Review of Three-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

Three-phase system

vbnvan vcn

Time

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.

Y-Connection-Source

a

b

c

a' b'

c' n Vaa

Vbb

Vcc a

a'

Vaa

b

b'

c

Vbb Vcc

c'

n

nVVVan 0

Van

Vbn

Vcn

Reference

Vcn Vbn Vann

cb

a

Phase voltage

a

c

b

VVVcn 120VVVbn 120

Van Vbn Vcn

GeneratorTransmission

Line

VVVVV phcnbnan

90V3120V120VVVV cnbnbc

Line-to-line voltage

30V3120V0VVVV bnanab

150V30V120VVVV ancnca

Vca Vbc

Vab

n

cb

a

Van Vbn Vcn

303

23

21

)120sin120(cos1

1200

ph

phph

ph

bnanab

V

jVV

jV

VVVVV Reference

VabVca

nVbn

Vcn

Van

Vbc

030

-Vbn

The other line to line voltages

oabca

oabbc

VV

VV

120

120

30V3120V0VVVV bnanab Line-to-line voltage is greater than phase voltage by

3 Line-to-line voltage leads phase voltage by

030

Main Conclusions

ExampleLet for a balanced three phase system

VVan 0240

Calculate the line-to-line voltages

oooabo

anab

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

Y-Connected System

Source LoadTransmission

Line

a

c b

n

+

Van Z

Z Z

Ia

Ib

Ic

VbnVcn

a

bcn

Ia Ia

Ic IcIb

Ib

Line current

Phase current

+

++ + +

Three Phase System

120

120

0

VVVVVV

cn

bn

an

)120(120

)120(120

0

o

c

cnc

o

b

bnb

o

a

ana

IZV

ZVI

IZ

VZVI

IZV

ZVI

Y- Connection

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

nVan

Vbn

Vcn

Ia

Ic

Ib

120

120

0

VVVVVV

cn

bn

an

)120(II

)120(II

II

oc

ob

oa

Reference

Neutral Current of Balanced Load

ZZZZ cba For balanced system

01201200 oocban IIIIIII

Ia

Ib

Ic

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

Ia

Ic

a

bcbcIb

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

AjZ

VI ooo

ana 13.53241.535

0120430120

87.6625)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

Delta () Connection: Loada

bc

Ia

Ib

Ic

IabIca

Ibc

+_

+_+

_

Ibc Iab

nVab

Vbc

Vca

Refere

Ica

Z

VIZ

VIZ

VI cacabcbcabab ,,

Ia

caaba III

IbcIab

nVab Refere

Ica

300

)30(I3I

30I3III

oaba

oabcaaba

- Ica

Line current lags phase current by

030

Main Conclusions for Delta-Connected Load

Line-to-line voltage is equal to phase voltage

)30(I3I oaba

3 Line current is greater than phase current by

Example

bc

Ia

Ic

a

bc

Ib

Calculate the phase currents of the load

VVan 40120 VZ 3010

Ibc

IabIca

abc

Ia

Ic

VanbcIb

Z

AZ

VI ooo

abab 757.41375

301203

oooabo

anab

V

VV

307.4153002403

303

Power of 3-phase circuits

Iphase

VphaseFor Single phase

)(sin

)(cos

phasephase

phasephase

IVQIVP

For 3-phase

)(sin3

)(cos3

phasephase

phasephase

IVQIVP

IMPORTANT

is the angle between phase voltage and phase current.

Use voltage as a reference

Real Power in Y Circuit

aphase

anphase

II

VV

aline

anabline

IIVVV

3

)(cos3)(cos3 aanphasephase IVIVP )(cos3)(cos3 linelineaab IVIVP

a

bc

Ia

Ib

Ic

Van

Vcn Vbn

Reactive Power in Y Circuit

aphase

aphase

IIVV

aline

aabline

IIVVV

3

)(sin3)(sin3 aaphasephase IVIVQ )(sin3)(sin3 linelineaab IVIVQ

a

bc

Ia

Ib

Ic

Va

Vc Vb

Real Power in Delta Circuita

bc

Ia

Ib

Ic

IabIca

Ibc

+_

+_+

_

abphase

abphase

IIVV

abaline

abline

III

VV

3

)(cos3)(cos3 ababphasephase IVIVP )(cos3)(cos3 linelineaab IVIVP

Reactive Power in Delta Circuita

bc

Ia

Ib

Ic

IabIca

Ibc

+_

+_+

_

abphase

abphase

IIVV

abaline

abline

III

VV

3

)(sin3)(sin3 ababphasephase IVIVQ )(sin3)(sin3 linelineaab IVIVQ

Review of Three-Phase Circuit Analysis Slide Number 2Three-phase systemWhy do we use 3-phase systems?Y-Connection-SourcePhase voltageSlide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Main ConclusionsExampleY-Connected SystemSlide Number 15Y-ConnectionSlide Number 17Neutral Current of Balanced LoadSlide Number 19Slide Number 20Slide Number 21Delta () Connection: LoadSlide Number 23Slide Number 24ExampleSlide Number 26Power of 3-phase circuitsSlide Number 28Slide Number 29Slide Number 30Slide Number 31