Transcript ee211_13 - University of Kentucky

Phasor Analysis
Phasors, Impedance, SPICE, and
Circuit Analysis
Kevin D. Donohue, University of Kentucky
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Impedance
The conversion of resistive, inductive, and capacitive elements
to impedance for a sinusoidal excitation at frequency  is
given by:
X L  L (Reactance)
1
(Reactance)
C
R  R (Resistance)
XC  
In general impedance is a complex quantity with a resistive
component (real) and a reactive component (imaginary):
X
Zˆ  R  jX  R 2  X 2  tan 1 
R
Kevin D. Donohue, University of Kentucky
2
Phasors
Sources can be converted to phasor notation as follows:
A cos(t   )  A
A sin(t   )  A  90 
This can be applied to all sources of the same frequency,
where  is used in the impedance conversion of the circuit.
If sources of different frequencies exist, superposition must
applied to solve for a given voltage or current:
1. Select sources with a common  and deactive all other sources.
2. Convert circuit elements to impedances.
3. Solve for desired voltage or current for selected .
4. Repeat steps 1 through 3 for new  until all sources have been applied.
5. Add together all time-domain solutions solutions obtain in Step 3.
Kevin D. Donohue, University of Kentucky
3
Loop Analysis Example
Determine the steady-state response for vc(t) when vs(t) =
5cos(800t) V
+ vc(t) 3 kW
vs(t)
114.86 nF
6 kW
Show:


Vˆc  2.5000 - j1.4434  2.8868   30   vc (t )  2.8868 cos 800 t   V
6

Kevin D. Donohue, University of Kentucky
4
Nodal Analysis Example
Find the steady-state value of vo(t) in the
circuit below, if vs(t) = 20cos(4t):
10 W
vs
ix
0.1 F
1H
2 ix
+
vo
0.5 H
-
Show: v0(t) = 13.91cos(4t + 198.3º)
Kevin D. Donohue, University of Kentucky
5
Multiple Source Example
Find io if is = 3cos(10t) and vs = 6cos(20t + 60º)
io
5W
vs
0.01 F
0.5 H
is
3
2
1
Amps
Show io = 0.54cos(20t+123.4º)+2.7cos(10t-153.4º)
0
-1
-2
Kevin D. Donohue, University of Kentucky
0
0.2
0.4
0.6
Seconds
0.8
1
1.2
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Equivalent Circuit Example
Find io steady-state using Norton’s Theorem, if
vs(t) = 2sin(10t):
10 W
vs
io
0.01 F
5W
.4 H
Show is(t)= .2sin(10t); Zth = 3-j = 3.2-18.4º;
io = 0.15cos(10t-153.4 º)
Kevin D. Donohue, University of Kentucky
7
Equivalent Circuit Example
Find vo steady-state using Thévenin’s Theorem, if
vs(t) = 20cos(4t):
10 W
vs
ix
1H
0.1 F
Iˆsc  53.333.7
Vˆoc  5.3  83.9
2 ix
+
vo
0.5 H
-
Zˆ th  0.1117 .6
vo (t )  5.5 cos(20t  85.3 )
Kevin D. Donohue, University of Kentucky
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