AC CircuitsThree-Phase Circuits

Contents

• What is a Three-Phase Circuit?

• Balance Three-Phase Voltages

• Balance Three-Phase Connection

• Power in a Balanced System

• Unbalanced Three-Phase Systems

• Application – Residential Wiring

2

A simple AC generators

Sinusoidal voltage sources

When a conductor rotates in a constant magnetic field, a sinusoidal wave is generated.

Single-phase system

When the loop is moving perpendicular to the lines of flux, the maximum voltage is induced.

When the conductor is moving parallel with the lines of flux, no voltage is induced.

Single-phase two-wire

Normal household: Single-phase three-wire

neutral

More pole pieces? More coils?

Three-phase circuit: A system produced by a generator consisting of three sources having the same amplitude and frequency but out of phase with each other by 120°.

4

Polyphase

Three sources with 120° out

of phase

Four wired system

Advantages of a three-phase circuit

• Most of the electric power is generated and distributed in three-phase:

– Less/more than 3-phase required…taken from 3-phase

• The instantaneous power in a three-phase system can be constant (not pulsating):

– uniform power transfer and less vibration of three-phase machine

• The amount of power, the three-phase system is more economical that the single-phase.

• In fact, the amount of wire required for a three-phase system is less than that required for an equivalent single-phase system.

5

Balance Three-Phase Voltages

6

• A three-phase generator consists of a rotating magnet (rotor) surrounded by a stationary winding (stator).

A three-phase generator The generated voltages

It can supply power to both single-phase and three-phase load

7

Balance Three-Phase Voltages

• Two possible configurations:

Three-phase voltage sources: (a) Y-connected ; (b) Δ-connected

Four wires Three wires

Phase voltage

Balanced voltages:

cnbnan

cnbnan

VVV

VVV

0

8

Balance Three-Phase Voltages• Balanced phase voltages are equal in magnitude and are out of phase

with each other by 120°.

• The phase sequence is the time order in which the voltages pass through their respective maximum values.

• A balanced load is one in which the phase impedances are equal in magnitude and in phase

3

2

3

4

3

2

j

p

j

p

j

p

p

eVeV

eV

V

bn

cn

an

V

V

V

pV cnbnan VVV3

2

3

4

3

2

j

p

j

p

j

p

p

eVeV

eV

V

cn

bn

an

V

V

V abc/positive sequence

acb/negative sequence

abc/positive sequence

acb/negative sequence

Y321 ZZZZ ZZZZ cba

YZZ 3

Transform:

9

Example

Example 1

Determine the phase sequence of the set of voltages.

)110cos(200

)230cos(200

)10cos(200

tv

tv

tv

cn

bn

an

Solution:

The voltages can be expressed in phasor form as

We notice that Van leads Vcn by 120° and Vcn in turn leads Vbn by 120°.Hence, we have an acb sequence.

V 110200V

V 230200V

V 10200V

cn

bn

an

10

Balance Three-Phase Connection

Four possible connections

1. Y-Y connection (Y-connected source with a Y-

connected load)

2. Y-Δ connection (Y-connected source with a Δ-

connected load)

3. Δ-Δ connection

4. Δ-Y connection

11

Balance Y-Y systemA balanced Y-Y system is a three-phase system with a balanced y-connected

source and a balanced y-connected load.

cabcabL

cnbnanp

pL

V

V

VV

VVV

VVV

where, 3

LlSY ZZZZ

6

3

2

32

1

2

33

2

3

2

11

j

j

pp

VpejVpjVp

eVV

bnannbanab VVVVVLine voltage:

3

4

3

23

2

,jj

j

eee

,

aca

Y

an

Y

bnb

Y

ana III

Z

V

Z

VI

Z

VI

Line current (KVL):

,0- ,0 cbancba II IIII I 0 nNV

Voltage across the neutral line is zero: the neutral line can be removed (earth acting as the neutral line).

The same as the phase

current in Y-Y

Phasor diagram?

12

Balanced Y-Y connection

Example 2

Calculate the line currents in the three-wire Y-Y system shown below:

A 2.9881.6I

A 8.14181.6I

A 8.2181.6I

Ans

c

b

a

13

Balance Y- Connection

• A balanced Y-Δ system is a three-phase system with a balanced y-connected source and a balanced Δ-connected load.

CABCABp

cbaL

pL

I

I

II

III

III

where, 3

Line current

Phase current

Line voltage = the voltage across the load

abAB VV

Phase current

Δ

ABAB

Z

VI

Line current

6

1

3

4

3jj

a ee

ABABABCAAB IIIIII

14

Balance Y- Connection: Example

Example 3

A balanced abc-sequence Y-connected source with V is connected to a Δ-connected load (8+j4) per phase. Calculate the phase and line currents.

Solution

Using single-phase analysis,

Other line currents are obtained using the abc phase sequence

10100Van

A 57.1654.3357.26981.2

10100

3/Z

VI an

a

15

Balance - Connection

• A balanced Δ-Δ system is a three-phase system with a balanced Δ -connected source and a balanced Δ -connected load.

Line voltage = the phase voltage

abAB VV

Phase currentΔΔ

ABAB

Z

V

Z

VI ab

Line current

6

1

3

4

3j

j

a

e

e

AB

ABABCAAB

I

IIIII

pL II 3

16

Balance - Connection: Example

Example 4

A balanced Δ-connected load having an impedance 20-j15 is connected to a Δ-connected positive-sequence generator having ( ). Calculate the phase currents of the load and the line currents.

Ans:

The phase currents

The line currents

V 0330Vab

A 87.1562.13IA; 13.812.13IA; 87.362.13I ABBCAB

A 87.12686.22IA; 13.11386.22IA; 87.686.22I cba

17

Balance -Y Connection

• A balanced Δ-Y system is a three-phase system with a balanced Y-connected source and a balanced Y-connected load.

Can you work out the relationship between line current and phase current?

18

Balance -Y Connection: Example

Example 5

A balanced Y-connected load with a phase impedance 40+j25 is supplied by a balanced, positive-sequence Δ-connected source with a line voltage of 210 V. Calculate the phase currents. Use Vab as reference.

Answer

The phase currents

A; 5857.2I

A; 17857.2I

A; 6257.2I

CN

BN

AN

Balanced three-phase: Summary of phase and line V/I

20

Power in a Balanced System• Instantaneous power (Y-load, ZY = Z)

• Comparing the power loss in (a) a single-phase system, and (b) a three-phase system

phase-single ,2'2

2

L

Lloss

V

PRP phase- three,''

2

2

L

Lloss

V

PRP

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

3

2cos2 ,

3

2cos2 ,cos2

tVvtVvtVv pCNpBNpAN

3

2cos2 ,

3

2cos2 ,cos2

tIitIitIi pcpbpa

cos3 ppcCNbBNaANcba IVivivivpppp θIV QIVPjQP pppp sin ,cos , :phase oneFor *

ppIVS

Time independent

21

Unbalanced Three-Phase Systems

• An unbalanced system is due to unbalanced voltage sources or an unbalanced load.

cban

C

CNc

B

BNb

A

ANa

IIII

,Z

VI ,

Z

VI ,

Z

VI

• To calculate power in an unbalanced three-phase system requires that we find the power in each phase.

• The total power is not simply three times the power in one phase but the sum of the powers in the three phases.

22

Unbalanced Three-Phase Systems: Example

Example 6

Determine the total average power, reactive power, and complex power

at the source and at the load

AnsAt the source:Ss = -(2087 + j834.6) VA

Pa = -2087W

Pr = -834.6VAR

At the load:SL = (1392 + j1113) VA

Pa = 1392W

Pr = 1113VAR

23

Application – Residential Wiring

A 120/240 household power system

24

Application – Residential Wiring (2)

Single-phase three-wire residential wiring

A typical wiring diagram of a room