chapter 8: three-phase circuits - oakland...

2/3/00 Electromechanical Dynamics 1

Chapter 8: Three-Phase Circuits


Three-phase circuits

• A 3-phase generator has three identical windings on the stator, placed at 120º to each other– at constant speed, the voltages

induced in the windings have the same effective value

– the voltage peaks occur at different times

-120

-80

-40

0

40

80

120

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2


Three-Phase Generation

• The generation of three-phase ac power can be viewed as three separate power sources, each with a 120º degree angular shift in time of the sinusoidal waveform with respect to the other sources.

( )( )( )ππ

πππ

32

32

602sin)(

602sin)(

602sin)(

+=−=

=

tVtv

tVtv

tVtv

mc

mb

ma

a

bcn

va(t)vc(t)

vb(t)


Three-Phase Waveforms

-120

-80

-40

0

40

80

120

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

va

vb

vc

n


Wye connected circuits

bcn

va(t)vc(t)

vb(t)

a

R

R

R

bc

a

clineccalineacca

blinebbclinecbbc

alineaablinebaab

iivvvv

iivvvv

iivvvv

−−

−−

−−

==−===−===−=


Delta connected circuits

bc

va(t)vc(t)

vb(t)

a

bc

a

clinebccaccalineca

blineabbcbbclinebc

alinecaabaablineab

iiiivv

iiiivv

iiiivv

−−

−−

−−

=−===−===−==

R

R R


Three-Phase Circuits

• Example– the 3-phase generator shown below is connected to three 20

ohm load resistors. The generator supplies a voltage of 120 V rms

– calculate• the power dissipated in each resistor and in the 3-phase load

• the peak power Pm and the ratio of Pm to the total load



• Example– a 3-phase, 60 Hz, wye connected generator has a line voltage

of 23900 V

– calculate• the line-to-neutral voltage

• voltage induced in the individual windings

• the time interval between the positive peak voltage of phases A and B

• the peak value of the line voltage



• Example– three identical impedances are connected in delta across a

550 V line. The current drawn is 10 A

– calculate• the current in each impedance

• the impedance values


Phase Sequence

• 3-phase systems have a property called phase sequence– it determines the direction of

rotation of 3-phase motors

– it means the order in which the three line voltages become successively positive

– two sequences: ABC & ACB

• positive sequence: ABC

• negative sequence: ACB

• Phase sequence is determined by a simple circuit– one lamp always burns brighter

– the sequence order is bright lamp, dim lamp, capacitor


Power Measurements

• Wattmeters measure active power– single-phase wattmeters have two pairs

of terminals: voltage & current

– each terminal pair has polarity markings

– polarity markings determine the direction of power flow

• 3-phase, 3-wire circuits– power is measured by two single-phase

wattmeters

– total power is the sum of the two meters

• 3-phase, 4-wire circuits– power is measured by three single-phase

wattmeters

– total power is the sum of the three meters


Power Measurements

• Example– a full-load test on a 10 hp, 3-phs motor yields the following:

• at VLine = 600 V, P1 = +5950 W; P2 = + 2380 W; ILine = 10A

– calculate the power factor of the motor

• Example– the same motor runs at no-load

• ILine drops to 3.6 A and P1 = +1295 W; P2 = - 845 W

– calculate the no-load losses and the power factor


Power Measurements

• Example– three identical resistors connected as a wye, dissipate a total

power of 3000 W when energized by a 550 V line

– calculate• the current in each line

• the resistor values


Power Measurements

• Example– A 3-phase, 550 V, 60 Hz line is connected to three identical

capacitors in a delta configuration which draws 22 A

– calculate the capacitance in each capacitor


Homework

• Homework Problems: 7.8, 7-12, 7-18, 7-22, 8-4, 8-12, 8-16, and 8-26

chapter 8: three-phase circuits - oakland...

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