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description
Phasor Method
Aug 24, 2011
USCUSC
Outline
• Review of analysis of DC (Direct Current) circuits• Analysis of AC (Alternating Current) circuits
– Introduction– Challenge of analysis of AC circuits
• Phasor method– Idea and concept– Advantage
• Conclusions• Next…
2
Review of Analysis of DC circuits
• DC circuits
3
L
CSU R
+
-
L
CSU R
+
-
dt
diLu
dt
duCi
Inductor:
Capacitor:
Resistor:R
ui
0u
0i
Short
Open
•Pure Resistive•Pure Resistive
t
u i
0
+
Review of Analysis of DC circuits
• Complete solution for DC circuits
4
E–+
G
R3 R4
R2R1
Unknown variable: 6 Voltages (b)
6 Currents (b)12 (2b)
Constraint Equations:
Elements: 6 (b)
Network:KCL: 4-1=3 (n-1)
KVL: 6-3=3 b-(n-1)
6 (b)
12 (2b)=12 (2b)
As number of braches grows:•Too many variables!•Too many equations!
As number of braches grows:•Too many variables!•Too many equations!
Review of Analysis of DC circuits
• Summary of DC circuits analysis methods– Circuit simplification
• Equivalent transformation of resistors• Equivalent transformation of sources
– General analytical methods• Node-voltage method (suitable for fewer nodes)• Mesh-current method (suitable of fewer meshs)
– Theorem• Superposition (linear circuits)• Thevenin and Norton equivalent
5
The purpose of circuit analysis method:•To reduce the number of variables and equations
The purpose of circuit analysis method:•To reduce the number of variables and equations
Introduction of AC circuits
• Why AC?– Generation, transmission, distribution
and consumption of electric energy are all in steady state sinusoidal.
6
t
u i
0+
• AC (Alternating current)Sinusoidal steady state analysis
– Any signal can be thought of as superposition of sinusoidal signals.
0
)sin()(n
nn naxf
Introduction of AC circuits• Challenge
7
dt
diLu
dt
duCi
Inductor:
Capacitor:
Resistor:R
ui
)sin()( ss tUtu
)()()(: tututuKVL CL
)sin()sin()sin( CCLLSS tUtUtU
)sin()sin()()( SiSSSCC tItUtituP
with analysis of AC circuitL
C)(tu R
+
-
)(tuL
)(tuC
+
+
-
- )sin()(
)sin()(
)sin()(
RRR
CCC
LLL
tUtu
tUtu
tUtu
The +,-,*,/ operation with trigonometric function is not easy!
The +,-,*,/ operation with trigonometric function is not easy!
Review of Analysis of DC circuits
• Summary of DC circuits analysis methods– Circuit simplification
• Equivalent transformation of resistors• Equivalent transformation of sources
– General analytical methods• Node-voltage method (suitable for fewer nodes)• Mesh-current method (suitable of fewer meshs)
– Theorem• Superposition (linear circuits)• Thevenin and Norton equivalent
8
Introduction of AC circuits
9
Phasor Method
10
)45sin(20)60sin(5)30sin(10 000 ttt
Hint:
Phasor Method
11
Charles Proteus Steinmetz
German-American mathematician and engineer(1865 – 1923)
•In 1893, he introduced the phasor method to calculation of AC circuits
GE required him to submit a itemized invoice. They soon received it. It included two items:1.Marking chalk "X" on side of generator: $1.2.Knowing where to mark chalk "X": $999.
Phasor Method
12
)sin( tU U
Trigonometric function Phasor Domain
)30sin(10 0t03010
0605
transform
Inversetransform
)60sin(5 0t
Phasor Method
13
Complex operation:
Sum/Subtraction:
)()()()( 21212211 bbjaajbajba
Multiplication/Division:
;21212211 FFFF
212
1
22
11
F
FF
F
Phasor Method
14
Sinusoidalexpression
Trigonometric calculation
Phasor（ Comple
x）
Result(Phasor)
ComplexOperation
transform
Inversetransform
Result (sinusoidal)
Time Domain Phasor Domain
Phasor Method
15
Trigonometric calculation
ComplexOperation
equivalent
)60sin(5)30sin(10 00 tt
00
00
60sin530sin10
60cos530cos10
b
a
00 30sincos1030cossin10 tt tsin)60cos530cos10( 00
tbta cossin )sin( tR
00 60sincos560cossin5 tt tcos)60sin530sin10( 00
Phasor Method
16
Trigonometric calculation
equivalent
R00
00
60sin530sin10
60cos530cos10
b
a
a
bbaR arctan;22
)60sin(5)30sin(10 00 tt 00 6053010
0000 60sin560cos530sin1030cos10 jj )60sin530sin10()60cos530cos10( 0000 j
jba
sincos jFFF
)sin( tR
ComplexOperation
Phasor Method
17
Example:
)76.10sin(75.25 0 t
)45sin(20)60sin(5)30sin(10 000 ttt 000 45206053010
)14.1414.14()33.45.2()566.8( jjj 81.43.25 j
076.1075.25
Conclusions
• The trigonometric function involved in the sinusoidal steady-state circuits is not convenient to calculation.
• By projecting trigonometric function to phasor domain, the calculation can be dramatically simplified.
18
Quiz 1- problem1
19
Convert the following instantaneous currents to phasors, using cos(wt) as the reference.Give your answer in polar form.(1).
2).
20
Review of Analysis of DC circuits
• Summary of DC circuits analysis methods– Circuit simplification
• Equivalent transformation of resistors• Equivalent transformation of sources
– General analytical methods• Node-voltage method (suitable for fewer nodes)• Mesh-current method (suitable of fewer meshs)
– Theorem• Superposition (linear circuits)• Thevenin and Norton equivalent
21
Review of Analysis of DC circuits
• Summary of DC circuits analysis methods– Circuit simplification
• Equivalent transformation of resistors• Equivalent transformation of sources
– General analytical methods• Node-voltage method (suitable for fewer nodes)• Mesh-current method (suitable of fewer meshs)
– Theorem• Superposition (linear circuits)• Thevenin and Norton equivalent
22
23
•For the circuit shown below, compute the voltage across the load terminals.
I=125 0° A
240 0 ° V LOAD LOAD
+
-
+
-
0.1Ω j0.5Ω
7.7851.05.01.0 j
7.7875.63
7.7851.0*0125
36.1593.235
5.625.227
5.625.12240
7.7875.630240
j
j
Power
Aug 24, 2011
USCUSC
Review of Phasor
25
Questions:
1. What is the main idea of Phasor method?
)30sin(10.2 0t
03010. a 06010. b060
2
10. c 030
2
10. d
Review of Phasor
26
L
C)(tu
R
+
-
)(tuL
)(tuC
+
+
-
-
)(tuR+ -
)sin()( ss tUtu
)sin()(
)sin()(
)sin()(
RRR
CCC
LLL
tUtu
tUtu
tUtu
Power
27
Instantaneous PowerAverage PowerReal PowerActive PowerReactive PowerComplex PowerApparent Power
Power
28
Power: Pure Resistive
29
Power: Pure Inductive
30
Power: Pure Capacitive
31
Average Power
32
Example 2.1
33
Complex Power
34
Power Triangle
35
Power Triangle
36