12. CA2

8/13/2019 12. CA2

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Lecture 12:

11. Balanced 3-Phase Circuits

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Analysis of Y-Δ circuit

The Δ connected load is converted into Y

For a balanced load, three Δ connectedimpedances are equal

Single-phase equivalent circuit can bemodelled with

c ba

c b1

ZZZ

ZZZ

3

ZZ Δ

Y

3

ZZ Δ

Y

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Single phase equivalent circuit

Zga

Va’n

a A

ZY

Zla

IaA

n N

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Δ load currents

The line currents are

calculated using the

equivalent circuit

Load currents in Δ arecalculated using the Δ

circuit

Phase voltage is the

same as line voltage

A

B

C

Z A Z C

ZB

IaA

IbB

IcC

I AB

IBC

ICA

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Line and phase currents

Line current is times the

phase current

For a positive phase

sequence, phase currents

lead line currents by 300

For a negative phase

sequence, phase currents

lag line currents by 300

IAB

IBC

ICA

IaA

IcC

IbB

3

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Exercise

Y connected source feeds a Δ connected load

Distribution line impedance 0.3 + j0.9 Ω/φ

Load impedance is 118.5 + j85.8 Ω/φ

a-phase internal voltage of generator is used as reference

1. Construct single-phase equivalent circuit

2. Calculate line currents IaA, IbB and IcC

3. Calculate phase voltages at the load terminals

4. Calculate the phase current of the load5. Calculate the line voltages at the source terminals

00120

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Exercise (cont’d)

Zga

Va’n

a A

ZY

Zla

IaA

n N

0.2+j0.50.3+j0.9

(118.5+j85.8)/3

39.5+j28.600120

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Exercise (cont’d)

IaA

IbB

IcC

VAN

VAB

VBC VCA

087.1564.2

00

Ylaga

0

87.364.230 j40

0120

ZZZ

0120

0

13.834.2

00 96.004.117)87.364.2)(6.28 j5.39( 0

AN

0 04.2972.202V303 0

96.9072.202 004.14972.202

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Exercise (continued..)

IAB

IBC

ICA

Van

Vab

Vbc

Vca0

68.14994.205

aA I

0303

1

087.639.1

087.12639.1

0

13.11339.1

0

an

0 68.2994.205V303

032.9094.205

00 32.090.118)87.364.2)(5.29 j8.39(

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Power calculations in balanced

three-phase circuits

Average power delivered to a

Y-connected load

P A = |VAN||IaA|Cos(θvA- θiA) PB = |VBN||IbB|Cos(θvB- θiB)

PC = |VCN||IcC|Cos(θvC- θiC)

Reference Chapter 10

V AN

VBN

VCN

A

NB

C

Z A

ZB

ZC

IaA

IbB

IcC

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In a balanced 3-phase system Vφ=|VAN| =|VBN| =|VCN|

Iφ=|IaA| =|IbB| =|IcC|

θφ= θvA- θiA = θvB- θiB = θvC- θiC

Power delivered to each phase of the load P A= PB= PC= Pφ=Vφ Iφcos θ

Total Average power delivered PT = 3Pφ=3Vφ Iφcos θφ

Total Power delivered in terms of line voltage andcurrent

φLLφLL

T θcosIV3θcosI

3

V3P

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Reactive Power in a Balanced Y load

Reactive Power in each phase

Total Reactive Power

Complex Power in a Balanced Y load

Sφ= V ANIaA* = VBNIbB

* = VCNIcC* = VφIφ

*

Sφ= Pφ+ jQφ = VφIφ*

ST

φφφφ θsinIVQ

φLLφT θsinIV3Q3Q

φLLφ θIV3S3

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Average power delivered to a Δ-

connected load

P A = |VAB||IAB|Cos(θvAB- θiAB) PB = |VBC||IBC|Cos(θvBC- θiBC)

PC = |VCA||ICA|Cos(θvCA- θiCA)

Reference Chapter 10

A

B

C

Z A Z

C

ZB

V AB

VBC

VCA

I AB

IBC

ICA

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In a balanced 3-phase system Vφ=|VAB| =|VBC| =|VCA|

Iφ=|IAB| =|IBC| =|ICA|

θφ= θvAB- θiAB = θvBC- θiBC = θvCA- θiCA

Power delivered to each phase of the load P A= PB= PC= Pφ=Vφ Iφcos θφ

Total Average power delivered PT = 3Pφ=3Vφ Iφcos θφ

Total Power delivered in terms of line voltage andcurrent

φLLφL

LT θcosIV3θcos3

IV3P

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Reactive Power in a Balanced Δ load

Reactive Power in each phase

Total Reactive Power

Complex Power in a Balanced Δ load

Sφ= V ABI AB* = VBCIBC

* = VCAICA* = VφIφ

*

Sφ= Pφ+ jQφ = VφIφ*

ST

φφφφ θsinIVQ

φLLφT θsinIV3Q3Q

φLLφ θIV3S3

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Instantaneous Power in 3-phase

circuits

p A = v ANiaA=Vm Im cos(ωt)cos(ωt - θφ)

pB = vBNibB =Vm Im cos(ωt -1200)cos(ωt – θφ -1200)

pC = vCNicC =VmIm cos(ωt +1200

)cos(ωt – θφ +1200

)

pT = p A + pB + pC = 1.5Vm Im cosθφ

Where θφis phase angle θvA- θiA

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Instantaneous Power in 3-phase

circuits

In 3-φ balanced system Power is invariant

with time

Torque developed by 3-φ motor is constant

Less vibration in machines powered by 3-φ

motors

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Measuring Average Power in 3-φ

circuits using two Watt-meters

V AN

VBN

VCN

A

NB

C

Z A

ZB

ZC

IaA

IbB

IcC

a

b

c

cc

cc

pc

pc

VAN

VBN

VCN

VAB

VCA

VBC

IaAIbB

IcC

300

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Measuring Average Power in 3-φ

circuits using two Watt-meters

W1 = |VAB||IaA|Cosθ1 = VLILCosθ1

W2 = |VCB||IcC|Cosθ2 = VLILCosθ2

θ1 = θ + 300 = θφ + 300

θ2 = θ - 300 = θφ – 300

W1 = VLILCos(θφ + 300)

W2 = VLILCos(θφ - 300)

PT = W1 + W2 = 2VLILCosθφCos300 = φLLT θcosIV3P

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Exercise (cont’d)

Wattmeter readings ?

Phase voltage at load

is 120 V

1. Zφ= 8 + j62. Zφ= 8 - j6

3. Zφ= 5 + j5√3

4. Zφ= 10-750

Verify total power for thefour cases

V AN

VBN

VCN

A

NB

C

Z A

ZB

ZC

IaA

IbB

IcC

a

b

c

cc

cc

pc

pc

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Exercise (cont’d)

Zφ= 8 + j6 = 1036.870

VL= 120√3 V IL= 120/10=12 A

W1= (120√3)(12)cos(36.870+ 300)=979.75 W

W2= (120√3)(12)cos(36.870- 300)=2476.25 W

Zφ= 8 - j6 = 10-36.870

VL= 120√3 V IL= 120/10=12 A

W1= (120√3)(12)cos(-36.870+ 300)=2476.25 W

W2= (120√3)(12)cos(-36.87

0

- 30

0

)=979.25 W

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Exercise (cont’d)

Zφ= 5(1 + j√3) = 10600

VL= 120√3 V IL= 120/10=12 A

W1= (120√3)(12)cos(600+ 300)=0 W

W2= (120√3)(12)cos(600- 300)=2160 W

Zφ= 10-750

VL= 120√3 V IL= 120/10=12 A

W1= (120√3)(12)cos(-750+ 300)=1763.63 W

W2= (120√3)(12)cos(-75

0

- 30

0

)=-645.53 W

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Exercise (cont’d)

PT= 3(12)2(8) = 3456 W

W1+ W2= 979.75 + 2476.25 = 3456 W

PT= 3(12)2(8) = 3456 W

W1

+ W2

= 2476.25 + 979.75 = 3456 W

PT= 3(12)2(5) = 2160 W

W1+ W2= 0 + 2160 = 2160 W

PT= 3(12)2(2.5882) = 1118.10 W

W1+ W2= 1763.63 - 645.53 = 1118.10 W

If p.f. > 0.5 both Watt meters read positive If p.f. = 0.5 one Watt meter reads zero

If p.f. < 0.5 one Watt meter reads negative

Reversing the phase sequence will interchange the

readings on the two Watt meters

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Summary

3-Phase Balanced Systems

3-Phase voltages

Y/ 3-Phase Balanced systems

3-phase system analysis Single-phase equivalent circuit

Y load Line voltages √3 times phase voltage

Line current same as phase current

Line voltages leading phase voltages by 300 in abcsequence

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Summary

load Line current √3 times phase current Line voltage same as phase voltage

Line currents lag phase currents by 300 in abc sequence

Average, Reactive and Complex power delivered in3-Phase systems

Instantaneous power invariant to time

Measuring Total Power using 2 Watt-meters

12. CA2

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