Presented by Byoung-Kuk Lee, Ph. D., Senior IEEE

Energy Mechatronics Lab. College of Information and Communication Eng.

Sungkyunkwan University

Tel: +82-31-299-4581 Fax: +82-31-299-4612 http://seml.skku.ac.kr EML: bkleeskku@skku.edu

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Technique of Circuit Analysis (Part II)

Circuit Analysis by Mesh-Current Method

1 1 3

2 3 2

( )

( )

a a b

b a b

v i R i i R

v i i R i R

Applying KVL around

two meshes leads to

1 1 3 3

2 3 2 3

( )

( )

a b

a b

v i R R i R

v i R i R R

• Collecting the coefficients

of ia and ib gives

1 2 3, ,

a b a bi i i i i i i

• Relation between mesh

and branch currents

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Technique of Circuit Analysis (Part II)

Mesh-Current Method

Ex. 4.4

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Technique of Circuit Analysis (Part II)

Simultaneous Equations in Matrix Form

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2 2

n n

n n

n n n n n

a x a x a x b

a x a x a x b

a x a x a x b

11 12 1 1 1

21 22 2 2 2

1 2

n

n

n n nn n n

a a a x b

a a a x b

a a a x b

BAX

11 12 1

21 22 2

1 2

n

n

n n nn

a a a

a a aA

a a a

1

2

n

x

xX

x

1

2

n

b

bB

b

The matrix equation can be put in a compact form as

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Technique of Circuit Analysis (Part II)

Solving Simultaneous Equations by Cramer's Rule

11 12 1

21 22 2

1 2

n

n

n n nn

a a a

a a a

a a a

1 12 1

2 22 2

1

2

n

n

n n nn

b a a

b a a

b a a

11 1 1

21 2 2

2

1

n

n

n n nn

a b a

a b a

a b a

11 12

21 22

1 2

n

n

n

n n n

a a b

a a b

a a b

1 2

1 2, , , n

nx x x

11 12 1 1 1

21 22 2 2 2

1 2

n

n

n n nn n n

a a a x b

a a a x b

a a a x b

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Technique of Circuit Analysis (Part II)

Determinants of 2x2 and 3x3 Matrices

11 12

11 22 12 21

21 22

a aa a a a

a a

11 12 13

22 23 12 13 12 13

21 22 23 11 21 31

32 33 32 33 22 23

31 32 33

a a aa a a a a a

a a a a a aa a a a a a

a a a

11 12 13

21 22 23

11 22 33 21 32 13 31 12 23

31 32 33

13 22 31 23 32 11 33 12 21

11 12 13

21 22 23

a a a

a a aa a a a a a a a a

a a aa a a a a a a a a

a a a

a a a

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Technique of Circuit Analysis (Part II)

Examples

1 2

1 2

4 3 17

3 5 21

x x

x x

1 2 3

1 2 3

1 2 3

25 5 20 50

5 10 4 0

5 4 9 0

x x x

x x x

x x x

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Technique of Circuit Analysis (Part II)

Solving Set of Liner Equations by MATLAB

1 2 3

1 2 3

1 2 3

25 5 20 50

5 10 4 0

5 4 9 0

x x x

x x x

x x x

A [25 -5 -20; -5 10 -4; -5 -4 9;]

B [50; 0; 0;]

X inv(A)*B

AX B

MATLAB (MATrix LABoratory): a special-purpose computer

program optimized to perform engineering and scientific

calculations

Originally designed to perform matrix mathematics

A very extensive library of predefined functions to make technical

programming tasks easier and more efficient

Matlab command Linear equations

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Technique of Circuit Analysis (Part II)

4.6 Mesh-Current Method & Dependent Sources

Three mesh-current equations are

• Branch current (i) controlling the dependent voltage source

Ex. 4.5

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Technique of Circuit Analysis (Part II)

4.7 Concept of Supermesh

100 3 6 0a b ai i v i

2 50 4 0c a c

v i i i

100 3 2 50 4 6 0a b c a c ai i i i i i

at mesh a

at mesh c

at supermesh a,c

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Technique of Circuit Analysis (Part II)

4.7 Concept of Supermesh

The supermesh eliminates the need for introducing the

unknown voltage across the independent current source

100 3 2 50 4 6 0a b c b c ai i i i i i

5c ai i

0 3 10 2b a b b ci i i i i

At supermesh a,c

Constraint Eq. at the current source

At mesh b

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Technique of Circuit Analysis (Part II)

Mesh-Current Analysis of Amplifier Circuit

super

mesh

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Technique of Circuit Analysis (Part II)

Homework Assignments

• Problems: 4.22, 4.26, 4.45, 4.50

• Due Date: next week class

• Where: 21502