Three-Phase AC CircuitsAppendix: A (p.681)

Significant Features of Three-Phase AC Circuits • Almost all ac power generation and transmission is in the form of three-

phase ac circuits

• AC power systems have a great advantage over DC systems in that their voltage levels can be changed with transformers to reduce transmission losses.

• Three-phase (3) ac power system consists of

– 3 ac generators

– 3 transmission lines

– 3 loads

• Advantages of having 3 power systems over 1ones:

– More power per pound of metal of electrical machines of 3.

– Power delivered to a 3 load is constant, instead of pulsating as it does in a 1 system.

Generation of 3 Voltages and Currents

A 3 generator consists of three 1 generators:

- voltage of all phases are equal in amagnitude- differing in phase angle from each aother by 120o.

Three-Phases of the Generator Connected to Three identical Loads

VA

VB

VC

Phasor diagram showing the voltages in each phase

Currents in the Three Phases and the Neutral

00

00

0

240240

120120

0

IZ

VI

IZ

VI

IZ

VI

c

b

a

0

240240

120120

240120

00

00

00

sinjcos

sinjcossinjcosI

III

IIII cbaN

Currents flowing in the three phases

It is possible to connect the negative ends of these three single phase generators and the loads together, so that they share a common return line, called neutral.

As long as the three loads are equal, the return current in the neutral is zero.

Balanced Power Systems

• In a balanced power system:

– Three generators have same voltage magnitude and phase difference is 120o.

– Three loads are equal and magnitude and angle.

• abc phase sequence: the voltages in the three phases peak in the order a, b and c. It is possible to have acb phase sequence.

Y and Connections

A connection of this sort is called Wye-connection.

Another possible connectionis delta-connection, in whichthe generators are connected

head to tail.

Z Z

Z

Z

Z

Z

Ic

Ib

Ia

In

Ia

Ib

Ic

Va

Va

Vb

Vb

Vc

Vc

+ +

+

-- -

-

-

-

+

+

+

Voltages and Currents in a Y-Connected 3 Circuit

Phase quantities: voltages and currents in a given phaseLine quantities: voltages between lines and current in the lines

Ib

Ic

Ia (=IL)VanVbn

Vcn

+ +

+

-- -

ResistiveLoad

n

I

Vca

Vab

Vbc

++

+

-

-

-

00

00

00

240120

120120

00

IIVV

IIVV

IIVV

ccn

bbn

aan

000 3031200 VVVVVV bnanab

Voltages and Currents in a Y-Connected 3 Circuit (cont’d)

VVLL 3

The relationship between the magnitude of the line-to-line voltage and the

line-to-neutral (phase) voltage in a Y-connected generator or load

IIL

In a Y-connected generator or load, the current in any line is the same as

the current in the corresponding phase.

Vab

Vbc

Vca

Van

Vcn

Vbn

Voltages and Currents in a -Connected 3 Circuit

VVLL

In a delta-connected generator or load, the line-to-line voltage between any two lines will be the same as the voltage in the corresponding phase.

IIL 3

In a delta-connected generator or load, the relationship between the magnitudes

of the line and phase currents:

IaIbcIb

Iab

Ic

Ica

Ibc

Ica

VA VB

VC

--

-

+

+

+

Ia

Iab

IbIc

Power Relationship in 3 Circuits

A balanced Y-connected load.

The 3 voltages applied to this load:

13

000

000

0

0

0

0

33

48022402402

24021201202

22

2402

1202

2

2402

1202

2

PcosVItPtPtPPtP

tcoscosVItsintsinVItitvtP

tcoscosVItsintsinVItitvtP

tcoscosVItsintsinVItitvtP

tsinIti

tsinIti

tsinIti

tsinVtv

tsinVtv

tsinVtv

aaatotal

ccnc

bbnb

aana

c

b

a

cn

bn

an

The 3 currents flowing in this load:

Instantaneous power supplied to each of the three phases:

Total power supplied to the 3 load:

3Power Equations Involving Phase Quantities

The 1 power equations:

sinIVQ

cosIVP

IVS

sinIVQ

cosIVP

IVS

3

3

3

3

3

3

1

1

1

The 3 power equations:

S

P=Scos

Q=Ssin

90o

3Power Equations Involving Line Quantities

VVandII LLL 3

cosIVcosI

VcosIVP

cosIVcosIV

cosIVP

LLLL

LL

LLLLLL

33

33

33

33

3

3

For a Y-connected load:

90o

For a delta-connected load: VVandII LLL 3

Therefore, regardless of the connection of the load:

LLL

LLL

LLL

IVS

sinIVQ

cosIVP

3

3

3

3

3

3

Analysis of Balanced 3 Systems

If a three-phase power system is balanced, it is possible to determine voltages and currents at various points in the circuit with a per

phase equivalent circuit.

• Neutral wire can be inserted, as no current would be flowing through it, thus, system is not affected.

• Three phases are identical except for 120o phase shift for each phases.

• It is thus possible to analyze circuit consists of one phase and neutral.

• Results would be valid for other two phases as well if 120o phase shift is included.

Wye-Delta Transformation

• Above analysis if OK for Y-connected sources and loads, but no neutral can be connected for delta-connected sources and loads.

• As a result, the standard approach is to transform the impedances by using the delta-wye transform of elementary circuit theory.