Chapter 2

Circuit Analysis

BASIC ELECTRICAL ENGINEERING

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 1

 Table of contents

● Thevenin’s Theorem
● Norton’s Theorem
● Maximum Power Transfer Theorem
● Reciprocity Theorem
● Assignment-5
● Tutorial-8
● Tutorial-9

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 2

 Thevenin’s Theorem
● Application of this theorem often comes useful when we want to determine the current flowing through any
branch or component of a network. We can conveniently determine the current through any component when
it is required that the component be replaced. The use of Kirchhoff’s laws to calculate the branch current for
the changed value of a resistor becomes time consuming as we have to repeat the calculations.
● Hence, Thevenin’s Theorem states that “any linear active network consisting of independent or dependent
voltage and current source and the network elements can be replaced by an equivalent circuit having a
constant voltage source Vth in series with a resistance Rth where, Vth is open circuited voltage across the open
circuited load terminals and Rth is the equivalent resistance of the network looking back into the open-
circuited terminals by short circuiting all voltage sources and open circuiting all current sources without
eliminating their internal resistance.”

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 3

 Continued…
When a particular load branch is removed from a network, an open circuit voltage appears across those
terminals of the network which is known as Thevenin’s equivalent voltage (Vth) and the equivalent resistance of
the network looking back into those terminals is known as Thevenin’s equivalent resistance (Rth).

Figure 2.56

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 4

 Continued…
The steps for solving any network by Thevenin’s Theorem are explained as follows:
1) Draw the given network by removing the load resistance RL or open circuiting the load terminals.
2) Remove all the sources by their internal resistance if given otherwise short-circuit the voltage source and
open-circuit the current source to find the equivalent resistance at the load terminals known as Thevenin’s
Resistance (Rth).
3) Calculate Thevenin’s equivalent resistance Rth by viewing back into the open-circuited load terminals.
4) Draw the initial network again with load resistance RL removed.
5) Calculate open-circuited voltage or Thevenin’s equivalent voltage, Vth by using KVL, KCL, Ohm’s law,
mesh analysis, nodal analysis, superposition theorem, etc. whichever methods are convenient.
6) Draw the Thevenin’s equivalent circuit by placing Vth, Rth and RL in series combination as shown in Figure
Vth
2.56 and determine the current flowing through load resistance RL as: IL = R
th +RL

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 5

 Norton’s Theorem
● Norton’s Theorem states that “any linear active network consisting of independent or dependent voltage and
current sources and the network elements can be replaced by an equivalent circuit having a constant current
source IN in parallel with a resistance RN where, IN is short circuited current flowing through load terminals
after short circuiting it known as Norton’s current and RN is the equivalent resistance of the network looking
back into the open-circuited terminals by short circuiting all voltage sources and open circuiting all current
sources without eliminating their internal resistance known as Norton’s equivalent resistance.”
● Norton’s Theorem is converse of Thevenin’s theorem in the sense that Norton equivalent circuit uses a
current generator instead of voltage generator and the resistance RN (which is the same as Rth) in parallel with
the generator instead of being in series with it.

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 6

 Continued…

Figure 2.57

The steps for solving any network by Norton's Theorem are explained as follows:
1) Draw the given network by removing the load resistance RL.
2) Remove all the sources by their internal resistance if given otherwise short-circuit the voltage source and
open-circuit the current source to find the equivalent resistance at the load terminals known as Norton’s
Resistance (RN).

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 7

 Continued…
3) Calculate Norton’s equivalent resistance RN as seen from the open-circuited load terminals.
4) Draw the initial network again replacing load resistance RL with short-circuit.
5) Calculate Norton’s current, IN by using Ohm’s law, KVL, KCL, mesh analysis, nodal analysis, superposition
theorem, etc. whichever methods are convenient.
6) Draw the Norton’s equivalent circuit by placing IN, RN and RL in parallel to each other and determine the
current flowing through load resistance RL using current division rule as

𝑅𝑁
𝐼𝐿 = × 𝐼𝑁
𝑅𝑁 + 𝑅𝐿

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 8

 Maximum Power Transfer Theorem
● Maximum power transfer theorem states that “for any loads connected directly to a dc voltage supply,
maximum power will be delivered to the load when the load resistance is equal to the internal resistance of
the source.”
● Let us consider a circuit with a voltage source or battery with emf ‘E’ and internal resistance ‘RS’ which is
connected in series to a load of resistance ‘RL’ as shown in Figure 2.58.

From Figure 2.58, we have

The current flowing through the load RL is given by

𝐸
𝐼𝐿 =
𝑅𝑆 + 𝑅𝐿
The power delivered to the load RL is given by

2
𝐸 Figure 2.58
𝑃𝐿 = 𝐼𝐿2 𝑅𝐿 = 𝑅𝐿 … … (1)
𝑅𝑆 + 𝑅𝐿

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 9

 Continued…
When R L = 0, PL = 0
When R L = ∞, PL = 0
For some intermediate value of RL, the power delivered to the load would be maximum. This value of RL can be
found out by taking its first derivative zero, so, we can write,

𝑑PL
= 0.
𝑑R L

2
E
d R +R RL
S L
=0
dR L

E 2 [(R S +R L )2 − 2R L R S + R L ]
𝑜𝑟, =0
RS + RL 4
𝑜𝑟, E 2 [(R S +R L )2 − 2R L R S + R L ] = 0

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 10

 Continued…
Since, E ≠ 0, so we have
(R S +R L )2 − 2R L R S + R L = 0
𝑜𝑟, R S + R L (R S +R L − 2R L ) = 0
𝑜𝑟, R S + R L (R S −R L ) = 0
𝑜𝑟, R2S − R2L = 0
𝑜𝑟, R2S = R2L
∴ RL= RS Figure 2.59
Thus, the power delivered to the load will be maximum when load resistance is equal to source internal
resistance.

2
E E2
From equation (1), we get PLmax = RS =
RS + RS 4R S

The Figure 2.59 shows the relationship between power delivered to the load, PL and the load resistance, RL

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 11

 Continued…
The following points are worth noting about maximum power transfer theorem
● The circuit efficiency at maximum power transfer is only 50% as one-half of the total power generated is
dissipated in the internal resistance RS of the source.

𝑃𝑜𝑤𝑒𝑟 𝑂𝑢𝑡𝑝𝑢𝑡 𝐼 2 𝑅𝐿 𝑅𝐿 1
𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 = = 2 = = = 50%
𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 𝐼 (𝑅𝐿 + 𝑅𝑆 ) 2𝑅𝐿 2
● Under the conditions of maximum power transfer, the load voltage is one-half of the open circuited voltage at
the load terminals.

𝑉 𝑉
𝐿𝑜𝑎𝑑 𝑉𝑜𝑙𝑡𝑎𝑔𝑒 = 𝐼𝑅𝐿 = 𝑅 =
𝑅𝐿 + 𝑅𝑆 𝐿 2

● Maximum power transferred across the load is given by

2
𝑉 𝑉2
𝑀𝑎𝑥. 𝑝𝑜𝑤𝑒𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟𝑟𝑒𝑑 = 𝐼 2 𝑅𝐿 = 𝑅𝐿 =
𝑅𝐿 + 𝑅𝑆 4𝑅𝐿

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 12

 Continued…
If the comparison is made between the circuit shown in Figure 2.58 and Thevenin’s equivalent circuit, the
following relation can be implemented for the maximum power delivered to the load, RL.

2
𝑉𝑡ℎ
𝑃𝐿𝑚𝑎𝑥 =
4𝑅𝑡ℎ
As Thevenin’s theorem and Norton’s theorem are converse of each other, we can also write,

𝐼𝑁2 𝑅𝑁
𝑃𝐿𝑚𝑎𝑥 =
4
Therefore,
𝑉𝑡ℎ = 𝐼𝑁 𝑅𝑁
From above discussions, the maximum power transfer theorem can be stated more conceptually as “the value of
load resistance that should be connected to any two terminals of the complex network should be equal to the
Thevenin’s equivalent resistance or Norton’s equivalent resistance for that load to achieve maximum power.”

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 13

 Applications of Maximum Power Transfer Theorem

Two important applications are listed below:

● In communication circuits, maximum power transfer is usually desirable. For instance, in a public address
system, the circuit is adjusted for maximum power transfer by making load (i.e. speaker) resistance equal to
source (i.e. amplifier) resistance. When source and load have the same resistance, they are said to be
matched. In most practical situations, the internal resistance of the source is fixed. Also, the device that acts
as a load has fixed resistance. In order to make RL = RS, we use a transformer. We can use the reflected-
resistance characteristic of the transformer to make the load resistance appear to have the same value as the
source resistance, thereby ‘‘fooling’’ the source into ‘‘thinking’’ that there is a match (i.e. RL = RS).This
technique is called impedance matching.

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 14

 Continued…
● Another example of maximum power transfer is found in starting of a car engine. The power delivered to the
starter motor of the car will depend upon the effective resistance of the motor and internal resistance of the
battery. If the two resistances are equal (when battery is fully charged), maximum power will be transferred
to the motor to turn on the engine. This is particularly desirable in winter when every watt that can be
extracted from the battery is needed by the starter motor to turn on the cold engine. If the battery is weak, its
internal resistance is high and the car does not start.

Note: Electric power systems are never operated for maximum power transfer because the efficiency under this
condition is only 50%. This means that 50% of the generated power will be lost in the power lines. This situation
cannot be tolerated because power lines must operate at much higher than 50% efficiency.

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 15

 Reciprocity Theorem
● Reciprocity Theorem states that “In any linear, bilateral network having only one independent source, the
ratio of response to excitation is constant even though the source is interchanged from the input terminals to
output terminals”.
● In other words, “In any linear bilateral network, if the positions of ideal voltage source and ideal ammeter are
interchanged, the reading of ammeter remains the same.”
● In other words, “If a source of emf E, located at one point in a network composed of linear, bilateral circuit
elements, produces a current I at a selected point in the network, the same source of emf E acting at the
second point will produce the same current at the first point.” In this process, the current in other branches
will not remain the same.”
● It can also be stated that “The ratio of V and I called the transfer resistance remains same when voltage
source and ammeter are interchanged.” This theorem holds true if both the voltage source and ammeter have
same internal resistance.

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 16

 Continued…

● Note: The Reciprocity theorem is applicable only to single-source networks and to the circuits containing
only independent sources (either current or voltage source).

Figure 2.60

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 17

 Assignment-5
1) State and explain Thevenin’s Theorem with suitable example.
2) How do you thevenize a given circuit? Explain the steps with a suitable example.
3) State and explain Norton’s Theorem with an appropriate example.
4) State and prove Maximum Power Transfer Theorem.
5) “Thevenin's theorem and Norton's theorem are dual of each other”. Justify the statement with suitable
example.
6) State and explain Reciprocity Theorem with suitable example.
7) Can Reciprocity Theorem be used in the following circuit? Justify your answer.

Figure 2.685

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 18

 Tutorial-8
1. Use Thevenin’s Theorem to solve each of the following questions:
a) Find the current flowing through 4Ω resistor as shown in the figure 2.85.
b) Find the current flowing through 6Ω resistor and voltage drop across it as shown in the figure 2.86.
c) Find the current flowing through 5Ω resistor and power dissipated in it as shown in the figure 2.87.

Figure 2.85 Figure 2.86 Figure 2.87

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 19

 Continued…
d) Find the current flowing through 2Ω resistor between terminals A and B as shown in the figure 2.88.
e) Find the current flowing through 4Ω resistor as shown in the figure 2.89.
f) Find the current flowing through 2Ω resistor as shown in the figure 2.90.

Figure 2.90

Figure 2.88

Figure 2.89

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 20

 Continued…
g) Find the current flowing through 4Ω resistor as shown in the figure 2.91.
h) Find the current flowing through 10Ω resistor between terminals A and B as shown in the figure 2.92.
i) Find the current flowing through 3Ω resistor between terminals A and B as shown in the figure 2.93.

Figure 2.91 Figure 2.93

Figure 2.92

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 21

 Continued…
j) Find the current flowing through 2Ω resistor in the unbalanced bridge as shown in the figure 2.94.
k) Find the current flowing through 10Ω resistor between terminals A and B as shown in the figure 2.95.
l) Find the current flowing through 10Ω resistor between terminals A and B as shown in the figure 2.96.

Figure 2.96

Figure 2.94 Figure 2.95

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 22

 Tutorial-8 Continued…
2. Use Norton’s Theorem to solve each of the following questions:
a) Calculate the current flowing through 15Ω resistor between terminals A and B as shown in the figure 2.97.
b) Calculate the current flowing through 5Ω resistor between terminals A and B.as shown in the figure 2.98.
c) Determine the current flowing through 3Ω resistor and power dissipated in it as shown in the figure 2.99.

Figure 2.97 Figure 2.98

Figure 2.99

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 23

 Continued…
d) Calculate the current flowing through 4Ω resistor and power dissipated in it as shown in the figure 2.100.
e) Calculate the current I flowing through 4Ω resistor as shown in the figure 2.101.
f) Determine the current flowing through 100Ω resistor as shown in the figure 2.102.

Figure 2.101

Figure 2.101 Figure 2.102

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 24

 Continued…
g) Calculate the current flowing through RL = 9Ω resistor as shown in the figure 2.103.
h) Calculate the current flowing through 4Ω resistor as shown in the figure 2.104.
i) Find the current flowing through 100Ω resistor between terminals A and B as shown in the figure 2.105.

Figure 2.9104 Figure 2.105

Figure 2.9103

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 25

 Continued…
j) Find the current which would flow in a 15Ω resistor connected between points A and B as shown in the
figure 2.106.
k) Calculate the current flowing through 6Ω resistor as shown in the figure 2.107.
l) Find the current flowing through 3Ω resistor between terminals A and B as shown in the figure 2.108.

Figure 2.9106 Figure 2.107 Figure 2.108

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 26

 Tutorial-9
1. Find the value of RL such that maximum power is transferred in the load resistance RL. Also, find the
maximum power that can be transferred to load resistance RL for the circuit shown in the figure 2.109.
2. What is the value of R such that maximum power is transferred to it? Find the value of this maximum power
for the circuit shown in the figure 2.110.
3. Find the value of R such that maximum power is transferred to it. Find the value of this maximum power for
the circuit shown in the figure 2.111.

Figure 2.109 Figure 2.110 Figure 2.111

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 27

 Continued…
4. Find the value of R such that maximum power is transferred in it. Also, find the value of this maximum
power for the circuit shown in the figure 2.112.
5. Find the value of RL such that maximum power is transferred in the load resistance RL. Also, find the
maximum power that can be transferred to load resistance RL for the circuit shown in the figure 2.113.
6. Find the value of R such that maximum power is transferred to it. Find the value of this maximum power for
the circuit shown in the figure 2.114.

Figure 2.112
Figure 2.114

Figure 2.113

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 28

 Continued…
7. Determine the value of R such that 4Ω resistor consumes maximum power. Also, find the maximum power
for the circuit shown in the figure 2.115.
8. Find the value of R such that the load resistance RL which is equal to 4Ω will deliver maximum power. Also
find the value of this maximum power for the circuit shown in the figure 2.116.
9. For the circuit shown in figure 2.117, what will be the value of RL to get maximum power. What is the
maximum power delivered to the load?

Figure 2.115 Figure 2.116 Figure 2.117

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 29

 Continued…
10. Find the value of R such that maximum power takes place from the current sources to the load R in the
figure 2.118. Also, obtain the amount of maximum power transfer.
11. In the network shown in figure 2.119, find the resistance RL connected between terminals A and B so that
maximum power is developed across load resistance RL. What is the maximum power?
12. Calculate the value of R which will absorb maximum power from the circuit shown below in the figure
2.120. Also find the value of maximum power.

Figure 2.120

Figure 2.118

Figure 2.119
9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 30
 Continued…
13. Verify reciprocity theorem in the circuit shown in the figure 2.121.
14. Verify reciprocity theorem in the circuit shown in the figure 2.122.
15. Verify reciprocity theorem in the circuit shown in the figure 2.123.
16. Verify reciprocity theorem in the circuit shown in the figure 2.124.
Figure 2.121

Figure 2.122 Figure 2.123 Figure 2.124

9/11/2023 Prepared by Er. Dhurba Karki, Lecturer, KhCE 31