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Transcript of Circuit Theoremssvbitec.wordpress.com1 Circuit Theorems VISHAL JETHAVA.
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Circuit Theorems svbitec.wordpress.com 1
Circuit Theorems
VISHAL JETHAVA
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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
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4.2 Linearity 4.2 Linearity PPropertyroperty
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Homogeneity property (Scaling)
iRvi kiRkvki
Additivity property
Rivi 222 Rivi 111
21212121 )( vvRiRiRiiii
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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
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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
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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
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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
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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
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4.3 Superposition4.3 Superposition
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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
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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
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Example 4.4Example 4.4
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Fig. 4.10
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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
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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
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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
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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
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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
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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
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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
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How to Find Thevenin’s How to Find Thevenin’s ResistanceResistance
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:inTh RR b.a atcircuitdeadtheofresistanceinput
circuitedopenba sourcestindependenalloffTurn
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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
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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
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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
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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
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Example 4.8Example 4.8
Circuit Theorems svbitec.wordpress.com 43Fig. 4.29
transformsource ely,Alternativ)3(
V302439612
24
32
THTHTH
THTH
VVV
VV
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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
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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
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Example 4.9Example 4.9For loop 1,
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2121 or0)(22 iiviiv xx
214But iivi x
21 3ii
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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
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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
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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
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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
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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
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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
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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.
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ocTh vV
NI
ThTh N
Th
VR R
R
sci
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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
Circuit Theorems svbitec.wordpress.com 59
:)(40.4Fig a
425
52020||5
)848(||5NRNRfindTo
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Example 4.11Example 4.11
Circuit Theorems svbitec.wordpress.com 60
NifindTo
.andterminalscircuitshort ba
))(40.4.Fig( b
:Mesh 0420,A2 2121 iiii
Nsc Iii A12
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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
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Example 4.11Example 4.11
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,HenceA14/4
Th
ThN
R
VI
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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
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Example 4.12Example 4.12
Circuit Theorems svbitec.wordpress.com 65
NIfindTo )(44.4. bFig
xiv 2||5||10||4 Parallel:
.5A,24
010 xi
A72(2.5)5
102 xxsc iii
7A NI
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4.8 Maximum Power 4.8 Maximum Power TrandferTrandfer
Circuit Theorems svbitec.wordpress.com 66
LL RRR
VRip
2
LTH
TH2
Fig 4.48
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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
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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
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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