Basic Concepts - Oakland University
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Transcript Basic Concepts - Oakland University
AC Analysis Using
Thevenin's Theorem
and Superposition
Discussion D11.2
Chapter 4
1
AC Thevenin's Theorem
2
AC Thevenin's Theorem
Thevenin’s theorem states that the two circuits given below are
equivalent as seen from the load ZL that is the same in both cases.
I
a
Zth
Linear
Circuit
ZL
b
AC
I
a
ZL
VTh
b
Zin
Zin
VTh = Thevenin’s voltage = Vab with ZL disconnected (= ) = the
open-circuit voltage = VOC
3
Thevenin's Theorem
I
a
Zth
Linear
Circuit
ZL
b
AC
I
a
ZL
VTh
b
Zin
Zin
ZTh = Thevenin’s impedance = the input impedance with all
independent sources turned off (voltage sources replaced by short
circuits and current sources replaced by open circuits). This is the
impedance seen at the terminals ab when all independent sources
are turned off.
4
Problem 4.57 in text: Solve Problem 4.40 using Thevenin's Thm.
-j1
+
j2
+
AC
60 V
-j1
20 A
-
+
VOC
-
AC
60 V
-
20 A
I0
+
8 j2
VOC 6 2(1 j ) 8 j 2
-j1
j2
AC
-j1
I0
I0
ZTH 1 j1
8 j2
8 j 2 8.246 14.04
1 j1 j 2 2 3 j1
3.16218.43
I0 2.608 32.47
5
AC Superposition
6
Superposition Principle
Because the circuit is linear we can find the response of the
circuit to each source acting alone, and then add them up to find
the response of the circuit to all sources acting together. This is
known as the superposition principle.
The superposition principle states that the voltage across (or
the current through) an element in a linear circuit is the
algebraic sum of the voltages across (or currents through) that
element due to each independent source acting alone.
7
Steps in Applying the Superposition Principle
1. Turn off all independent sources except
one. Find the output (voltage or current)
due to the active source.
2. Repeat step 1 for each of the other
independent sources.
3. Find the total output by adding
algebraically all of the results found in
steps 1 & 2 above.
In some cases, but certainly not all, superposition can simplify
the analysis.
8
Example
30sin 5t
+
+
v0(t)
-
-
AC
0.2F
2 cos10t
1H
Note that the voltage source and the current source have two
different frequencies. Thus, if we want to use phasors, the
only way we've solved sinusoidal steady-state problems, we
MUST use superposition to solve this problem. We will
consider each source acting alone, and then find v0(t) by
superposition.
Remember that sin t cos t 90
9
Example
30sin 5t
+
+
v0(t)
-
-
AC
0.2F
2 cos10t
1H
Consider first the 30sin 5t acting alone.
Since, 30sin 5t 30 cos 5t 90 ,we have = 5 and
ZC
30 90 AC
1
1
j1
jC j 5(0.2)
+
+
V01
-
-
-j1
j5
O.C.
Z L j L j 5
10
Example
Use voltage division
30 90 AC
Z1
+
VS AC
ZC
1
1
j1
jC j 5(0.2)
+
+
V01
-
-
-j1
j5
Z L j L j 5
+
V0
Z2
-
-
O.C.
( j1)( j 5) 5
Z2
j1.25
j1 j5
j4
Z2
V
VS
Z1 Z 2
1
0
Z1 8
j1.25
1.25 90
V
30 90
30 90
8 j1.25
8.097 8.881
1
0
V01 4.631171.1
v01 (t ) 4.631cos 5t 171.12 4.631sin 5t 81.12
11
Example
30sin 5t
+
+
v0(t)
-
-
AC
0.2F
2 cos10t
1H
Now consider first the 2 cos10t acting alone.
We have = 10 and
S
YC jC j10(0.2) j 2
+
V02
j2
20
-j/10
-
YL
1
j L
1
j 10
j10
12
Example
S
+
2
0
V
Yeq
YC jC j10(0.2) j 2
+
I
V02
-
j2
20
-j/10
-
I
V
YV I
Y
For a parallel combination of Y's we have
2
0
YL
1
j L
1
j 10
j10
Yeq Yi 1 8 j 2 j0.1 0.125 j1.90
Yeq 1.90486.24
V02
20
1.05 86.24
1.90486.24
v02 (t ) 1.05cos 10t 86.24
13
Example
30sin 5t
+
+
v0(t)
-
-
AC
0.2F
v01 (t ) 4.631sin 5t 81.12
v02 (t ) 1.05cos 10t 86.24
By superposition
2 cos10t
1H
v0 (t ) v01 (t ) v02 (t )
v0 (t ) 4.631sin 5t 81.12 1.05cos 10t 86.24
14