2 modeling of dc machines

8/10/2019 2 Modeling of DC Machines

http://slidepdf.com/reader/full/2-modeling-of-dc-machines 1/23

Modeling of DC MachinesBy

Dr. Ungku Anisa Ungku AmirulddinDepartment of Electrical Power Engineering

College of Engineering

Dr. Ungku Anisa, July 2008 1EEEB443 - Control & Drives



Outline Introduction

Theory of Operation

Field Excitation Separately Excited DC Motor

State-Space Modeling

Block Diagrams and Transfer FunctionsMeasurement of Motor Constants

References

Dr. Ungku Anisa, July 2008 EEEB443 - Control & Drives 2



IntroductionDC motor in service for more than a century

Dominated variable speed applications before

Power Electronics were introduced

Advantage:

Precise torque and speed control without

sophisticated electronics




Introduction Some limitations:

High maintenance (commutators & brushes)

Expensive Speed limitations

Sparking

Commonly used DC motors Separately excited

Series (mostly for traction applications)




DC Machine – Theory of Operation Field winding - on stator pole

i f produces f Armature winding –on rotor

i a produces a f and a mutually

perpendicular maximum torque

Rotor rotates clockwise

For unidirectional torque and

rotation i a must be same polarity under

each field pole

achieved using commutatorsand brushes




DC Machine – Field Excitation Depends on connections of field winding relative to

armature winding

Types of DC machines: Separately Excited

Shunt Excited

Series Excited

Compounded Permanent Magnet




DC Machine – Field Excitation Separately Excited

Field winding separated from armature winding

Independent control of i f ( f ) and i a (T )




DC Machine – Field Excitation Shunt Excited

Field winding parallel toarmature winding

Variable-voltage operationcomplex

Coupling of f (i f ) and T (i a)production

T vs characteristic almostconstant

AR = armature reaction

(as T , i a , armature fluxweakens main flux f , )




DC Machine – Field Excitation Series Excited

Field winding in series witharmature winding

Variable-voltage operationcomplex

Coupling of f (i f ) and T (i a)production

T i a

2 since i f = i

a

High starting torque

No load operation must beavoided (T = 0, )




DC Machine – Field Excitation Compounded

Combines best feature of

series and shunt

Series – high starting torque

Shunt – no load operation

Cumulative compounding

shunt and series field

strengthens each other.

Differential compounding

shunt and series field

opposes each other.


Long-shunt

connection

Short-shuntconnection



DC Machine – Field Excitation Permanent Magnet

Field provided by magnets

Less heat

No field winding resistivelosses

Compact

Armature similar toseparately excited

machine Disadvantages:

Can’t increase flux

Risk of demagnetisationdue to armature reaction




Lf Rf

if

aaaaaa edt

di Li Rv

+

ea

_

La Ra

ia +

vt

_

+

vf

_

Separately Excited DC Machine


dt di Li Rv f

f f f f

abae i K i K T Electromagnetic torque

ba K K e Armature back e.m.f.

Armaturecircuit

Fieldcircuit



Separately Excited DC Motor Motor is connected to a

load.

Therefore,

where

T L= load torque

J = load inertia (kg/m2)

B = viscous friction

coefficient (Nm/rad/s)


Le T Bdt

d J T



DC Machine - State-Space

Modeling DC motor dynamic equations:

Therefore,


aa

aaaa edt

di Li Rv

Le T Bdt

d J T

a

ba

a

a

a

aa

L K v

Li

L R

dt di 1

Lab T

J J

Bi

J

K

dt

d 1

abae i K i K T

ba K K e (1) (2)

(3) (4)

(5)

(6)




Modeling From (5) and (6), the dynamic equations in state-space

form:

where s = differential operator with respect to time

This can be written compactly as:


L

aaa

b

a

b

a

a

a

T v

J

Li

J B

J K

L

K

L

R

s si

10

01

BUAXX

(7)

(8)




Modeling Comparing (7) and (8):


J B J

K

L

K

L

R

b

a

b

a

a

A

vector variablestate-----T

ai X

vector input-----T

La T vU

J

La

10

01

B




Modeling The roots of the system are the eigenvalues of matrix A

1 and 2 always have negative real part, i.e. motor isstable on open-loop operation.


J

B J

K L

K

L

R

b

a

b

a

a

A

a

b

a

a

a

a

a

a

JL

K

JL

B R

J

B

L

R

J

B

L

R 22

21 4

2

1

2

1, (9)



DC Machine – Block Diagrams

and Transfer Functions Taking Laplace transform of (1) and (3) and neglecting initial

conditions:

These relationships can be represented in the following block

diagram


aa

b

L R

K

s

sωsVsI aa

J B

K b

s

sTsIsω

La

(10) (11)

aa L R s

1

J B s

1

K b

TL(s)

Te(s)Ia(s)

Va(s)

K b

(s)+

+

-

-



DC Machine – Block Diagrams

and Transfer Functions From the block diagram, the following transfer functions can be derived:

Since the motor is a linear system, the speed response due to simultaneous

V a input and T L disturbance is:

The Laplace inverse of (14) gives the speed time response (t).


22a

ωV

sssV

sωsG

a

baaaa

b

K BR JR BL JL

K

(12)

(13)

22L

ωT

ss

s

sT

sωsG

L

baaaa

aa

K BR JR BL JL

L R

sTsGsVsGsω LωTaωV La (14)



DC Machine – Measurement of

Motor Constants To analyse DC motors we need values for Ra, La and K b

Armature Resistance Ra

DC voltage applied at armature terminals such that rated i aflows

This gives the dc value for Ra

Need to also correct for temperature at which motor is

expected to operate at steady state

Similar procedure can be applied to find R f of field circuit


rated,

resistancecontact

a

brushdca

i

V V V R




Motor Constants Armature Inductance La

Apply low AC voltage throughvariac at armature terminals

Measure i a

Motor must be at standstill(i.e. = 0 and e = 0)

f = supply frequency in Hz

Ra = ac armature resistance

Similar procedure can beapplied to find L f of field circuit


f

R I

V

L

a

a

a

a 2

2

2

(variac)




Motor Constants EMF Constant K b = K

Rated field voltage appliedand kept constant

Shaft rotated by another dcmotor up to rated speed

Voltmeter connected toarmature terminals givesvalue of E a

Get values of ea at differentspeeds

Plot E a vs.

Slope of curve = K b

Units of K b = [V/rads-1]


E a (V)

(rad/s)



References Krishnan, R., Electric Motor Drives: Modeling, Analysis and

Control , Prentice-Hall, New Jersey, 2001.

Chapman, S. J., Electric Machinery Fundamentals, McGraw Hill,

New York, 2005.

Nik Idris, N. R., Short Course Notes on Electrical Drives,

UNITEN/UTM, 2008.

Ahmad Azli, N., Short Course Notes on Electrical Drives,

UNITEN/UTM, 2008.

Dr Ungku Anisa July 2008 23EEEB443 - Control & Drives

2 modeling of dc machines

Documents