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CHAPTER 2

MODELING OF DC MOTORS AND DC-DC CONVERTER

2.1 INTRODUCTION

The block diagram of the developed system with hybrid intelligent

controller is shown in Figure 2.1. The system consists of DC-DC buck

converter to drive the DC motor. The DC motor may be a DC series motor or

a DC separately excited motor or a PMDC motor. The DC-DC buck converter

(DC chopper) switch can be a Power Transistor or SCR or GTO or IGBT or

Power MOSFET or similar switching device. In order to get high switching

frequency (upto 100 KHz), the power MOSFET may be taken as a switching

device for the DC-DC converter. When the gate pulse is applied, the device is

turned on and the input supply is connected with the motor. When the gate

pulse is removed, the device is turned off and the motor is disconnected from

the input supply.

Voltage control using the switch-mode concept utilises solid state

components, including high power transistors driven by PWM (Pulse Width

Modulation). The benefits of using this method for motor control are many, in

particular;

The ability to smoothly control voltage (and current) delivered to

the motor, thus controlling its speed and torque.

High efficiency losses can be minimised and an efficiency of 80%

or higher is easily obtainable.

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Easily implemented into digital circuits, allowing high levels of

controllability and it has large feature sets to be implemented in software.

Figure 2.1 Block diagram of Hybrid Intelligent controller based DC

motor drive

From the general equivalent circuit of DC motors the voltage and

torque equations are obtained. Using the voltage and torque equations the

mathematical modeling of DC motor has been developed for simulation of the

system using MATLAB/Simulink. Such an equation modeling is more

effective than the transfer function model. In transfer function model, it was

mandatory to develop different model for every input and output parameter

changes. In equation modeling, the voltage and load torque may be the input

parameters the output parameters may be speed, current and deflecting torque

etc.

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As mentioned the system has two loops, namely an inner ON/OFF

current control loop and an outer Hybrid Intelligent control based speed

control loop. The current control loop is used to blocks the PWM signal

whenever the motor current exceeds the reference current (ILref). In outer

speed control loop, the actual speed (k) is sensed by speed sensor and it is

used as a speed feedback. The error signal e(k) is obtained by comparing

actual speed (k) with reference speed r(k). The change in error e(k) can

be calculated from the present error e(k) and pervious error epervious(k). The

error and change in error are given as input to the speed controller. The output

of the controller is denoted as duty cycle dc(k). The change in duty cycle

dc(k) can be calculated from the new duty cycle dc(k) and previous duty

cycle dcprevious(k).

The PWM signal is generated, by comparing the carrier signal and

the duty cycle from the speed controller. Then the PWM signal controls the

output voltage of DC-DC buck converter. The output voltage of the DC-DC

buck converter is varied from zero to maximum input voltage applied, so

wide range of speed control is possible from zero to the rated speed. The

input and output gain of the speed controller can be estimated by simulation.

The Hybrid Intelligent control based speed controller can reduce the error to

zero by changing the duty cycle of the switching signal.

2.2 TRANSFER FUNCTION MODEL OF DC SERIES MOTOR

The stability of the system was done using transfer function model

of DC motor and other analysis of the system was done using equation

modeling of DC motor and DC-DC buck converter. In a DC series motor, the

armature current and field current are the same. Figure 2.2 shows the

equivalent circuit of DC series motor. From the equivalent circuit consider

Ra=Rarm+Rse and La=Larm+Lse+2M

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Figure 2.2 Equivalent Circuit of DC Series Motor

Then the voltage equation is written as

(2.1)

The torque equation is

(2.2)

Let us consider that is the armature current and is the motor speed.

Thus it can be written as

eb and ia (i.e Before saturation)

eb ia

eb= Kaf ia

eb= Kaf ia (2.3)

Armature Voltage Constant

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Figure 2.3 shows the chopper controlled DC series motor circuit.

The chopper gain (Duty Cycle) is considered as .

Figure 2.3 Chopper Controlled DC Series Motor

From the Figure 2.3 the motor voltage i.e chopper output is

(2.4)

The non-linear equations 1 and 2 can be simulated with

MATLAB/simulink. A nonlinear controller is desired to control the speed of

the modeled DC motor. The intelligent based controller is the one of the best

suited non-linear controller, to control the DC motor.

The torque developed is

T ia and ia (i.e Before saturation)

(2.5)

Kt = Torque Constant

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Also from equation (2.2) Torque equation is

The transfer function for DC series motor in this case could no

longer be valid. In order to obtain the transfer function let us consider the

operating point as armature current Ia0, armature voltage Va0, load torque TL0,

back emf Eb0 and residual emf Eres0.

Consider small perturbations around the operating points as ia,

va, td, , vs, tl, eb and eres respectively around the operating points

we can write

(2.6)

Since the quantities ( ia)2 and ( ia) ( ) are very small the above

equations (2.1) to (2.5) can be linearized to

(2.7)

(2.8)

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(2.9)

(2.10)

(2.11)

Applying Laplace Transform to the equation (2.7)-(2.11),

(2.12)

(2.13)

(2.14)

(2.15)

(2.16)

Also

(2.17)

Substituting equation (2.13) and (2.14) in equation (2.15)

(2.18)

))(())(()( 00 resaafafaaaa kIkskSLRsIsV

))(()())(( 00 resaafaafaaa kIkssVkSLRsI

(2.19)

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Put (2.19) in equation (2.16)

(2.20)

By equating the equation (2.20) and (2.17)

(2.21)

(2.22)

(2.23)

Assume that no changes in the load torque

)]()2()[()(2

)()(

00002

0

resaafaafafaaafaa

aaf

a kIkIkBKBRSBLJkJRSJLIk

sVs

(2.24)

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The equation (2.24) is the open loop transfer function of DC series

motor. The motor is fed by DC-DC converter, therefore the transfer function

of chopper fed DC series motor is become,

(2.25)

From equation (2.12)

(2.26)

Substituting the equation (2.24) and (2.26) in (2.25),

(2.27)

(2.28)

The equation (2.28) is the generalized transfer function for chopper

fed DC series motor

where,

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2.3 EQUATION MODELING OF DC SERIES MOTOR

From the voltage equation (2.1) and torque equation (2.2) of DC

series motor the equation modeling of DC series motor is obtained.

Replacing eb and eres in equation (2.1),

(2.29)

(2.30)

Replacing T in equation (2.4),

(2.31)

(2.32)

The simulink model DC series motor was modeled using MATLAB

with the modeling equations (2.30) and (2.32) and shown in Figure 2.4.

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Figure 2.4 Simulink model of DC series motor

Equation (2.28) shows the generalized transfer function for chopper

fed DC series motor and the Table 2.1 shows the DC series motor

specifications. By substituting the parameter in the transfer function model,

the transfer function model of DC series motor was developed with

MATLAB/Simulink. Similarly from the Equation (2.30) and (2.32) with the

same motor parameters the equation model of DC series motor was developed

with MATLAB/Simulink. Figure 2.5 shows the combined open loop transfer

function model and equation model of DC series motor using MATLAB

/Simulink. From the Figure 2.5 the open loop speed response of DC series

motor for transfer function model and equation model was obtained. The open

loop response for both the model is given in Figure 2.6. The response shows

that the both the model gives the same result.

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Table 2.1 220V DC Series Motor Specifications

DC motor Parameters Value

Motor Rating

DC supply voltage

Motor rated Current

Inertia constant J

Damping constant B

Armature resistance Ra

Armature inductance La

Motor Speed

Armature voltage constant Kaf

Residual magnetism voltage const. Kres

5HP

220 V

18 A

0.0465 Kg-m2

0.005 N.m.Sec./rad

1

0.032 H

1800 rpm

0.027 H

0.027 V.Sec./rad

Figure 2.5 DC series motor model using MATLAB /Simulink

(a)Transfer function model and (b)Equation Model

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Figure 2.6 Open loop speed response of DC series motor

2.4 EQUATION MODELING OF DC SEPARATELY EXCITED

MOTOR

Figure 2.7 Equivalent Circuit of DC Separately Excited Motor

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Figure 2.7 shows the equivalent circuit of DC Separately Excited

motor. From the equivalent circuit, the voltage and torque equations are

obtained, which is given in Equation (2.33) and (2.34) respectively.

(2.33)

(2.34)

By rearranging the Equation (2.33) and (2.34), the following

equations were obtained.

(2.35)

(2.36)

The simulink model DC separately excited motor was modeled

using MATLAB with the modeling equations (2.35) and (2.36) and shown in

Figure 2.8.

Figure 2.8 Simulink model of DC separately excited motor

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2.5 MODELING OF DC-DC CONVERTER

The DC-DC converter switch can be a Power Transistor, SCR,

GTO, IGBT, Power MOSFET or similar switching device. In order to get

high switching frequency (upto 100 KHz) the Power MOSFET may be taken

as a switching device. Normally on state drop in the switch is small and it is

neglected.

When the gate pulse is applied the device is turned on. During the

period the input supply connects with the load. When the gate pulse is

removed the device is turned off and the load disconnected from the input

supply. The circuit and waveform of DC-DC converter is shown in

Figure 2.9.

Figure 2.9 DC-DC converter circuit and waveform

The model equation for DC-DC converter is given as

(2.37)

where, and

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The operating modes of DC-DC buck converter is given in Table

2.2. From the table it is infer that the motor is running in forward direction.

Load current i0 is the motor current and V0 is the applied voltage to the motor

voltage. The i0 and V0 are shown in Fig. 2.9. Here Mode 1 is forward

motoring mode and mode 2 is freewheeling mode. During mode 1 the motor

voltage is Vs and mode 2 is 0. The entry +ve is mentioned to represent that the

motor is taking positive current when the motor is running in both forward

motoring mode and freewheeling mode.

Table 2.2 DC-DC buck converter switching operation

Operating mode

Switch Position Converter

Output voltage V0

Load (Motor) Current

io Motoring (Mode 1)

Freewheeling (Mode 2)

Mode1 Mode2

Forward motoring

MOSFET (Q) ON

Diode (DF) ON

Vs 0 + ve

2.6 CONCLUSION

The transfer function model and equation modeling of DC series

motor, equation modeling of DC separately excited motor and equation

modeling of DC-DC converter were developed. These models are to be

utilized for the simulation of the system using MATLAB/Simulink.