network theorems with example.

1. NETWORK SYSTEM ANALYSIS
UNIT- 1
NETWORK THEOREMS
The first theorem to be introduced is the superposition theorem, followed
by Thévenin’s theorem, Norton’s theorem, and the maximum power
transfer theorem. The chapter concludes with a brief introduction to
Millman’s theorem and the substitution and reciprocity theorems.
LONG QUESTION ANSWER
Question no.1 Explain superposition theorem with example?
Ans. The superposition theorem states the following:
The current through, or voltage across, any element of a network is equal
to the algebraic sum of the currents or voltages produced independently
by each source.
In other words, this theorem allows us to find a solution for a current or
voltage using only one source at a time. Once we have the solution for
each source, we can combine the results to obtain the total solution. The
term algebraic appears in the above theorem statement because the
currents resulting from the sources of the network can have different
directions, just as the resulting voltages can have opposite polarities.
If we are to consider the effects of each source, the other sources
obviously must be removed. Setting a voltage source to zero volts is like
placing a short circuit across its terminals.
Therefore, when removing a voltage source from a network schematic,
replace it with a direct connection (short circuit) of zero ohms. Any

2. internal resistance associated with the source must remain in the network.
Setting a current source to zero amperes is like replacing it with an open
circuit. Therefore, when removing a current source from a network
schematic, replace it by an open circuit of infinite ohms. Any internal
resistance associated with the source must remain in the network.
The above statements are illustrated in Fig. 9.1.
Since the effect of each source will be determined independently, the
number of networks to be analyzed will equal the number of sources.
Example-
a. Using the superposition theorem, determine the current through resistor
R2 for the network in Fig. 9.2.
b. Demonstrate that the superposition theorem is not applicable to power
levels.

3. Solution-
a. In order to determine the effect of the 36 V voltage source, the current
source must be replaced by an open-circuit equivalent as shown in Fig.
9.3. The result is a simple series circuit with a current equal to
Examining the effect of the 9 A current source requires replacing the 36
V voltage source by a short-circuit equivalent as shown in Fig. 9.4. The
result is a parallel combination of resistors R1 and R2. Applying the
current divider rule results in

4. Since the contribution to current I2 has the same direction for each
source, as shown in Fig. 9.5, the total solution for current I2 is the sum
of the currents established by the two sources. That is,
b. Using Fig. 9.3 and the results obtained, we find the power delivered to
the 6 Ω resistor
Using Fig. 9.4 and the results obtained, we find the power delivered to the
6 Ω resistor
Using the total results of Fig. 9.5, we obtain the power delivered to the 6
Ω resistor
It is now quite clear that the power delivered to the 6 Ω resistor using the
total current of 8 A is not equal to the sum of the power levels due to each
source independently. That is,

5. Question 2. Explain thevenin’s theorem with example?
Ans. Thévenin’s theorem states the following: Any two-terminal dc
network can be replaced by an equivalent circuit consisting solely of a
voltage source and a series resistor as shown in Fig. 9.23.
EXAMPLE- 9.6 Find the Thévenin equivalent circuit for the network in
the shaded area of the network in Fig. 9.26. Then find the current through
RL for values of 2 Ω, 10 Ω, and 100 Ω.
Solution: Steps 1 and 2: These produce the network in Fig. 9.27. Note that
the load resistor RL has been removed and the two “holding” terminals
have been defined as a and b. Step 3: Replacing the voltage source E1
with a short-circuit equivalent yields the network in Fig. 9.28(a), where

6. Step 4: Replace the voltage source (Fig. 9.29). For this case, the open
circuit voltage Eth is the same as the voltage drop across the 6 Ω
resistor. Applying the voltage divider rule gives..
Step 5: (Fig. 9.31):

7. Question no 3 Explain Norton’s theorem with example?
Ans. The theorem states the following: Any two-terminal linear bilateral
dc network can be replaced by an equivalent circuit consisting of a
current source and a parallel resistor, as shown in Fig. 9.65.
EXAMPLE-Find the Norton equivalent circuit for the network in the
shaded area in Fig. 9.67

8. Solution- Steps 1 and2: See Fig. 9.68.
Step 3: See Fig. 9.69, and
Step 4: See Fig. 9.70, which clearly indicates that the short-circuit
connection between terminals a and b is in parallel with R2 and eliminates
its effect. IN is therefore the same as through R1, and the full battery
voltage appears across R1 since.

9. Step 5: See Fig. 9.71. This circuit is the same as the first one considered
in the development of Thévenin’s theorem. A simple conversion indicates
that the Thévenin circuits are, in fact, the same (Fig. 9.72).
Question no. 4 Explain maximum power theorem with example?
Ans. maximum power transfer theorem, which states the following:
A load will receive maximum power from a network when its resistance
is exactly equal to the Thévenin resistance of the network applied to the
load. That is,
Proof-

10. The original two terminal circuit is replaced with a Thevenin’s equivalent
circuit across the variable load resistance. The current through the load
for any value of load resistance is
Form the above expression the power delivered depends on the values of
RTH and RL. However, the Thevenin’s equivalent is constant, the power
delivered from this equivalent source to the load entirely depends on the
load resistance RL. To find the exact value of RL, we apply differentiation
to PL with respect to RL and equating it to zero as-
The maximum power delivered to the load is,

11. Example-EXAMPLE 9.16 The analysis of a transistor network resulted
in the reduced equivalent in Fig. 9.92.
a. Find the load resistance that will result in maximum power transfer to
the load, and find the maximum power?
a. Replacing the current source by an open-circuit equivalent results in
Restoring the current source and finding the open-circuit voltage at the
output terminals results in
For maximum power transfer to the load,
with a maximum power level of

12. Question no 5 Explain millman’s theorem with example?
Ans. Through the application of Millman’s theorem, any number of
parallel voltage sources can be reduced to one. In Fig. 9.97, for example,
the three voltage sources can be reduced to one. This permits finding the
current through or voltage across RL without having to apply a method
such as mesh analysis, nodal analysis, superposition, and so on. The
theorem can best be described by applying it to the network in Fig. 9.97.
Basically, three steps are included in its application.
Step 1: Convert all voltage sources to current sources as outlined in
Section 8.2. This is performed in Fig. 9.98 for the network in Fig. 9.97.
Step 2: Combine parallel current sources as described in Section 8.2.
The resulting network is shown in Fig. 9.99, where

13. Step 3: Convert the resulting current source to a voltage source, and the
desired single-source network is obtained, as shown in Fig. 9.100.
In general, Millman’s theorem states that for any number of parallel
voltage sources
The plus-and-minus signs appear in Eq. (9.8) to include those cases
where the sources may not be supplying energy in the same direction.

14. EXAMPLE 9.19 Using Millman’s theorem, find the current through and
voltage across the resistor RL in Fig. 9.101.
Sol.
The minus sign is used for E2/R2 because that supply has the opposite
polarity of the other two. The chosen reference direction is therefore that
of E1 and E3. The total conductance is unaffected by the direction, and

15. The resultant source is shown in Fig. 9.102, and
Question no.6 Explain reciprocity theorem with example?
Ans.The reciprocity theorem is applicable only to single-source networks.
It is, therefore, not a theorem used in the analysis of multisource networks
described thus far. The theorem states the following:
The current I in any branch of a network due to a single voltage source E
anywhere else in the network will equal the current through the branch in
which the source was originally located if the source is placed in the
branch in which the current I was originally measured.
In other words, the location of the voltage source and the resulting current
may be interchanged without a change in current. The theorem requires
that the polarity of the voltage source have the same correspondence with
the direction of the branch current in each position.
In the representative network in Fig. 9.112(a), the current I due to the
voltage source E was determined. If the position of each is interchanged
as shown in Fig. 9.112(b), the current I will be the same value as indicated

16. In the representative network in Fig. 9.112(a), the current I due to the
voltage source E was determined. If the position of each is interchanged
as shown in Fig. 9.112(b), the current I will be the same value as indicated
To demonstrate the validity of this statement and the theorem, consider
the network in Fig. 9.113, in which values for the elements of Fig.
9.112(a) have been assigned. The total resistance is

17. For the network in Fig. 9.114, which corresponds to that in Fig.
9.112(b), we find
Question no. Explain tellegen’s theorem and verify tellegen’s theorem?
Ans. The Tellegen’s theorem is one of the most general theorem used in
network analysis.It is applicable to any n/w made up of lumped two
terminal elements may be linear or non linear active or passive,time
varying or time invariant.
It is based on the two kirchoff law i.e KVL +KCL.
STATEMENT OF TELLEGENS THEOREM-
“For any given time ,the sum of power delivered to each branch of any
electric n/w is zero”
Thus for kTh branch ,this theorem states that