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Circuit Theorems•Introduction•Linearity Property•Superposition•Source Transformation•Thevenin’s Theorem•Norton’s Theorem•Maximum Power Transfer•Summary
Introduction
•To develop analysis techniques applicable tolinear circuits.
•To simplify circuit analysis and help handlethe complexity.
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Linear Circuits•A linear circuit is one whose output is linearly
related (or directly proportional) to its input•Linear circuit consist of• linear elements• linear dependent sources• independent sources
Example 4.1•For the circuit in fig 4.2 find I0 when vs=12V
and vs=24V.
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•KVL
Eqs(4.1.1) and (4.1.3) we get
0412 21 svii03164 21 sx vvii
12ivx becomes)2.1.4(
01610 21 svii
(4.1.1)(4.1.2)
(4.1.3)
2121 60122 iiii
Example 4.1
Eq(4.1.1), we get
When
When
Showing that when the source value is doubled, I0doubles.
76076 22
ss
vivi
A7612
20 iIV12sv
A7624
20 iIV24sv
Example 4.1
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Linearity Property•Homogeneity (Scaling) + Additivity•Homogeneity property:
•Additivity property:
•A linear circuit follows the relationship below:
)()( kvRkiiRv
)(
Applyingand
212121
21
2211
vvRiRiRiiviii
RivRiv
InputConstantOutput
Superposition
),,0;0,,0(...)0,,0,,0;0,,0()0,,0,;0,,0(
)0,,0;,0,,0()0,,0;0,,0,,0()0,,0;0,,0,(
),,;,,(
)0,,0,,,0;0,,0(
)0,,0;0,,0,,,0(
)equations.lineargivelawsOhm'andKCL,KVL(
sourcestindependen:,
),,;,,(
bygiveniscurrent)(orvoltageDCthesystemlinearaFor*
21
21
11
,
,
1,
1,11
M
N
MN
mmIm
nnVn
mn
M
mmmI
N
nnnVMN
IvIvIv
VvVvVv
IIVVv
IAIv
VAVv
IV
IAVAIIVVv
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Superposition•Based on linearity property•Statement–The voltage across (current through) an element is
the algebraic sum of the voltage across (currentthrough) that element due to each independentsource acting alone.
•A turned-off voltage source = a short circuit•A turned-off current source = an open circuit
),...,0;0,...,0(...)0,...,;0,...,0(
)0,...,0;,...,0(...)0,...,0;0,...,(
),...,;,...,(
1
1
11
M
N
MN
IvIv
VvVv
IIVVv
Superposition
•Superposition involves more work but simplercircuits.•Superposition is not applicable to the effect on
power.
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Comments on Superposition
SS
SS
II
VV
CCaSSaCSaCS
CS
aCS
RiivvRivRiv
0:sourceCurrent
0:sourceVoltage
sources,tindependenFor
)()(
.signalcontrol:
signalcontrolled:whereletsources,dependentFor
)()(,resistorsFor
element.circuiteachinpropertydesiredKeep1.:tsrequiremenTwo
212122
11
212122
11
Input 1 LinearCircuit
Input 2 LinearCircuit
00
0
,KVLFor
,KCLForsatisfied.bemustlawscircuitTwo.2
2,1,2,
1,
2,1,2,1,2,2,
1,1,
RTRTRT
RT
leavingleavingenteringenteringleavingentering
leavingentering
VVV
V
IIIIII
II
Comments on Superposition
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Example 1
=
(V)10828141
13
124
6
21
vvv+
Example 2
+=
"0
'00 iii
Keep dependent sources!
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Example 3
321 iiii
Source Transformation
21,arbitraryFor iiVab
•Transformations between voltage source andcurrent source.•Equivalence means identical i-v characteristics.
i1 i2
Vab
+
_Vab
+
_
1
2
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Cont’d
22
11
RV
ii
RVv
i
abS
abS
Rv
iRRR
RRRv
i
RRV
Rv
i
Vii
SS
SS
abS
S
ab
and
011
and0
)11
()(
.arbitraryforLet
21
121
121
21
i1 i2
Vab
+
_Vab
+
_
1
2
Applicable to Dependent Sources
RivRv
i SSS
S or
i1 i2
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Source Transformation•Impossible source Transformation:•Ideal voltage source (R = 0)•Ideal current source (R = )
Example 1
34
43
12/6
12/3
4-2
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Example 2
V5.7A5.4
(2)013
(1)01853
x
x
x
vi
vi
vi
xv25.04
6/2 13
I-V Characteristic•For a linear two-terminal network, its I-V curve (DC)
must be a straight line in the I-V plane.
i
v
i
v
R
VS(VS||R)
R VS
R||IS
IS(IS R)
: Series connection||: Parallel connection
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Equivalent Circuits
i
vd
Slope=1/c
(i)i
+v_
(ii)i
+v_ c
dcv
i
dciv
dciv
-d/c
d
d/c
c
c
•Equivalent circuit:same i-v relation at theterminals
Thevenin’s TheoremA variableelement
Thevenin equivalent circuit
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Cont’d
ThLTh
LLLL
LTh
ThL
VRR
RIRV
RRV
I
= A simple voltage divider.
Proof of Thevenin’s Theorem
iv
R
iRiAv
B
Vvvi
iAvABV
AR
ViRBiA
iAvAiAv
iiivvv
oc
mmi
M
mnnv
N
n
mmi
M
mnnv
N
n
MN
Th
Th0
0
Th
,1,10Th
0Th
ThTh00
,1,10
2121
.0haveweoff,turnedsourcestindependeninternalallWhen*
0When*
where
ion,superpositBy*
,,,,,,sourcestindependenwithcircuitlinearA*
=
RTh
VTh
identical i-vcharacteristic
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Finding VTh and RTh
•If no dependent sources included.–RTh is the equivalent input resistance of the
resistor network .
A resistornetwork
Cont’d•If dependent sources
included, two methodscan be applied todetermine RTh.–External voltage source
method.
–External current sourcemethod.
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Example 1
=RTh
VTh
Example 2
=RTh
voc
1
0iRTh
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2
RTh
VTh RN= IN
Norton’s Theorems
Th
Th
eqTh
ation,transformsourceBy
RV
I
RRR
N
N
Norton’s Theorem
Norton equivalent circuitinTh RRRN
ThThN RVI
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Example
= RNIN
V
IR = 0
R =
•Req = 0–External voltage source (vo = 1V)
may violate the requirement vo= 0V.–Use external current source to find
RTh .
•Req = –External current source (io = 1A)
may violate the requirement io= 0A.–Use external voltage source to find
RTh .
•For others–Both exist.
More Comments
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Maximum Power Transfer
Th
2Th
max
Th
3Th
Th2Th
4Th
Th2
Th2Th
2
2
4
0
2
sfer,power tranmaximumachieveTo*
bygivenistodeliveredpowerThe*
RV
p
RR
RRRR
V
RRRRRRR
VdRdp
RRR
VRip
R
L
L
L
L
LLL
L
LLTh
ThL
L
A linear circuit
Applications: Source Modeling
sLs
LL v
RRR
v
sLs
LL v
RRR
v
sLs
LL v
RRR
v
Voltage source
Current source
sLs
LL v
RRR
v
sLp
pL i
RR
Ri
0sR
pR
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Resistance Measurement
21
3132
32
21
21
0
or
,0When
RRR
RRRRR
vRR
Rvv
RRR
v
i
xx
x
x
The Wheatstone bridge
i0
2
31If
RR
RR
x