The Mesh Analysis Theorem provides a procedure for electric circuit analysis using mesh current as of the circuit variable. The mesh analysis makes use of Kirchhoff’s Voltage Law is a basic key to analyze the circuit. In contrast to Nodal analysis, it uses loop current as a variable rather than element current, so it reduces the number of equations and complexity.

Mesh is a loop that does not contain any other loop. Referring to the figure below; ABEF and BCDE both are mesh but ACDF is not a mesh because it does contain another loop inside it.

## Mesh Analysis Method :

- Assign a name to each mesh current like i
_{1}, i_{2,}and i_{3}… - Apply KVL to each mesh and use ohm’s law to express the voltage drop in each circuit element.
- The second step will give you n number of simultaneous equations, where n is a number of meshes.
- Use any method to solve these simultaneous equations for n mesh current.

## Mesh Analysis Example:

Suppose we know the following parameter of the given circuit.

V_{1}= 12v, V_{2}= 8v

R_{1}= 5Ω, R_{2}= 6Ω

R_{3}= 10Ω

### Solution:

#### Assign Mesh Currents:

First, I start by:

- Assigning the name of I
_{1}to the current in mesh ABEF in the direction specified in the following diagram. - And I
_{2}to the current in mesh BCDE in the direction specified in the following diagram.

#### Write Mesh Equations :

In step 2: I start by writing an equation for each mesh using KVL

For mesh 1:

$V_{1}= R_{1}I_{1}+ R_{3} (I_{1}- I_{2}) – (1) $

For mesh 2:

$V_{2}= R_{2}I_{2}+ R_{3} (I_{2}-I_{1}) – (2)$

Note firstly that R_{3} is common in both meshes and is counted twice (for each equation). Secondly, in resistance R_{3}, the current I_{1} is flowing downward and current I_{2} is flowing upward (opposite of each other). For the mesh 1 equation, I followed the current I_{1} direction and subtracted current I_{2} from current I_{1}. For the mesh 2 equation, I followed the current I_{2} direction and subtracted current I_{1} from current I_{2}.

Note:

Remember a small mistake at this level will lead to a huge problem, so before proceeding to the next step make sure your equations are correct.

### Solve Equations :

We had two mesh, that’s why we got two simultaneous equations. If you have written equations correctly, you are done. At this stage, you can use various software to calculate these mesh currents for you. Here I am converting these simultaneous equations to the matrix and find a reduced echelon form of it.

$ (R_{1}+ R_{3})I_{1}- R_{3} I_{2}=V_{1}$

$-R_{3}I_{1}+(R_{2}+R_{3}) I_{2}=V_{2}$

Putting the appropriate values will get us:

$(5+ 10)I_{1}- 10 I_{2}=12$

$-10I_{1}+(6+10) I_{2}=8$

Converting to the augmented matrix for further operation:

$\lbrack \begin{matrix}

15 & -10 & \\

-10 & 16 & \\

\end{matrix}

\rbrack \lbrack \begin{matrix}

I_{1} & \\

I_{2} & \\

\end{matrix}

\rbrack = \lbrack \begin{matrix}

12 & \\

8 & \\

\end{matrix}

\rbrack $

Dividing Row 1 by 3:

$\lbrack \begin{matrix}

5 & -\frac{10}{3} & \\

-10 & 16 & \\

\end{matrix}

\rbrack \lbrack \begin{matrix}

I_{1} & \\

I_{2} & \\

\end{matrix}

\rbrack = \lbrack \begin{matrix}

4 & \\

8 & \\

\end{matrix}

\rbrack $

R2 + 2R1:

$\lbrack \begin{matrix}

5 & -\frac{10}{3} & \\

0 & \frac{68}{3} & \\

\end{matrix}

\rbrack \lbrack \begin{matrix}

I_{1} & \\

I_{2} & \\

\end{matrix}

\rbrack = \lbrack \begin{matrix}

4 & \\

16 & \\

\end{matrix}

\rbrack $

This means that:

$\frac{68}{3}I_{2}=16 $

OR

$I_{2}=\frac{48}{68}=0.7 A$

Now using the value of I$_{2}$:

$5I_{1}-\frac{10}{3}I_{2}=4$

OR

$5I_{1}-\frac{10}{3} (0.7)=4$

OR

$5I_{1}=4+\frac{7}{3}=\frac{19}{3}$

OR

$I_{1}=\frac{19}{15}=1.26 A$

Finally, we got both values of I_{1} and I_{2}. Now, we can know the current of any resistor in the circuit, and use Ohm’s law we can find the voltage drop across any resistor. Suppose I want to know the current in R_{3}, and I take the direction of I_{2} as a reference means downward.

$I_{2}-I_{1}=0.7-1.26= -0.56A Downward$

If I take the direction of I$_{1}$ as a reference, means upward, so the equation will become:

$I_{1}-I_{2}=1.26 -0.7= 0.56A upward$

In both cases, the expression means that the current is flowing upward with the magnitude of 0.56 A.

## Mesh Analysis with Current Source:

Mesh analysis theorem uses KVL, in the current source we don’t know the voltage of the source rather the current. There are two possible cases for current sources.

#### Case 1 :

When the current source and voltage source are in separate mesh, as shown in the figure. Such type of circuit is less complex than voltage source circuits because it reduces the number of equations. Suppose if the current source is 5A.

#### Case 2 :

When the voltage source and current source both are in the same mesh, as shown in the figure. Write the KCL equation for the node near the current source and replace the current source with an open circuit that leads to super mesh. Now write the mesh equation.

## Mesh Analysis Theorem Limitations:

- Mesh analysis theorem is limited to planner circuits
- Where planner circuits mean that no branch crosses another branch if the circuit is drawn over a plan.

Is there an algorithm to select R-L-C network tree such that R, L mesh matrices are non-singular? Or an algorithm to transform any given tree such same condition is met?