ene 104 electric circuit theory -...
TRANSCRIPT
ENE 104
Electric Circuit Theory
Lecture 11:
Polyphase Circuits
http://webstaff.kmutt.ac.th/~dejwoot.kha/
(ENE) Mon, 26 Mar 2012 / (EIE) Wed, 4 Apr 2012
Week #12 : Dejwoot KHAWPARISUTH
Objectives : Ch12
Week #12
Page 2
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
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
(a) The definition of the
voltage Vab.
(b) Vad = Vab + Vbc+ Vcd
= Vab + Vcd.
Double-Subscript Notation:
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ENE 104 Polyphase Circuits
.V0100 anV
.V120100 bnV
.V240100 cnV
Double-Subscript Notation:
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ENE 104 Polyphase Circuits
.V302.173)6.8650(100
.V1201000100
j
bnannbanab VVVVV
Practice: 12.1
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ENE 104 Polyphase Circuits
Let 𝐕𝑎𝑏 = 100∠0° V, 𝐕𝑏𝑑 = 40∠80° V, and
𝐕𝑐𝑎 = 70∠200° V. Find (a) 𝐕𝑎𝑑; (b) 𝐕𝑏𝑐; (c) 𝐕𝑐𝑑
Practice: 12.1
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ENE 104 Polyphase Circuits
Practice: 12.2
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
Single-Phase Three-Wire System:
Week #12
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ENE 104 Polyphase Circuits
Single-Phase Three-Wire System:
Week #12
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ENE 104 Polyphase Circuits
Since Van = Vnb
Then
And InN = IBb + IAa = IBb - IaA = 0 p
nbBb
p
anaA
Z
VI
Z
VI
Consider:
Example:
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ENE 104 Polyphase Circuits
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:
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ENE 104 Polyphase Circuits
0)(100)(30115
0)(50)(100)1020(
0)(3)(500115
32313
12322
31211
IIIII
IIIII
IIIII
j
Example:
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ENE 104 Polyphase Circuits
.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
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ENE 104 Polyphase Circuits
.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:
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ENE 104 Polyphase Circuits
The transmission efficiency, :
%8.892372086
2086
generatedpowertotal
loadtodeliveredpowertotal
Practice: 12.3
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
Practice: 12.3
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ENE 104 Polyphase Circuits
Three-Phase Y-Y Connection:
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ENE 104 Polyphase Circuits
A Y-connected three-phase four-wire source.
A balanced three-phase system:
may be defined as having
and
cnbnan VVV
0 cnbnan VVV
Three-Phase Y-Y Connection:
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ENE 104 Polyphase Circuits
Positive, or abc, phase sequence.
Negative, or cba, phase sequence.
Three-Phase Y-Y Connection:
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ENE 104 Polyphase Circuits
Line-to-Line Voltage:
303
1200
pab
pp
bnanab
V
VV
V
VVV
Three-Phase Y-Y Connection:
Week #12
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ENE 104 Polyphase Circuits
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.
Three-Phase Y-Y Connection:
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ENE 104 Polyphase Circuits
p
anaA
Z
VI
120
120
aA
p
an
p
bnbB
I
Z
V
Z
VI
240aAcC II
Therefore:
0 cCbBaANn IIII
Example:
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ENE 104 Polyphase Circuits
Find the currents, voltages and the total power
dissipated in the load
Example:
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ENE 104 Polyphase Circuits
“Per-phase”:
.V0200 anV
.V120200 bnV
.V240200 cnV
2103 pca VV
903 pbc VV
303 pab VV 602p
anaA
Z
VI
3002cCI
1802bBI
Example:
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ENE 104 Polyphase Circuits
“Per-phase”:
)0120cos(2200 tvAN
)60120cos(22 tiAN
Example:
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ENE 104 Polyphase Circuits
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:
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ENE 104 Polyphase Circuits
the instantaneous power absorbed by the total
load is therefore
Wtptptptp CBA 600)()()()(
Practice: 12.4
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
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:
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ENE 104 Polyphase Circuits
.3
300 VVp
.4003
1200 W
Example:
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
A balanced 600-W lighting load is added (in
parallel) to the previous example. Determine the
new line current.
per-phase circuit
Example:
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ENE 104 Polyphase Circuits
9.3660pZ
89.22 I
The amplitude of the lighting current is
determined by
So assume
And the line current is
Example:
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
Practice: 12.6
Week #12
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ENE 104 Polyphase Circuits
Where and
The Delta (∆) Connection:
Week #12
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ENE 104 Polyphase Circuits
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
The Delta (∆) Connection:
Week #12
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ENE 104 Polyphase Circuits
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
The Delta (∆) Connection:
Week #12
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ENE 104 Polyphase Circuits
CAABaA III
CABCABpI III
cCbBaALI III
pL II 3
∆-Connected:
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ENE 104 Polyphase Circuits
total power,
cos3
cos
cos
LL
pL
ppp
IV
IV
IVP
cos3
3
LL
p
IV
PP
Y-connected:
Week #12
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ENE 104 Polyphase Circuits
total power,
cos3
cos
cos
LL
Lp
ppp
IV
IV
IVP
cos3
3
LL
p
IV
PP
Practice: 12.7
Week #12
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
Example 12.5:
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ENE 104 Polyphase Circuits
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:
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
Power Measurement:
Week #12
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ENE 104 Polyphase Circuits
wattmeter
Power Measurement:
Week #12
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ENE 104 Polyphase Circuits
IVIV angangP 22 cos
Practice: 12.9
Week #12
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
Practice: 12.9
Week #12
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ENE 104 Polyphase Circuits
Power Measurement:
Week #12
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ENE 104 Polyphase Circuits
The wattmeter in a Three-Phase System:
Power Measurement:
Week #12
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ENE 104 Polyphase Circuits
The wattmeter in a Three-Phase System:
dtivT
P aA
T
AxA 0
1
dtivivivT
PPPP cCCxbBBxaA
T
AxCBA )(1
0
NxCNCx
NxBNBx
NxANAx
vvv
vvv
vvv
Power Measurement:
Week #12
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ENE 104 Polyphase Circuits
The wattmeter in a Three-Phase System:
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
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ENE 104 Polyphase Circuits
The Two-Wattmeter Method:
aAABaAAB angangP IVIV cos1
cCCBcCCB angangP IVIV cos2
Power Measurement:
Week #12
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ENE 104 Polyphase Circuits
The Two-Wattmeter Method:
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
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ENE 104 Polyphase Circuits
The Two-Wattmeter Method:
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
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
60230acV
W13891.10560cos554.8230
cos1
aAacaAac angangP IVIV
0230abV
1.105554.8
154
30
154
3230
jj
anaA
VI
120230caV
Example:
Week #12
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ENE 104 Polyphase Circuits
W5.5129.134120cos554.8230
cos2
bBbcbBbc angangP IVIV
1.105554.8aAI
120230bcV
9.134554.8bBI
Practice: 12.10
Week #12
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ENE 104 Polyphase Circuits
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
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ENE 104 Polyphase Circuits
Practice: 12.10
Week #12
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ENE 104 Polyphase Circuits
Example: Final 2/47 Page 71
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 Page 72
ENE 104 AC Circuit Power Analysis
+_
+_
+_
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 Page 73
ENE 104 AC Circuit Power Analysis
+_
+_
+_
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 Page 74
ENE 104 AC Circuit Power Analysis
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
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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: Page 76
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: Page 77
ENE 104 AC Circuit Power Analysis Week #12
Reference: Page 78
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