ENE 104 Electric Circuit Theorywebstaff.kmutt.ac.th/~dejwoot.kha/.../L11_Polyphase... · ENE 104...

ENE 104

Electric Circuit Theory

Lecture 11:

Polyphase Circuits

http://webstaff.kmutt.ac.th/~dejwoot.kha/

Week #12 : Dejwoot KHAWPARISUTH



Objectives : Ch12

Week #12

ENE 104 Polyphase Circuits

• single-phase and polyphase systems

• Y- and Δ- connected three-phase system

• per-phase analysis of three-phase systems

Introduction:

Week #12


An example set of three voltages, each of which is 120o out of

phase with the other two. As can be seen, only one of the voltages

will be zero at a particular instant.

Double-Subscript Notation:

Week #12


(a) The definition of the

voltage Vab.

(b) Vad = Vab + Vbc+ Vcd

= Vab + Vcd.


Week #12


.V0100 anV

.V120100 bnV

.V240100 cnV


Week #12


.V302.173)6.8650(100

.V1201000100

j

bnannbanab VVVVV

Practice: 12.1

Week #12


Let 𝐕𝑎𝑏 = 100∠0° V, 𝐕𝑏𝑑 = 40∠80° V, and

𝐕𝑐𝑎 = 70∠200° V. Find (a) 𝐕𝑎𝑑; (b) 𝐕𝑏𝑐; (c) 𝐕𝑐𝑑

Practice: 12.1

Week #12


Practice: 12.2

Week #12


Refer to the circuit of figure below and let

𝐈𝑓𝑗 = 3 A, 𝐈𝑑𝑒 = 2 A, and 𝐈ℎ𝑑 = −6 A. Find (a),

𝐈𝑐𝑑; (b) 𝐈𝑒𝑓; (c) 𝐈𝑖𝑗

Practice: 12.2

Week #12


Single-Phase Three-Wire System:

Week #12


Single-Phase Three-Wire System:

Week #12


Since Van = Vnb

Then

And InN = IBb + IAa = IBb - IaA = 0 p

nbBb

p

anaA

Z

VI

Z

VI

Consider:

Example:

Week #12


Effect of Finite Wire Impedance:

Analyze the system below and determine the power

delivered to each of the three loads as well as the power

lost in the neutral wire and each of the two lines.

Example:

Week #12


0)(100)(30115

0)(50)(100)1020(

0)(3)(500115

32313

12322

31211

IIIII

IIIII

IIIII

j

Example:

Week #12


.A83.1924.111 I

.A47.24389.92 I

.A80.2137.103 I

.W206502

2150 IIP

.W1171002

23100 IIP

The average power drawn by each load:

.W1763202

21020 IjP

Example:

Week #12


.W12612

1 IaAP

.A83.1924.111 I

.W10812

3 IbBP

.A47.24389.92 I

.A80.2137.103 I

The loss in each of the wires:

.W312

13 IInNP

Example:

Week #12


The transmission efficiency, :

%8.892372086

2086

generatedpowertotal

loadtodeliveredpowertotal

Practice: 12.3

Week #12


Modified Figure below by adding a 1.5 Ω

resistance to each of the two outer lines, and a

2.5 Ω resistance to the neutral wire. Find the

average power delivered to each of the three

loads.

Practice: 12.3

Week #12


Three-Phase Y-Y Connection:

Week #12


A Y-connected three-phase four-wire source.

A balanced three-phase system:

may be defined as having

and

cnbnan VVV

0 cnbnan VVV


Week #12


Positive, or abc, phase sequence.

Negative, or cba, phase sequence.


Week #12


Line-to-Line Voltage:

303

1200

pab

pp

bnanab

V

VV

V

VVV


Week #12


Line-to-Line Voltage:

2103 pca VV

903 pbc VV

303 pab VV

A balanced three-phase system, connected

Y-Y and including a neutral.


Week #12


p

anaA

Z

VI

120

120

aA

p

an

p

bnbB

I

Z

V

Z

VI

240aAcC II

Therefore:

0 cCbBaANn IIII

Example:

Week #12


Find the currents, voltages and the total power

dissipated in the load

Example:

Week #12


“Per-phase”:

.V0200 anV

.V120200 bnV

.V240200 cnV

2103 pca VV

903 pbc VV

303 pab VV 602p

anaA

Z

VI

3002cCI

1802bBI

Example:

Week #12


“Per-phase”:

)0120cos(2200 tvAN

)60120cos(22 tiAN

Example:

Week #12


even though phase voltages and currents have

zero value at specific instants in time, the

instantaneous power delivered to the total load is

never zero.

)]60240cos(400200

)]60240cos()60[cos(400

)60120cos()120cos(800)(

t

t

ttivtp ANANA

)]300240cos(400200)( ttpB

)]180240cos(400200)( ttpC

Example:

Week #12


the instantaneous power absorbed by the total

load is therefore

Wtptptptp CBA 600)()()()(

Practice: 12.4

Week #12


A balanced three-phase three-wire system has a

Y-connected load. Each phase contains three

loads in parallel: −j100 Ω, 100 Ω, and 50 + j50 Ω.

Assume positive phase sequence with 𝐕𝑎𝑏 =

400∠0° V. Find (a) 𝐕𝑎𝑛; (b) 𝐈𝑎𝐴; (c) the total power

drawn by the load

Practice: 12.4

Week #12


A balanced three-phase system with a line

voltage of 300V is supplying a balanced Y-

connected load with 1200W at a leading PF of

0.8. Find the line current and the per-phase load

impedance.

The phase voltage is

The per-phase power is

Example:

Week #12


.3

300 VVp

.4003

1200 W

Example:

Week #12


The line current is

The phase impedance is

.3

300 VVp .4003

1200 WPp

)cos()cos(2

1 effeffmm IVIVP

89.2

3

300

L

p

pI

VZ

.89.28.0

400

)cos(3

300A

V

PI

p

p

L

Example:

Week #12


The PF is 0.8 leading, so the impedance phase

angle is

thus 9.3660pZ

9.36)8.0(cos 1

Practice: 12.5

Week #12


A balanced three-phase three-wire system has a

line voltage of 500 V. Two balanced Y-connected

loads are present. One is a capacitive load with

7 − j2 Ω per phase, and the other is an inductive

load with 4 + j2 Ω per phase. Find (a) the phase

voltage; (b) the line current; (c) the total power

drawn by the load; (d) the power factor at which

the source is operating.

Practice: 12.5

Week #12


A balanced 600-W lighting load is added (in

parallel) to the previous example. Determine the

new line current.

per-phase circuit

Example:

Week #12


9.3660pZ

89.22 I

The amplitude of the lighting current is

determined by

So assume

And the line current is

Example:

Week #12


155.11 I .A0155.11 I

0cos200 13

300 I

)cos( effeff IVP

.A9.3689.22 I

.A6.2687.321 IIIL

Practice: 12.6

Week #12


Three balanced Y-connected loads are installed on a

balanced three-phase four-wire system. Load 1

draws a total power of 6kW at unity PF, load 2

requires 10kVA at PF=0.96 lagging, and load 3

needs 7kW at 0.85 lagging. If the phase voltage at

the load is 135 V, if each line has a resistance of

0.1 Ω, and if the neutral has a resistance of 1 Ω, find

(a) the total power drawn by the load; (b) the

combined PF of the load; (c) the total power lost in

the four lines; (d) the phase voltage at the source;

(e) the power factor at which the source is operating.

Practice: 12.6

Week #12


Where and

The Delta (∆) Connection:

Week #12


A balanced D-connected load is present on a three-wire

three-phase system. The source happens to be Y-

connected. cn

bn

anpV

V

V

V

ca

bc

abLV

V

V

V

pL VV 3 303 pab VV

The phase current

the line current:

and


Week #12


CAABaA III p

caCA

p

bcBC

p

abAB

Z

VI

Z

VI

Z

VI

CABCABpI III cCbBaALI III

pL II 3

An example with an inductive load


Week #12


CAABaA III

CABCABpI III

cCbBaALI III

pL II 3

∆-Connected:

Week #12


total power,

cos3

cos

cos

LL

pL

ppp

IV

IV

IVP

cos3

3

LL

p

IV

PP

Y-connected:

Week #12


total power,

cos3

cos

cos

LL

Lp

ppp

IV

IV

IVP

cos3

3

LL

p

IV

PP

Practice: 12.7

Week #12


Each phase of a balanced three-phase Δ-

connected load consists of a 0.2 H inductor in

series with the parallel combination of a 5 𝜇F

capacitor and a 200 Ω resistance. Assume zero

line resistance and a phase voltage of 200 V at

𝜔 = 400 rad/s. Find (a) the phase current; (b)

the line current; (c) the total power absorbed by

the load

Practice: 12.7

Week #12


Example 12.5:

Week #12


Determine the amplitude of the line current in a

three-phase system with a line voltage of 300 V

that supplies 1200 W to a ∆-connected load at a

lagging PF of 0.8

cos3

cos

cos

LL

pL

ppp

IV

IV

IVP

9.36180.,89.2 pL AI Z

Example 12.6:

Week #12


Determine the amplitude of the line current in a

three-phase system with a line voltage of 300 V

that supplies 1200 W to a Y-connected load at a

lagging PF of 0.8

cos3

cos

cos

LL

Lp

ppp

IV

IV

IVP

9.3660.,89.2 pL AI Z 3

DZ

ZY

Practice: 12.8

Week #12


A balanced three-phase three-wire system is

terminated with two Δ-connected load in

parallel. Load 1 draws 40kVA at lagging PF of

0.8, while load 2 absorbs 24 kW at a leading PF

of 0.9. Assume no line resistance, and let

𝐕𝑎𝑏 = 440∠30° V. Find (a) the total power drawn

by the load; (b) the phase current 𝐈𝐴𝐵1 for the

lagging load; (c) 𝐈𝐴𝐵2; (d) 𝐈𝑎𝐴.

Practice: 12.8

Week #12


Power Measurement:

Week #12


wattmeter

Power Measurement:

Week #12


IVIV angangP 22 cos

Practice: 12.9

Week #12


Determine the wattmeter reading in Figure

below, state weather or not the potential coil had

to be reversed in order to obtain an upscale

reading, and identify the device or devices

absorbing or generating this power. The (+)

terminal of the wattmeter is connected to: (a) x;

(b) y; (c) z.

Practice: 12.9

Week #12


Power Measurement:

Week #12


The wattmeter in a Three-Phase System:

Power Measurement:

Week #12



dtivT

P aA

T

AxA 0

1

dtivivivT

PPPP cCCxbBBxaA

T

AxCBA )(1

0

NxCNCx

NxBNBx

NxANAx

vvv

vvv

vvv

Power Measurement:

Week #12



dtivivivT

dtiiivT

dtivivivT

dtivivivT

PPPP

cCCNbBBNaA

T

AN

T

cCbBaANxcCCNbBBNaA

T

AN

cCCxbBBxaA

T

AxCBA

)(1

)(1

)(1

)(1

0

00

0

NxCNCx

NxBNBx

NxANAx

vvv

vvv

vvv

Power Measurement:

Week #12


The Two-Wattmeter Method:

aAABaAAB angangP IVIV cos1

cCCBcCCB angangP IVIV cos2

Power Measurement:

Week #12



in the case of a

balance load:

we can find PF

30cos

cos1

LL

aAABaAAB

IV

angangP IVIV

30cos

cos2

LL

cCCBcCCB

IV

angangP IVIV

Power Measurement:

Week #12



in the case of a

balance load:

we can find PF

30cos

30cos

2

1

P

P

12

123tanPP

PP

Example 12.7:

Week #12


The balanced load is fed by a balanced three-

phase system having and positive

phase sequence. Find the reading of each

wattmeter and the total power drawn by the load.

120230caV

120230bcV

0230abV

0230abV

Example:

Week #12


60230acV

W13891.10560cos554.8230

cos1

aAacaAac angangP IVIV

0230abV

1.105554.8

154

30

154

3230

jj

anaA

VI

120230caV

Example:

Week #12


W5.5129.134120cos554.8230

cos2

bBbcbBbc angangP IVIV

1.105554.8aAI

120230bcV

9.134554.8bBI

Practice: 12.10

Week #12


For the circuit of Figure below, let the loads be

𝐙𝐴 = 25∠60°Ω, 𝐙𝐵 = 50∠ − 60°Ω, 𝐙𝐶 = 50∠60°Ω,

𝐕𝐴𝐵 = 600∠0° V rms with (+) phase sequence, and

locate point x at C. Find (a) 𝑃𝐴; (b) 𝑃𝐵; (c)𝑃𝐶

Practice: 12.10

Week #12


Example: Final 2/47

ENE 104 AC Circuit Power Analysis

เม่ือ แหล่งจ่ายในวงจร เป็นแบบ balanced และ positive phase sequence จงหา , , และ the total complex power supplied by the source

+_

+_

+_

a

b

c

480 0o Vrms

A

C

B

10 ohm

-j10 ohm

j5 ohm

aAI bBI cCI

Week #12

Example: Final 2/47


+_

+_

+_

a

b

c

480 0o Vrms

A

C

B

10 ohm

-j10 ohm

j5 ohm

301.8310

abAB

VI

3.1665

9038.831

jBCI

1201.83

10

15038.831

jCAI

.V0480 anV

V120480 bnV

.V240480 cnV

21038.831caV

904803bcV

303 pab VV

Week #12

Example: Final 2/47


+_

+_

+_

a

b

c

480 0o Vrms

A

C

B

10 ohm

-j10 ohm

j5 ohm

3.166BCI

301.83ABI

1201.83CAI

CAABaA III

ABBCbB III

BCCAcC III

***

CACABCBCABABtotal IVIVIVS

Week #12

Example: Final 2/46


เม่ือ Wattmeters ในวงจร อ่านค่าได ้ W. และ W. เม่ือค่า the magnitude ของ the line voltage มีค่า 4160 V. จงหา

54.297,371 W

31.196,1392 W

Z

12

123tanPP

PP

Week #12

ENE 104

+_

+_

+_

a

b

c

480 0o Vrms

120o

480

480 -120o

Wm1

+- +-

Wm2

+-+-

ZA

ZB

ZC

A

B

C

ให้หา VAB, VBC, IAB, IBC, IaA, IcC และค่าที่อ่านได้ที่

wattmeter ทั้งสอง

ZA= 60 -30˚

ZB= 24 30˚

ZC= 80 0˚

Ex:

AC Circuit Power Analysis Week #12

Ex:

ENE 104

+_

+_

+_

a

b

c

480 0o Vrms

120o

480

480 -120o

Wm1

+- +-

Wm2

+-+-

ZA

ZB

ZC

A

B

C

AC Circuit Power Analysis Week #12

Hw:

ENE 104 AC Circuit Power Analysis Week #12

Reference:

ENE 104

W.H. Hayt, Jr., J.E. Kemmerly, S.M. Durbin, Engineering Circuit Analysis, Sixth Edition.

Copyright ©2002 McGraw-Hill. All rights reserved.

Circuit Analysis and Electrical Engineering

ENE 104 Electric Circuit Theorywebstaff.kmutt.ac.th/~dejwoot.kha/.../L11_Polyphase... · ENE 104...

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