Three-Phase System 1 by Dr Rosemizi Abd Rahim Click here to watch the three phase animation video

Three-Phase System

1

by Dr Rosemizi Abd Rahim

Click here to watch the three phase animation video

http://rmz4567.blogspot.my/2013/02/electrical-engineering.html

http://rmz4567.blogspot.my/2013/02/electrical-engineering.html

COURSE OUTCOME (CO)

CO1: Ability to define and explain the concept of single-phase and three-phase system.

2

3

• A sinusoid is a signal that has the form of the sine or cosine function.

• A general expression for the sinusoid,

where

Vm = the amplitude of the sinusoidω = the angular frequency in radians/sФ = the phase

Revision

)sin()( tVtv m

4

Revision

A periodic function is one that satisfies v(t) = v(t + nT), for all t and for all integers n.

2

T

HzT

f1

f 2

• Only two sinusoidal values with the same frequency can be compared by their amplitude and phase difference.

• If phase difference is zero, they are in phase; if phase difference is not zero, they are out of phase.

5

Revision

Example 1

Given a sinusoid, , calculate its amplitude, phase, angular frequency, period, and frequency.

)604sin(5 ot

6

Revision

Example 1

Given a sinusoid, , calculate its amplitude, phase, angular frequency, period, and frequency.

Solution:

Amplitude = 5, phase = –60o, angular frequency = 4 p rad/s, Period = 0.5 s, frequency = 2 Hz.

)604sin(5 ot

7

Revision

Example 2

Find the phase angle between and , does i1 lead or lag i2?

)25377sin(41oti

)40377cos(52oti

8

Revision

Example 2

Find the phase angle between and , does i1 lead or lag i2?

)25377sin(41oti

)40377cos(52oti

Solution:

Since sin(ωt+90o) = cos ωt

therefore, i1 leads i2 155o.

)50377sin(5)9040377sin(52ooo tti

)205377sin(4)25180377sin(4)25377sin(41oooo ttti

9

RevisionImpedance transformation

YZ3Z Z3

1ZY

Single-Phase Circuit

Three wired system

• same magnitude

• same phase

A single phase circuit consists of a generator connected through a pair of wires

to a load

Two wire system

Aa

Two-Phase Circuit

Three wired system

Second source with 90° out

of phase

Three wired system

• same magnitude

• different phase

• It is 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°.

12

What is a Three-Phase Circuit?

Three sources with 120° out

of phaseFour wired

system

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

13

Balance Three-Phase Voltages

A three-phase generator The generated voltages

• Two possible configurations:

14


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

15

Phase sequences

a) abc or positive sequence b) acb or negative sequence


16

If the voltage source have the same amplitude and frequency ω and are out of phase with each other by 120o, the voltage are said to be balanced.

0VVV cnbnan

cnbnan VVV

Balanced phase voltages are equal in magnitude and out of phase with each other by 120o


17

0p

0pcn

0pbn

0pan

120V240VV

120VV

0VV

abc sequence or positive sequence:

acb sequence or negative sequence:

0p

0pbn

0pcn

0pan

120V240VV

120VV

0VV

pV is the effective or rms value


Example 1

Determine the phase sequence of the set of voltages.

)110cos(200

)230cos(200

)10cos(200

tv

tv

tv

cn

bn

an

18


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

19


20

Two possible three-phase load configurations:

a) a wye-connected load b) a delta-connected load


21

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

For a balanced wye connected load:

Y321 ZZZZ

ZZZZ cba

For a balanced delta connected load:

YZ3Z Z3

1ZY


• 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

22

Balance Three-Phase Connection

Balance Y-Y Connection

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

24

Y

L

s

Z

Z

Z

Z

Source impedance

Line impedance

Load impedance

Total impedance per phase

LY ZZ


LsY ZZZZ

Since all impedance are in series, Thus


Y

ana Z

VI

26

Applying KVL to each phase:

Y

ana Z

VI

0a

Y

0an

Y

bnb 120I

Z

120V

Z

VI

0a

Y

0an

Y

cnc 240I

Z

240V

Z

VI

0IIII ncba

0IZV nnnN



0pca

0pbc

0pab

210V3V

90V3V

30V3V

Line to line voltages or line voltages:

Magnitude of line voltages:

pL V3V

cabcabL VVVV

cnbnanp VVVV

Example 2 Calculate the line currents in the three-wire Y-Y system

shown below:

28


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

29


30

Balance Y-Δ Connection •A balanced Y-Δ system is a three-phase system with

a balanced y-connected source and a balanced Δ-connected load.

Balance Y-Δ Connection

A single phase equivalent circuit

3

ZZY

3/

Z

V

Z

VI an

Y

ana



CA0

pca

BC0

pbc

AB0

pab

V210V3V

V90V3V

V30V3V

Line voltages:

Balance Y-Δ Connection A single-phase equivalent circuit of a balanced Y- circuit

903

1503

303

ABBCCAc

ABABBCb

ABCAABa

IIII

IIII

IIII

Line currents:

Z

VI

Z

VI

Z

VI

CACA

BCBC

ABAB

Phase currents:

Balance Y-Δ Connection A single-phase equivalent circuit of a balanced Y- circuit

)24011(IIII

240II0

ABCAABa

0ABCA

0ABa 303II

pL II 3

cbaL IIII

CABCABp IIII

Magnitude line currents:

Example 3 A balanced abc-sequence Y-connected source with ( ) 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

35


36

Balance Δ-Δ Connection

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

37

Z

VI

Z

VI

Z

VI

CACA

BCBC

ABAB

CAca

BCbc

ABab

VV

VV

VV

Line voltages: Line currents:

3II pL

Magnitude line currents:

3

ZZY

Total impedance:

Phase currents:

903

1503

303

ABBCCAc

ABABBCb

ABCAABa

IIII

IIII

IIII


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

38

A 87.12686.22IA; 13.11386.22IA; 87.686.22I cba


39

Balance Δ-Y Connection

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

40


Applying KVL to loop aANBba:

Y

pba Z

VII

00

From:

0ab 120II

0aba 303III

Y

p

a Z

V

I

0303

Line currents:

41

Replace Δ connected source to equivalent Y connected source.

Phase voltages:

0pcn

0pbn

0pan

903

VV

1503

VV

303

VV


42


Y

p

Y

ana Z

V

Z

VI

0303


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 210V. Calculate the phase currents. Use Vab as reference.

Answer

The phase currents

A; 5857.2I

A; 17857.2I

A; 6257.2I

CN

BN

AN

43


phase-single ,2'2

2

L

Lloss V

PRP

phase- three,'3

'3''3'2

2

2

22

L

L

L

LLloss V

PR

V

PRRIP

44

Power in a Balanced System•Comparing the power loss in (a) a single-phase

system, and (b) a three-phase system

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

phase-single ,222

22

L

LLloss V

PRRIP

45

)120cos(2

)120cos(2

cos2

0

0

tVv

tVv

tVv

pCN

pBN

pAN

For Y connected load, the phase voltage:

Power in a Balanced System

46

Phase current lag phase voltage by θ.

ZZYIf

)120cos(2

)120cos(2

)cos(2

0

0

tIi

tIi

tIi

pc

pb

pa

The phase current:


47

Total instantaneous power:

cCNbBNaANcba ivivivpppp

cosIV3p pp

Average power per phase:

Apparent power per phase:

Reactive power per phase:

Complex power per phase:

sinIVQ ppp cosIVP ppp

*

ppppp IVjQPS ppp IVS


48

Total average power:

cosIV3cosIV3P3P LLppp

Total reactive power:

sinIV3sinIV3Q3Q LLppp

Total complex power:

*

p

2

p

p

2

p

*

ppp Z

V3ZI3IV3S3S

LLIV3jQPS


49

Power loss in two wires:

2

L

2

L2

Lloss V

PR2RI2P

Power loss in three wires:

2

2

2

22

333

L

L

L

LLloss V

PR

V

PRRIP PL : power absorbed by the load

IL : magnitude of line currentVL : line voltageR : line resistance


50

Example 6A three-phase motor can be regarded as a balanced Y-load. A three-phase motor draws 5.6 kW when the line voltage is 220 V and the line current is 18.2 A. Determine the power factor of the motor.

51

Example 6A three-phase motor can be regarded as a balanced Y-load. A three-phase motor draws 5.6 kW when the line voltage is 220 V and the line current is 18.2 A. Determine the power factor of the motor.

The apparent power is

VAIVS LL 13.6935)2.18)(220(33

The real power is WSP 5600cos

The power factor is8075.0cos

S

Ppf

52

Exercise 6

Calculate the line current required for a 30-kW three-phase motor having a power factor of 0.85 lagging if it is connected to a balanced source with a line voltage of 440 V.

53

Exercise 6

Calculate the line current required for a 30-kW three-phase motor having a power factor of 0.85 lagging if it is connected to a balanced source with a line voltage of 440 V.

Answer : 31.46LI

54

Exercise 7

For the Y-Y circuit in Exercise 2, calculate the complex power at the source and at the load.

Documents

Three-Phase System 1 by Dr Rosemizi Abd Rahim Click here to watch the three phase animation video