Ee132 Lec 1 Polyphase Circuits

EE 132EE 132Electric Circuit Electric Circuit Theory IITheory II

Lecture 1

Polyphase Circuits

Review: 1-Review: 1-φφ circuits circuitsReview: 1-Review: 1-φφ circuits circuits

Two-wire type

Three-wire type

VVp p = magnitude of the source voltage

φφ = phase of source voltage

Polyphase CircuitsPolyphase CircuitsPolyphase CircuitsPolyphase Circuits

Circuits or systems in which the ac sources operate at the same frequency but different phases

Two-phase, three-wire system

The source is a generator with two coils placed in perpendicular to each other so that the voltage generated by one lags the other by 90°.

Polyphase CircuitsPolyphase CircuitsPolyphase CircuitsPolyphase Circuits

Circuits or systems in which the ac sources operate at the same frequency but different phases

Three-phase, four-wire system

The source is a generator consisting of three sources having the same amplitude and frequency but out of phase with each other by 120°.

Why 3-Why 3-φφ systems? systems?Why 3-Why 3-φφ systems? systems? All electric power is generated and distributed in 3-φ, at

the operating frequency of 60 Hz (or ω = 377 rad/s) or 50 Hz (or ω = 314 rad/s)

The instantaneous power can be constant (not pulsating) Uniform power transmission & less vibration of 3-φ

machines

For the same amount of power, 3-φ is more economical Less volume of wire needed

Balanced 3-Balanced 3-φφ sources sourcesBalanced 3-Balanced 3-φφ sources sources

Three voltages sources connected to loads by 3 or 4 wires

Equivalent to three (3) single phase circuits

Can be connected in WYE (Y) or DELTA (Δ)

Balanced 3-Balanced 3-φφ sources sources

Balanced phase voltagesBalanced phase voltages are equal in magnitude and are out of phase by 120°.

Y-connected source Δ-connected source

Vp = phase voltageVL = line voltage

0an bn cn

an bn cn

V V V

V V V

Balanced 3-Balanced 3-φφ sources sources

The phase sequencephase sequence is the time order in which the phase voltages reach their peak values wrt time.

abc (positive) Phase sequence

acb (negative) Phase sequence

0

120

240 120

an p

bn p

cn p p

V

V

V V

V

V

V

0

120

240 120

an p

bn p

cn p p

V

V

V V

V

V

V

Balanced 3-Balanced 3-φφ load load

A balanced loadbalanced load is one in which the phase impedances are equal in magnitude and in phase. Otherwise, load is unbalanced.

Y-connected load Δ-connected load

Balanced 3-Balanced 3-φφ load load

Y-connected load

1 2 3 Y Z Z Z Z

A balanced loadbalanced load is one in which the phase impedances are equal in magnitude and in phase. Otherwise, load is unbalanced.

Balanced 3-Balanced 3-φφ load load

Δ-connected load

A B C Z Z Z Z

A balanced loadbalanced load is one in which the phase impedances are equal in magnitude and in phase. Otherwise, load is unbalanced.

Balanced 3-Balanced 3-φφ load load

Y-connected load Δ-connected load

or3 Y Z Z 1

3Y Z Z

Balanced 3-Balanced 3-φφ systems systems

We can have four possible combinations:

Y-Y connection (Y-connected source, Y-connected load)

Y-Δ connection Easy to remove and add loads connected in delta

Δ -Δ connection

Δ-Y connectionNot common because of the circulating current that will result in the delta windings of the source if the phase voltages are slightly unbalanced

source impedance

line impedance

load impedance

neutral impedance

Balanced Y-Y connectionBalanced Y-Y connectionA balanced Y-Y systembalanced Y-Y system is a 3-φ system with a balanced

Y-connected source & a balanced Y-connected load.

total load impedance per phaseY

S l L

Z

Z Z Z

Balanced Y-Y connectionBalanced Y-Y connectionA balanced Y-Y systembalanced Y-Y system is a 3-φ system with a balanced

Y-connected source & a balanced Y-connected load.

= total load impedance per phase

Balanced Y-Y connectionBalanced Y-Y connection

3 30

3 90

3 210

ab p

bc p

ca p

V

V

V

V

V

V

Assuming positive phase sequence:

The phase voltages are

The line voltages are

0

120

120

an p

bn p

cn p

V

V

V

V

V

V

0 120

1 31 3 30

2 2

3 90

3 210

ab an nb an bn p p

p p

bc bn cn p

ca cn an p

V V

V j V

V

V

V V V V V

V V V

V V V

3L pVV

Balanced Y-Y connectionBalanced Y-Y connection

Where

and

3L pVV

p an bn cnV V V V

L ab bc caV V V V

3 30

3 90

3 210

ab p

bc p

ca p

V

V

V

V

V

V

0

120

120

an p

bn p

cn p

V

V

V

V

V

V

0a b c I I I

0n a b c I I I I

0nN n n V Z I L PI I

Balanced Y-Y connectionBalanced Y-Y connection

Where

And

Define:

For the Y-Y connection:

3L pVV

p an bn cnV V V V

L ab bc caV V V V

3 30

3 90

3 210

ab p

bc p

ca p

V

V

V

V

V

V

0

120

120

an p

bn p

cn p

V

V

V

V

V

V

L PI I

IIPP = phase current

= current in each phase of the source/load

IILL = line current

= current in each line

Examples:1. Calculate the line currents in

the circuit shown.

Balanced Y-Y connectionBalanced Y-Y connection

6.81 21.8

6.81 141.8

6.81 98.2

a

b

c

I

I

I

2. A Y-connected balanced three-phase generator with an impedance of 0.4 + j0.3 Ω per phase is connected to a Y-connected balanced load with an impedance of 24 + j19 Ω. The line joining the generator and the load has an impedance of 0.6 + j0.7 Ω per phase. Assuming a positive sequence for the source voltages and that Van = 120∟30° V, find: (a) the line voltages; (b) the line currents.

Balanced Y-Balanced Y-ΔΔ connection connectionA balanced Y-balanced Y-ΔΔ system system consists of a balanced Y-

connected source feeding a balanced Δ-connected load.

3 30

3 90

3 210

ab p

bc p

ca p

V

V

V

V

V

V

0

120

120

an p

bn p

cn p

V

V

V

V

V

V

The phase currents are:

Balanced Y-Balanced Y-ΔΔ connection connectionA balanced Y-balanced Y-ΔΔ system system consists of a balanced Y-

connected source feeding a balanced Δ-connected load.

3 30

3 90

3 210

ab p

bc p

ca p

V

V

V

V

V

V

0

120

120

an p

bn p

cn p

V

V

V

V

V

V

To get the line currents, apply KCL at nodes A, B & C:

a AB CA

b BC AB

c CA BC

I I I

I I I

I I I 3L PI I

Balanced Y-Balanced Y-ΔΔ connection connectionA balanced Y-balanced Y-ΔΔ system system consists of a balanced Y-

connected source feeding a balanced Δ-connected load.

3 30

3 90

3 210

ab p

bc p

ca p

V

V

V

V

V

V

0

120

120

an p

bn p

cn p

V

V

V

V

V

V

To get the line currents, apply KCL at nodes A, B & C:

a AB CA

b BC CA

c CA BC

I I I

I I I

I I I 3L PI I

Example:

Balanced Y-Balanced Y-ΔΔ connection connection

Balanced Y-Balanced Y-ΔΔ connection connectionExample:

Example:

Balanced Y-Balanced Y-ΔΔ connection connection

Balanced Balanced ΔΔ - -ΔΔ connection connectionA balanced balanced ΔΔ - -ΔΔ system system is one in which both the

balanced source and balanced load are Δ-connected.

0

120

120

ab p

bc p

ca p

V

V

V

V

V

V

The phase currents are:

Assuming no line impedances,

ab AB

bc BC

ca CA

V V

V V

V V

The line currents are:

3L PI I

Example:

Balanced Balanced ΔΔ - -ΔΔ connection connection

Example:

Balanced Balanced ΔΔ - -ΔΔ connection connection

Example:

Balanced Balanced ΔΔ - -ΔΔ connection connection

Balanced Balanced ΔΔ -Y connection -Y connectionA balanced balanced ΔΔ - -YY system system consists of a balanced Δ -

connected source feeding a balanced Y-connected load.

0

120

120

ab p

bc p

ca p

V

V

V

V

V

V

These are also the line voltages.

To obtain the line currents, we can apply KVL to loop aANBba i.e

0

0

0

ab Y a Y b

Y a b ab p

pa b

Y

V

V

V Z I Z I

Z I I V

I IZ

Balanced Balanced ΔΔ -Y connection -Y connectionA balanced balanced ΔΔ - -YY system system consists of a balanced Δ -

connected source feeding a balanced Y-connected load.

0

120

120

ab p

bc p

ca p

V

V

V

V

V

V

These are also the line voltages.

To obtain the line currents, we can apply KVL to loop aANBba i.e

The line currents are:

0

0

0

ab Y a Y b

Y a b ab p

pa b

Y

V

V

V Z I Z I

Z I I V

I IZ

But for the abc phase sequence,Thus

120 ,b a I I

1 1 120 ,

3 30

a b a

a

I I I

I

3 30pa

Y

V I

Z

120

120

b a

c a

I I

I I

Balanced Balanced ΔΔ -Y connection -Y connectionA balanced balanced ΔΔ - -YY system system consists of a balanced Δ -

connected source feeding a balanced Y-connected load.

0

120

120

ab p

bc p

ca p

V

V

V

V

V

V

These are also the line voltages.

To obtain the line currents, we can apply KVL to loop aANBba i.e

0

0

0

ab Y a Y b

Y a b ab p

pa b

Y

V

V

V Z I Z I

Z I I V

I IZ

L PI I

Balanced Balanced ΔΔ -Y connection -Y connectionA balanced balanced ΔΔ - -YY system system consists of a balanced Δ -

connected source feeding a balanced Y-connected load.

Alternatively, to obtain the line currents, we can also transform the ΔΔ-connected load into a Y-connected load.

SummarySummary

SummarySummary

Example:

Balanced Balanced ΔΔ -Y connection -Y connection

Example:

Balanced Balanced ΔΔ -Y connection -Y connection

Example:

Balanced Balanced ΔΔ -Y connection -Y connection