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2.1 Branch-Current Analysis2.2 Mesh Analysis2.3 Nodal Analysis2.4 Mesh with Current Sources2.5 Nodal with Voltage Sources2.6 Bridge Network

Chapter 2: Circuit Analysis

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Introduction

Consider this network• Circuit simplification is difficult because there are

2 or more sources, and they are neither in series nor parallel.

• There will be an interaction between sources that will not permit the reduction techniques that is used to find quantities such as the effective resistance.

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Steps required for Branch-Current Analysis: -

1. Assign a distinct current of arbitrary direction to each branch of the circuit.

2. Add the polarities for each voltage drop across resistor.

3. Apply KVL for each mesh.

4. Apply KCL to a node that includes all the branch currents.

5. Solve the equations for branch currents.

2.1 Branch-Current Analysis

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Example :

Determine the current in each branch of the network using branch-current analysis

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Solution:• Assign distinct current

of arbitrary direction to each branch of the network (one branch one current)

• Add the polarities for each R

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Solution(continue…):Kirchhoff’s Voltage Law:

Loop 1:

- E1 + V1 + V3 = 0

I1R1 + I3R3 = E1

Loop 2:

- V3 – V2 + E2 = 0

I2R2 + I3R3 = E2

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Solution(continue…):Kirchhoff’s Current Laws

I1 + I2 = I3

Solve the equations

2I1 + 0I2 + 4I3 = 2 ---(1)

0I1 + I2 + 4I3 = 6 ---(2)

I1 + I2 - I3 = 0 ---(3)

Observation: 3 branches, 3 equations

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• Mesh Analysis – defines a unique array of currents (Mesh or Loop current) to the network using KCL

• Steps required for Mesh Analysis: -1. Assign current in clockwise direction to each closed loop

of network.2. Insert polarities for each resistor.3. Apply KVL to each closed loop.4. Solve the resulting equations.

Note: if a resistor has two or more current passing through it, the netcurrent = the mesh current of the closed loop + mesh

currents from other loops in same direction - mesh currents from other loops in opposite direction.

2.2 Mesh Analysis

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Example : find the branch currents I1, I2, I3.

1.Assign current in clockwise direction

123

51015

01010515

:1mesh For

21

21

211

ii

ii

iii

12or 12

102010

0461010

:2mesh For

2121

21

2212

iiii

ii

iiii

Solution:+ -

+

-

+ -

+

-

2. Insert polarities for each R according to the direction of current.

3. Apply KVL to each closed loop in clockwise direction.

-

+

Observation: 2 meshes, 2 equations.

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2.3 Nodal Analysis• Nodal analysis - provides nodal voltages by using KCL.

• Steps required for Nodal Analysis: -1. Determine the no. of nodes (junction of 2 or more

branches).2. Select a reference node (Ground), and label all other

nodes.3. Apply KCL at each node (except the reference node).4. Solve the resulting equations.

Note: A network of N nodes require (N-1) equations to find (N-1) nodal voltages where by the Reference node is eliminated.

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Example : Calculate the node voltages.

1. Determine the no. of nodes.2. Select a reference node (Ground),

and label all other nodes. 3. Apply KCL at each node except the

reference node.4. Solve the resulting equations.

203

522

0

45

:1

21

121

121

321

vv

vvv

vvv

iii

nodeFor

Solution:

6053

260120336

0510

4

:2

21

221

221

5142

vv

vvv

vvv

iiii

nodeFor

Observation: (3-1) nodes, 2 equations.

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Example:

For the circuit below, obtain v1 and v2.

Observation: 4 meshes, (3-1) nodes

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At node 1,

60 = - 8v1 + 5v2 (1)

At node 2,

36 = - 2v1 + 3v2 (2)

Solving (1) and (2),

v1 = 0 V, v2 = 12 V

2

vv6

5

v

10

v 2111

2

vv63

4

v 212

Solution:

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2.4 Mesh with Current SourcesCase 1: If the current source exists only in one of the mesh, find the

current directly.Case 2: If the current source exists between two or more meshes, it

forms a supermesh. Exclude the current source (replace with an open circuit) and the elements that connected in series with it. Apply KVL in the supermesh and KCL in the nodes of the excluded

elements.

Consider the following case 1

Applying Mesh analysis in usual wayLoop 1:

06410 211 iii

Loop 2:

A 52 i A 21 i

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Supermesh is formed when two meshes have a (dependent or independent) current source in common.

Consider the following case 2

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Apply KVL on the supermesh loop figure (b):

20146

0410620

21

221

ii

iii

Applying KCL:612 ii

Solving Equation:

A 8.2

A 2.3

2

1

i

i

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Consider the following circuitFind the nodal voltages forthe given circuit.

2.5 Nodal with Voltage SourcesCase 1: If the voltage source is connected to a reference node, set the

voltage equal to the voltage source.Case 2: If the voltage source is connected between two non-reference

nodes, it forms a supernode. Exclude the voltage source and replace with a short circuit. Apply KCL in the supernode and KVL to determine the nodes voltages.

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• Define the node voltage• Replace voltage source with short circuit • Apply KCL to the new network

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244-6

:SupernodetheFor

21

2121

232131

VV

IIII

IIIIII

SS

SS

2112

:circuit original theRefer to

VV

12

250250

:equations Resulting

21

21

V V

V.V.

*challenge: use source conversion or mesh analysis

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Summary: Mesh-Nodal Analysis

(i) Mesh• To find mesh currents • Suitable to be used with

circuit with many series-connected elements, voltage sources & Supermesh

Supermesh• Replace current sources

with an open circuits

(ii) Nodal• To find node voltages • Suitable to be used with

circuit with many parallel-connected elements, current sources & Supernode

Supernode• Replace voltage sources

with a short circuits

Mesh Analysis vs. Nodal Analysis

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2.6 Bridge Network

• Belong to the family of complex networks because the elements are neither in series nor in parallel.

• Bridge configuration may be analyzed by using either mesh or nodal analysis.

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052152

054254

2024243

213

312

321

III

III

III

0852

05114

20249

321

321

321

III

III

III

A 667.2

A 667.2

A 4

3

2

1

I

I

I

Note that:A 0325 III R

A bridge network is said tobe balanced, if the current or voltage through the bridgearm is 0A or 0V.