Circuits IIEE221Unit 12

Instructor: Kevin D. Donohue

Three Phase Circuits, Balanced Y-Y, Y-, and - Three-Phase Circuits

Polyphase Circuits

Polyphase circuits contain multiple sources at the same frequency but different phases. Power is distributed over the power grid in the form of three-phase sinusoids.

Advantages of Three-Phase power distribution include: (Constant Power) Instantaneous power can be constant in a

three phase system. (More Economical) For equivalent power, the 3-Phase systems

are more economical than single-phase (can be driven with lower currents and voltages, and fewer wires required because of a common neutral connection between the phases).

(Flexible) Single phase service can be extracted from the 3-phase systems or phases manipulated to create additional phases.

Balanced 3-Phase Voltages

Balanced phase voltage are equal in magnitude and separate by 120 degrees in phase.

Voltages generated from a 3-phase generator can have 2 phase sequence possibilities depending on direction of the rotor:

Positive sequence (Counter Clockwise Rotation):

Negative sequence (Clockwise Rotation):

Show that the sum of all phase voltages in a balanced system is zero.

120240ˆ

120ˆ

0ˆ

ppcn

pbn

pan

VVV

VV

VV

120ˆ

120240ˆ

0ˆ

pcn

ppbn

pan

VV

VVV

VV

Single and 3-Phase Circuit Comparison

Consider the phase voltages of equal amplitude

Show that the line voltages are given by:

In general:

cnbnanp VVVV ˆˆˆˆ

0º

pbcacab VVVV ˆ3ˆˆˆ

2103ˆ

903ˆ

303ˆ

pca

pbc

pab

VV

VV

VV

Balanced 3-Phase Voltage Connections

There are 2 ways to connect a Balanced set of sources:

Y (wye)-Connected

(delta)-Connected

Balanced Loads

Balanced loads are equal in magnitude and phase.There 2 ways to connect balanced loadsY (wye)-Connected

(delta)-Connected

Show that for equivalent loads Z = 3ZY

A

B

N

C

A

B

C

Load-Source Connections

There are 4 possible ways balanced sources and loads can be connected:

Y Source to Y Load (Y-Y) Source to Load (-) Y Source to Load (Y-) Source to Y Load (-Y)

If not specified, the voltages on the sources will be assumed to be in RMS values.

Balanced Y-Y ConnectionThe complete Y-Y connection is shown below with impedances listed separately for the source (subscript s), line (subscript l), and load (subscript L).

cabcabL

anananp

VVVV

VVVV

ˆˆˆ

ˆˆˆ

For a positive sequence with , it can be shown that

2103ˆ

903ˆ

303ˆ

pca

pbc

pab

VV

VV

VV

0ˆpan VV

Balanced Y-Y ConnectionShow that the current in each phase can be expressed as:

,

and that

240ˆˆ ,120ˆˆ ,ˆ

ˆ acabY

ana IIII

Z

VI

Because of the symmetry of a balanced 3 phase system, the neutral connection can be dropped and the system analyzed on a per phase basis. In a Y-Y connected system, the phase (source or load) and line currents are the same.

aI

cIbI

nIYZ

YZYZ

0ˆˆˆ ˆ ncba IIII

Balanced Y- ConnectionIn this case the line voltages are directly across each load. It can be shown that:

and the load currents and phase currents are related by:

240ˆˆ ,120ˆˆ ,ˆˆˆ3ˆ

ABCAABBC

ABabpAB IIII

Z

V

Z

V

Z

VI

Note the –connected load can be converted to a Y-connected load through:

30-3ˆ ˆ ABa II

3

ˆˆ

ZZY

Balanced - Connection

In this case the line voltages are the phase voltages and are directly across each load. It can be shown that:

The line currents can be obtained from the phase currents

240ˆˆ ,120ˆˆ ,ˆˆ

ˆ

ABCAABBCABab

AB IIIIZ

V

Z

VI

30-3ˆ ˆ ABa II

Balanced -Y Connection

In this case the phase voltages are across the lines. It can be shown that:

the line current is related to the phase voltage by:

120ˆ ,120ˆ ,0ˆ pcapbcpab VVVVVV

Note the –connected source can be converted to a Y-connected source through:

30-3

ˆ Y

aba

Z

VI

303

ˆˆ aban

VV

Power in Balanced System

Show that the instantaneous power absorbed by a load in a balanced Y-Y system is a constant given by:

where the impedance in a single phase is given by:

The complex power per phase is

Note that average power or real power is the same as the instantaneous power for the 3-phase system.

ZZYˆ

)cos(3)( ppIVtp

)exp( jIVS pp