Maxwell’s Loop Current Method or Mesh Analysis

Generally, we call Maxwell’s Loop Current Method as Mesh Analysis. We very widely use this circuit analysis technique. In Maxwell’s Loop Current Method, first, we identify all the meshes of the circuit. We imagine one loop current for every mesh. The direction of the loop currents may be arbitrary. But generally, we consider co – directional loop currents for the shake of our calculations only.

James Clerk Maxwell
James Clerk Maxwell

Step by Step Methods of Mesh Analysis

In Maxwell’s Loop Current Method, we have to construct an equation for each mesh using Kirchhoff’s Voltage Law. For that, we normally follow the direction of loop current of that mesh. The imaginary loop current may enter into the negative terminal of a voltage source and leaves the positive terminal. Then we consider the voltage as gain.

voltage gain

And denote the voltage with a positive sign in the loop equation.

Again, the loop current may enter into the positive terminal of a voltage source and leaves from the negative terminal. In that condition, we consider the voltage as a drop.

Voltage Drop Across a Voltage Source

Hence, we denote it with a negative sign in the equation.

Again the loop current also may pass through a resistance. Then we consider the entry point as the positive and leaving point as the negative.

voltage drop across a resistance

We also denote the voltage drop across the resistance with a negative sign in the equation. Since the current enters through the positive end of the resistance.

Now we come to a common branch of the mesh. We mean the branch is common to two adjacent meshes. Then we have to consider the branch current not the loop current alone. In this case, we take the direction of the branch current as the direction of the own loop current. The magnitude of the current is the difference between the own loop current and adjacent loop current. Considering this direction of the branch current we have to express the voltage gain or drop across the branch.

In this way, we construct a linear equation for each mesh.

By solving these equations we can calculate the circuit parameters.

Example of Maxwell’s Loop Current Method

For further understanding the method let us consider the following circuit.

Maxwell's Loop Current Method
Maxwell’s Loop Current Method

Mesh 1

Here let us first consider the mesh 1. Now also we consider arbitrarily the direction of the loop current I1 in clockwise. Hence, here E1 is positive or gain and I1R1 is negative or drop. (I1 – I2)R4 is also negative or drop. As per Kirchhoff’s Voltage Law, the sum of all three voltages is zero. Hence we can write

Mesh 2

Now we come to the second mesh 2. Again, here we will take the direction of the loop current I2 arbitrarily. For simplicity, we consider the direction as same as that of first loop current. Here the branch current of the common branch between mesh 1 and 2 is (I2 – I1). The voltage drop across the branch is (I2 – I1)R4. The voltage drop across the resistance R2 is R2I2. The current in the branch between mesh 2 and 3 would be (I2 – I3). The voltage drop across the resistance of this common branch would be R5(I2 – I3). Since in this second mesh, all voltages are dropping voltage, all the voltages will have a negative sign in the equation.

Mesh 3

Now we will come to the last or third mesh. Here voltage drop across the common branch would be R5(I3 – I2), the voltage drop across the resistance R3 would be R3I3. Here in the third mesh, the loop current I3 enters through the positive terminal of the voltage source E2. So, we will consider it as a voltage drop too. Now considering the polarity or sing of all voltages of the mesh. Accordingly, we can write the KVL equation of the mesh as.

By solving these equations [(i), (ii) and (iii)] we get the value of each loop current. If the loop current becomes negative, the actual direction of the current is opposite of the direction we have considered. From the loop currents, we can also find out all the branch currents of the circuit.

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