E E 2415

Lecture 02 -Mesh Current Analysis

Introduction to Mesh Current Method

• More direct than branch equations• Fewer equations to solve• Express all variables in terms of mesh

currents• Solution is set of mesh currents• Solution completely defines the circuit• Most Convenient Method to Model

Magnetic Coupling (E E 2446 Topic)

Mesh Current Example 1 (1/2)

KVL at Mesh 1:

KVL at Mesh 2:

Using Ohm’s Law:

Vs1 Vs2

Ra Rb

RcI1I2

+ va - + vb -+vc-

10 s a cV v v

20 c b sv v V

1 1 1 2s a cV R I R I I

2 1 2 2s c bV R I I R I

Mesh Current Example 1 (2/2)

Above linear equations can be solved for mesh currents I1 and I2.

Vs1 Vs2

Ra Rb

RcI1I2

+ va - + vb -+vc-

1 1

2 2

s a c c

s c b c

V R R R I

V R R R I

Mesh Current Example 1a (1/2)

120 V 64 V

8

24 I1 I2

6

KVL at Mesh 1:

KVL at Mesh 2:

1 1 20 120 6 24I I I 2 1 20 64 24 8I I I

1

2

120 30 24

64 24 32

I

I

Solve:1

2

6

2.5

I A

I A

Mesh Current Example 2 (1/2)

KVL @ Mesh 1:

KVL @ Mesh 2:

But:

ix Vs1

Ra Rb

RcI2 I1ix

1 1 2 10 s c bV R I I R I

2 2 10 x a ci R I R I I

2 1xi I I

Mesh Current Example 2 (2/2)

Solve for I1

and I2:

ix Vs1

Ra Rb

RcI2 I1ix

1 1 2s b c cV R R I R I

1 20 c a cR I R R I

Mesh Current Example 2a (1/2)

4ix 120 V

10 8

24 I2 I1ix

KVL @ Mesh 1:

KVL @ Mesh 2:

But:

1 2 10 120 24 8I I I

2 2 10 4 10 24xi I I I

2 1xi I I

Mesh Current Example 2a (2/2)

4ix 120 V

10 8

24 I2 I1ix

1

2

120 32 24

0 20 30

I

I

Solve for I1

and I2:

1 27.5 5I A I A

Forced Mesh (1/2)

• No KVL equation possible for mesh 2• But I2 is known: I2 = Is

Vs Is

Ra Rb

RcI1 I2

Forced Mesh (2/2)

KVL for mesh 1:

Substitute and Solve:

Vs Is

Ra Rb

RcI1 I2

1 1 20 s a cV R I R I I

1s c s

a c

V R II

R R

Forced Mesh Example 3a

108 V 5 A

6 8

20 I1 I2

KVL for mesh 1:

Substitute and Solve:

1 10 108 6 20 5I I

1

108 100

6 20I

1 8I A

Supermesh Example (1/5)

• No KVL possible for meshes 1 or 2• Use Supermesh (dotted loop) for KVL

Vs1 Vs2

Is

Ra Rb

Rc Rd

Re

I1 I2

I3

Supermesh Example (2/5)

Supermesh KVL:

Mesh 3 KVL:

Vs1 Vs2

Is

Ra Rb

Rc Rd

Re

I1 I2

I3

1 1 2

2 2 3

1 3

0

( )

( )

s a b

s d

c

V R I R I

V R I I

R I I

3 1 3 2 30 ( ) ( )c d eR I I R I I R I

Supermesh Example (3/5)

Also:

Vs1 Vs2

Is

Ra Rb

Rc Rd

Re

I1 I2

I3

2 1 2 1s sI I I I I I

1 2 1 1

1 3 1 3

( )

( ) ( )s s a b s

d s c

V V R I R I I

R I I I R I I

Subst for I2:

Supermesh Example (4/5)

And:

Rearranging the equations:

Vs1 Vs2

Is

Ra Rb

Rc Rd

Re

I1 I2

I3

3 1 3 1 30 ( ) ( )c d s eR I I R I I I R I

Supermesh Example (5/5)

Vs1 Vs2

Is

Ra Rb

Rc Rd

Re

I1 I2

I3

1 2 1

3

s s b d s a b c d

c d

V V R R I R R R R I

R R I

1 3d s c d c d eR I R R I R R R I

Supermesh with Numbers (1/3)

200 V 120 V

20 A

4 6

12 8

20

I1 I2

I3

Supermesh with Numbers (2/3)

1

3

40 30 20

160 20 40

IV

IV

1

3

6

7

I A

I A

2 20 6 26I A A A

Supermesh with Numbers (3/3)

200 V 120 V

20 A

4 6

12 8

20

6A 26A

7A

+24V- +156V-

-152V++12V-

-140V+