DC Drives - K13 - DC motor model .pdf

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SEPARATELY ECXITED DC MOTOR

Applied Newtonian mechanics to find the differential equations for mechanical systems.

Using Newton’s second law:

Electromagnetic torque developed by separately excited DC motor:

Viscous torque :

Load torque : TL

dt

dJJT

J : equivalent moment

of inertia

afafe iiLT

rmviscous BT

Equivalent circuit for separately excited DC motors

VOLTAGE

SUPPLY

LOAD

rfafa iLE

+

-

er T,LT

+

-

ar

ai

arr

aL

frr

fi

fr

fu

fL

auaxisquadrature

axisdirect

armature

field

SEPARATELY EXCITED DC MOTORS

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LrmfaafLviscouse

r TBiiLJ

TTTJdt

d

11

a

a

rf

a

af

a

a

aa uL

iL

Li

L

r

dt

di 1

f

f

f

f

ffu

Li

L

r

dt

di 1

J

T

J

Bii

J

L

dt

d Lr

mfa

afr

From Newton’s Second Law, Torsional-Mechanical equation is given as

The nonlinear differential equation for separately excited DC motor which is

found using Kirchhoff’s Voltage Law


Using Newton’s second law :

Dynamics of rotor angular displacement :

The derived three first order differential equations are rewritten in the

s-domain

LrmfaafLviscouse

r TBiiLJ

TTTJdt

d

11

rr

dt

d

)()()(

1)( sussiL

rsLsi arfaf

aa

a

)(

1)( su

rsLsi f

ff

f

Lfaaf

m

r TsisiLBJs

s

)()(1

)(

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x

x

aa rsL

1

ff rsL

1

mBsJ

1

afL

afL

fu

auai

eT

LT

fi

r

SEPARATELY ECXITED DC

GENERATOR

pmrmfaafpmviscouse

r TBiiLJ

TTTJdt

d

11

a

a

rf

a

af

a

a

aa uL

iL

Li

L

r

dt

di 1

f

f

f

f

ffu

Li

L

r

dt

di 1

J

T

J

Bii

J

L

dt

d Lr

mfa

afr


The nonlinear differential equation for separately excited DC generator which is


The expression for the voltage at the load terminal must be used.

For the resistive load LR

au

aLa iRu

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Analysis of eqn(3) indicates that the angular velocity of the separately excited motor can be regulated by changing the applied voltages to the armature and field windings.

The flux is a function of the field current in the stator winding, and higher angular velocity can be achieved by field weakening by reducing the stator current [eqn(3)]

However, there exists a mechanical limit imposed on the maximum angular velocity. The maximum allowed (rated) armature current is specified as well, one concludes that the electromagnetic torque is bounded.

afafe iiLT

fi

fuau

)3(

2

e

faf

a

faf

a

faf

aaar T

iL

r

iL

u

iL

iru

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A separately excited, 2 kW DC motor with rated armature current 20 A and angular velocity 200 rad/s operates at the constant voltages and . The motor parameters are: , , , and .

Calculate:

The steady state angular velocity at the minimum and maximum load conditions, Nm and Nm.

The armature current at the minimum and maximum load conditions, Nm and Nm.

Vua 100 Vu f 20 18.0ar 5.3fr 1.0afL

radNmsBm /007.0

0min LT

0min LT

10max LT

10max LT

Example

Steady state condition Le TT

f

f

fr

ui

)3(

2

e

faf

a

faf

a

faf

aaar T

iL

r

iL

u

iL

iru

NmTL 0min

NmTL 10max

rr 007.07.51.0

18.0

7.51.0

1002

rr 007.010

7.51.0

18.0

7.51.0

1002

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Steady state condition Le TT

faf

rmL

faf

ea

iL

BT

iL

Ti

NmTL 0min

NmTL 10max

faafe iiLT

7.51.0

007.0 min

r

faf

ea

iL

Ti

7.51.0

007.010 min

r

faf

ea

iL

Ti

Example

Plot the torque-speed characteristic curves for a

separately excited, 2-kW DC motor if the

rated (maximum) armature voltage is

and the field voltage is . The

motor parameters are: , ,

, and

The load characteristic if

Vua 100max

Vu f 20

rmLL BTT 0

18.0ar 5.3fr

1.0afL radNmsBm /007.0

NmTL 50

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% parameters of separately-exited motor

ra=0.18; Laf=0.1; Bm=0.007; If=5.7; Tl0=5;

Te=0:1:10;

for ua=11:10:100;

wr=ua/(Laf*If)-(ra/((Laf*If)^2))*Te;

wrl=0:1:200; Tl=Tl0+Bm*wrl;

plot(Te,wr,'-',Tl,wrl,'-');hold on;

axis([0, 10, 0, 160]);

end; disp('End')


(cont~)

%transient dynamics of a separately excited dc motor

function yprime=difer(t,y);

ra=0.18; rf=3.5; La=0.0062; Lf=0.0095; Laf=0.1; J=0.04; Bm=0.007;

T1=0;

%T1=10;

ua=100; uf=20;

yprime=[(-ra*y(1,:)-Laf*y(2,:)*y(3,:)+ua)/La;...

(-rf*y(2,:)+uf)/Lf;...

(Laf*y(1,:)*y(2,:)-Bm*y(3,:)-T1)/J];

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(cont~)

%transient dynamics of a separately excited dc motor

clc

t0=0; tfinal=0.4; tol=1e-7; trace=1e-7; y0=[0 0 0]';

[t,y]=ode45('CHP5_1mdno',t0,tfinal,y0,tol,trace);

subplot(2,2,1); plot(t,y(:,1),'r-');

xlabel('Time (seconds)'); title('Armature Current ia, [A]');

subplot(2,2,2); plot(t,y(:,2),'g-.');

xlabel('Time (seconds)'); title('Field Current if, [A]');

subplot(2,2,3); plot(t,y(:,3),'b-');

xlabel('Time (seconds)'); title('Angular Velocity wr, [rad/s]');

subplot(2,2,4);plot(t,y(:,1),'r-',t,y(:,2),'g-.',t,y(:,3),'b-')

xlabel('Time (seconds)'); title('LAB 1');


(cont~)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-100

0

100

200

300

X: 0.03529Y: 270.5

Time (seconds)

Armature Current ia, [A]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

1

2

3

4

5

6

Time (seconds)

Field Current if, [A]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

50

100

150

200

250

Time (seconds)

Angular Velocity wr, [rad/s]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-100

0

100

200

300

Time (seconds)

LAB 1

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(cont~)

tl and te

field current

combine

armature current

angular velocity

1

0.04s+0.007

Transfer Fcn2

1

0.0095s+3.5

Transfer Fcn1

1

0.0062s+0.18

Transfer Fcn

combine

To Workspace

Step1

Step

SignalGenerator

Product1

Product

1

Gain5

1

Gain4

1

Gain3

1

Gain2

0.1

Gain1

0.1

Gain

SHUNT CONNECTED DC MOTOR

The armature and field windings are connected in parallel

VOLTAGE

SUPPLY

LOAD

rfafa iLE

+

-

er T,LT

+

-

ar

ai

arr

aL

frr

fifr

fu

fL

auaxisquadrature

axisdirect

armature

field

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SHUNT CONNECTED DC MOTOR

LrmfaafLviscouse

r TBiiLJ

TTTJdt

d

11

a

a

rf

a

af

a

a

aa uL

iL

Li

L

r

dt

di 1

;1

f

f

f

f

ffu

Li

L

r

dt

di

J

T

J

Bii

J

L

dt

d Lr

mfa

afr


The nonlinear differential equation for separately excited DC motor which is


fa uu

Steady state condition fa uu

f

af

r

ui

a

rfafa

ar

iLui

Substituting the currents equation into torque equation, gives

faafe iiLT

21 a

f

raf

fa

af

e ur

L

rr

LT

It shows that

The electromagnetic torque is a linear function of the angular velocity

The electromagnetic torque varies as the square of the armature voltage applied

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SHUNT CONNECTED DC

MOTOR (Example)

A shunt connected motor, drives a fan.

Given

When one applies the angular velocity is 150rad/s. For steady state operating condition and assuming the viscous friction is negligibly small, find the developed electromagnetic torque and the currents in the armature and field windings

,12.0,23,0,15.0 affraraf rrrrL

Vua 100

SHUNT CONNECTED DC

MOTOR (cont~)

21 a

f

raf

fa

af

e ur

L

rr

LT

mNTe .8.1110023

15015.01

2312.0

15.0 2

Ar

ui

f

f

f 35.423

100

AiL

Ti

faf

ea 1.18

35.415.0

8.11

faafe iiLT

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SERIES CONNECTED DC MOTOR

The armature and field windings are connected in series

VOLTAGE

SUPPLY

LOAD

rfafa iLE

+

-

er T,LT

+

-

ar

fa ii

arr

aL

fr

fL

auaxisquadrature

axisdirect

armature

field

Steady state condition 0dt

dia

dt

diLLirriLu a

faafaraafa

Then, currents equation

2

aafe iLT

21 a

f

raf

fa

af

e ur

L

rr

LT

It shows that

The developed electromagnetic torque is proportional to the square of the current

Saturation effect should be taken into account

The nonlinear differential equation for series connected DC motor which is


faraf

aa

rrL

ui

Substituting the currents equation into torque equation, gives

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SERIES CONNECTED DC

MOTOR

LrmfaafLviscouse

r TBiiLJ

TTTJdt

d

11

a

fa

ra

fa

af

a

fa

faa uLL

iLL

Li

LL

rr

dt

di

1

J

T

J

Bi

J

L

dt

d Lr

ma

afr 2


The nonlinear differential equation for series connected DC motor which is


04/01/2013

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Documents

DC Drives - K13 - DC motor model .pdf