Dc Machine

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


LrmfaafLviscouser TBiiL

JTTT

Jdt

d 11

aa

rfa

afa

a

aa uL

iL

Li

L

r

dt

di 1

ff

ff

ff uL

iL

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


JTTT

Jdt

d 11

rr

dt

d

)()()(1

)( sussiLrsL

si arfafaa

a

)(1

)( sursL

si fff

f

Lfaafm

r TsisiLBJs

s

)()(1

)(


x

x

aa rsL 1

ff rsL 1

mBsJ 1

afL

afL

fu

auai

eT

LT

fi

r

SEPARATELY ECXITED DC GENERATOR

pmrmfaafpmviscouser TBiiL

JTTT

Jdt

d 11

aa

rfa

afa

a

aa uL

iL

Li

L

r

dt

di 1

ff

ff

ff uL

iL

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 found using Kirchhoff’s Voltage Law

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

LRau

aLa iRu

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


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 conditionLe TT

f

ff r

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

Steady state conditionLe 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 therated (maximum) armature voltage isand the field voltage is . Themotor 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

% parameters of separately-exited motorra=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')

SEPARATELY ECXITED DC MOTOR (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];

SEPARATELY ECXITED DC MOTOR (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');

SEPARATELY ECXITED DC MOTOR (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

SEPARATELY ECXITED DC MOTOR (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+0.18Transfer 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

SHUNT CONNECTED DC MOTOR


JTTT

Jdt

d 11

aa

rfa

afa

a

aa uL

iL

Li

L

r

dt

di 1

;1

ff

ff

ff uL

iL

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 found using Kirchhoff’s Voltage Law

fa uu

Steady state conditionfa uu

f

af r

ui

a

rfafaa r

iLui

Substituting the currents equation into torque equation, gives

faafe iiLT

21 af

raf

fa

afe u

r

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

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 af

raf

fa

afe u

r

L

rr

LT

mNTe .8.1110023

15015.01

2312.0

15.0 2

Ar

ui

f

ff 35.4

23

100

AiL

Ti

faf

ea 1.18

35.415.0

8.11

faafe iiLT

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

2aafe iLT

21 af

raf

fa

afe u

r

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 found using Kirchhoff’s Voltage Law

faraf

aa rrL

ui

Substituting the currents equation into torque equation, gives

SERIES CONNECTED DC MOTOR


JTTT

Jdt

d 11

afa

rafa

afa

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 found using Kirchhoff’s Voltage Law

Documents

Dc Machine