Power Engineering Foundation

By Fuad Latip 2006 1

3-phase (φ) Circuit

A 3- φ cct power system consists of 3-φ generators, transmission lines and loads.

AC power systems have a great advantage over DC system in that their voltage

levels can be changed with transformers.

3-φ power systems have 2 major advantages over single phase AC power system:

• It is possible to get more power per kilogram of metal from 3-φ machine.

• The power delivered to a 3-φ load is constant at all times, instead of pulsing

as it does in single-φ system.

Generation of 3-φ voltages and Currents

The current flowing to each load can be found from equation

ZVI =

Current flowing in 3-φ

θθ

θθ

θθ

−∠=∠

∠=

−∠=∠

∠=

−∠=∠∠

=

oo

c

oo

B

o

A

IZ

VI

IZ

VI

IZVI

240240

120120

0

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A three phase generator, consisting of three single-phase sources

equal in magnitude and 120o apart in phase.

The voltages in each phase of the generator.

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The three phases of the generator connected to

three identical loads

Phasor diagram showing the voltages

in each phase.

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Phase sequence

The phase sequence of a 3- φ power system is the order in which the voltages in the

individual phases peak. The 3-φ power system is said to have phase sequence abc, since

the voltages in the 3-φ peak in order, a,b,c. It is also possible to connect the 3-φ of power

system so that the voltages in the phases peak in order a,c,b. This type of power system is

said to have phase sequence acb.

Voltages and Currents in a 3-φ Circuit

a) Y Connection

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-Vbn

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oab

oobnanab

ocn

obn

oan

VV

jV

VjV

VjVV

VV

VVV

VV

VV

VV

303

21

233

23

23

23

21

1200

240

120

0

∠=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−=

−∠−∠=

−=

−∠=

−∠=

∠=

φ

φ

φφ

φφφ

φφ

φ

φ

φ

Thus

φφ

IIVV

L

LL

== 3 Y-Connection

b) Δ – Connection

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oa

oocaaba

oca

obc

oab

II

jI

IjI

IjII

II

III

II

II

II

303

21

233

23

23

23

21

2400

240

120

0

−∠=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+−−=

−∠−∠=

−=

−∠=

−∠=

∠=

φ

φ

φφ

φφφ

φφ

φ

φ

φ

Thus

-Ica

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φφ

VVII

LL

L

== 3 Δ-Connection

Summary

Y- Connection Δ-Connection

Voltage magnitudes φVVLL 3= φVVLL =

Current Magnitudes φII L = φIIL 3=

abc phase sequence Vab leads Va by 30o Ia lags Iab by 30o

Acb phase sequence Vab lags Va by 30o Ia leads Iab by 30o

Δ= φφ VV Y3

Δ= φφ II Y 3

3Δ=φφ ZZ Y

Power Relationships in 3-φ Circuits

a. The 3-φ Power Equations Involving Phase Quantities.

The real, reactive and apparent powers supplied to a balanced 3-φ load for Y and Δ

connection are given by:

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ZIS

ZIQ

ZIP

IVSIVQIVP

2

2

2

3

sin3

cos3

3sin3cos3

φ

φ

φ

φφ

φφ

φφ

θ

θ

θ

θ

=

=

=

=

=

=

b. The 3-φ Power Equations Involving Line Quantities.

LLL

LLL

LLL

LLL

LL

L

IVS

IVQ

IVP

IVP

inngSubstituti

VV

IIConnectionYin

IVP

3

sin3

cos3

cos3

3

)1(

3

)1(cos3

=

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

=

−−−−−−−=

θ

θ

θ

θ

φ

φ

φφ

IMPORTANT to realize that the cosθ and sinθ term are the cosine and sine of the

angle between phase voltage and the phase current. NOT the angle between the line-

to-line voltage and the line current.

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Analysis of Balanced 3-φ Systems

1. Determine the voltages, currents and powers at various points in the circuit with a

per-phase equivalent circuit.

2. Draw the per-phase equivalent circuit.

3. Solve it as common circuit in circuit theory.

Analysis Δ-connected sources and loads in power system.

• The standard approach is to transform the impedances by the Y- Δ

transform of elementary circuit theory.

• For the special case of balanced loads, the Y – Δ transformation states that

the a Δ-connected load consisting of three impedances, each of value Z, is

totally equivalent to a Y-connected load consisting of three impedances,

each value of .3Z

• This equivalence means that the voltages, currents and powers supplied to

the two loads cannot be distinguished in any fashion by anything external

to the load itself.

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Example 1:

A 208-V, 3-Φ power system is shown in figure below. It is consists

• Ideal 208-V Y-connected 3-Φ generator

• 3-Φ Transmission line has an impedance 0.06 + j.012Ω per phase.

• Load has an impedance of 12 + j9 Ω per phase

For the simple power system, find

a. The magnitude of the line current IL

b. The magnitude of the load's line and phase voltages VLL and VφL

c. The real, reactive and apparent powers consumed by the load

d. The power factor of the load

e. The real, reactive and apparent powers consumed by the transmission line

f. The real, reactive and apparent powers supplied by the generator

g. The power factor of generator

(ANSWER WILL BE DISSCUSS IN A CLASS)

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Example 2

From the figure above, answer all question same as example 1.

(ANSWER WILL BE DISSCUSS IN A CLASS)

Example 3

Figure above shows a one-line diagram of a small 480V industrial distribution system.

The impedance of the distribution line is negligible.

a. Find the overall power factor of the distribution system

b. Find the total line current supplied to the distribution system.

(ANSWER WILL BE DISSCUSS IN A CLASS)