Unit - 1

Network Theorems

1.1 Superposition theorem

This theorem states that in a linear network containing several independent sources, the overall response at any point in the network equals the sum of responses due to each independent source considered separately with all other independent sources made inoperative.

Question 1. Find the current through resistance.

Fig 1 Circuit Diagram

Solution:

Fig 2 Short circuit 2V

1 = 0

Fig 3 Open circuit 6A

2   =

1 + 2 

Key takeaway

Special Case

Fig 4 Circuit with two voltage sources

Since two voltage sources with different magnitudes in parallel cannot be connected as in a single branch, two different currents are not possible (if 5V then I = zero).

Question 2: Find Value of current I

Fig 5 Circuit Diagram

Sol: Applying Superposition Theorem

Fig 6 Circuit with 2A open circuit

Fig 7 Circuit with 1V short circuit

1 =

2 =

=

1 + 2

  =

1.2 Thevenin theorem

Thevenin’s Theorem states that a two-terminal network containing sources and passive elements is equivalent to a voltage source in series with the network impedance with all independent sources removed.

Fig 8 Two-port network

Fig 9 Thevenin’s equivalent of A

Fig 10 Norton’s equivalent of A

1.3 Norton theorem

Norton’s Theorem states that a two-terminal network containing sources and passive elements is equivalent to an independent current source in parallel with the network impedance with all independent sources removed. The current of the current source has the same magnitude and reference direction as that of the current which would exist at the terminal in the original network if the terminals were short-circuited.

The short circuit current is then given as

sc = Vth/Rth

Norton’s equivalent is obtained by source conversion of Thevenin’s equivalent circuit.

CONDITIONS FOR APPLICATION

Firstly, open circuit terminal A and B.

2.     If network A is operating with independent and dependent sources:

3.     If the network is operating with only dependent sources:

Connect generation between A and B

First, open circuit terminal A and B.

Find out the voltage between A and B this is Vth

Fig 11 Terminal where we find Vth

Isc =

Fig 12 Finding Isc

Question: Find Vth and Rth for the circuit shown below?

Fig 13 Circuit Diagram

Answer:

Fig 14 Circuit for finding Rth

Fig 15 Finding Vth across a and b

Finding Isc from circuit directly:

Fig 16 Finding Isc

By KCL,

Question: Find Vth and Rth across a and b?

Fig 17 Circuit Diagram

Sol: Finding Rth

Fig 18 Circuit for Rth

Also, clear from circuit that Vth = 1V.

Fig 19 Circuit for Isc

By applying KVL we get,

1-3Isc=0

Isc=A

Que: Find Isc and Rth across terminal a and b?

Fig 20 Circuit Diagram

Ans;

Fig 21 Circuit for finding RTh

Rth=3k+2k=5k

By applying KVL we get

Therefore,

Question: Find the value of voltage Vx?

Fig 22 Circuit Diagram

Solution: For Rth

Fig 23 Circuit for finding Rth

By KCL,

But,

By KVL,

Question: Find value of current I in the circuit?

Fig 24 Circuit diagram

Solution: Since, no independent source is present so,

 Isc = 0

And we know that,

Since Rth cannot be zero

Fig 25 Circuit for finding Rth

But

Question: Find out the Norton’s equivalent

Fig 26 Circuit diagram

Solution:

Fig 27 Circuit for finding Isc

Fig 28 Circuit for finding Vth

Since, there is no significance of current source

A

1.4 Maximum power transfer theorem

The maximum power transfer theorem states that to obtain maximum external power from a source with a finite internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals.

Fig 29 Network With load RL

where Rth is Thevenin’s equivalent resistance across a and b.

Fig 30 Circuit Diagram

Maximum power is absorbed by ZL when

Condition:

Comparing real and imaginary parts

OR

Maximum power absorbed by ZL is

Key takeaway

For maximum power transfer

Question: Find out the value of load resistance if power absorbed is maximum.

Fig 31 Circuit Diagram

Solution: find Thevenin’s equation

Fig 32 Finding Rth

Question: Find maximum power delivered is RL if its value is

Ω

Solution

Fig 33 Circuit for finding Rth

Therefore,

1.5 Reciprocity theorem

In a voltage source V (current source i) located at one point in a network produces current i (voltage v) at the second point in the network then the same voltage source V (current source) acting at the second point if produces the same current i at the first point the network is called as Reciprocal Theorem.

Fig 34 Network from which reciprocity is described

When the excitation is a voltage and response is current then short circuit conditions are implied at both ends. In this theorem, we can also apply the current source at one end and measure the open-circuit voltage at the other end, and open circuit conditions are implied at both ends. The assumptions for the theorem to apply to the network are

i)                   All initial conditions are zero.

ii)                 There is only one source of excitation in the network.

iii)              The network is linear.

iv)               The dependent sources are excluded in the network even if they are linear.

Q) Verify the Reciprocity theorem for the given network?

Fig 35 Circuit Diagram

Sol: The value of current Ix flowing due to 20V source is

Fig 36 When 20V is at its place

Ix = = 2A

By current Division

Ik = Ix = 1A

Now the voltage source of 20V is connected to branch cd. The current source now will be given as

Fig 37 When 20V is connected to the cd

Iy= = 3/2 A

Then Ij = Iy = 1A

From above we see that Ik = Ij = 1A

The reciprocity theorem is verified.

1.6 Compensation theorem

According to this theorem, any resistance in the network can be replaced by a voltage source that has zero internal resistance and a voltage equal to the voltage drop across the replace resistance due to the current which was flowing through it.

Fig 38 Compensation Theorem

As seen from the figure above the resistance R carrying current, I can be replaced by a voltage V=IR. This is the basic concept of compensation theorem.

1.7 Analysis with dependent current and voltage sources

Voltage and current sources

1. Voltage Source: A voltage source always have a resistance in series. 

Ideal source

R = 0

Practical = 2 - 5Ω

I = 0

Fig 39 Voltage source

2. Current Source: Always have R parallel with the source.

Practical = Mega Ω

Ideal =     I =

Fig 40 Current Source

Dependent and Independent sources

Fig 40 Independent Source

few either

AC or DC 

dc
 

Fig 41 Independent Source

Key takeaway

For voltage source R is always in series

For current source R is always in parallel

1.8 Node and Mesh Analysis

NODE ANALYSIS

For these, we assume every node as a voltage point and write the current equation for every element. For the current source, the current entering is negative.

Fig 42 Circuit for Node analysis

For node V1

For node V2

For V

Example: Using nodal analysis find the voltage across 5resistor.

Solution:

For V1

For V2

Solving 1 and 2:

For 5 voltage =

= -50.9 + 57.27

= 6.37V

Key takeaway

We use Nodal analysis in those circuits where we need to find the voltage. The loop analysis is applied to the circuits when we want to calculate the branch or loop currents.

1.9 Concept of duality and dual networks

The two networks are said to be dual if the mesh equation of one network is equal to the node equation of the other.

Let us consider the RLC circuit

Fig 43 RLC Circuit

Applying KVL we have

v(t) = ri(t) + L + 

So, dual of the above network will be

Fig 44 Dual of RLC circuit

i(t) = Gv(t) + C +

Steps for construction of Dual Network: -

i)          In each loop place a dot. This internal dot corresponds to an independent node in the dual network.

ii)        Outside network place a dot. This is called the datum node is a dual network

iii)     Connect all the adjacent loops by dashed lines crossing common branches.

iv)     In a dual network, this common branch is replaced by its dual element.

Fig 45 Circuit whose dual is to make

Fig 46 Dual of the above circuit

Key takeaway

Some important duals are given below.

References:

Education, 2013.

4.     C. K. Alexander and M. N. O. Sadiku, “Electric Circuits”, McGraw Hill Education, 2004.

5.     K. V. V. Murthy and M. S. Kamath, “Basic Circuit Analysis”, Jaico Publishers, 1999.