ee 132 - lecture 1

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EE 132

LECTURE 1

Prepared by:

Engr. Rogelio F. Bersano Jr.

Polyphase Circuits

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Topic Outline

Polyphase Systems

Single-Phase Three-wire Systems

Three Phase Balanced Circuit

The Delta ( ) & Wye (Y) Connection

Power Calculation in Balanced circuit

Wattmeter Method

Three Phase Unbalanced Circuit

Unbalanced Delta and Wye Connected Loads

Power Calculation in Unbalanced circuit

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Course Objective

1. Learn the distinction between single-phase and

polyphase systems.

2. Familiarize with working with both Y- and -

connected three-phase sources and three-phasenetworks.

3. Master the technique of per-phase analysis of

three-phase systems.

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Single Phase System

1. A generator connected through a pair of wire to a

loadSingle Phase Two Wire.

2. Vpis the magnitude of the source voltage, and

is the phase.

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Polyphase System

Circuit or system in which AC sources operate at

the same frequency but different phases are

known as polyphase.

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Polyphase System

Two Phase System:

A generator consists of two coils placed

perpendicular to each other

The voltage generated by one lags the other by90.

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Polyphase System

Three Phase System:

A generator consists of three coils placed 120

apart.

The voltage generated are equal in magnitudebut, out of phase by 120.

Three phase is the most economical polyphase

system.

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Polyphase System

Three Phase System:

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Ideal Y- and- connected voltage

sources

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Real Y- and- connected voltage

sources

Internal impedance of a generator is usually

inductive(due to the use of coils).

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Balanced three-phase source

Three sinusoidal voltages of the same amplitude,

frequency, but differing by 120 phase

difference with one another.

There are two possible sequences:

1. abc (positive) sequence : vb(t) lags va(t) by

120.

2. acb (negative) sequence: vb(t) leads va(t) by120.

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3-phase Advantages

1. The horsepower rating of three-phase motors

and the kVA rating of three-phase transformers

are 150% greater than single-phase motors or

transformers of similar frame size.

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3-phase Advantages

2. The power delivered by a single-phase system

pulsates and falls to zero. The three-phase

power never falls to zero. The power delivered to

the load in a three-phase system is the same atany instant. This produces superior operating

characteristics for three-phase motors.

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3-phase Advantages

3. A three-phase system needs three conductors;

however, each conductor is only 75% the size of

the equivalent kVA rated single-phase

conductors.

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3-phase: Basic Properties

Three-phase systems have either three or four

conductors.

These three-phase conductors are identified as

A, B, and C. The three phases are 120 degrees out of phase

with each other (360 divided by 3).

There is sometimes a fourth conductor, which is

the neutral.

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3-phase: Wye Connection

The wye, or star, connection is made by

connecting one end of each of the phase

windings together in a common node.

Each line terminals has a voltage drop known asthe phase voltage.

The line voltage is measured from phase

conductor to a different phase conductor.

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In a wye system, the line voltage is higher than

the phase voltage by a factor of the square root

of 3 (1.732).

ELine= EPhasex 1.732EPhase= ELine/ 1.732

In a wye system, the line current is equal to the

phase current.ILine= IPhase

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Wye Connected Source

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

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

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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

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Line and phase voltages in a wye connection.

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Line and phase currents in a wye connection.

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Vector sum of typical wye system voltages.

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3-phase: Delta Connection

In a delta system, the line current is higher than

the phase current by a factor of the square root

of 3 (1.732).

ILine= IPhasex 1.732

IPhase= ILine/ 1.732

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In a delta system, the line voltage is equal to the

phase voltage.

ELine= EPhase

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Delta Connected Source

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

Vab

Vbc

Vca

Vab

Vbc

Vca

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3 30

3 30

3 30

a AB CA

b BC AB

c CA BC

AB

BC

CA

I I I

I I I

I

I

I II

I

3

L a b c

p AB BC CA

L p

I I I I

I I I I

I I

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Delta system voltage and current relationships.

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Balanced Three-phase Load

A Balanced load has equal impedances on allthe phases

a) Wye-connected load b) Delta-connected load

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Balanced Three-phase Load

1 2 3

Conversion of Delta circuit to Wye or Wye to Delta.Balanced Impedance Conversion:

Y

a b c

Z Z Z Z

Z Z Z Z

1Z 3 Z

3Y YZ Z

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Balanced Three-phase Network

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Balanced Y-Y Connection

Source

Impedance

Line Impedance

Load

Impedance

Line current Inadd up to zero.

Neutral current is zero:

In= -(

Ia+

Ib+

Ic)= 0

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Balanced Y-Y Connection

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Single Phase Equivalent of Balanced

Y-Y Connection

Balanced three phase circuits can be analyzed on perphasebasis..

We look at one phase, say phase A and analyze the

single phase equivalent circuit.

Because the circuit is balanced, we can easily obtainother phase values using their phase relationships.

ana

Y

VI

Z

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Example 1

Obtain the line currents in the 3-phase circuit of the figureshown below

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Balanced Y-Connection

Three phase sources are usually Wye connected andthree phase loads are Delta connected.

There is no neutral connection for the Y-system.

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Balanced Y-Connection

3

L a b c

p AB BC CA

L p

I I I I

I I I I

I I

Phasor diagram of phase and line currents

Single phase equivalent circuit of the balanced Wye-delta

connection

3

Z

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Example 2

Solve for the line currents in the Y- circuit of the figureshown below. Take Z= 6045

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Balanced-Connection

Both the source and load are Delta connected andbalanced.

, ,BC CAABAB BC CAV VV

I I IZ Z Z

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

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Example 3

Refer to the -circuit of the figure shown below. Find theline and phase currents. Assume that the load impedance is

12 + j9 per phase.

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Balanced-Y Connection

30

3

pV

Transforming a Delta connected source

to an equivalent Wye connectionSingle phase equivalent of Delta Wye connection

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Example 4

In the circuit, if Vab = 44010 , Vbc = 440250 , Vca =440130.Find the line currents.

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Power in a Balanced System The total instantaneous power in a balanced three phase system is

constant.

=

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Power in a Balanced System The total instantaneous power in a balanced three phase system is

constant.

=

=

=

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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

2

p p p p

Complex power for each phas

3 cos

1 1= cos = sin

3 3

S V I

3 3 cos 3 cos

3 3 sin 3 sin

S=3S 3V

e

I 3

p p

p p p p p p p

a b c p p p L L

p p p L L

p

p V I

P p V I Q p V I S V I

P jQ

P P P P P V I V I

Q Q V I V I

I Z

p L

2

p

p , , and are

Total

all rm

complex powe

s values, is the load impedance angl

3

3

r

eL

p

L L

V

Z

P jQ

I

V

V I

I

V

S

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Power in a Balanced System2

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

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

VLVL

VL

Vp Vp

Vp

Ip

Ip Ip

VL

Vp

Ip

VL

VL Vp

Vp

Ip

Ip

Y connected load. connected load.

3L p L pV V I I 3L p L pV V I I

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Example 5

VL=840 V (Rms)

Capacitors for pf

Correction

IL

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Unbalanced Three Phase Systems An unbalanced system is due to unbalanced voltage sources or

unbalanced load.In an 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

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Example 5

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Power Measurements: The Wattmeter

The wattmeter is a four-terminal device that measures powerdelivered to the network if connected as shown:

the wattmeter

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Measuring Three-Phase Power

The sum of the

powers measured

by the wattmeters

is the total power

delivered.

The common node

xis arbitrary.

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The Two-Wattmeter Method

One wattmeter

can be

eliminated if the

point x ismoved to a line

(as shown, B).

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Example 6

A three-phase system has a line-to-line voltage of 4000 V rms and abalanced Delta-connected load with Z = 40 + j30 ohms. The phase

sequence is abc. Use the two wattmeters connected to lines A and C,

with line B as the common line for the voltage measurement.

Determine the total power measurement recorded by the wattmeters.

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An unbalanced delta-connected load is connected across the lines of a

4-wire balanced three-phase supply with neutral connection. Two

wattmeter W1, W2 are connected in lines 1 and 3 with the common

potential connection in the neutral wire. These wattmeters read 1796.5

and 3000 watts, respectively. The current in lines 1 and 3 is 20 A. If the

common potential connection is transferred to line 2, what will be thereading of the two wattmeters? Solve the power factor of the system.

The system voltage per phase is 208 V. Assume a phase sequence of

1-2-3 with V1nas reference.

Example 7

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END

ee 132 - lecture 1

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