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© The McGraw-Hill Companies, Inc. 2000McGraw-Hill1
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
C H A P T E R
3Resistive Network Analysis
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill2
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.2 Use of KCL in nodal analysis
i1
R1va
v b
i 3
vc
vd
R3
R2
i 2
By KCL : i1 – i 2 – i 3 = 0. In the node voltage method, we express KCL by
v a – v b
R 1
–v b – v c
R 2
–v b – v d
R 3
= 0
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill3
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.3 Illustration of nodal analysis
i1
va vb
i 3
vc = 0
i 2
R 1i S R 3
R 2Node a Node b
Node c
R 1 R 3
R2
iS
iS
Va/R1+(Va-Vb)/R2 =Is
Vb/R3+(Vb-Va)/R2=0
Or
Va(1/R1+1/R2)+Vb(-1/R2)=Is
Va(-1/R2) +Vb(1/R2+1/R3)=0
or, in matrix form
03/12/12/1
2/12/11/1 Is
Vb
Va
RRR
RRR
0322
221 Is
Vb
Va
GGG
GGG
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill4
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.5
I 2I1
R4R1
R2
R3
Node 1
I2I1
R4R1
R2
R3
Node 2
Example 3.1
R1=1K, R2=2K, R3=10K,R4=2K
I1=10mA, I2=50mA,
V1/R1+(V1-V2)/R2+(V1-V2)/R3=I1
V2/R4+(V2-V1)/R2+(V2-V1)/R3=-I2
Or
(1/R1+1/R2+1/R3)V1+ (-1/R2-1/R3)V2=I1
(-1/R2-1/R3)V1 + (1/R2+1/R3+1/R4)V2= I2
Plugging the numbers
1.6 V1- 0.6 V2=10
-0.5V1 +1.1 V2=-50
By solving the above Eq.
V1=-13.57
V2=-52.86
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill5
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.8 Nodal analysis with voltage sources
R2
R1
vS R4
va vc
iS
R3
+_
vb
Va=Vs
(Vs-Vb)/R1-vb/R2-(Vb-Vc)/R3=0
(Vb-Vc)/R3+Is-Vc/R4=0
Or
(1/R1+1/R2+1/R3)Vb+(-1/R3)Vc=Vs/R1
(-1/R3)Vb+ (1/R3+1/R4)Vc=Is
Or in Matrix form
Is
RVs
Vc
Vb
RRR
RRRR 1/
4/13/13/1
3/13/12/11/1
Is
RVs
Vc
Vb
GGG
GGGG 1/
433
3321
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill6
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.13 Assignment of currents and voltages around mesh 1
R 3
R4vS
R 1
R2+_ i1 i2v2
v 1+ –
+
–
Mesh 1: KVL requires thatv S – v 1 – v 2 = 0, where v1 = i1 R1 ,v 2 = ( i1 – i2 ) R1 .
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill7
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.14 Assignment of currents and voltages around mesh 3
R 3
R 4v S
R 1
R 2+_ i 1 i 2v 2
v 3+ –
+
–
v 4
+
–
Mesh 2: KVL requires that
v 2 + v 3 + v 4 = 0
where
v 2 = ( i 2 – i 1 ) R 2 ,
v 3 = i 2 R 3 ,
v 4 = i 2 R 4
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill8
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.12 A two-mesh circuit
R3
R4vS
R1
R2+_ i1 i2
I1R1+(I1-I2)R2=Vs
(I2-I1)R2 + I2R3 + I2R4=0
Or
02
1
4322
221 Vs
I
I
RRRR
RRR
The advantage of Mesh Current Method is that it uses resistances in the equations, rather than conductances.
But Node Voltage Method is physically more sensible.
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill9
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.18 Mesh analysis with current sources
2
4 10 V
5
2 A
i1vx i2
+_
+
–
5I1 +Vx =10
-Vx+2I2+4I2=0
I1-I2=2
Adding Eqs. 1 and 2 will delete Vx
5I1 +6 I2 =10
I1-I2=2
I1=2 A
I2=0
P3.1-3.20
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill10
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.26 The principle of superposition
RvB2+_
+_vB1i = R
+_vB1iB1
The net current throughR is the sum of the in-dividual source currents:i = iB1 + iB2.
RvB2+_
iB2+
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill11
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.27 Zeroing voltage and current sources
iS
R1
+_vS
A circuit
iS
R1
R2
The same circuit with vS = 0
iS
R1
+_vS R2
R2
A circuit
R1
R2
The same circuit with iS = 0
+_vS
1. In order to set a voltage source equal to zero, we replace it with a short circuit.
2. In order to set a current source equal to zero, we replace it with an open circuit.
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill12
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.28 One-port network
Linear
network
i
v
+
–
i
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill13
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.29 Illustration of equivalent-circuit concept
R3+_vS R2
i
v
+
–
R1
LoadSource
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill14
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.31 Illustration of Thevenin theorum
ii
Loadv+
–Source Loadv
+
–
+_
RT
vT
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill15
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.32 Illustration of Norton theorem
v+–
RNiN
i
v+Source
––
i
Load Load
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill16
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.34 Equivalent resistance seen by the load
R 2
a
b
R3
R 1
a
b
R 3
R 1||R 2 RT
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill17
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.35 An alternative method of determining the Thevenin resistance
R 2
a
b
R 3
R 1 vx
+
–
iS
R 3
RT = R 1 || R 2 + R 3
R 1 iSR 2 i S
What is the total resistance the current iS will encounter in flowing around the circuit?
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill18
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.46
R2
R1
+_
vS RL
R 3
iL
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill19
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.47
R1
+
_
v S
R 3
R 2
v O C
+
–
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill20
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.48
R 1
+_
vS
R3
R2 v OC
+
–
vOC
+
–
+ –0 V
i
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill21
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.49 A circuit and its Thevenin equivalent
R2
R1
+_vS RL
R3
iL
R2
R1 + R2
vS
R3 + R1 || R2
+_ RL
iL
A circuit Its Thévenin equivalent
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill22
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.57 Illustration of Norton equivalent circuit
iSCiNRT = R N
i SCOne-portnetwork
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill23
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.58 Computation of Norton current
R2
R 1
+_
vS
R 3
iS C
i1
i2
Short circuitreplacing the load
v
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill24
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.63 Equivalence of Thevenin and Norton representations
vT
RT
One-port
network iN RT+_
Thévenin equivalent Norton equivalent
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill25
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.64 Effect of source transformation
R 2
R1
vS
R 3
iSC+_
R 3
R2 vS iSCR 1 R1
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill26
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.65 Subcircuits amenable to source transformation
i S+_
The évenin subcircuits
R
R
vS+_
iS Ror
Node a
Node b
a
b
a
b
vSor
Norton subcircuits
a
b
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill27
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.71 Measurement of open-circuit voltage and short-circuit current
a
b
rm
A
a
b
rmV
a
b
+
–
Unknownnetwork
Unknownnetwork
Unknownnetwork
Load
An unknown network connected to a load
Network connected for measurement of short-circuit current
Network connected for measurement of open-circuit voltage
“ iSC ”
“ vO C ”
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill28
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.73 Power transfer between source and load
vT
RT
RL+_
iL
Source equivalent
Practical source
Load
R L
Given vT and RT, what value of R L
will allow for maximum power transfer?
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill29
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.74 Source loading effects
vT
v i n t
+_ R L
+ –
R T
i
i N v R L
+
–
i i n t
R T
Source Load
Source Load
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill30
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.77 Representation of nonlinear element in a linear circuit
RT
+_
i x
vTvx
+
–
Nonlinearelement
Nonlinear element as a load. We wish to solve for vx and ix.
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill31
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.78 Load line
iX
vx
1RT
Load-line equation: ix = –vT
RTvx +
vT
–1RT
vT
RT
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill32
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.79 Graphical solution equations 3.48 and 3.49
i x
vx
i = Ioe v,v > 0
i-v curve of “exponential resistor”
Solution
1RT
Load-line equation: ix =vT
RTvx +
vT
RT
vT
© The McGraw-Hill Companies, Inc. 2000McGraw-Hill33
PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERINGTHIRD EDITION
G I O R G I O R I Z Z O N I
Figure 3.80 Transformation of nonlinear circuit of Thevenin equivalent
ix
vx
+
–
Linearnetwork load
R
Nonlinear
T
+_vT vx
+
–
ix
loadNonlinear
