ee lecture

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Circuit Analysis Techniques and

Network Theorems

A. Mesh Analysis

Mesh

It is a loop which does not contain any other

loop within it.

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In mesh analysis, Kirchhoff·s Voltage Law (KVL) is

used.

Steps:

1. Draw a current loop for each mesh. All

mesh

currents must be in the same or uniform indirection (clockwise or counterclockwise).

2. Formulate mesh equations following the

pattern. For an n-mesh network, we have n-

mesh equations.3. Solve for the unknown mesh current by

applying techniques involved in evaluating

linear equations.

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Equation Pattern:

V1

= R11

I1

- R12

I2

- . . . . . . - R1N

IN

.

V2 = -R21I1 + R22I2 - . . . . . . . - R2NIN .

.

.

VN

= -RN1

I1

- RN2

I2

- . . . . . . + RNN

IN

.

where:

V1 , V2 . . . . Vn = algebraic sum of all voltage

sources in a mesh

R11 , R22 . . .RNN = sum of all resistance in a mesh

R12 , R22 . . .RN = common or mutual resistance

between two meshes

I1 , I2 . . . . . . .IN = mesh currents

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V1

V2

V3

R1 R2

R3

1 2

I1 I2

Mesh 1: V1 - V2 = (R1 + R3) I1 - R3 I2

Mesh 2: V2 - V3 = - R3 I1 + (R2 + R3) I2

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1 2

I1 I2

10V

2 3

4

6V

3V

;

; ;

 4 A 5 ohms

10 V

40 ohms

1 ohm4 ohms

2 ohms

10 A

io

1. Determine mesh currents I1 and I2.

2. Determine io by mesh analysis.

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3. Determine the mesh currents.

2

2

3 4

10V

15V

10A 1

;

;

;

;

;

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B. Nodal Analysis

I3

I1 I2

R3

R1 R2

Node

A point at which two or more elements have a

common connection.

Nodal analysis is a method that uses Kirchhoff·s Current Law

(KCL) to obtain a solution of simultaneous equations that 

when manipulated will provide a means of solving for the

voltages on each node and for every branch.

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Steps:

1. Identify the nodes of the given circuit 

(including the reference node). All nodeshave corresponding voltages (except the

reference node) and must have the same

voltage sign convention where the given

node is positive and the reference node isnegative.

2. Formulate node equations following the

pattern. For an n - node network, we have

n -1 node equations.3. Solve for the unknown mesh current by

applying techniques involved in evaluating

linear equations

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Equation Pattern:

I1 = G11 V1 - G12 V2 - . . . . . . - GN1 VN

I2 = G21 V1 - G22 V2 - . . . . . . - G2N VN

.

.

IN

= -GN1

V1

- GN2

V2

- . . . . . .+ GNN

VN

where:

I1 , I2 , IN = algebraic sum of all current sources

connected to a node.

G11 , G22 , GNN = sum of all conductance connectedto a node.

G12 , G21 , GN = common conductance between two

nodes.

V1 , V2, VN = unknown node voltages

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Node 1 :

I1

- I3

= (G1

+ G3) V

1- G

3V

2----1

Node 2 :

I3

+ I2

= - G3

V1

+ (G3

+ G2) V

2---2

2

3

1

31

31

111V 

 RV 

 R R I  I  ¹¹

 º

 ¸©©ª

¨!

1

32

1

3

23

111V 

 R RV 

 R I  I  ¹¹

 º

 ¸©©ª

¨!

I3

I1 I2

R3

R1 R2

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Sample Problems:

Determine the node voltages.5A

10A 3A

4S

2S 3S

Determine Vx.

10A

5A0.5Vx1;

2;

3;

- Vx +

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Find i using nodal analysis.

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B. Superposition theorem

Find v using superposition theorem.

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Solution: To get v1, we set the current source to zer o.

Applying KVL,

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Find i. Answer: 2A

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i3

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Thevenin Equivalent Circuit

(TEC)

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Obtain the TEC. Determine the current thru R L if R L = 6 ohms, 16 ohms, 36 ohms

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Applying mesh analysis;

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The TEC is shown  below. 

The current thru R L is now calculated :

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TH  N R R !

TH 

TH 

 N 

 R

V  I  !

Norton Equivalent Circuit (NEC)

To get INNOTE:

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