# Maxwell’s Mesh Current Method

### Network Theorems and Techniques

Many times we encounter some complex electrical circuits that can not be solved using Ohm’s law and Kirchhoff’s law.

To overcome this difficulty, other network theorems and techniques have been developed which are very useful. By using these theorems and techniques, it is possible either to simplify the network or render the analytical solution easy. The network theorems and techniques basically depend upon the type of network arrangement.

#### Maxwell’s Mesh Current Method

The mesh current method uses simultaneous equations, Kirchhoff’s voltage law (KVL), and Ohm’s law to determine unknown currents in a network.

- In this method, Kirchhoff’s voltage law is applied to a network to write mesh equation in terms of mesh currents instead of branch currents.
- Each mesh is assigned a separate mesh current.
- This mesh current is assumed to flow clockwise around the perimeter of the mesh without splitting at a junction into branch currents.
- Kirchhoff’s voltage law is then applied to write equations in terms of unknown mesh currents.
- The branch currents are then found by taking the algebraic sum of the mesh currents which are common to that branch.

##### Explanation

Maxwell’s mesh current method consists of following steps :

##### Step 1: Assign Mesh Current

Each mesh is assigned a separate mesh current. For convenience, all mesh currents are assumed to flow in clockwise direction.

For example, in Fig.1, mesh ABDA has assigned mesh current I_{1} and mesh BCDB has been assigned mesh currents I_{2}.

Fig.1

**Step 2 :**

If two mesh currents are flowing through a circuit element, the actual current in the circuit element is equal to the algebraic sum of the two.

Thus, in Fig.1, there are two mesh currents I_{1} and I_{2} flowing in R_{2}.

If we go from B to D, current is I_{1}– I_{2} and if we go in the other direction (i.e. from D to B ), current is I_{2}– I_{1} .

**Step 3 : Apply Kirchhoff’s Voltage Law to Write Mesh Equation**

Now, apply Kirchhoff’s voltage law to write equation for each mesh in terms of mesh currents.

Note that while writing mesh equation, rise in potential is assigned positive sign and fall in potential is assigned negative sign.

Now let’s apply Kirchhoff’s voltage law to Fig.1,

Thus, we have,

**Mesh ABDA :**

-I_{1} R_{1}– ( I_{1} – I_{2} ) R_{2} + E_{1} = 0

Or, I_{1 }( R_{1} + R_{2}) – I_{2} R_{2 }= E_{1 }………………… Eq.(1)

**Mesh BCDB :**

-I_{2 }R_{3} – E_{2} – ( I_{2} – I_{1} ) R_{2} = 0

Or, – I_{1} R_{2} + (R_{2} +R_{3} ) I_{2} = -E_{2} ……………. Eq.(2)

Solving Eq.(1) and Eq.(2) simultaneously, mesh currents I1 and I2 can be found out. Once the mesh currents are known, the branch currents can be readily obtained.

##### Step 4 :

If the value of any mesh current comes out to be negative in the solution, it means that true direction of that mesh current is anticlockwise i.e., opposite to the assumed clockwise direction.

The advantage of this method is that it usually reduces the number of equations to solve a network problem.

Example : Consider the circuit shown in Fig.2 below.

Fig.2

Here, let’s consider 2 meshes ABDA and BCDB with mesh currents I_{1 }and I_{2 }respectively.

By applying KVL in mesh 1 (ABDA), we get,

– 20 I_{1 } – 80 ( I_{1} – I_{2}) + 20 = 0

Or I_{1 }( 20 + 80 ) – 80 I_{2 }= 20

Or 100 I_{1 }– 80 I_{2 }= 20 ………………. Eq.(1)

By applying KVL in mesh 2 (BCDB), we get,

-40 I_{2 } – 40 – ( I_{2} – I_{1} ) 80 = 0

Or, – 80 I_{1} + (40 + 80 ) I_{2} = -40

Or, – 80 I_{1} + 120 I_{2} = -40 ……………..Eq. (2)

Solving Eq.(1) and Eq.(2), we get

I_{1 }= – 0.142 A

I_{2 }= -0.429 A