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R
2
R
6
R
4
R
5
R
3
+
+
–
–
Fig. 4.11-5 

---

Problems 
4.47
6. Find V
1
, V
2
and V
3
such that v
1
=
10 V, v
2
=
10 V and v
3
=
20 V in the circuit in Fig. 4.11-6. With 
these values of V
1
, V
2
and V
3
, find the power delivered by all voltage sources and power dissipated 
by all resistors. Use nodal analysis. 
V
2
V
3
V
1
v
1
v
2
v
3
20 
Ω
10 
Ω
10 
Ω
10 
Ω
5 
Ω
5 
Ω
R
+
+
+
–
–
–
Fig. 4.11-6 
7. (i) The nodal conductance matrix of the circuit in Fig. 4.11-7 is given as 
2 
Ω
+
–
2 
Ω
10 
Ω
R
k
1
i
x
i
x
v
x
i
y
k
3
v
x
k
2
i
y
I
5 
Ω
5 
Ω
Fig. 4.11-7 
Y
=
−
−
−










0 1
0 3 0 13
0 2
0 8
0 47
0 1
0 6
0 7
.
.
.
.
.
.
.
.
.
S. Find k
1
, k
2
and k
3
and solve the circuit completely by nodal analysis if I 
=
1 A.
8. Find k
1
, k
2
and k
3
such that the nodal conductance matrix for the circuit in Fig. 4.11-8 is lower triangular. 
Find the power delivered by independent sources and dependent sources. Use nodal analysis.
+
+
+
–
–
–
5 V
10 V
10 
Ω
R
k
1
i
x
i
x
i
y
v
x
k
2
v
x
k
3
i
y
4 
Ω
20 
Ω
5 
Ω
2 
Ω
5 
Ω
Fig. 4.11-8 
9. Find all dependent source coefficients such that the 
Y
-matrix of the circuit in Fig. 4.11-9 is 
diagonal.
+
+
+
+
+
+
–
–
–
–
–
–
1 
Ω
1 
Ω
R
k
1
i
y
v
x
v
z
v
y
k
2
v
z
k
6
v
z
k
5
v
y
k
3
v
x
k
4
v
z
4 
Ω
5 
Ω
5 
Ω
5 
Ω
Fig. 4.11-9 

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4.48
Nodal Analysis and Mesh Analysis of Memoryless Circuits
10. Find k such that 
v
is zero in the circuit in Fig. 4.11-10. Solve the circuit completely with this value 
of k. Use nodal analysis.
1 
Ω
4 V
10 V
2 
Ω
2 
Ω
2 
Ω
1 
Ω
v
x
v
kv
x
+
+
+
++
–
–
–
––
Fig. 4.11-10 
11. Find k such that 
v
is zero in the circuit in Fig. 4.11-11. Solve the circuit completely for this value 
of k. Use nodal analysis.
5 V
2 A
1 A
10 V
2 
Ω
1 
Ω
2.5 
Ω
v
x
v
kv
x
+
+
+
+
+
–
–
–
–
–
Fig. 4.11-11 
12. Find the node voltages and resistor currents in the circuit in Fig. 4.11-12 by nodal analysis.
2 
Ω
6 
Ω
2 
Ω
5 
Ω
10 
Ω
7 A
R
i
x
3
i
x
i
y
2
i
y
10 V
+
+
–
–
Fig. 4.11-12 
13. The nodal conductance matrix of the circuit in Fig. 4.11-13 is given below. Find the values of all 
resistances in the circuit.
i
3
i
1
i
2
Z 
= 
Ω
14 
–4 
–6 
12 
–4 
–3 
–6 
–3 
13 
Fig. 4.11-13 
14. Express all the resistor currents in the directions as marked in the form linear combinations of V
1
, 
V
2
and V
3
for the circuit shown in Fig. 4.11-14. Use Mesh Analysis.
4 
Ω
4 
Ω
5 
Ω
6 
Ω
5 
Ω
+
+
– ––
3 
Ω
V
2
V
1
V
3
Fig. 4.11-14 

---

Problems 
4.49
15. (i) Express 
v
in the circuit in Fig. 4.11-15 as a linear combination of V
1
, V
2
and I. (ii) Find I such 
that 
v
=
0 if V
1
=
V
2
=
5 V. (iii) Solve the circuit completely with these source values. Use Mesh 
Analysis.
2 
Ω
2 
Ω
2 
Ω
2 
Ω
+
+
+
–
–
–
3 
Ω
3 
Ω
V
2
v
I
V
1
Fig. 4.11-15 
16. All resistors in the circuit in Fig. 4.11-16 are of 2
W
. Find currents in all resistors and voltage 
across current sources by mesh analysis. 
1 A
1 A
Fig. 4.11-16 
17. Find the current delivered to the 12.5 V source, power delivered by all voltage sources and power 
dissipated in all resistors by Mesh Analysis on the circuit in Fig. 4.11-17. 
0.1
Ω
+
+
+
+
–
–
–
–
13.5 V
13.6 V
13.7 V
0.15
 
Ω
0.2
Ω
0.2
Ω
12.5 V
Fig. 4.11-17 
18. Repeat the above problem (Problem 17) by nodal analysis.
19. Repeat Problem 5 by using mesh analysis.
20. Solve the circuit in Fig. 4.11-18 completely.
2
Ω
2
Ω
2
Ω
1
Ω
3
Ω
3
Ω
3
Ω
2 A
2 A
1 A
1 A
Fig. 4.11-18 
21. Can the circuit in Fig. 4.11-19 be solved uniquely? If yes, find the solution. If no, find at least two 
solutions.

---

4.50
Nodal Analysis and Mesh Analysis of Memoryless Circuits
1 A
1 A 2 A
2 
Ω
2 
Ω
1 
Ω
3 
Ω
3 
Ω
Fig. 4.11-19
22. Solve Problem 11 by mesh analysis.
23. Express the resistor currents as linear combinations of V
1
, V
2
in the circuit in Fig. 4.11-20. Use 
mesh analysis.
2
Ω
+
+
+
+
+
–
–
–
–
–
3
Ω
3
Ω
–3
i
x
v
x
v
x
i
x
V
1
V
2
I
Fig. 4.11-20

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C i r c u i t T h e o r e m s
CHAPTER OBJECTIVES
• To derive Superposition Theorem from the property of linearity of elements.
• To explain the two key theorems – Superposition Theorem and Substitution Theorem in
detail.
• To derive other theorems like Compensation Theorem, Thevenin’s Theorem, Norton’s
Theorem, Reciprocity Theorem and Maximum Power Transfer Theorem from these two
key principles.
• To provide illustrations of applications of circuit theorems in circuit analysis through solved 
examples.
• To emphasise the use of Compensation Theorem, Thevenin’s Theorem and Norton’s Theorem 
in circuits containing dependent sources as a pointer to their applications in the study of 
electronic circuits.
IntroductIon
Circuit Analysis involves determination of element voltages and currents for all elements of the 
circuit using element equations and interconnection equations. Kirchhoff’s Current Law equations 
at all nodes and Kirchhoff’s Voltage Law equations in all loops along with element v
-
i relationship 
equations will yield the necessary set of equations.
However, we need systematic procedures for exploiting these equations. Node analysis and mesh 
analysis were two such systematic procedures we took up for detailed study in the last chapter. In this 
chapter, we discuss some circuit theorems and circuit transformations that increase our efficiency in 
solving circuits and that render further insight into certain features of a linear circuit. These theorems 
constitute a basic set of tools that enhance the analyst’s efficiency in solving circuits.
The previous chapter showed the following:
(1) All the element voltages and element currents in a circuit can be obtained from its node 
voltages that are governed by a matrix equation