MODELING OF CONTROLLED DC MOTOR WITH A CHOPPER …asmarya.edu.ly/journal/wp-content/uploads/2016/08/7.pdf · Figure 1 shows the schematic diagram of the open loop controlled DC motor

مجلة الجامعة األسمرية للعلوم األساسية والتطبيقية

6112، يونيو (1، العدد )( 1) السنة

44

MODELING OF CONTROLLED DC MOTOR WITH A

CHOPPER CIRCUIT

Abdulmajed Elbkosh Almergib University, Faculty of Engineering, Department of Electrical and Computer Engineering

Alkhoms, Libya, [email protected]

ABSTRACT

To design a control system for any system, it is required to model the system's dynamic

behaviour. The modeling and analysis of these systems can be either numerical or analytical.

This paper studies the analytical methods of modeling DC motors mainly continuous-time

averaging technique and discrete-time approach. It shows that the discrete-time approach

provides clear expressions representing the operation of DC motors and thus can help

designers gain a better understanding of the performance of the motor. The results are shown

using numerical simulations.

KEY WORDS

DC motor, discrete map, linearization, nonlinear dynamic.

1. Introduction

The electrical motor is a motor that convert electrical enegy into mechanical energy. In

general there are two types of motors which are AC motors and DC motors. A simple motor

uses electricity and magnetic field for producing torque which rotate the motor.

DC motors outperform to AC motors in terms of providing better speed control on high

torque loads, use in wide indestrial applications, and more useble as it designed to use with

batteries and solar cells energy sources which provide protabillity where it is required and

thus provide cost effective solution as it is not possible to have an AC power supply in every

place.

Usually DC motors show responses at both voltage and current, the applied voltage

describes the speed of the motor while the current in the armature windings shows the torque.



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Generally it can be said that the applied voltage effect the speed of the motor while the torque

is controlled by the current. One of the most effective method of controlling the speed of the

motor is using a chopper circuit which has been used in this paper.

The analysis of controlled DC motor with a chopper circuit, which are nonlinear systems,

requires the use of nonlinear mathematical techniques. However, unlike linear systems where

the solutions can be written in closed-form in terms of the system’s eigenvalues and

eigenvectors, most nonlinear systems are known to be very hard to solve analytically i.e. there

is no general method for obtaining a closed-form solution for these systems. To work around

this problem, power electronics engineers normally use the technique of averaging for

determining the stability and dynamic behaviour of switching systems [5-7]. In this method,

the actual nonlinear system is linearised around a steady-state operating point to yield a linear

model. This averaging method gives a simple and accurate model at low operating

frequencies. Another approach to study the stability of switching systems is based on the

construction of a sampled data model [7-11]. Stability can be achieved by explicitly deriving

a discrete map that describes the evolution of the state from one clock instant to the next and

then locally linearising the map around its steady-state operating point.

In this paper, the controlled DC motor with a chopper circuit will be modeled using two

approaches, the continuous-time averaging approach and discrete-time iterative map.

2. Basic operation of the DC motor

Figure 1 shows the schematic diagram of the open loop controlled DC motor. It contains of a

source of voltage E, a chopper circuit which consists of a diode D and a switch S controlled

by a pulse-width modulated signal. The required output voltage of the chopper circuit is

achieved by setting the switch's duty cycle d., and an equivalent circuit of DC motor which is

connected to the chopper circuit.

The simple DC motor has the following a physical parameters:

R: The armature resistance, (ohm)

L: The armature inductance, (H)

J: The moment inertia of the motor and load, (Kg m2 /s

2)

B: The damping ratio of the mechanical system, (Nms)



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TL: The load torque, (N.m)

:ω The speed of the shaft and the load (angular velocity), (rad/s)

iL: The armature current, (A)

Te: The electrical torque, (N.m)

In addition the DC motor has the following constants:-

:K t The torque factor constant, (Nm/Amp)

:Ke The back emf constant , (V/rad)

DC motorChopper Circuit

L

D

S

E

R

J

B

KE ω

KT i

TL

Figure 1 Open loop DC motor

During the interval when the switch is ON, the diode is reverse biased and the input voltage

provides energy to the load and the inductor. During the second interval when the switch is

OFF, the inductor current flows through the diode transferring some of its stored energy to the

load.

During the first interval ton, when the switch is closed, the equations that represent the DC

motor operating in continuous conduction mode are:

LtL iKBωTJ

1

dt

dω (1)

LeL RiωKE

L

1

dt

di (2)

These equations can be written in state space form as:

UBtxAtx ONON (3)



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TTL xxi 21 x is the state vector, and TL ETU is the input vector. A and B are the

system matrices that contain the system parameters, defined as:

R/L/LK

/JKJ

B

A

e

t

ON ,

1/L0

0J

1

BON (4)

During the interval toff, when the switch is open, the equations that represent the DC motor

operating in continuous conduction mode are:

LtL iKBωTJ

1

dt

dω (5)

LeL RiωK

L

1

dt

di (6)

These can be described in state space form as:

Ett OFFOFF BxAx (7)

where

R/L/LK

/JKJ

B

A

e

t

OFF ,

00

0J

1

BOFF (8)

The ratio of ton to T is the duty cycle d. It is defined as Ttd on / where offon ttT is the

period of one cycle.

In this paper, the following parameters of the dc motor are used to investigate the behaviour

of the circuit.

L = 53.7mH, , R = 2.8Ω , B=0.000275 Nms, J=0.000557 Kg m2 /s

2, TL=0.38N.m , Ke=0.1356

V/rad, and Kt=0.1324 Nm/Amp

3. Modeling of DC MOTORS

To design a control system for a DC motor, it is necessary to model the motor's dynamic

behaviour. The modeling and analysis of these systems can be either numerical or analytical.



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In numerical techniques, various algorithms are used to produce quantitative results. These

methods are accurate, powerful and easy to use, however, they fail to provide the design

insight needed to understand the behaviour of the converters i.e. they are not ideal for tasks

such as stability analysis or controller design [3-7]. On the other hand, analytical techniques

provide clear expressions representing the operation of dc motors and thus can help designers

gain a better understanding of the performance of the system.

Analytically, there are two main approaches to model and analyze switching systems.

The first approach is the state space averaging model proposed by Middlebrook and Ćuk

[5], also known as the continuous-time averaging model. This method is widely used to

analyze and design circuits at low operating frequencies.

The sampled data model or discrete-time iterative map, first reported by Lee [8] and

Verghese [10], is another approach to the modeling of switching systems. In this analysis, the

switching continuous system is replaced by a discrete system that describes the states of the

system at the switching frequency.

3.1 Continuous-time averaging approach

The conventional way of modelling any switching system is to take an average of the state

variables over one switching cycle. In this model, the system is linearised around a steady-

state operation to yield a linear model. This makes it possible to use a Laplace transform

domain analysis which is useful for control theory. In addition this method approximates the

original circuits by continuous-time systems which make it easier for the designer to design a

feedback controller.

In the DC motor, the state space averaged model can be derived by taking the average of

the states during ON and OFF intervals. This can be made by multiplying Equation (3) by dT

(the time when the switch is ON) and Equation (7) by Td (the time when the switch is OFF)

and then summing the result to take the average over one cycle.

This yields the following time-varying continuous system:

Et

Eddtddt

BxA

BBxAAx

ONOFFONOFF

(9)



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where d is the ratio of toff to T and related to the duty ratio as d = 1- d.

The model of the converter becomes a nonlinear time invariant model. Perturbing the above

system around the steady-state point EEP BAx1 and linearizing the system around that

point, a small signal model can be obtained. The small signal model is very useful for:

Studying the dynamic behaviour of the system.

Designing a suitable feedback controller using any standard linear technique such as the

Nyquist criterion or root-locus method.

The simulation results of the open loop DC motor modeled using the averaging approach are

shown in Figure 2 and Figure 3.

Figure 2 Inductor current waveform at input voltage of 20V and d =0.5;

Figure 3 Output voltage waveform at input voltage of 20 V and d =0.5;

From the figures it can be seen that the state space averaging model smoothed out the ripple

and captures only the basic dynamics.

This approach discards the switching details and hence it fails to predict the fast-scale

dynamics that are higher than the switching frequency of the system and can capture only the

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

1

2

3

4

5

6

7

Time(sec)

Cu

rren

t(A

mp

)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

20

40

60

80

100

120

140

Time (sec)

Sp

eed

(ra

d/s

ec

)



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slow-scale dynamics that are much slower than the switching frequency of the system [8, 12].

Therefore a more comprehensive method of analysis is needed to obtain all necessary

information about the system’s behaviour.

3.2 Discrete-time iterative map approach

In order to explore the nonlinear phenomena such as sub-harmonic and chaos which may

appear across a wide range of frequencies, an alternative modeling approach which is based

on deriving the exact discrete model must be used [3, 16]. The aim of this approach is to

transform a continuous-time system into a discrete-time system by sampling its states at

chosen instants. This modeling technique can predict the fast-scale dynamics as well as the

slow-scale dynamics. Different types of maps have been proposed in the literature for

switched systems, but the most common maps are the stroboscopic map where the states are

sampled at the beginning of the switching cycle [11] and the impact map where the states are

sampled at the switching instants [12].

In this paper, the stroboscopic map is used to obtain the discrete model i.e. the system

states are periodically sampled at fixed time instants t= nT as shown in Figure 4. The state

variables x n+1 can be calculated from the previous sampling instant values x n by the

following map:

nnn df ,1 xx (10)

t

x1

xn x n+1 x n+2

nT (n+1)T (n+2)T

x2

Figure 4 Discrete model of a periodic dynamical system

For instance, the exact discrete time model of the buck converter operating in continuous

conduction mode can be obtained by stacking the consecutive solutions of (3), (7) over a



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switching period. The result is a discrete time difference equation which can be written in

state space form as:

Endnndn )()(1 ΓxΦx (11)

where

)()()( OFFON ndndnd ΦΦΦ (12)

and

d

d)()(

ON0 ON

OFF'

0 OFFON

E

Endnd

dT

Td

BΦ

BΦΦΓ

(13)

ФON and ФOFF are the state transition matrices during ON and OFF intervals, given by:

Tded

'ON

ONAΦ and dT

ed OFFOFF

AΦ (14)

ФON and ФOFF can be calculated by the series:

1

ON'

!ON

m

mTd

m

Tde

AI

A (15)

Equation (11) describes the dynamic of the DC motor however, it is complex and

transcendental in form; therefore, there is no easy method for analyzing the behaviour of the

system [14-20].

The output waveforms of the open loop DC motor modeled by the sampled data approach

are shown in Figure 5 and Figure 6

Figure 5 Inductor current waveform at input voltage of 20V and

d =0.5;

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

1

2

3

4

5

6

7

8

9

Time(sec)

Cu

rren

t(A

mp

)



444

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-20

0

20

40

60

80

100

120

140

160

Time (sec)

Sp

eed

(ra

d/s

ec)

Figure 6 Output voltage waveform, at input voltage of 20V and d =0.5;

The waveforms show that the discrete time model captures the complete dynamics of the

DC motor. Hence it can predict the fast scale dynamics as well as the slow scale dynamics.

4 CONCLUSION

This paper outlined the basic topology of controlled DC motor by a chopper circuit and the

conventional methods of modeling switching systems were presented.

It was shown that, modeling the system using the continuous-time averaging technique is

easier when compared to the discrete-time approach, however, the former method discards the

switching details and hence it fails to predict the fast-scale dynamics. Therefore, the later

method of analysis is needed to obtain all necessary information about the system’s

behaviour.

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Documents

MODELING OF CONTROLLED DC MOTOR WITH A CHOPPER …asmarya.edu.ly/journal/wp-content/uploads/2016/08/7.pdf · Figure 1 shows the schematic diagram of the open loop controlled DC motor