• Meshes and Loops

• Steps of Mesh Analysis

• Supermesh

• Examples

Lecture 6. Mesh Analysis

1

2

Circuit Analysis – A Systematic Approach

• Mesh Analysis is another general method that is almost as powerful as Nodal Analysis.

- Nodal analysis was developed by applying KCL at each non-reference node.

- Loop analysis is developed by applying KVL around loops in the circuit.

- Loop analysis results in a system of linear equations which must be solved for unknown currents.

3

Learning by Examples: A Summing Circuit

• The output voltage V of this circuit is proportional to the sum of the two input voltages V1 and V2.

• This circuit could be useful in audio applications or in instrumentation.

• The output of this circuit would probably be connected to an amplifier.

+

-

Vout

1k

1k

1k

V1

+

-V2

+

-

Vout = (V1 + V2)/3

4

Mesh Analysis: The Recipe

1. Identify mesh (loops).

2. Assign a current to each mesh.

3. Apply KVL around each loop to get an equation in terms of the loop currents.

4. Solve the resulting system of linear equations.

5

Mesh 2

1k

1k

1k

Step 1: Identifying the Meshes

V1

+

-V2

+

-Mesh 1

6

Mesh Analysis: The Recipe

1. Identify mesh (loops).

2. Assign a current to each mesh.

3. Apply KVL around each mesh to get an equation in terms of the mesh currents.

4. Solve the resulting system of linear equations.

7

1k

1k

1k

Step 2: Assigning Mesh Currents

V1

+

-V2

+

-I1 I2

8

Mesh Analysis: The Recipe

1. Identify mesh (loops).

2. Assign a current to each mesh.

3. Apply KVL around each mesh to get an equation in terms of the mesh currents.

4. Solve the resulting system of linear equations.

9

Step 3: Voltages from Mesh Currents

R

I1

+ -VR

VR = I1 R

R

I1

+ -VRI2

VR = (I1 - I2 ) R

10

1k

1k

1k

V1

+

-V2

+

-I1 I2

KVL Around Mesh 1

-V1 + I1 1k + (I1 - I2) 1k = 0

I1 1k + (I1 - I2) 1k = V1

11

1k

1k

1k

V1

+

-V2

+

-I1 I2

KVL Around Mesh 2

(I2 - I1) 1k + I2 1k + V2 = 0

(I2 - I1) 1k + I2 1k = -V2

12

Mesh Analysis: The Recipe

1. Identify mesh (loops).

2. Assign a current to each mesh.

3. Apply KVL around each loop to get an equation in terms of the loop currents.

4. Solve the resulting system of linear equations.

13

Step 4: Solve the Equations

• The two equations can be combined into a single matrix/vector equation.

2

1

2

1

k1k1k1

k1k1k1

V

V

I

I

I1 + (I1 - I2) 1k = V1

(I2 - I1) 1k + I2 1k = -V2

• Re-organize the equations:

- 1kI1 + (1k + 1k ) I2 = -V2

(1k + 1k )I1 - 1k I2 = V1

14

Using MATLAB

>> A = [1e3+1e3 -1e3;

-1e3 1e3+1e3];

>> v = [7; -4];

>> i = inv(A)*v

i =

0.00333

-0.00033 I1 = 3.33mAI2 = -0.33mA

Vout = (I1 - I2) 1k = 3.66V

15

Another Example

1k

2k

2k

12V+

-4mA

2mA

I0

16

Mesh 2

Mesh 3

Mesh 1

Identify Mesh’s

1k

2k

2k

12V+

-4mA

2mA

I0

17

Assign Mesh Currents

1k

2k

2k

12V+

-4mA

2mA

I0

I1 I2

I3

18

How to Deal with Current Sources

• The current sources in this circuit will have whatever voltage is necessary to make the current correct.• We can’t use KVL around the loop because we don’t know the voltage. What to do?

• The 4mA current source sets I2:

I2 = -4mA

• The 2mA current source sets a constraint on I1 and I3:

I1 - I3 = 2mA

• We have two equations and three unknowns. Where is the third equation?

19

1k

2k

2k

12V+

-4mA

2mA

I0

I1 I2

I3

The Supermesh surrounds this source!

The Supermesh

does not include this

source!

SuperMesh

20

KVL Around the Supermesh

-12V + I3 2k + (I3 - I2)1k + (I1 - I2)2k = 0

I3 2k + (I3 - I2)1k + (I1 - I2)2k = 12V

21

Solve the Equations

• The three equations can be combined into a single matrix/vector equation.

V12

mA2

mA4

1k2k2k1k2k

101

010

3

2

1

I

I

I

22

Solve Using MATLAB

>> A = [0 1 0; 1 0 -1;

2e3 -1e3-2e3 2e3+1e3];

>> v = [-4e-3; 2e-3; 12];

>> i = inv(A)*v

i =0.0012

-0.0040

-0.0008I1 = 1.2mAI2 = -4mAI3 = -0.8mA

I0 = I1 - I2 = 5.2mA