GALIZA, A-JAY N. 3ECE-B

EE306B Homework THREE-PHASE CURCUITS

There are two types of system available in electric circuit, single phase and three phase system. In single phase circuit, there will be only one phase, i.e the current will flow through only one wire and there will be one return path called neutral line to complete the circuit. So in single phase minimum amount of power can be transported. Here the generating station and load station will also be single phase. This is an old system using from previous time.

The three phases can be used as single phase each. So if the load is single phase, then one phase can be taken from the three phase circuit and the neutral can be used as ground to complete the circuit. Three-phase systems are commonly used in generation, transmission and distribution of electric power. Power in a three-phase system is constant rather than pulsating and three-phase motors start and run much better than single-phase motors.

Three phase circuit is the polyphase system where three phases are send together from the generator to the load. Each phase are having a phase difference of 120, i.e 120 angle electrically. So from the total of 360, three phases are equally divided into 120 each. The power in three phase system is continuous as all the three phases are involved in generating the total power.

There are three phase voltages present in the circuit; Va , Vb, Vc represented by:

,240tcosVv

120tcosVv

tcosVv

mc

mb

ma

The phasors corresponding to each voltage are:

.eVV

eVV

VV

240j

mc

120j

mb

ma

BALANCED THREE PHASE VOLTAGES

In a balanced system, three sinusoidal voltages form a set of balanced voltages when they have the same amplitudes and frequency. These voltages are shifted in phase by 120o with each other. The standard practice is to name those phases by a, b and c and use phase a as reference. These voltages represent phase a voltage, phase b voltage and phase c voltage.

Van + Vbn + Vcn = 0

|Van| = |Vbn| = |Vcn |

The order in which the three voltage in the phase reach their maximum positive value is called the phase sequence. There are two possible sequences:

1. abc (positive) sequence: Vbn lags Van by 120o.

2. acb (negative) sequence: Vbn leads Van by 120o.

A three phase system where the individual impedances are identical is called a balanced system. Which is indicated by the resulting equality, Z1= Z2 = Z3 = Z .Such a load is called a balanced load and is described by equations

0

120

240

an p

bn p

cn p

V V

V V

V V

0

120

240

an p

bn p

cn p

V V

V V

V V

Z

VI

Z

VI

Z

VI cba

321 .

Using KCL, we have

3211

VVVZ

IIII cban ,

02

3

2

1

2

3

2

11

240sin240cos120sin120cos1

1 240120321

jjV

jjV

eeVVVV

m

m

jj

m

Setting the above result into the previous equations we obtain

In 0 .

It can is determined that the current flowing through n is zero. Therefore, the wire connected to it

can be disregarded.

The figure depicts a Y system or a star system because of the configuration and connection of the

voltage and the load branches. Point n is called the neutral point of the generator and point n is

called the neutral point of the load.Each branch of the generator or load is called a phase. The wires

connecting the supply to the load are called the lines. In the Y-system shown in the figure each line

current is equal to the corresponding phase current, whereas the line-to-line voltages (or simply

line voltages) are not equal to the phase voltages.

THREE-PHASE CIRCUIT CONNECTIONS

In three phase circuit, connections can be given in two types:

Star connection Delta connection

STAR OR Y OR WYE CONNECTION

In star connection, there is four wire, three wires are phase wire and fourth is neutral which is taken from the star point. Star connection is preferred for long distance power transmission because it is having the neutral point. In this we need to come to the concept of balanced and unbalanced current in power system.

When equal current will flow through all the three phases, then it is called as balanced current. And when the current will not be equal in any of the phase, then it is unbalanced current. In this case, during balanced condition there will be no current flowing through the neutral line and hence there is no use of the neutral terminal. But when there will be unbalanced current flowing in the three phase circuit, neutral is having a vital role. It will take the unbalanced current through to the ground and protect the transformer. Unbalanced current affects transformer and it may also cause damage to the transformer and for this star connection is preferred for long distance transmission.

Using Kirchhoffs Voltage Law, we can obtain the relationships of each line voltages. The line

voltages Vab, Vbc, Vca form a symmetrical set of phasors leading by 30 the set representing the phase

voltages and they are 3 times greater, thus, V V V Vab bc ca a 3 .

DELTA CONNECTION

In delta connection, there are three wires alone and no neutral terminal is taken. Normally delta connection is preferred for short distance due to the problem of unbalanced current in the circuit. The figure is shown below for delta connection. In the load station, ground can be used as neutral path if required.

In delta connection, the line voltage is same with that of phase voltaage. And the line current is 3 times of phase current. It is shown as VLine = VPhase and ILine = 3IPhase.

DELTA to WYE and Wye to DELTA TRANSFORMATION

In converting a circuit from delta to wye or wye to delta, the following expressions can be used:

Zdelta = Za = Zb = Zc

ZY = Z1 = Z2 = Z3

Zdelta = 3 ZY and ZY = 1/3 (Zdelta)

Vca

c Vbc

Vab

n

a Va

Vc

b Vb

Vc

VbcVb

30 Va

Vab

30

30

Vca

Both the three phase source and the three phase load can be connected either Wye or DELTA. We have 4 possible connection types: Y-Y connection, Y- connection, - connection, -Y connection. Balanced connected load is more common and Y connected sources are more common.

BALANCED WYE-WYE (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.

In the notation, we let E as V (voltage).

Phase voltages are: Va, Vb and Vc. The three conductors connected from a to A, b to B and c to C are called LINES. The voltage from one line to another is called a LINE voltage Line voltages are: Vab, Vbc and Vca Magnitude of line voltages is 3 times the magnitude of phase voltages. VL= 3 Vp

3

0 , 120 ,

30

3 90

3 21

120

0

an p bn p cn p

ab an nb an bn

bc bn cn

ca cn an

p

p

an bn p

V

V V V V V V

V V V V V

V V V

V V V

V

V VV

BALANCED WYE-DELTA CONNECTION

BALANCED DELTA-DELTA CONNECTION

BALANCED DELTA-WYE CONNECTION

ABAB

BCBC

CACA

VI

Z

VI

Z

VI

Z

3

L a b c

p AB BC CA

L p

I I I I

I I I I

I I

, ,a AB CA b BC AB c CA BCI I I I I I I I I

, ,BC CAABAB BC CAV VV

I I IZ Z Z

POWER CALCULATIONS IN BALANCED THREE-PHASE CIRCUITS Average Power in a balanced Wye Load

Effective Power, P = VrmsIrms cos(V i) For a three-phase circuit (rms), PA = |VAN| |IaA| cos(VA iA), where VA and iA are phase angles of voltage and current.

PB = |VBN| |IbB| cos(VB iB) PC = |VCN| |IcC| cos(VC iC)

For a balanced load,

V = |VAN| =|VBN| =|VCN| I = |IaA| = |IbB| = |IcC| = VA iA= VB iB = VC iC P = PA = PB = PC = I V cos

Total power delivered to the three-phase load is PT = 3P

For line voltage VL and current IL in rms values, PT = 3 IL VL cos Complex Power in a balanced Wye Load Reactive Power, Q = VrmsIrms sin(V i) For Balanced Load, Q = I V cos

Total Reactive Power, QT = 3Q = 3 IL VL sin

For Complex Power, S = P + j Q = I* V

Total Complex Power, ST = 3ST = 3 IL VL *

Average Power in a balanced Delta Load The calculations are basically the same as the Wye For a three-phase circuit (rms),

PA = |VAB| |IAB| cos(VAB iAB) PB = |VBC| |IBC| cos(VBC iBC) PC = |VCA| |ICA| cos(VCA iCA)

For a balanced load,

V = |VAB| =|VBC| =|VCA| I = |IAB| = |IBC| = |ICA| = VAB iAB= VBC iBC = VCA iCA P = PA = PB = PC = I V cos

Total power PT = 3P

PT = 3 IL VL cos QT = 3Q = 3 IL VL sin

ST = 3ST = 3 IL VL * Instantaneous Power in Three Phase Circuits PA = vAN iaA =Vm Im cos(t - ) PB = vBN ibB =Vm Im cos(t 120o) cos(t - 120o) PC= vCN icC =Vm Im cos(t + 120o) cos(t - + 120o) Total Instantaneous Power PT = PA = PB = PC = 1.5 Vm Im cos

EXAMPLES

1.

2. If Vab = 400 V in a balanced Y-connected three-phase generator, find the phase voltages,

assuming the phase sequence is: (a) abc (b) acb

3.

4. For the Y-Y circuit of Fig. 12.41, find the line currents, the line voltages, and the load

voltages.

5. Obtain the line currents in the three-phase circuit: