chapter-2 network theorems. content 1. kirchhoff’s laws, voltage sources and current sources. 2....

CHAPTER-2

NETWORK THEOREMS

CONTENT

1. Kirchhoff’s laws, voltage sources and current sources.

2. Source conversion, simple problems in source conversion.

3. Superposition theorem, simple problems in super position theorem.

4. Thevenin’s theorem, Norton’s theorem, simple problems.

5.Reciprocity theorem, Maximum power transfer theorem, simple problems.

6. Delta/star and star/delta transformation.

Gustav Robert Kirchhoff

Definitions

• Circuit – It is an interconnection of electrical elements in a closed path by conductors(wires).

• Node – Any point where two or more circuit elements are connected together

• Branch –A circuit element between two nodes• Loop – A collection of branches that form a closed path returning to

the same node without going through any other nodes or branches twice

Example

• How many nodes, branches & loops?

+ -

Vs Is

R1

R2 R3

+

Vo

-

Example-Answer

• Three nodes

+ -

Vs Is

R1

R2 R3

+

Vo

-

Example-Answer

• 5 Branches

+ -

Vs Is

R1

R2 R3

+

Vo

-

Example-Answer

• Three Loops, if starting at node A

+ -

Vs Is

R1

R2 R3

+

Vo

-

A B

C

Example

DC

1 2

3 4 5

6

7

2A

How many nodes, branches & loops?bn

l

9

5

5

Kirchhoff's Current Law (KCL)

Also called Kirchhoff's Point law and Kirchhoff's First rule..


Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 + total volume of water flowing through pipe 3)


Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 + total current leaving the node through the wire 3)


"The algebraic sum of all currents entering and leaving a node

must equal zero"

∑ (Entering Currents) = ∑ (Leaving Currents)

Established in 1847 by Gustav R. Kirchhoff


It states that, in any linear network the algebraic sum of the

current meeting at a point (junction) is zero.

∑ I (Junction) = 0

∑ I (Entering) = ∑ I (Leaving)

∑ I (Entering) - ∑ I (Leaving) =0


Assign positive signs to the currents entering the node and

negative signs to the currents leaving the node, the KCL can be re-

formulated as:

All currents at the node) =


Example

I1= 1 A

I2= 3 A

I3= 0.5 A

Find the current I4 in A

Kirchhoff's Voltage Law (KVL)

Also called Kirchhoff's loop rule and Kirchhoff's second rule..


“The algebraic sum of voltages around each loop is zero”.

Σ voltage rise - Σ voltage drop = 0Or

Σ voltage rise = Σ voltage drop


It states that, in any linear bilateral active closed network the

algebraic sum of the product of the current and resistance in each of

the conductors in any closed path (mesh) in the network plus the

algebraic sum of e.m.f in the path is zero.

∑ IR + ∑ e.m.f = 0

The sign of each voltage is the polarity of the terminal first encountered

in traveling around the loop.The direction of travel is arbitrary.

Clockwise:

Counter-clockwise:

0 1 2 0V V V

2 1 0 0V V V

0 1 2V V V V0

I

R1

R2

V1

V2

A +

+

-

-

Sign Convention

Example

• Kirchoff’s Voltage Law around 1st Loop

+ -

Vs Is

R1

R2 R3

+

Vo

-

A B

C

I2

I1

+

I2R2

-

+ I1R1 -

Assign current variables and directions

Use Ohm’s law to assign voltages and polarities consistent with passive devices (current enters at the + side)

Example


+ -

Vs Is

R1

R2 R3

+

Vo

-

A B

C

I2

I1

+

I2R2

-

+ I1R1 -

Starting at node A, add the 1st voltage drop: + I1R1

Example


+ -

Vs Is

R1

R2 R3

+

Vo

-

A B

C

I2

I1

+

I2R2

-

+ I1R1 -

Add the voltage drop from B to C through R2: + I1R1 + I2R2

Example


+ -

Vs Is

R1

R2 R3

+

Vo

-

A B

C

I2

I1

+

I2R2

-

+ I1R1 -

Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0

Notice that the sign of each term matches the polarity encountered 1st

Source Conversion

DCsv

sRabv

+

-

abv

+

-

sRs

s

v

R

ai 'ai

Voltage Source to Current Source

Current Source to Voltage Source

Convert to Current Source

Answer-1

Convert to Voltage Source

Answer-2

Superposition Theorem

STATEMENT-

In a network of linear resistances containing more than one

generator (or source of e.m.f.), the current which flows at any point is

the sum of all the currents which would flow at that point if each

generator were considered separately and all the other generators

replaced for the time being by resistances equal to their internal

resistances.


STATEMENT-

In a linear circuit with several sources the voltage and

current responses in any branch is the algebraic sum of the voltage

and current responses due to each source acting independently with

all other sources replaced by their internal impedance.


Replace a voltage source with a short circuit.


Replace a current source with an open circuit.


Step-1:

Select a single source acting alone. Short the other voltage

source and open the current sources, if internal impedances are not

known. If known, replace them by their internal resistances.


Step-2:

Find the current through or the voltage across the required

element, due to the source under consideration, using a suitable

network simplification technique.


Step-3:

Repeat the above two steps far all sources.


Step-4:

Add all the individual effects produced by individual sources, to

obtain the total current in or voltage across the element.

Explanation


Consider a network, having two voltage sources V1 and V2.

Let us calculate, the current in branch A-B of network, using

superposition theorem.

Step-1:

According to Superposition theorem, consider each source

independently. Let source V1 is acting independently. At this time,

other sources must be replaced by internal resistances.


But as internal impedance of V2 is not given, the source V2 must

be replaced by short circuit. Hence circuit becomes, as shown.

Using any of the network reduction techniques, obtain the

current through branch A-B i.e. IAB due to source V1 alone.


Step 2:

Now Consider Source V2 volts alone, with V1 replaced by short circuit,

to obtain the current through branch A-B. Hence circuit becomes, as shown.

Using any of the network reduction techniques, obtain the current

through branch A-B i.e. IAB due to source V2 alone.


Step 3:

According to the Superposition theorem, the total current

through branch A-B is sum of the currents through branch A-B

produced by each source acting independently.

Total IAB = IAB due to V1 + IAB due to V2

Example

Find the current in the 6 Ω resistor using the principle of

superposition for the circuit.

SolutionStep-1:Replace Current Source with open circuit

Step-2:Replace Voltage Source with Short circuit

Step-3:Current through 6 Ω resistor is

Thevenin’s theorem


Statement

“Any linear circuit containing several voltages and

resistances can be replaced by just a Single Voltage VTH in series with a

Single Resistor RTH “.


VTH

Req or RTH

Thevenin’s Equivalent Circuit

Steps to be followed for Thevenin’s TheoremStep 1:

Remove the branch resistance through which current is to be

calculated.

Step 2:

Calculate the voltage across these open circuited terminals, by

using any of the network simplification techniques. This is VTH.

Steps to be followed for Thevenin’s TheoremStep 3:

Calculate Req as viewed through the two terminals of the branch

from which current is to be calculated by removing that branch

resistance and replacing all independent sources by their internal

resistances. If the internal resistance are not known, then replace

independent voltage sources by short circuits and independent current

sources by open circuits.

Steps to be followed for Thevenin’s Theorem

Step 4:

Draw the Thevenin’s equivalent showing source VTH, with the resistance

Req in series with it, across the terminals of branch of interest.

Step 5:

Reconnect the branch resistance. Let it be RL. The required current

through the branch is given by,

Example- Find VTH, RTH and the load current flowing through and

load voltage across the load resistor in fig by using Thevenin’s

Theorem.

Step 1- Open the 5kΩ load resistor

Step 2-Calculate / measure the Open Circuit Voltage. This is the

Thevenin's Voltage (VTH).

Step 3-Open Current Sources and Short Voltage Sources

Step 4-Calculate /measure the Open Circuit Resistance. This is

the Thevenin's Resistance (RTH)

Step 5-Connect the RTH in series with Voltage Source VTH and re-

connect the load resistor. i.e. Thevenin's circuit with load resistor. This

the Thevenin’s equivalent circuit.

Thevenin’s Equivalent Circuit=VTH

RTH =

Step 6- Calculate the total load current & load voltage

Norton’s theorem

Norton’s theorem

STATEMENT-

Any Linear Electric Network or complex circuit with Current and

Voltage sources can be replaced by an equivalent circuit containing of

a single independent Current Source IN and a Parallel Resistance RN.

Norton’s theorem

IINN

RRNN

Norton’s Equivalent Circuit

Norton’s theorem

Steps to be followed for Norton’s Theorem

Step 1:

Short the load resistor

Step 2:

Calculate / measure the Short Circuit Current.

This is the Norton Current (IN)


Step 3:

Open Current Sources, Short Voltage Sources and Open Load

Resistor.

Calculate /Measure the Open Circuit Resistance.

This is the Norton Resistance (RN)


Step 4

Now, Redraw the circuit with measured short circuit Current (IN)

in Step (2) as current Source and measured open circuit resistance (RN)

in step (4) as a parallel resistance and connect the load resistor which

we had removed in Step (3).

This is the Equivalent Norton Circuit.

Example 1-Find RN, IN, the current flowing

through and Load Voltage across the load resistor

in fig (1) by using Norton’s Theorem.

Step 1-Short the 1.5Ω load resistor

Step 2-Calculate / measure the Short Circuit Current. This is the

Norton Current (IN).

Step 3-Open Current Sources, Short Voltage Sources and Open

Load Resistor.

Step 4-Calculate /measure the Open Circuit Resistance. This is

the Norton Resistance (RN)

Step 5- Connect the RN in Parallel with Current Source INand

re-connect the load resistor.

Step 6-Now apply the last step i.e. calculate the load current

through and Load voltage across load resistor

Maximum Power Transfer Theorem


Statement:

In an active resistive network, maximum power transfer to the

load resistance takes place when the load resistance equals the

equivalent resistance of the network as viewed from the terminals of

the load.

Steps to be followed for MPTT

Example-In the network shown, find the value of RL such that

maximum possible power will be transferred to RL. Find also the value

of the maximum power.

Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH

Thevenin’s Equivalent Circuit

Reciprocity Theorem

Statement:

In any linear bilateral network, if a source of e.m.f E in any

branch produces a current I in any other branch, then the same e.m.f

E acting in second branch would produce the same current I in the

first branch.

Reciprocity Theorem

Example-In the network given below, find (a) ammeter

current when battery is at A and ammeter at B and (b) when battery is

at B and ammeter at point A.

What is STAR Connection?

If the three resistances are connected in such a manner that one

end of each is connected together to form a junction point called STAR

point, the resistances are said to be connected in STAR.

Star or Y or T Network

What is DELTA Connection?If the three resistances are connected in such a manner that one

end of first is connected to first end of second, the second end of

second to first end of third and so on to complete a loop then the

resistances are said to be connected in DELTA.

Delta or π Network

STAR to DELTA

DELTA to STAR

To convert a STAR to DELTA

To convert a DELTA to STAR

Example 1-Convert given DELTA into STAR

Answer

Example 2-Convert given STAR into DELTA

Answer

=R12

=R23

R31 =

Example 3-Calculate the effective resistance between points A & B

Answer-Step 1

Answer-Step 2

Answer-Step 3

Answer-Step 4 & 5

RAB = 3.69 Ω

Example 4-Find the equivalent resistance between P & Q in the ckt

P

Q

Solution

Req =14.571Ω

Example 5-In the circuit shown, find the resistance between M and N.

Solution- Step 1

Solution- Step 2

Solution- Step 3

Solution- Step 4 & 5

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