3. ARMATURE VOLTAGE AND GOVERING EQUATIONS

1). Armature Voltage（电枢中的感应电势）The induced voltage in a conductor is

）（1lvBe xx Then the induced voltage across armature terminal is

)2(2

1

2

1

2

1

aN

x

aN

x

aN

xa BlvlvBeE

Let Bp express average flux density under one pole.

0)3()(

1dxxBBp

)4(2

2

1p

aN

x Ba

NB

From Eqs. (2) and (4),

)5(pa Ba

NlvE

Let )6(60

pnv

)7(lA

Bp

Substituting Eqs. (6) and (7) into Eqs. (5) and simplifying,

)8(nCE ea

a

pNCe 60

where --- constant

nCE ea

Question : If saturation appears in magnetic field, will Ea become larger or smaller?

Note: Ea exists both in generator and in motor.

In generator, Ea produces armature current. It is called generated emf or positive voltage.

In motor, supply voltage U produces armature current and Ea opposes current to flow into armature winding. In this case it is called back emf or negative voltage.

2). Governing Equations of steady-state operation

Generator :

Example: Separately excited generator

• Governing equation of armature circuit

saaa URIEU 2where aR --- resistant of armature winding

VU s 22 --- voltage drop caused by brush contact resistant

• Features : (1). UEa (2). Ia is produced by Ea and has the same direction as Ea.

Ea is positive voltage.(3). Torque T is called negative torque because it oppose n.

• The total equations are

II

IRU

IRU

URIEU

a

L

fff

saaa 2

MotorExample: Separately excited motor

• Features : (1). UEa (2). Ia is produced by U and has the opposite direction from Ea.

Ea is negative voltage.

(3). Torque T is called positive torque because it produces n.

• Governing equation of armature circuit

saaa URIEU 2• Total equations

II

IRU

URIEU

a

fff

saaa 2

Summary

Before writing governing equations, please draw schematic diagram according to excitation modes and then give correct reference direction in terms of operation modes.

Example: Shunt generator and shunt motor

4. POWER FLOW AND EFFICIENCY

1). Losses

• Copper loss: armature circuit loss

field circuit loss

brush loss

aaa RIp 2)( 22

sasfff RIpRIp

asb IUp 2

• Core Loss fep 3.12 fB

• Mechanical loss --- against windage and friction

mp

• Additional loss (stray loss)

--- cannot be computed accurately

2)%1~5.0( Pp

2). Power Flowing diagram

Example: compound dc machine Compound generator

inputP

mp Fep pMP

fp ap sp bpoutputP

Mechanical power electrical power

Developed power (electromagnetic power): aaM IEP --- It is the total electrical power developed by or converted from the mechanical energy.

powermechanicalPPinput 1

UIPPoutput 2

bsafM

bsafM

FemM

ppppPpPP

PppppP

pppPP

12

2

1

• Power equations from power flow diagram:

Compound motor

inputP

mp Fep pMP

fp ap bp

outputP

sp

electrical power Mechanical power

In motors, PM is the total mechanical power converted from input electrical power.

(3). Efficiency

%1001

2 P

P

5. DEVELOPED TORQUE (ELECTROMAGNETIC TORQUE)

The electromagnetic force acting on a conductor is

cxx liBF where ic is the current of conductor.

The force on the total conductors under each pole is

p

N

cxp liBF1

The force on all conductors of armature winding is

)1(1

pN

cxp liBppFF

The developed torque produced by armature winding is

)2(2

DFT

where D is the outer diameter of rotor.

Let

)4(

)3(

1p

pN

x

ac

Bp

NB

a

Ii

)6(

)5(

p

D

lBp

Substituting Eqs. (1), (3), (4), (5) and (6) into Eqs. (2) and simplifying, we obtain

aT ICT

a

pNCT 2

Where is a constant.

Questions

1. What is the affect of armature reaction to T if core involves in saturation ?

2. If load torque becomes larger, how about Ia under constant external voltage in shunt motor?

3. In shunt motor, if external voltage and load torque are all kept constant, when we put an external resistance into armature circuit, how will Ia change ?

Useful Expression

TIa

pN

Ia

pNIn

a

pNIEP

a

aaaaM

2

6026060

MPT --- calculating T from power

Governing Equation for torque

In generator,

01 TTT

pppT Fem

0 --- torque under no-load operation

1

1

PT --- input mechanical torque

T is called an opposed torque. T1 has to overcome T to make prime mover deliver its mechanical energy into generator.

In motor,

02 TTT

22

PTwhere

T is called an active torque. It drives motor and load to rotate.