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UNIT-2 UNIT-2 NETWORK ANALYSISNETWORK ANALYSIS

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2.1 KIRCHHOFFS CURRENT LAW (KCL):-

It states that the algebraic sum of all currents entering a node is zero. Mathematically:

Currents are positive if entering a node Currents are negative if leaving a node. Example:

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Applying Kirchhoff's current law:

I1 + I2 + I3 + I4 = 0 (the negative sign in I2 indicates that I2 has a magnitude of 3A and is flowing in the direction opposite to that indicated by the arrow) Substituting: 5 - 3 + I3 + 2 = 0 Therefore, I3 = - 4A (ie 4A leaving node)

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2.2 KIRCHHOFFS VOLTAGE LAW (KVL):-

It states that the algebraic sum of the voltage drops around any loop, open or closed, is zero. Mathematically

Example:

Going round the loop in the direction of the current, I, Kirchhoff's Voltage Law gives:

10- 2I - 3I = 0

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- 2I and - 3I are negative, since they are voltage drops i.e. represent a decrease in potential on proceeding round the loop in the direction of I. For the same reason + 10V is positive as it is a voltage rise or increase in potential. Concluding: 5 I = 10 Therefore, I = 2A

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2.3 Loop Analysis:-• Loop analysis is systematic method of network analysis. • It is a general method and can be applied to any electrical

network, howsoever complicated it may be.• It is based on writing KVL equations for independent

loops.• A loop is a closed path in a network.• A node or a junction is a point in the network where three

or more elements have a common connection.

• Before the loop analysis can be applied to a network, we must first check that it has only voltage sources (independent or dependent).

• Any current source must be transformed into its equivalent voltage source.

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Example:-• Find the currents i1 and i2 in the circuit given below.

Solution : Applying KVL to the two loops,

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and

Example• Find a single voltage source equivalent of the

following circuit.

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Solution : We first replace the current source by its equivalent voltage source. We then apply Thevenin’s Theorem.

Now Applying KVL, we get

V42

636636

A1

01212

0366432216

IV

I

I

III

PQ

10To find Thevenin’s equivalent resistance, we reduce the network as shown.

To find Thevenin’s equivalent resistance, we reduce the network as shown.

Equivalent voltage source is

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2.4 Mesh Analysis:- • Mesh is a loop which contains no other loop within it.• Mesh analysis is applicable only to a planar network.• But most of the networks we shall need to analyze are

planar.• Once a circuit has been drawn in planar form, it often looks

like a multi-paned window.• Each pane is a mesh.• Meshes provide a set of independent equations.• By definition, a mesh-current is that current which flows

around the perimeter of a mesh. It is indicated by a curved arrow that almost closes on itself.

• Branch-currents have a physical identity. They can be measured.

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•Mesh-currents are fictitious.•The mesh analysis not only tell us the minimum number of unknown currents, but it also ensures that the KVL equations obtained are independent. EXAMPLE:-

Solution:-

Now Applying KVL, we get

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Resistance Matrix

Mesh current matrix

Source matrix

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Applying Crammer’s rule :

The current in 3-ohm resistor is I1 – I2 = 6 – 4 = 2A

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Example: Three-mesh Network

• Write the three equations for the three meshes and put them in a matrix form.

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Solution:- Now Applying KVL, we get

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Self-resistance of mesh 1

Mutual resistance between mesh 1 and 2.

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For The Resistance Matrix

• It is symmetrical about the major diagonal, as R12 = R21, R13 = R31, etc.

• All the elements on the major diagonal have positive values.

• The off-diagonal elements have negative values.• The mutual resistance between two meshes will be zero,

if there is no resistance common to them.

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Mesh Analysis Limitations:-• It is applicable only to those planar networks which contain

only independent voltage sources.• If there is a practical current source, it can be converted to

an equivalent practical voltage source.

2.5 Planar Network:-• If a network can be drawn on sheet of paper without

crossing lines, it is said to be planar.

• Is it a planar network ?

EXAMPLE:-

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• Yes, it is. Because it can be drawn in a plane, as shown in the figure.

This is definitely

non-planar network.

EXAMPLE:-

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2.6 Procedure for Mesh Analysis:-1. Make sure that the network is planar.

2. Make sure that it contains only independent voltage sources.

3. Assign clockwise mesh currents.

4. Write mesh equations in matrix form by inspection. An element on the principal diagonal is the self-resistance of the mesh. These elements are all positive. An element off the major diagonal is negative (or zero), and represents the mutual resistance.

5. Check the symmetry of resistance matrix about the major diagonal.

6. An element of the voltage source column matrix on the right side represents the algebraic sum of the voltage

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sources that produce current in the same direction as the assumed mesh current.

7. Solve the equations to determine the unknown mesh currents, using Cramer’s rule.

8. Determine the branch currents and voltages.Example: Determine the currents in various resistances of the network shown. •

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Writing the mesh equations by inspection,

Solving, we get I1 = 2.55 A, I2 = 3.167 A

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Example:-Find the current drawn from the source in the network, using mesh analysis.

Solution:-