NETWORK THEOREMS

KAMIL HUSSAIN

NETWORK THEOREMS

• KIRCHHOFFS LAWS• MESH ANALYSIS• NODAL ANALYSIS• NORTAN• SUPERPOSITION• THEVENIN• MAXIMUM POWER TRANSFER

Kirchhoff's Laws

Kirchhoff's circuit laws are two equalities that deal with the conservation of charge and energy in electrical circuits.

There basically two Kirchhoff's law :-

1. Kirchhoff's current law (KCL) – Based on principle of conservation of electric charge.

2. Kirchhoff's voltage law (KVL) - Based on principle of conservation of energy.

Kirchhoff's current law (KCL)

This law is also called Kirchhoff's first law, Kirchhoff's point rule, Kirchhoff's junction rule (or nodal rule), and Kirchhoff's first rule.

The principle of conservation of electric charge implies that: At any node (junction) in an electrical circuit, the sum

of currents flowing into that node is equal to the sum of currents flowing out of that node, or The algebraic sum of currents in a network of conductors meeting at a point is zero.

Strictly speaking KCL only applies to circuits with steady currents (DC).However, for AC circuits having dimensions much smaller than a wavelength, KCL is also approximately applicable.

The current entering any junction is equal to the current leaving that junction. i1 + i4 =i2 + i3

Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be stated as:

0I

Kirchhoff's voltage law (KVL)

This law is also called Kirchhoff's second law, Kirchhoff's loop (or mesh) rule, and Kirchhoff's second rule.

The principle of conservation of energy implies that The directed sum of the electrical potential

differences (voltage) around any closed circuit is zero, or More simply, the sum of the emfs in any closed loop is

equivalent to the sum of the potential drops in that loop Strictly speaking KVL only applies to circuits with steady

currents (DC). However, for AC circuits having dimensions much smaller than

a wavelength, KVL is also approximately applicable.

The algebraic sum of the products of the resistances of the conductors and the currents in them in a closed loop is equal to the total emf available in that loop. Similarly to KCL, it can be stated as:

OR RIVemfVn

loop 0KVL:

The sum of all the voltages around the loop is equal to zero. v1 + v2 + v3 - v4 = 0

Mesh Analysis

Mesh analysis (or the mesh current method) is a method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in the circuit. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other.

Mesh analysis works by arbitrarily assigning mesh currents in the essential meshes. An essential mesh is a loop in the circuit that does not contain any other loop.

Steps to Determine Mesh Currents:1. Assign mesh currents i1, i2, .., in to the n meshes.

Current direction need to be same in all meshes either clockwise or anticlockwise.

2. Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of the mesh currents.

3. Solve the resulting n simultaneous equations to get the mesh currents

ExampleA circuit with two meshes.

Apply KVL to each mesh. For mesh 1,

For mesh 2,

123131

213111

)(

0)(

ViRiRR

iiRiRV

223213

123222

)(

0)(

ViRRiR

iiRViR

Solve for the mesh currents.

Use i for a mesh current and I for a branch current. It’s evident from Fig. 3.17 that

2

1

2

1

323

331

VV

ii

RRRRRR

2132211 , , iiIiIiI

Nodal Analysis

In electric circuits analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage (potential difference) between "nodes" (points where elements or branches connect) in an electrical circuit in terms of the branch currents.

Nodal analysis is possible when all the circuit elements branch constitutive relations have an admittance representation.

Kirchhoff’s current law is used to develop the method referred to as nodal analysis

STEPS FOR NODAL ANALYSIS:-• Note all connected wire segments in the circuit. These are

the nodes of nodal analysis.• Select one node as the ground reference. The choice does not

affect the result and is just a matter of convention. Choosing the node with most connections can simplify the analysis.

• Assign a variable for each node whose voltage is unknown. If the voltage is already known, it is not necessary to assign a variable.

• For each unknown voltage, form an equation based on Kirchhoff's current law. Basically, add together all currents leaving from the node and mark the sum equal to zero.

• If there are voltage sources between two unknown voltages, join the two nodes as a super node. The currents of the two nodes are combined in a single equation, and a new equation for the voltages is formed.

• Solve the system of simultaneous equations for each unknown voltage.

1. Reference Node

The reference node is called the ground node where V = 0

+

–

V 500W

500W

1kW

500W

500WI1 I2

Example

V1, V2, and V3 are unknowns for which we solve using KCL

500W

500W

1kW

500W

500WI1 I2

1 2 3

V1 V2 V3

Steps of Nodal Analysis1. Choose a reference (ground) node.2. Assign node voltages to the other nodes.3. Apply KCL to each node other than the reference

node; express currents in terms of node voltages.4. Solve the resulting system of linear equations for

the nodal voltages.

Currents and Node Voltages

500W

V1

500WV1 V2

50021 VV

5001V

3. KCL at Node 1

500W

500WI1

V1 V2

500500

1211

VVVI

3. KCL at Node 2

500W

1kW

500W V2 V3V1

0500k1500

32212

VVVVV

3. KCL at Node 3

2323

500500I

VVV

500W

500W

I2

V2 V3

Superposition Theorem• It is used to find the solution to networks with two or more

sources that are not in series or parallel• The current through, or voltage across, an element in a linear

bilateral network is equal to the algebraic sum of the currents or voltages produced independently by each source.

• For a two-source network, if the current produced by one source is in one direction, while that produced by the other is in the opposite direction through the same resistor, the resulting current is the difference of the two and has the direction of the larger

• If the individual currents are in the same direction, the resulting current is the sum of two in the direction of either current

Superposition Theorem• The total power delivered to a resistive element must be

determined using the total current through or the total voltage across the element and cannot be determined by a simple sum of the power levels established by each source

For applying Superposition theorem:-• Replace all other independent voltage sources with a short

circuit (thereby eliminating difference of potential. i.e. V=0, internal impedance of ideal voltage source is ZERO (short circuit)).

• Replace all other independent current sources with an open circuit (thereby eliminating current. i.e. I=0, internal impedance of ideal current source is infinite (open circuit).

Example:- Determine the branches current using Superposition theorem.

Solution:

• The application of the superposition theorem is shown in Figure 1, where it is used to calculate the branch current. We begin by calculating the branch current caused by the voltage source of 120 V. By substituting the ideal current with open circuit, we deactivate the current source, as shown in Figure 2.

120 V 3

6

12 A4

2

i1i2

i3i4

Figure 1

• To calculate the branch current, the node voltage across the 3Ω resistor must be known. Therefore

120 V 3

6

4

2

i'1 i'2i'3 i'4

v1

Figure 2

42

v

3

v

6

120v 111

= 0

where v1 = 30 V

The equations for the current in each branch,

6

30120 = 15 A

i'2 = 3

30= 10 A

i'3 = i'

4 =

6

30= 5 A

In order to calculate the current cause by the current source, we deactivate the ideal voltage source with a short circuit, as

shown

3

6

12 A4

2

i1"

i2"

i3"

i4"

i'1 =

To determine the branch current, solve the node voltages across the 3Ω dan 4Ω resistors as shown in Figure 4

The two node voltages are

3

6

12 A4

2

v3v4

+

-

+

-

2634333 vvvv

124

v

2

vv 434

= 0

= 0

• By solving these equations, we obtain

v3 = -12 V

v4 = -24 V

Now we can find the branches current,

To find the actual current of the circuit, add the currents due to both the current and voltage source,

Thevenin's theorem

Thevenin's theorem for linear electrical networks states that any combination of voltage sources, current sources, and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R.

Any two-terminal, linear bilateral dc network can be replaced by an equivalent circuit consisting of a voltage source and a series resistor

Thévenin’s Theorem The Thévenin equivalent circuit provides an equivalence at

the terminals only – the internal construction and characteristics of the original network and the Thévenin equivalent are usually quite different

• This theorem achieves two important objectives:– Provides a way to find any particular voltage or current

in a linear network with one, two, or any other number of sources

– We can concentration on a specific portion of a network by replacing the remaining network with an equivalent circuit

Calculating the Thévenin equivalent

• Sequence to proper value of RTh and ETh • Preliminary

– 1. Remove that portion of the network across which the Thévenin equation circuit is to be found. In the figure below, this requires that the load resistor RL be temporarily removed from the network.

– 2. Mark the terminals of the remaining two-terminal network. (The importance of this step will become obvious as we progress through some complex networks)

– RTh:– 3. Calculate RTh by first setting all sources to zero

(voltage sources are replaced by short circuits, and current sources by open circuits) and then finding the resultant resistance between the two marked terminals. (If the internal resistance of the voltage and/or current sources is included in the original network, it must remain when the sources are set to zero)

• ETh:– 4. Calculate ETh by first returning all sources to their

original position and finding the open-circuit voltage between the marked terminals. (This step is invariably the one that will lead to the most confusion and errors. In all cases, keep in mind that it is the open-circuit potential between the two terminals marked in step 2)

• Conclusion:– 5. Draw the Thévenin

equivalent circuit with the portion of the circuit previously removed replaced between the terminals of the equivalent circuit. This step is indicated by the placement of the resistor RL between the terminals of the Thévenin equivalent circuit

Insert Figure 9.26(b)

Another way of Calculating the Thévenin equivalent

• Measuring VOC and ISC– The Thévenin voltage is again determined by

measuring the open-circuit voltage across the terminals of interest; that is, ETh = VOC. To determine RTh, a short-circuit condition is established across the terminals of interest and the current through the short circuit Isc is measured with an ammeter

– Using Ohm’s law:

RTh = Voc / Isc

Example:- find the Thevenin equivalent circuit.

Solution

• In order to find the Thevenin equivalent circuit for the circuit shown in Figure1 , calculate the open circuit voltage, Vab. Note that when the a, b terminals are open, there is no current flow to 4Ω resistor. Therefore, the voltage vab is the same as the voltage across the 3A current source, labeled v1.

• To find the voltage v1, solve the equations for the singular node voltage. By choosing the bottom right node as the reference node,

25 V20

+

-

v13 A

5 4 +

-

vab

a

b

• By solving the equation, v1 = 32 V. Therefore, the Thevenin voltage Vth for the circuit is 32 V.

• The next step is to short circuit the terminals and find the short circuit current for the circuit shown in Figure 2. Note that the current is in the same direction as the falling voltage at the terminal.

0320

v

5

25v 11

25 V20

+

-

v23 A

5 4 +

-

vab

a

b

isc

Figure 2

04

v3

20

v

5

25v 222

Current isc can be found if v2 is known. By using the bottomright node as the reference node, the equationfor v2 becomes

By solving the above equation, v2 = 16 V. Therefore, the short circuitcurrent isc is

The Thevenin resistance RTh is

Figure 3 shows the Thevenin equivalent circuit for the Figure 1.

Figure 3

Norton theorem

Norton's theorem for linear electrical networks states that any collection of voltage sources, current sources, and resistors with two terminals is electrically equivalent to an ideal current source, I, in parallel with a single resistor.

Any two linear bilateral dc network can be replaced by an equivalent circuit consisting of a current and a parallel resistor.

Calculating the Norton equivalent

• The steps leading to the proper values of IN and RN

• Preliminary– 1. Remove that portion of the network across

which the Norton equivalent circuit is found– 2. Mark the terminals of the remaining two-

terminal network

• RN:– 3. Calculate RN by first setting all sources to zero

(voltage sources are replaced with short circuits, and current sources with open circuits) and then finding the resultant resistance between the two marked terminals. (If the internal resistance of the voltage and/or current sources is included in the original network, it must remain when the sources are set to zero.) Since RN = RTh the procedure and value obtained using the approach described for Thévenin’s theorem will determine the proper value of RN

Norton’s Theorem

• IN :– 4. Calculate IN by first returning all the sources to

their original position and then finding the short-circuit current between the marked terminals. It is the same current that would be measured by an ammeter placed between the marked terminals.

– Conclusion:– 5. Draw the Norton equivalent circuit with the

portion of the circuit previously removed replaced between the terminals of the equivalent circuit

Example Derive the Norton equivalent circuit

Solution Step 1: Source transformation (The 25V voltage

source is converted to a 5 A current source.)

25 V20 3 A

5 4 a

b

20 3 A5

4 a

b

5 A

4 8 A

4 a

b

Step 3: Source transformation (combined serial resistance to produce the Thevenin equivalent circuit.)

8

32 V

a

b

Step 2: Combination of parallel source and parallel resistance

• Step 4: Source transformation (To produce the Norton equivalent circuit. The current source is 4A (I = V/R = 32 V/8 ))

Norton equivalent circuit.

8 Ω

a

b

4 A

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 be equal to the resistance of the source as viewed from the output terminals. A load will receive maximum power from a linear bilateral

dc network when its total resistive value is exactly equal to the Thévenin resistance of the network as “seen” by the load

RL = RTh

Resistance network which contains dependent and independent sources

L

2Th

R4

V 2

L

L2

Th

R2

RVpmax = =

• Maximum power transfer happens when the load resistance RL is equal to the Thevenin equivalent resistance, RTh. To find the maximum power delivered to RL,

Application of Network Theorems

• Network theorems are useful in simplifying analysis of some circuits. But the more useful aspect of network theorems is the insight it provides into the properties and behaviour of circuits

• Network theorem also help in visualizing the response of complex network.

• The Superposition Theorem finds use in the study of alternating current (AC) circuits, and semiconductor (amplifier) circuits, where sometimes AC is often mixed (superimposed) with DC