lecture #11 egr 272 – circuit theory ii

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Lecture #11 EGR 272 – Circuit Theory IIRead: Chapter 11 in Electric Circuits, 6th Edition by Nilsson

Chapter 11 - Three Phase (3) CircuitsSo far we have only considered single phase (1) circuits.

Transmission of power• Power is transmitted using AC since voltage levels are easily stepped up for

transmission and down for customers through the use of transformers. As voltage increases, current decreases, so line losses (I2R losses) are reduced.

• AC voltages are generated using 3 phase voltages (three voltages that are separated in phase by 120°).

• Businesses and industry commonly use 3 phase voltages.• The 3 phases are split up so that a residential neighborhood uses only one of the 3

phases.• A single phase voltage of 240V is provided to a house along with 120V formed

using a center tap transformer (illustrate below).

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Lecture #11 EGR 272 – Circuit Theory II

Advantages of 3-phase circuits:• more efficient (smaller I2R losses)• less vibration in machinery• smaller conductors

Polyphase circuits • AC voltages can be generated using various numbers of phases.• 3 phase is the most common• 6 and 12 phase are sometimes seen

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Lecture #11 EGR 272 – Circuit Theory IISingle-phase (1 ) and 3-phase (3 ) generators

Generator coil representation

+ V -

Waveform:

t

V

Waveforms (abc phase sequence)

+ V -

Single-phase generator

+ Van - + Vbn -

+ Vcn -

3- phase generator

Generator coil representation

+ Van -+ Vbn -

+ Vcn -

t

Van Vbn Vcn

(one set of windings on the rotor)

(3 sets of windings on the rotor, where each are physically staggered by 120º)

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Lecture #11 EGR 272 – Circuit Theory IITCC Engineering students touring the Dominion Virginia Power

Nuclear Power Station in Surry, VA

One of the 800 MW 3-phase steam turbine generators is shown on the right. Nuclear energy is used to heat water and create steam to power the

generators.

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Lecture #11 EGR 272 – Circuit Theory IITCC Engineering students touring the Dominion Virginia Power

Nuclear Power Station in Surry, VA

Huge transformers are used to increase the voltage to 500,000 V before it leaves the power plant. Increases the voltage reduces the

current, resulting in lower line losses (I2R) during transmission.

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Three small transformers are mounted on the wall in the old electronics lab (Lynnhaven Building, Room 144). Each transformer carries a separate phase.

The outputs of the transformers were used for 3-phase experiments.

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3 transformers on a pole carry 3-phase power along Indian River

Road in Virginia Beach

1 transformer on a pole carries 1 single phase into a nearby

neighborhood in Virginia Beach

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an p

bn p

cn p

Time-domain expressions:

v (t) = V cos(wt)

v (t) = V cos(wt - 120 )

v (t) = V cos(wt 120 )

an p

bn p

cn p p

Phasor expressions (abc):

V V 0

V V 120

V V 240 V 120

1) abc (or positive) phase sequence

an p

bn p

cn p

Time-domain expressions:

v (t) = V cos(wt)

v (t) = V cos(wt + 120 )

v (t) = V cos(wt - 120 )

an p

bn p

cn p p

Phasor expressions (acb):

V V 0

V V 120

V V 240 V 120

2) acb (or negative) phase sequence

Phase sequencesThe three phases may be arranged in two possible phase sequences:

t

Van Vbn Vcn

t

Van Vcn Vbn

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180o

Phasor Diagram (abc phase sequence)

0o

90o

270o

Van

Vbn

Vcn

120o

120o

120o

180o

Phasor Diagram (acb phase sequence)

0o

90o

270o

Van

Vcn

Vbn

120o

120o

120o

Phase sequencesThe two possible phase sequences are sometimes represented by phasor diagrams as shown below.

Note: A practical illustration of the effect of phase sequence is that if a 3 motor is connected with an abc phase sequence it will turn in one direction and if it is connected with an acb phase sequence it will turn in the other direction.

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Lecture #11 EGR 272 – Circuit Theory IIGenerator ConnectionsThere are two basic types of generator connections:• Wye (Y) generator• Delta () generator

Wye (Y) generator

Van

+

_ _

Vcn Vbn

+ +

_

a

b

c

n

Vbc c

_ a

+

b

Vab Vca

+

+

_

_

a

b

c

Delta () generator

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Lecture #11 EGR 272 – Circuit Theory IIThe wye and delta generators shown on the previous page might also be represented using voltage sources, as shown in the PSPICE schematics below.

Wye Generator

a

Vcn Vbn

Van

c

b

Vca

b

Vab

a

cVbc

Delta Generator

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Balanced versus unbalanced generatorsBalanced generators have the same magnitude for each phase and exactly 120o of phase shift between each phase. Systems with balanced generators are easier to analyze.

Example: For each case shown below, is the generator balanced? What is the phase sequence? Circle the correct responses.

an

bn

cn

Case 1:

V 240 0 V RMS

V 225 120 V RMS

V 235 120 V RMS Balanced? Yes NoPhase sequence? abc acb

an

bn

cn

Case 2:

V 480 30 V RMS

V 480 150 V RMS

V 480 90 V RMSBalanced? Yes NoPhase sequence? abc acb

an

bn

cn

Case 3:

V 120cos(377 ) V

V 120cos(377 110 ) V

V 120cos(377 125 ) VBalanced? Yes NoPhase sequence? abc acb

t

t

t

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Lecture #11 EGR 272 – Circuit Theory II3 Y generatorDefine the line voltages Vab , Vbc , and Vca .

Show how to calculate line voltages in general using KVL.Show that for a balanced Y generator:

Wye (Y) generator

Van

+

_ _

Vcn Vbn

+ +

_

a

b

c

Vab

+

+

_

_

Path #1 n

Vbc

Vca

+

_

L pV 3 V

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Lecture #11 EGR 272 – Circuit Theory II3 generatorShow that the line voltages are equal to the generator phase voltages.In general, for a Y generator:

Vbc c

_ a

+

b

Vab Vca

+

+

_

_

a

b

c

Delta () generator

L pV V

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Lecture #11 EGR 272 – Circuit Theory IILoad connectionsThere are two common ways to connect the three phase loads:1) Wye (Y) connection2) Delta () connection

Wye (Y) Load:

Wye (Y) load

ZBN

ZAN

ZCN IbB

IcC

IaA

IAN

ICN IBN

A

B

C

N

aA

AN

aA AN

Wye load:

I = line current

I = load current

I = Ior in general

Line current = load current

Note: A wye load is balanced if ZAN = ZBN = ZCN.

Discussion: When would loads be balanced?

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Delta Load:an

ab

V = generator phase voltage

V = line voltageDelta () load

ZCA ZAB

ZBC IbB

IcC

IaA

IBC

IAB

ICA

A

B C

Note: A delta load is balanced if ZAB = ZBC = ZCA.

aA bB cCShow how to find I , I , and I using KCL

AB BC CAp p p

Line load

Show that if I = I 0 , I = I 120 , I = I 120 , then

I = 3 I for a balanced generator and load

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Lecture #11 EGR 272 – Circuit Theory IIGenerator-load configurations There are several possible generator andload configurations, including:1) Y-Y

a) 4-wireb) 3-wire

2) Y- --Y

3-PhaseGenerator

3-PhaseLoad

a

b

n

c

A

B

N

C

N

n

I nN

ZBN

ZAN

ZBN

4-wire Y-Y system

Y-Generator Y-Load

nN

an AN bn BN cn CN

Notes:1) If the 4-wire Y-Y system has a balanced generator and load, I = 0.

2) With the 4 wire system V V , V V , and V V .

3) If there is no neutral wire (i.e., a 3-wire system)

an AN bn BN cn CNthen in general V V , V V , and V V .

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Lecture #11 EGR 272 – Circuit Theory IIExample: A 4-wire Y-Y system has a balanced generator with Van = 480 V and a positive phase sequence.

A B nN

C

C

If Z Z 2 j2 , then find I if :

A) Z 2 j2 (i.e., the load is balanced)

B) Z 2 - j2 (i.e., the load is unbalanced)

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a

b

c

A

C B

n

IaA

Z CA Z AB

Z BC

_

+

IbB

IcC

ICA

IBC

IAB

Vbc

Vab

+

_

Y- configuration

Y- system:Note that the Y- system is significantly more difficult to analyze that the Y-Y system.

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Example: Determine all three line currents in a Y - system that has a balanced generator with Van = 240 V and a negative phase sequence.

AB BC CAUse Z 6 j8, Z 6 - j8, and Z 6.

lecture #11 egr 272 – circuit theory ii

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