1. Basic Laws of Electric Circuits
Mesh Analysis
Lesson 7

2. Basic Circuits Mesh Analysis: Basic Concepts: 
In formulating mesh analysis we assign a mesh
current to each mesh. 
Mesh currents are sort of fictitious in that a particular
mesh current does not define the current in each branch
of the mesh to which it is assigned.

3. Basic Circuits Mesh Analysis: Basic Concepts: Eq 7.1
Figure 7.2: A circuit for illustrating mesh analysis.
Around mesh 1:
Eq 7.1

4. Basic Circuits
Mesh Analysis: Basic Concepts:
Eq 7.2
Eq 7.3
Eq 7.4

5. Basic Circuits Mesh Analysis: Basic Concepts:
We are left with 2 equations: From (7.1) and (7.4)
we have,
Eq 7.5
Eq 7.6
We can easily solve these equations for I1 and I2.

6. Basic Circuits Mesh Analysis: Basic Concepts:
The previous equations can be written in matrix form as:
Eq (7.7)
Eq (7.8)

7. Basic Circuits Mesh Analysis: Example 7.1. Eq (7.9)
Write the mesh equations and solve for the currents I1, and I2.
Figure 7.2: Circuit for Example 7.1.
Eq (7.9)
Mesh 1
4I1 + 6(I1 – I2) =
Eq (7.10)
Mesh 2
6(I2 – I1) + 2I2 + 7I2 =

8. Basic Circuits Mesh Analysis: Example 7.1, continued.
Simplifying Eq (7.9) and (7.10) gives,
10I1 – 6I2 = 8
-6I I2 = 22
Eq (7.11)
Eq (7.12)
» % A MATLAB Solution
» R = [10 -6;-6 15];
» V = [8;22];
» I = inv(R)*V
I =
2.2105
2.3509
I1 =
I2 =

9. Basic Circuits Mesh Analysis: Example 7.2
Solve for the mesh currents in the circuit below.
Figure 7.3: Circuit for Example 7.2.
The plan:
Write KVL, clockwise, for each mesh. Look for a
pattern in the final equations.

10. Basic Circuits Mesh Analysis: Example 7.2
Mesh 1: I1 + 10(I1 – I3) + 4(I1 – I2) =
Eq (7.13)
Mesh 2: (I2 – I1) + 11(I2 – I3) + 3I2 =
Eq (7.14)
Mesh 3: I3 + 11(I3 – I2) + 10(I3 – I1) =
Eq (7.15)

11. Basic Circuits Mesh Analysis: Example 7.2 WE NOW MAKE AN IMPORTANT
Clearing Equations (7.13), (7.14) and (7.15) gives,
Standard Equation form
In matrix form:
20I1 – 4I2 – 10I3 = 30
-4I1 + 18I2 – 11I3 = -18
-10I1 – 11I2 + 30I3 = 20
WE NOW MAKE AN IMPORTANT
OBSERVATION!!

12. Basic Circuits Mesh Analysis: Standard form for mesh equations R11 =
Consider the following:
R11 =
of resistance around mesh 1, common to mesh 1 current I1.
R22 =
of resistance around mesh 2, common to mesh 2 current I2.
R33 =
of resistance around mesh 3, common to mesh 3 current I3.

13. Basic Circuits Mesh Analysis: Standard form for mesh equations
R12 = R21 = - resistance common between mesh 1 and 2
when I1 and I2 are opposite through R1,R2.
R13 = R31 = - resistance common between mesh 1 and 3
when I1 and I3 are opposite through R1,R3.
R23 = R32 = - resistance common between mesh 2 and 3
when I2 and I3 are opposite through R2,R3.
= sum of emf around mesh 1 in the direction of I1.
= sum of emf around mesh 2 in the direction of I2.
= sum of emf around mesh 3 in the direction of I3.

14. Basic Circuits Mesh Analysis: Example 7.3 - Direct method.
Use the direct method to write the mesh equations for the following.
Figure 7.4: Circuit diagram for Example 7.3.
Eq (7.13)

15. Basic Circuits Mesh Analysis: With current sources in the circuit
Example 7.4: Consider the following:
Figure 7.5: Circuit diagram for Example 7.4.
Use the direct method to write the mesh equations.

16. Basic Circuits Mesh Analysis: With current sources in the circuit
This case is explained by using an example.
Example 7.4: Find the three mesh currents in the circuit below.
Figure 7.5: Circuit for Example 7.4.
When a current source is present, it will be directly related to
one or more of the mesh current. In this case I2 = -4A.

17. Basic Circuits Mesh Analysis: With current sources in the circuit
Example 7.4: Continued. An easy way to handle this case is to
remove the current source as shown below. Next, write the mesh
equations for the remaining meshes.
Note that I 2 is retained for writing the equations through the
5  and 20  resistors.

18. Basic Circuits Mesh Analysis: With current sources in the circuit
Example 7.4: Continued.
Equation for mesh 1:
10I1 + (I1-I2)5 = 10 or
15I1 – 5I2 = 10
Equations for mesh 2:
Constraint Equation
2I3 + (I3-I2)20 = 20
I2 = - 4A or
- 20I2 + 22I3 = 20

19. Basic Circuits Mesh Analysis: With current sources in the circuit
Example 7.4: Continued. Express the previous equations in
Matrix form:
I1 = A
I2 = - 4 A
I3 = A

20. circuits
End of Lesson 7
Mesh Analysis