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ELECTRICAL AND ELECTRONIC ELECTRICAL AND ELECTRONIC TECHNOLOGYTECHNOLOGY

((BEX17003BEX17003) )

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Chapter 3(b): Direct Current Circuit

Analysis (II)

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Lecture Contents

Nodal Analysis Nodal Analysis

Mesh/Loop AnalysisMesh/Loop Analysis

Methods of AnalysisMethods of Analysis

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Nodal and Mesh/Loop

Analysis

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Nodal Analysis Nodal analysis – based on the systematic

application of KCL. [important technique – pay attention]

We can analyze any linear circuit by:obtaining a set of simultaneous equationssolved to obtain the required values (voltage or current)we can solve the simultaneous equation either using Cramer’s Rule or any other software such as MATLAB or MathCAD.

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Cont… Nodal analysis provides a general procedure

for analyzing circuit using node voltages as a circuit variables.

Important key idea resistance is a passive element, by the passive sign convention, current must always flow from a higher potential to a lower potential

Rvv lowerhigheri …..( Eq. 3.21)

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Example 3.12

I1

I2

R2

R1 R3v1v2

Consider the figure below using nodal analysis.

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Solution 3.12 1. Label the nodes (including the reference node)

I1

R2

R1R3v1

v2

v = 0

I2

+ +

- -

1 2

i1

i2

i3

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Cont…

Rvlowervhigheri

I1R2

R1 R3

v1 v2

v = 0

I21 2

i1

i2

i3

1

11

0Rvi

2

212 R

vvi

3

23

0Rvi

# Current flows from a higher potential to a lower potential in a resistor

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2. Write the KCL equation for node 1 and 2.

KCL at node 1.I1 = I2 + i1 +i2

KCL at node 2.I2 +i2 = i3

2

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1

121 R

vvRvII

3

2

2

212 R

vR

vvI

R2

R1 R3v1v2

v = 0

I2

+ +

- -

1

i1

i2

i3

2

I1

I2

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Example 3.13 Find the node voltages for the following circuit.

10A

5 A

R1=2 Ω R3=6 Ωv1

v2

1 2R2=4 Ω

v=0V

v3

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Solution 3.13

20v3vvv2v20:(x4)

4vv

20v5

ii5

21

211

211

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KCL at node 1.

KCL at node 2.

v1

605v3v2v601203v3v:(x12)

60v5104

vvi510i

21

221

221

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V1

5 A

10A

R1=2 Ω R3=6 Ωv2

V2R2=4 Ω

5A

i2

i1

10A

i3

5A

v=0V

v3(1)

(2)

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Using the elimination technique

Substituting

20Vv804v :(2) (1)

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20v2

13.33V340v20203v 11

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To use Cramer’s rule, we put the equation in matrix form

Vv

Vv

baba

abab

aaaa

vv

vvvv

2012240

33.1312

160

24060180)20)(3(603603203

16060100)1)(60(520560120

12315)1)(3(535313

6020

5313

6053203

22

11

221

1112

222

2111

2221

1211

2

1

21

21

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Consider the following circuit with current and voltage source. Look for Vb.

[2].................R

0-VbIR

VbVaIII

b, Node[1]................VVa

a, Node

3B

2

3B2

B

∑I entering the node = ∑I leaving the node

I1

I2

I3

IB

VB IB

2R1

R3R

a b

c

Va Vb

Vc=0

Example 3.14

1632

2

BB

b

2

BB

32b

2

BB

3

b

2

b

B3

b

2

b

2

B

B3

b

2

b

2

B

3

bB

2

bB

R1

R1

RVI

V

RVI

R1

R1V

RVI

RV

RV

IRV

RV

RV-

IRV

RV

RV

RVI

RVV

[2] into [1] substitute

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Example 3.15Determine the voltages for the nodes in the following figure.

4

2 8

43A 2ix

1 2 3

ix

0

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Solution 3.15

-(3)------------- 032:)3/8(2

)(284

v-v 2

3, nodeAt -(2)--------------- 0v-7v-4v:8)(

40-v

8v-v

2v-v

2, nodeAt (1)------------- 1223:)4(

243 3

1, nodeAt

321

21323121

321

2322132

321

21311

vvv

vvvviii

iii

vvv

vvvvii

x

x

x

4

843A

2ixV1 V2 V3

1

i1

I=3A

ix 2 i2

i3

3i1

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(5)-------2vv 04v2v-

(3) and Eq.(2) Adding(4)2.45

12v-v

or125v-5v

2121

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:Eq(3) and (1) Eq Addingtechnique, neliminatio the Using

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-2.4Vv 2.4V,v 4.8V, vThus,

-2.4V-v4v-3v2v-3vvget, we (3) Eq. From

4.8V2vv 2.4,v 2.4v-2vyield (4) into Eq.(5) ngSubstituti

321

222123

21222

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Nodal Analysis with Voltage Sources

Important Supernode – formed by enclosing a

(dependent or independent) voltage source connected between two non-reference nodes and any elements connected in parallel with it.

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Mesh/Loop Analysis

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Mesh/Loop Analysis Mesh analysis provides another general

procedure for analyzing circuits. Recall that a loop is a closed path with

no node passed more than once. A mesh is a loop that does not contain

any other loop within it. Mesh analysis apply KVL to find

unknown currents

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Cont… Mesh analysis is not quite as general as

nodal analysis, because it is only applicable for a circuit that is planar.

A planar circuit is one that can be drawn in a plane with no branches crossing one another; otherwise it is nonplanar.

For better understanding, refer to Sadiku and Alexander, 2nd Edition page 90-91.

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Example 3.17

Write the mesh equation for the following circuit.

1VB

2R1R

5R 2VB4R

3R

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Solution 3.17

)1(V)(-RI)R(RI

0)RI-(IRIV-0VVV-

: 1 loopfor KVL0 clockwise V

loop, allFor

B142411

42111B1

R4R1B1

1VB

R31R

R42VB

R2

R5

1I

2I 3I

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1VB

R31R

R42VB

R2

R5

1I

2I 3I

-(2)---------0)(-RI)RR(RI)(-RI0)RI-(IRI)RI-(I

0VVV: 2 loopfor KVL

53542241

53222412

R5R2R4

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1VB

R31

R

R42

VB

R2

R5

1I

2I 3

I

)3(V)R(RI)(-RIVRI)RI-(I

0VVV: 3 loopfor KVL

B253352

B233523

B2R3R5

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2

1

3

2

1

535

55424

441

B253352

53542241

B142411

00

0matrix,In

)3(V)R(RI)(-RI-(2)---------0)(-RI)RR(RI)(-RI

)1(V)(-RI)R(RI

B

B

V

V

III

RRRRRRRR

RRR

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Example 3.18Find Vo for the following circuit.

4A

2 6 5V

873A Vo

313.04V(8)IV I

8 (21)I 0 8I4)-6(I5-3)7(I08I)I-6(I5-)I-7(I

0 VV5-V 0 clockwise V

:3 Loop forKVL 3AI

:Loop2 For4A I

:Loop1 For

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3

3

3 33

31323

867

2

1

A38.0

4A

2 6 5V

873A Vo

I1

I2 I3

Solution 3.18

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Mesh Analysis with Current SourcesCase 1: When a current source exists only in one mesh

Loop 2: I2= -5A

10V

4Ω 3Ω

6ΩI1 I2 5AI1= -2A

-10+4i1+6(i1-i2) = 0

Loop 1:

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Nodal vs. Mesh AnalysisHow do we know which method is

better or more efficient?

Nodal analysis is normally used when a circuit has fewer node equations than mesh equations.

Mesh analysis is normally used when a circuit has fewer mesh equations than node equations.

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Nodal vs. Mesh Analysis(Nature of network)

Nodal AnalysisNetworks that contain many :series-connected elementsvoltage sourcessupernodes

Mesh AnalysisNetworks that contain

many : parallel-connected elements current sources supermeshes

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Nodal vs. Mesh Analysis(Information required)

If node voltages are required, it maybe expedient to apply nodal analysis

If branch or mesh currents are required, it maybe expedient to apply mesh analysis