Chapter 12Three-Phase Circuit Analysis

Ia

Ib

Ic

N S

N

S

N

S

N

S

NS

N

S

Va

Vb

Vc

N

S

N

S

A three phase generator has three coils each represent a phase

Vc

Vb

Va

2

Three-phase system

3

Why do we use 3-phase systems?• Three-phase system produces rotating magnetic

field.– Three-phase motors can start without the need for

extra equipment.• For the same physical size, a three-phase

generator produces more power than a single phase generator.

• Three-phase lines transmit more power.• Three-phase lines are more reliable.

– In distribution circuit, you can operate the system with one missing phase.

4

Y-Connection-Source

a

b

c

a' b'

c'

n Vaa’

Vbb’

Vcc’ a

a'

Vaa’

b

b'

c

Vbb’ Vcc’

c'

n

5

n

VVVan 0

Van

Vbn

Vcn

Reference

Vcn Vbn Vann

c

b

a

Phase voltage

a

c

b

VVVcn 120

VVVbn 120

Van Vbn Vcn

GeneratorTransmission

Line

VVVVV phcnbnan

6

7

8

9

90V3120V120VVVV cnbnbc

Line-to-line voltage

30V3120V0VVVV bnanab

150V30V120VVVV ancnca

Vca Vbc

Vab

n

c

b

a

Van VbnVcn

10

303

2

3

2

1

)120sin120(cos1

1200

ph

phph

ph

bnanab

V

jVV

jV

VVVVV Reference

VabVca

n

Vbn

Vcn

Van

Vbc

030

-Vbn

The other line to line voltages

oabca

oabbc

VV

VV

120

120

11

30V3120V0VVVV bnanab

• Line-to-line voltage is greater than phase voltage by

3

• Line-to-line voltage leads phase voltage by

030

Main Conclusions

12

Example

Let for a balanced three phase system

VVan 0240

Calculate the line-to-line voltages

oooab

oanab

V

VV

307.4153002403

303

Reference030

van

vab

oobc

oabbc

V

VV

907.415120307.415

120

ooca

abca

V

VV

1507.415120307.415

120

The other voltages can be computed by the balanced system relationship

13

Y-Connected System

Source LoadTransmission

Line

a

c b

n

+

Van Z

Z Z

Ia

Ib

Ic

Vbn Vcn

a

b c

n

Ia Ia

Ic Ic Ib

Ib

Line current

Phase current

+

++ + +

14

Three Phase System

15

120

120

0

VV

VV

VV

cn

bn

an

)120(120

)120(120

0

o

c

cnc

o

b

bnb

o

a

ana

IZ

V

Z

VI

IZ

V

Z

VI

IZ

V

Z

VI

Y-Connecti

on

ZZZZ cba For balanced system

a

c b

n

+

Van Z

Z Z

Ia

In

Ib

Ic

Vbn Vcn

a

b c

n

Ia Ia

Ic Ic Ib

Ib

16

nV

an

Vbn

Vcn

Ia

Ic

Ib

120

120

0

VV

VV

VV

cn

bn

an

)120(II

)120(II

II

oc

ob

oa

Reference

17

Neutral Current of Balanced Load

ZZZZ cba For balanced system

01201200 oocban IIIIIII

Ia

Ib

Ic

18

30V3120V0VVVV bnanab

• Line-to-line voltage is greater than phase voltage by

3

• Line-to-line voltage leads phase voltage by 030

Main Conclusions for Y-Connected Load

• Line current equals phase current

19

Ia

Ic

a

bc

bcIb

Example: For the three phase system shown , findThe load currents of each phaseThe neutral currentThe magnitude of the line to line voltage

VVan 0120 43 jZ

20

AjZ

VI

o

ooan

a 13.53241.535

0120

43

0120

87.6624)13.53120(24)120(

13.17324)13.53120(24)120(

13.5324

ooc

oob

ooa

II

II

II

The other currents can be computed by the balanced system relationship

087.662413.1732413.5324 oocban IIII

The load currents of each phase

The neutral current

The magnitude of the line to line voltage

VVV phL 20812033 21

Delta () Connection: Loada

bc

Ia

Ib

Ic

IabIca

Ibc

+

_

+_

+

_

Ibc Iab

nV

ab

Vbc

Vca

Reference

Ica

Z

VI

Z

VI

Z

VI ca

cabc

bcab

ab ,,

22

Ia

caaba III

IbcIab

nV

ab Reference

Ica

300

)30(I3I

30I3III

oaba

oabcaaba

- Ica

23

Main Conclusions for Delta-Connected Load

• Line-to-line voltage of the source is equal to phase voltage (across load).

• The line current (coming from the source) is greater than phase current (of the load) by

•Line current lags phase current by

24

)30(3 oaba II

3

030

Example

bc

Ia

Ic

a

bc

Ib

Calculate the phase currents of the load

VVan 0120

VZ 3010

Ibc

IabIca

25

a

bc

Ia

Ic

Van

bcIb

Z

AZ

VI o

o

oab

ab 079.203010

301203

oooab

oanab

V

VV

309.2073001203

303

26

ooooaba II 3036301079.203303

Power of 3-phase circuits

Iphase

VphaseFor Single phase

)(sin

)(cos

phasephase

phasephase

IVQ

IVP

For 3-phase

)(sin3

)(cos3

phasephase

phasephase

IVQ

IVP

IMPORTANT

is the angle between phase voltage and phase current.

Use voltage as a reference 27

Real Power in Y Circuit

aphase

anphase

II

VV

aline

anabline

II

VVV

3

)(cos3)(cos3 aanphasephase IVIVP

)(cos3)(cos3 linelineaab IVIVP

a

bc

Ia

Ib

Ic

Van

Vcn Vbn

28

Reactive Power in Y Circuit

aphase

aphase

II

VV

aline

aabline

II

VVV

3

)(sin3)(sin3 aaphasephase IVIVQ

)(sin3)(sin3 linelineaab IVIVQ

a

bc

Ia

Ib

Ic

Va

Vc Vb

29

Real Power in Delta Circuit

a

bc

Ia

Ib

Ic

IabIca

Ibc

+

_

+_

+

_

abphase

abphase

II

VV

abaline

abline

III

VV

3

)(cos3)(cos3 ababphasephase IVIVP

)(cos3)(cos3 linelineaab IVIVP 30

Reactive Power in Delta Circuit

a

bc

Ia

Ib

Ic

IabIca

Ibc

+

_

+_

+

_

abphase

abphase

II

VV

abaline

abline

III

VV

3

)(sin3)(sin3 ababphasephase IVIVQ

)(sin3)(sin3 linelineaab IVIVQ 31

Three Phase Power Measurement Two-meter method for measuring three-phase power