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Chapter 12 Three Phase Circuits

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

Payam zarbakhsh

EElE301 Circuit Theory II

Department of Electrical and Electronic Engineering

Cyprus International university

‹#› Eeng 224

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 p

pP V I

pP

Payam zarbakhsh

EElE301 Circuit Theory II

Department of Electrical and Electronic Engineering

Cyprus International university

‹#› Eeng 224

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

3

1= 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

Payam zarbakhsh

EElE301 Circuit Theory II

Department of Electrical and Electronic Engineering

Cyprus International university

‹#› Eeng 224

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

2

p 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

p

p

L L

LV I V I

VZ

Z

P jQ V I

S

Payam zarbakhsh

EElE301 Circuit Theory II

Department of Electrical and Electronic Engineering

Cyprus International university

‹#› Eeng 224

Power in a Balanced System

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

2

p2

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

p

p

L L

V

VI Z

Z

P

V

V I

I

jQ

I

S

VL VL

VL

Vp Vp

Vp Ip

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 Payam zarbakhsh

EElE301 Circuit Theory II

Department of Electrical and Electronic Engineering

Cyprus International university

‹#› Eeng 224

Payam zarbakhsh

EElE301 Circuit Theory II

Department of Electrical and Electronic Engineering

Cyprus International university

‹#› Eeng 224

VL=840 V (Rms)

Capacitors for pf

Correction

IL

Payam zarbakhsh

EElE301 Circuit Theory II

Department of Electrical and Electronic Engineering

Cyprus International university

‹#› Eeng 224

7365050.68A

3 3840

Without Pf Correction

L

L

SI

V

‹#› Eeng 224

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

Payam zarbakhsh

EElE301 Circuit Theory II

Department of Electrical and Electronic Engineering

Cyprus International university

‹#› Eeng 224