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Transcript of Circuit Theoremssvbitec.wordpress.com1 Circuit Theorems VISHAL JETHAVA.
Circuit Theorems svbitec.wordpress.com 1
Circuit Theorems
VISHAL JETHAVA
Chap. 4 Circuit TheoremsChap. 4 Circuit TheoremsIntroductionLinearity propertySuperpositionSource transformationsThevenin’s theoremNorton’s theoremMaximum power transfer
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4.1 Introduction4.1 Introduction
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A largecomplex circuitsA largecomplex circuits
Simplifycircuit analysisSimplifycircuit analysis
Circuit TheoremsCircuit Theorems
‧Thevenin’s theorem ‧ Norton theorem‧Circuit linearity ‧ Superposition‧source transformation ‧ max. power transfer
‧Thevenin’s theorem ‧ Norton theorem‧Circuit linearity ‧ Superposition‧source transformation ‧ max. power transfer
4.2 Linearity 4.2 Linearity PPropertyroperty
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Homogeneity property (Scaling)
iRvi kiRkvki
Additivity property
Rivi 222 Rivi 111
21212121 )( vvRiRiRiiii
A linear circuit is one whose output is linearly related (or directly proportional) to its input
Fig. 4.1
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vV0
I0
i
Linear circuit consist of ◦linear elements ◦linear dependent sources◦independent sources
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mA1mV5
A2.0V1
A2V10
iv
iv
iv
s
s
s
nonlinearRv
Rip :2
2
Example 4.1Example 4.1For the circuit in fig 4.2 find I0
when vs=12V and vs=24V.
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Example 4.1Example 4.1KVL
Eqs(4.1.1) and (4.1.3) we get
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0412 21 svii
03164 21 sx vvii
12ivx becomes)2.1.4(
01610 21 svii
(4.1.1)(4.1.2)
(4.1.3)
2121 60122 iiii
Example 4.1Example 4.1Eq(4.1.1), we get
When
When
Showing that when the source value is doubled, I0 doubles.
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76 076 22
ss
vivi
A7612
20 iI
V12sv
A7624
20 iIV24sv
Example 4.2Example 4.2Assume I0 = 1 A and use linearity
to find the actual value of I0 in the circuit in fig 4.4.
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Example 4.2Example 4.2
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A,24/
V8)53(thenA,1If
11
010
vI
IvI
A3012 III
A27
,V14682 23212 VIIVV
A5234 III A5SI
A510 SIAI
A15A30 SII
4.3 Superposition4.3 Superposition
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Circuit Theorems svbitec.wordpress.com 13
How to turn off independent How to turn off independent sourcessourcesTurn off voltages sources = short
voltage sources; make it equal to zero voltage
Turn off current sources = open current sources; make it equal to zero current
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Superposition involves more work but simpler circuits.
Superposition is not applicable to the effect on power.
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Example 4.3Example 4.3Use the superposition theorem to
find in the circuit in Fig.4.6.
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Example 4.3Example 4.3
Since there are two sources, letVoltage division to get
Current division, to get
Hence
And we findCircuit Theorems svbitec.wordpress.com 17
21 VVV
V2)6(84
41
V
A2)3(84
83
i
V84 32 iv
V108221 vvv
Example 4.4Example 4.4Find I0 in the circuit in Fig.4.9
using superposition.
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Example 4.4Example 4.4
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Fig. 4.10
Example 4.4Example 4.4
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Fig. 4.10
4.5 Source Transformation4.5 Source TransformationA source transformation is the
process of replacing a voltage source vs in series with a resistor R by a current source is in parallel with a resistor R, or vice versa
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Fig. 4.15 & 4.16Fig. 4.15 & 4.16
Rv
iRiv ssss or
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Equivalent CircuitsEquivalent Circuits
R
v
R
vi
viRv
s
s
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i i
++
--
vv
v
i
vs-is
Arrow of the current source positive terminal of voltage source
Impossible source Transformation◦ideal voltage source (R = 0)◦ideal current source (R=)
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Example 4.6Example 4.6Use source transformation to find
vo in the circuit in Fig 4.17.
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Example 4.6Example 4.6
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Fig 4.18
Example 4.6Example 4.6
we use current division in Fig.4.18(c) to get
and
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A4.0)2(82
2
i
V2.3)4.0(88 ivo
Example 4.7Example 4.7Find vx in Fig.4.20 using source
transformation
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Example 4.7Example 4.7
Applying KVL around the loop in Fig 4.21(b) gives (4.7.1)Appling KVL to the loop containing only the 3V voltage source, the resistor, and vx yields (4.7.2)
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01853 xvi
1
ivvi xx 3013
Example 4.7Example 4.7
Substituting this into Eq.(4.7.1), we obtain
Alternatively thus
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A5.403515 ii
A5.40184 iviv xx
V5.73 ivx
4.5 Thevenin’s Theorem4.5 Thevenin’s TheoremThevenin’s theorem states that a
linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTh in series with a resistor RTh where VTh is the open circuit voltage at the terminals and RTh is the input or equivalent resistance at the terminals when the independent source are turn off.
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Property of Linear CircuitsProperty of Linear Circuits
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i
v
v
i
Any two-terminalLinear Circuits
+
-Vth
Isc
Slope=1/Rth
Fig. 4.23Fig. 4.23
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How to Find Thevenin’s Voltage How to Find Thevenin’s Voltage
Equivalent circuit: same voltage-current relation at the terminals.
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:Th ocvV ba atltagecircuit voopen
How to Find Thevenin’s How to Find Thevenin’s ResistanceResistance
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:inTh RR b.a atcircuitdeadtheofresistanceinput
circuitedopenba sourcestindependenalloffTurn
CASE 1 If the network has no dependent
sources:◦Turn off all independent source.◦RTH: can be obtained via
simplification of either parallel or series connection seen from a-b
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Fig. 4.25Fig. 4.25CASE 2If the network has
dependent sources◦Turn off all independent
sources.◦Apply a voltage source vo at
a-b
◦Alternatively, apply a current source io at a-b
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o
o
iv
R Th
o
oTh i
vR
The Thevenin’s resistance may be negative, indicating that the circuit has ability providing power
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Fig. 4.26Fig. 4.26Simplified circuit
Voltage divider
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LL RR
VI
Th
Th
ThTh
VRR
RIRV
L
LLLL
Example 4.8Example 4.8Find the Thevenin’s equivalent
circuit of the circuit shown in Fig 4.27, to the left of the terminals a-b. Then find the current through RL = 6,16,and 36 .
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Find RFind Rthth
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shortsourcevoltageV32:Th R
opensourcecurrentA2
4116
124112||4ThR
Find VFind Vthth
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analysisMesh)1(
:ThV
A2,0)(12432 2211 iiiiA5.01 i
V30)0.25.0(12)(12 21Th iiV
AnalysisNodal ely,Alternativ)2(12/24/)32( ThTh VV
V30Th V
Example 4.8Example 4.8
Circuit Theorems svbitec.wordpress.com 43Fig. 4.29
transformsource ely,Alternativ)3(
V302439612
24
32
THTHTH
THTH
VVV
VV
Example 4.8Example 4.8
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:getTo Li
LLL RRR
Vi
430
Th
Th
6LR A310/30 LI16LR A5.120/30 LI
A75.040/30 LI36LR
Example 4.9Example 4.9Find the Thevenin’s equivalent of
the circuit in Fig. 4.31 at terminals a-b.
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Example 4.9Example 4.9(independent + dependent
source case)
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Fig(a) :findTo ThR
0sourcetindependen intactsourcedependent
,V1ovoo
o
iiv
R1
Th
Example 4.9Example 4.9For loop 1,
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2121 or0)(22 iiviiv xx
214But iivi x
21 3ii
Example 4.9Example 4.9
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:3and2Loop
0)(6)(24 32122 iiiii
012)(6 323 iii
gives equations theseSolving
.A6/13 i
A61
But 3 iio
61
ThoiV
R
Example 4.9Example 4.9
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0)(22 23 iivx
51 i
Fig(b):getTo ThV
23 iivx
analysisMesh
06)(2)(4 21212 iiiii 02412 312 iii
.3/102 i
V206 2Th ivV oc
xvii )(4But 21
Example 4.10Example 4.10Determine the Thevenin’s
equivalent circuit in Fig.4.35(a).
Solution
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)caseonlysourcedependent(
o
o
iv
R Th0Th V
:anaysisNodal4/2 oxxo viii
Example 4.10Example 4.10
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220 oo
x
vvi But
4424oooo
xo
vvvvii
oo iv 4or
:4Thus Th o
o
iv
R powerSupplying
Example 4.10Example 4.10
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Example 4.10Example 4.10
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4.6 4.6 Norton’s TheoremNorton’s TheoremNorton’s theorem states that a
linear two-terminal circuit can be replaced by equivalent circuit consisting of a current source IN in parallel with a resistor RN where IN is the short-circuit current through the terminals and RN is the input or equivalent resistance at the terminals when the independent source are turn off.
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Fig. 4.37Fig. 4.37
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v
i
Vth
-IN
Slope=1/RN
How to Find Norton How to Find Norton CurrentCurrent
Thevenin and Norton resistances are equal:
Short circuit current from a to b :
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ThRRN
Th
Th
RV
iI scN
Thevenin or Norton equivalent Thevenin or Norton equivalent circuit :circuit :
The open circuit voltage voc across terminals a and b
The short circuit current isc at terminals a and b
The equivalent or input resistance Rin at terminals a and b when all independent source are turn off.
Circuit Theorems svbitec.wordpress.com 57
ocTh vV
NI
ThTh N
Th
VR R
R
sci
Example 4.11Example 4.11Find the Norton equivalent circuit
of the circuit in Fig 4.39.
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Example 4.11Example 4.11
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:)(40.4Fig a
425
52020||5
)848(||5NRNRfindTo
Example 4.11Example 4.11
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NifindTo
.andterminalscircuitshort ba
))(40.4.Fig( b
:Mesh 0420,A2 2121 iiii
Nsc Iii A12
Example 4.11Example 4.11
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NIformethodeAlternativ
Th
ThN
R
VI
voltagecircuitopen: ThV ba and
:))(40.4( cFig
:analysisMesh
012425,2 343 iiAi
A8.04 i
terminalsacross
V45 4 iVv Thoc
Example 4.11Example 4.11
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,HenceA14/4
Th
ThN
R
VI
Example 4.12Example 4.12Using Norton’s theorem, find RN
and IN of the circuit in Fig 4.43 at terminals a-b.
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Example 4.12Example 4.12
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NRfindTo )(44.4. aFig
shortedresistor4Parallel:2||||5 xo iv
Hence, 2.05/15/ ox vi
52.0
1
o
oN
iv
R
Example 4.12Example 4.12
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NIfindTo )(44.4. bFig
xiv 2||5||10||4 Parallel:
.5A,24
010 xi
A72(2.5)5
102 xxsc iii
7A NI
4.8 Maximum Power 4.8 Maximum Power TrandferTrandfer
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LL RRR
VRip
2
LTH
TH2
Fig 4.48
Fig. 4.49Fig. 4.49Maximum power is transferred to
the load when the load resistance equals the Thevenin resistance as seen the load (RL = RTH).
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Circuit Theorems svbitec.wordpress.com 68
TH
TH
THL
LTHLLTH
LTH
LLTHTH
LTH
LTHLLTHTH
L
RV
p
RR
RRRRR
RRRRR
V
RRRRRRR
VdRdp
4
)()2(0
0)(
)2(
)()(2)(
2
max
32
4
22
Example 4.13Example 4.13Find the value of RL for maximum
power transfer in the circuit of Fig. 4.50. Find the maximum power.
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Example 4.13Example 4.13
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918
126512632THR
Example 4.13Example 4.13
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WRV
p
RR
VVVii
Aiii
L
TH
THL
THTHi
44.1394
224
9
220)0(231612
2 ,121812
22
max
2
221