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Chapter 12Three Phase Circuits

Huseyin BilgekulEeng224 Circuit Theory II

Department of Electrical and Electronic Engineering Eastern Mediterranean University

Chapter Objectives: Be familiar with different three-phase configurations and

how to analyze them. Know the difference between balanced and unbalanced

circuits Learn about power in a balanced three-phase system Know how to analyze unbalanced three-phase systems Be able to use PSpice to analyze three-phase circuits Apply what is learnt to three-phase measurement and

residential wiring

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Power in a Balanced System The total instantaneous power in a balanced three phase system is constant.

2 cos( ) 2 cos( 120 )

2 cos( 120 )

2 cos( ) 2 cos( 120 )

2 cos( 120 )

cos( )cos( ) cos( 120 )cos( 120 )2

cos( 120 )co

AN p BN p

CN p

a p b p

c p

a b c AN a BN b CN c

p p

v V t v V t

v V t

i I t i I t

i I t

p p p p v i v i v i

t t t tp V I

t

Using the above identity and simplifying, =2 t- we obtain

s( 120 )

1cos cos [cos( ) cos( )]

2 that:

13cos

2

cos 2 oos 3c c sp pp p

t

A B A B A B

VV I Ip

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Power in a Balanced System The important consequences of the instantenous power equation of a balanced three phase system are:

The instantenous power is not function of time.

The total power behaves similar to DC power.

This result is true whether the load is Y or connected.

T

3 cos p pp V I

he per phase is obtained as .3

AVERAGE POWE

R

cos 3

p

p p ppP V I

pP

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Power in a Balanced System The complex power per phase is Sp. The total complex power for all phases is S.

p

p

p p p p p

3 cos

1= cos

31

= sin

(Total Instantenous Power)

(Average Power per phase)

(Reactive Pow 3

er per phase)

(Apparent Power per pha

S V I

s

e

)

p p

p p

p p

p p p

p V I

P p V I

Q p V I

S V I

P jQ

p p

and refer to magnitude val

C

ues whereas

V and I refer to phasor

omplex power for e

values (Both magn

ach pha

itude a

se

nd phase)

p pV I

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Power in a Balanced System

The complex power per phase is Sp. The total complex power for all phases is S.

p p p p p

p p p

p p p

Complex power for each phase

Total Complex power for three phas

S V I

S= 3S 3V I

3 3 cos 3 cos

3 3 sin 3 sin

S=3S 3V

e

I 3

a b c p p p L L

a b c p p p L L

P jQ

P jQ

P P P P P V I V I

Q Q Q Q Q V I V I

I

2

p

p p L

2p Total complex power

Total complex power using

, , and are all rms values, is the load imp

3

line val

edance

3

angl

s u e

e

pp

L L

LV I V I

VZ

Z

P jQ V I

S

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Power in a Balanced System

Notice the values of Vp, VL, Ip, IL for different load connections.

2p2

p p p

p

p

p L, , and are all rms values, is the load impe

To

da

3S=3S 3V I 3

nce

al c

an

3

omplex ower

gle

p

L

pp

L L

V

VI Z

Z

P

V

V I

I

jQ

I

S

VLVL

VL

Vp Vp

VpIp

Ip Ip

VL

Vp

Ip

VL

VL Vp

Vp

Ip

Ip

Y connected load. Δ connected load.

3L p L pV V I I 3L p L pV V I I

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Power in a Balanced System

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Single versus Three phase systems Three phase systems uses lesser amount of wire than single phase systems for the same line voltage VL and same power delivered.

a) Single phase system b) Three phase system

2 2

'2 '2

Wire Material for Single phase 2( ) 2 2(2) 1.33

Wire Material for Three phase 3( ) 3 3

r l r

r l r

If same power loss is tolerated in both system, three-phase system use only 75% of materials of a single-phase system

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VL=840 V (Rms)

Capacitors for pf Correction

IL

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7365050.68A

3 3840

Without Pf Correction

L

L

SI

V

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Unbalanced Three Phase Systems An unbalanced system is due to unbalanced voltage sources or unbalanced load. In a unbalanced system the neutral current is NOT zero.

Unbalanced three phase Y connected load.

Line currents DO NOT add up to zero.

In= -(Ia+ Ib+ Ic) ≠ 0

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

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Residential Wiring

Single phase three-wire residential wiring