle: ') f3 = n-f2; plot (t,f3) hold on Vin = input('enter the peak2peak value of the pwm pulse: ') f4 = Vin/2*sign(f3); plot(t,f4) title ('pulse width modulted wave generation') xlabel('time') ylabel('voltage') grid on

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% duty cycle duty_cycle = (8 + n)*6.25 Ton = duty_cycle/100; % average voltage Vav = (Vin*Ton/2-(Vin*(1-Ton)/2)) Output form the code

fs_min = 28.2514

enter variable between (-8,8) to set the duty cycle: 5

enter the peak2peak value of the pwm pulse: 10

Vin = 10

duty_cycle = 81.2500

Vav = 3.1250

Fig. 6.2. Pulse width modulated wave with

81.25% duty cycle and Vav = 3.125 V

Fig. 6.3. Pulse width modulated wave with

18.75% duty cycle and Vav = -3.125 V

b. Affects of Switching Frequency on the Armature

Current

The switching frequency, fs, is

PWM amplifiers. The following discussion provides guidelines for selecting the

The switching frequency rates must be higher so that the servo system does not respond to it.

Thus, if the motion requirements are such that a bandwidth or the

fbw, then the switching frequency must be

frequencies of the loop.

Also, the switching period must be much longer than the transistor switching delay time Td.

Otherwise the amplifier may have a significant non

Also, the power dissipation in the amplifier transistors is directly proportional to

order to minimize those losses, we need to select the

frequency also affects the current variations which are having an opposite effect on the losses.

The following .m file simulates

frequency, in order to determine an optimum switching frequency,

of the above guidelines.

pwm_current.m

% start of code

clc

% include constants

constants

fs = zeros(1,81);

del_I_max = zeros(1,81);

del_I_fs = zeros(1,81);


Z = tf([1],[L R]);

% dynamic tranfer function

I = tf([1],[J B]);


Tf = feedback(Z*Kt*I, Ke);

% break or cut off frequency

wn = sqrt((R*B+Ke*Kt)/L/J)

Affects of Switching Frequency on the Armature

is probably the most important parameter in the

PWM amplifiers. The following discussion provides guidelines for selecting the


Thus, if the motion requirements are such that a bandwidth or the band width of the system is

, then the switching frequency must be ten times the fbw plus all the significant


he amplifier may have a significant non-linearities. Thus,

Also, the power dissipation in the amplifier transistors is directly proportional to

order to minimize those losses, we need to select the fs as small as possible. But the switchi


simulates the variation of current variations with respect to the

frequency, in order to determine an optimum switching frequency, fs,and to conform the basis



% break or cut off frequency - system bandwidth in rad/s

wn = sqrt((R*B+Ke*Kt)/L/J)

32

Affects of Switching Frequency on the Armature

the design of the

PWM amplifiers. The following discussion provides guidelines for selecting the fs


band width of the system is

plus all the significant


Also, the power dissipation in the amplifier transistors is directly proportional to fs. Therefore in

as small as possible. But the switching


the variation of current variations with respect to the

and to conform the basis

33

% lowest safer limit for the switching frequency

wn_safe = 10*wn

fs_safe = wn_safe/(2*pi)

wn2 = wn+10*80;

wn3 = [wn:10:wn2]

% peak to peak motor current variation

for i = 1:81

% lowest frequency limits in Hz

fs(i) = wn3(i)/(2*pi);

% duty cycle

D = 80;

n = (D*1.559/50-1.559);

% magnitude of the output pwm pulse

Vs = 5;

% time on

m = 1.559/((1/fs(i))/4);

x1 = -n/m;

x2 = -1*( - 2*1.559 - n)/m;

Ton = x2-x1;

% peak to peak current

del_I_fs(i) = 2*Vs*Ton/L*(1-Ton*(1/fs(i)));

% max peak current

del_I_max(i) = Vs*(1/fs(i))/2/L;

end

fs

del_I_fs

del_I_max

m = 1.559/((1/fs_safe)/4);

x1 = -n/m;

x2 = -1*( - 2*1.559 - n)/m;

Ton = x2-x1;

del_I_fs_min_safe = 2*Vs*Ton/L*(1-Ton*(1/fs_safe))

y = 0:.01:13;

x = 0:.01:140;

hold on

34

figure(1),plot(fs,del_I_fs,'-')

plot(fs_safe,y,'-')

text(fs_safe+1,8,'fs safe')

plot(x,del_I_fs_min_safe,'-')

text(1,0.9,'I_p_-_p at fs safe')

title('peak to peak current variation with frequency')

xlabel('frequency')

ylabel('peak to peak current')

grid off

figure(2),plot(fs,del_I_max,'-')

title('max. peak current variation with frequency')

xlabel('frequency')

ylabel('max. peak current')

grid on

Output from the pwm_current.m file

wn = 35.4355 % break frequency (rad/s)

wn_safe = 354.3553 % min pwm switching frequency in rad/s

fs_safe = 56.3974 % min PWM switching frequency in Hertz

Fig. 6.4. Peak to Peak current variation with Frequency

Peak to peak variation with

Frequency

1. It is evident from the

characteristic that the peak to

peak current decreases the as

switching frequency of the

input pulse is increased.

2. Therefore the switching

frequency is only limited by

the fact that the transistor

losses are higher at higher

frequencies.

35

Fig. 6.5. Max. peak current variation with frequency

Max. peak current variation with

Frequency

1. The same behavior is

applicable to the max. Peak

current as this also decreases

with the increase in the

frequency.

The above current vs. frequency characteristics shows that the frequency should be high enough

from the system bandwidth, so that it doesn’t respond to the switching frequency, which conform

the fact previously discussed in the guidelines provided for selection of the fs.

But since the transistor losses are directly proportional to the frequency therefore the frequency

is optimally chosen so that the power losses are minimized.

Thus, there are reasons to increase the switching frequency and others to decrease it. The final

selection is a compromise between those contradicting conditions.

36

c. Analysis using SIMULINK®

The pwm.m file previously discussed does not provide the actual over view of the PMDC Motor

response over the time scale. To accomplish this task SIMULINK®

’s help was taken and the

bipolar PWM model was created with the same idea discussed in the previous section.

The PWM wave thus generated was provided as the input to the system as shown in the

following blocks prepared for the detailed simulation of the PMDC Motor. The results are

discussed below.

a. Analysis of System Response with No-Load

Firstly, the simulation was carried out with the no load conditions as shown in the fig. 4. A PWM

wave of 55 volts was given to the system. The output response from the scope is shown in the

fig. 5.

Fig.6.6. PWM Generator

Fig. 6.7. For No Load notice the switch is not

connected to the load

37

Fig.6.8. DC Motor operated with a PWM Input with no-load

The output response from the scope and the characteristic graphs in fig. 6 and fig. 7 highlights

the following important observations which back the results and the design methodology applied

before when writing the .m files for the accomplishment of the same task.

The various observations are discussed as follows:

1. The angular speed response of the system is same as the application of a step input of

average value of the PWM wave. Therefore, the PWM wave just acts as an step input of

the average value of the input PWM wave, with benefit of the change in the average

value of the PWM wave thus changing the overall angular speed of the system.

2. The second major observation is the formation of the ripples in the current wave form.

These ripples in the current are frequency dependent and as the frequency is increased

these ripples start die out. This observation was previously discussed in detail

3. The next observation applies to the application of torque on the system. Since No Load is

applied over here, the fig. 6 and fig. 7, shows that the current and the angular speed

remain constant at the steady state. With no variation in the torque applied or the change

in the voltage the speed and the current remains unchanged too.

4. The application of No Load causes the PMDC Motor to run at its full rated speed and at

no load current.

38

Fig.6.9. Simulation Results showing variations of Armature Current, load torque and angular speed with

time on the application of PWM input without load acting on the system.

39

Fig.6.10. Current (y axis) vs Torque (x axis)

Fig.6.11. Angular speed (y axis) vs. Torque (x axis)

40

b. Analysis of system Response with Load

The same simulation is carried in order to study the PMDC Motor behavior with one change and

that is the application of load at 0.5 seconds. The nature of the load is ramp i.e. an increasing

load with time. The load is switched as shown below with following specification of the load

torque as ramp’s slope is set to 0.085 Nm/s and the switch is switched after 0.5 seconds from no

load.

Fig.6.12. For Load notice that the Ramp load is

connected to the switch

The following observations were being made:

1. The response is again similar to the step input response simulated earlier in the SIMULINK®

platform. With the exception that the current response is having ripples in it which is evident

from the introduction of the pulsed input to the system.

2. The load is switched at 0.5 sec which is increasing at the rate of .085 Nm/s. With the application

of this increasing load the current starts to increase at same rate.

3. The angular speed starts decreasing after the load is applied. The decrease in the speed is the

negative rate of the applied load torque.

4. Fig.4.16 and 4.17 describes the above observed behavior between the speed vs. torque and current

vs. torque respectively.

41

Fig.6.13. Simulation Results showing variations of Armature Current, load torque and angular speed with

time on the application of PWM input with load acting on the system.

42

Fig.6.14. Angular speed (y axis) vs. Torque (x axis)

Fig.6.15. Current (y axis) vs. Torque (x axis)

43

7. Conclusion

The following concluding points are laid depending upon the various observations taken during

the progress of the project. They are summarized as follows:

1. The PMDC motor system design should accomplish the fact that the parameters it going

to be employ should result a desired response based upon the need and the availability of

the raw materials and sophistication of the system required.

2. In general, a under damped response is most desirable. This under damped response is

such that it should have minimum rise time and minimum settling times, with damping

ratio near to one.

3. The motor bandwidth should be such that it totally responds to the servo commands, in a

servo system.

4. The switching frequency should be high enough (>10 times the system bandwidth) so

that the system doesn’t responds to the switching signals and should be less enough to

prevent the switching losses.

5. The motor current is highest at zero speed and so is the electromagnetic torque generated.

44

8. APPENDIX 1

Open Loop System Analysis – A Comparative Approach .m files.

1. variation_R.m clc constants tf R = [7:12]; wm_R = zeros(1,6); zeta_R = zeros(1,6); i = 1; for t = 1:6 % bandwidth frequency of the motor wm_R(t) = sqrt((R(t)*B+Ke*Kt)/L/J); zeta_R(t) = (J*R(t)+B*L)/2/(sqrt(L*J*(R(t)*B+Ke*Kt))); end wm_R zeta_R grid on figure(1),plot(R,wm_R) title('R vs wm') xlabel('R(ohm)') ylabel('wm break_frequency (rad/s)') grid on figure(2),plot(R,zeta_R) title('R vs \zeta') xlabel('R(ohm)') ylabel('\zeta')

Output fig. 4.a. (5, 6)

2. variation_L.m

clc constants tf % variation in inductance wm_L = zeros(1,6); zeta_L = zeros(1,6); L = [.12:.01:.17] for t = 1:6 % bandwidth frequency of the motor wm_L(t) = sqrt((R*B+Ke*Kt)/L(t)/J); zeta_L(t) = (J*R+B*L(t))/2/(sqrt(L(t)*J*(R*B+Ke*Kt))); end wm_L zeta_L grid on figure(3),plot(L,wm_L)

45

title('L vs wm') xlabel('L (Henry)') ylabel('wm break_frequency (rad/s)') grid on figure(4),plot(L,zeta_L) title('L vs \zeta') xlabel('L (Henry)') ylabel('\zeta')

Output fig. 4.b. (5, 6)

3. variation_J.m clc constants tf % variation in the motor inertia wm_J = zeros(1,6); zeta_J = zeros(1,6); J = [1.6e-6:.2e-6:2.6e-6]; for t = 1:6 % bandwidth frequency of the motor wm_J(t) = sqrt((R*B+Ke*Kt)/L/J(t)); zeta_J(t) = (J(t)*R+B*L)/2/(sqrt(L*J(t)*(R*B+Ke*Kt))); end wm_J zeta_J grid on figure(5),plot(J,wm_J) title('J vs wm') xlabel('J Nm/A') ylabel('wm break_frequency (rad/s)') figure(6),plot(J,zeta_J) title('J vs \zeta') xlabel('J Nm/A') ylabel('\zeta') Output fig. 4.c. (5, 6)

4. variation_B.m clc constants tf % variation in the damping coefficient of the motor wm_B = zeros(1,6); zeta_B = zeros(1,6); B = [6.04e-6:.2e-6:7.04e-6] for t = 1:6 % bandwidth frequency of the motor

46

wm_B(t) = sqrt((R*B(t)+Ke*Kt)/L/J); zeta_B(t) = (J*R+B(t)*L)/2/(sqrt(L*J*(R*B(t)+Ke*Kt))); end wm_B zeta_B figure(7),plot(B,wm,'-') title('B vs wm') xlabel('B') ylabel('wm break_frequency (rad/s)') figure(8),plot(B,zeta,'-') title('B vs \zeta') xlabel('B') ylabel('\zeta') grid on Output fig. 4.d. (5, 6)

5. variable_change_R.m clc clear all Kt = .0141; Ke = .0141; R = [7:10:57]; L = [.12:.02:.22]; J = [1.6e-6:.2e-6:2.6e-6]; B = [6.04e-6:.2e-6:7.04e-6]; tr = zeros(1,6); tp = zeros(1,6); Mp = zeros(1,6); ts = zeros(1,6); for n = 1:6 % electrical transfer function Z = tf([1],[L(1) R(n)]); % dynamic tranfer function I = tf([1],[J(1) B(1)]); % closed loop transfer function T = feedback(Z*Kt*I, Ke); % closeb current transfer function C = feedback(Z,Ke*Kt*I); % natural frequency wm = sqrt((R(n)*B(1)+Ke*Kt)/L(1)/J(1)); % damping ratio zeta = (J(1)*R(n)+B(1)*L(1))/2/(sqrt(L(1)*J(1)*(R(n)*B(1)+Ke*Kt))); % damped natural frequency wd = wm*sqrt(1-zeta^2); % rise time tr(n) = (pi-(wd/zeta/wm))/wd; % peak time tp(n) = pi/wd; %maximum overshoot Mp(n) = exp(-zeta*pi/sqrt(1-zeta^2)); % settling time

47

ts(n) = 4/zeta/wm; graphs end tr tp Mp ts figure(5),subplot(2,2,1) plot(R,tr,'-') title('tr vs R') subplot(2,2,2) plot(R,tp,'-') title('tp vs R') subplot(2,2,3) plot(R,Mp,'-') title('Mp vs R') subplot(2,2,4) plot(R,ts,'-') title('ts vs R') Output fig. 4.a. (1, 2, 3, 4, 7)

6. variable_change_L.m clc clear all Kt = .0141; Ke = .0141; R = [7:10:57]; L = [.12:.02:.22]; J = [1.6e-6:.2e-6:2.6e-6]; B = [6.04e-6:.2e-6:7.04e-6]; tr = zeros(1,6); tp = zeros(1,6); Mp = zeros(1,6); ts = zeros(1,6); for n = 1:6 % electrical transfer function Z = tf([1],[L(n) R(1)]); % dynamic tranfer function I = tf([1],[J(1) B(1)]); % closed loop transfer function T = feedback(Z*Kt*I, Ke); % closeb current transfer function C = feedback(Z,Ke*Kt*I); % na tural frequency wm = sqrt((R(1)*B(1)+Ke*Kt)/L(n)/J(1)); % damping ratio zeta = (J(1)*R(n)+B(1)*L(n))/2/(sqrt(L(n)*J(1)*(R(1)*B(1)+Ke*Kt))); % damped natural frequency wd = wm*sqrt(1-zeta^2); % rise time tr(n) = (pi-(wd/zeta/wm))/wd; % peak time

48

tp(n) = pi/wd; %maximum overshoot Mp(n) = exp(-zeta*pi/sqrt(1-zeta^2)); % settling time ts(n) = 4/zeta/wm; graphs end tr tp Mp ts figure(5),subplot(2,2,1) plot(L,tr,'-') title('tr vs L') subplot(2,2,2) plot(L,tp,'-') title('tp vs L') subplot(2,2,3) plot(Mp,L,'-') title('Mp vs L') subplot(2,2,4) plot(L,ts,'-') title('ts vs L') Output fig. 4.b. (1, 2, 3, 4, 7)

7. variable_change_J.m clc clear all Kt = .0141; Ke = .0141; R = [7:10:57]; L = [.12:.02:.22]; J = [1.6e-6:.2e-6:2.6e-6]; B = [6.04e-6:.2e-6:7.04e-6]; tr = zeros(1,6); tp = zeros(1,6); Mp = zeros(1,6); ts = zeros(1,6); for n = 1:6 % electrical transfer function Z = tf([1],[L(1) R(n)]); % dynamic tranfer function I = tf([1],[J(n) B(1)]); % closed loop transfer function T = feedback(Z*Kt*I, Ke); % closeb current transfer function C = feedback(Z,Ke*Kt*I); % na tural frequency wm = sqrt((R(1)*B(1)+Ke*Kt)/L(1)/J(n)); % damping ratio zeta = (J(1)*R(1)+B(1)*L(1))/2/(sqrt(L(1)*J(n)*(R(1)*B(1)+Ke*Kt))); % damped natural frequency

49

wd = wm*sqrt(1-zeta^2); % rise time tr(n) = (pi-(wd/zeta/wm))/wd; % peak time tp(n) = pi/wd; %maximum overshoot Mp(n) = exp(-zeta*pi/sqrt(1-zeta^2)); % settling time ts(n) = 4/zeta/wm; graphs end tr tp Mp ts figure(5),subplot(2,2,1) plot(J,tr,'-') title('tr vs J') subplot(2,2,2) plot(R,tp,'-') title('tp vs J') subplot(2,2,3) plot(J,Mp,'-') title('Mp vs J') subplot(2,2,4) plot(J,ts,'-') title('ts vs J') Output fig. 4.c. (1, 2, 3, 4, 7)

8. variable_change_B.m

clc clear all Kt = .0141; Ke = .0141; R = [7:10:57]; L = [.12:.02:.22]; J = [1.6e-6:.2e-6:2.6e-6]; B = [6.04e-6:.2e-6:7.04e-6]; tr = zeros(1,6); tp = zeros(1,6); Mp = zeros(1,6); ts = zeros(1,6); for n = 1:6 % electrical transfer function Z = tf([1],[L(1) R(1)]); % dynamic tranfer function I = tf([1],[J(1) B(n)]); % closed loop transfer function T = feedback(Z*Kt*I, Ke); % closeb current transfer function C = feedback(Z,Ke*Kt*I); % na tural frequency wm = sqrt((R(1)*B(n)+Ke*Kt)/L(1)/J(1));

50

% damping ratio zeta = (J(1)*R(1)+B(n)*L(1))/2/(sqrt(L(1)*J(1)*(R(1)*B(n)+Ke*Kt))); % damped natural frequency wd = wm*sqrt(1-zeta^2); % rise time tr(n) = (pi-(wd/zeta/wm))/wd; % peak time tp(n) = pi/wd; %maximum overshoot Mp(n) = exp(-zeta*pi/sqrt(1-zeta^2)); % settling time ts(n) = 4/zeta/wm; hold on figure(1),step(T) title('step response') hold on figure(2),bode(T) title('frequency response') hold on figure(3),step(C) title('response of current to step') hold on figure(4),bode(C) title('frequency response of current') end tr tp Mp ts figure(5),subplot(2,2,1) plot(B,tr,'-') title('tr vs B') subplot(2,2,2) plot(B,tp,'-') title('tp vs B') subplot(2,2,3) plot(B,Mp,'-') title('Mp vs B') subplot(2,2,4) plot(B,ts,'-') title('ts vs B') Output fig. 4.d. (1, 2, 3, 4, 7)

51

9. References

1. DC Motors, Speed Controls, Servo Systems – An Engineering Handbook, 5th

ed.,

Electro-Craft Corporation, Hopkins, MN, 1980.

2. Mohan, Undeland, Robbins; Power Electronics, Converters, Application and Design, 2nd

ed., John Wiley and Sons, Inc., 1992.

3. MATLAB® Official Webpage (www.mathworks.com)

4. Oguntoyinbo, O. J. “PID Control of Brushless DC motor and Robot Trajectory Planning

and Simulation with MATLAB®

/SIMULINK®

”, University of Applied Sciences.

5. en.wikipedia.org/wiki/PID

6. Ogata, K., Modern Control Engineering, 4th

ed., Pearson Education International, New

Jersey, 2002

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