The article introduces the Mesh Current Analysis method, a systematic technique for calculating loop currents in electrical circuits using Kirchhoff’s Voltage Law (KVL). It outlines the procedure for assigning loop currents, applying KVL, solving simultaneous equations, and emphasizes its applicability to both AC and DC resistive circuits.
What is a Mesh Current Analysis Method
Mesh Current Analysis is a method used in electrical circuit analysis to calculate the currents flowing in the loops (or “meshes”) of a circuit. It is particularly useful for planar circuits (circuits that can be drawn on a plane without any crossing wires). This technique is based on Kirchhoff’s Voltage Law (KVL), which states that the sum of voltage drops around any closed loop in a circuit is zero.
Key Concepts of Mesh Current Analysis
Mesh Definition. A “mesh” is a loop in the circuit that does not contain any other loops within it.
Assumptions. Mesh currents are hypothetical currents assumed to flow around the meshes. Each mesh is assigned a current, typically in the clockwise direction for simplicity.
Application of KVL. For each mesh, KVL is applied to write an equation where the sum of all voltage drops (including those across resistors, sources, and other elements) is set to zero.
System of Equations. The equations for all meshes are solved simultaneously (using algebra or matrix methods) to determine the mesh currents.
The circuit shown in the following figure has two loops:
We can assign currents to represent each loop, such as IA to represent loop 1, IB to represent loop 2. These currents are called loop currents. Loop currents provide us with a mathematical solution for the branch currents and thus the voltage drop in a network.
Steps to Perform Mesh Current Analysis
A general procedure to analyze network by Mesh or loop analysis method is as follows:
1. Assign a loop current for each independent closed path. The number of independent loops in a network is given by the expression.
$T-N+1$
Where T is the total number of elements in the network (both active and passive) and N the number of nodes in the network. Although the direction of loop currents is arbitrary. We shall only assign clockwise direction for the sake of uniformity.
2. Indicate the polarity markings across each passive element due to the direction of loop current in that closed path. The polarities of voltage sources are determined by their own positive and negative terminals.
3. Apply Kirchhoff’s voltage law (KVL) to each closed path. If we are applying KVL to one loop and a passive element has two or more loop currents flowing through it, the total voltage drop equals the sum of the voltage drops due to each loop current separately. If the loop currents are in the same directions, the voltage drops add. However, if the loop currents flow in the opposite directions, the voltage drops subtract from one another.
4. Solve the set of simultaneous linear equations for the loop currents.
Mesh Current Analysis Example
Find branch currents I1, I2, and I3 using mesh current method.
Loop 1 Equation
$-15+5{{i}_{1}}+10+10\left( {{i}_{1}}-{{i}_{2}} \right)=0$
$-2{{i}_{2}}+3{{i}_{1}}=1~~\text{ }\cdots \text{ }~~\left( 1 \right)$
Loop 2 equation
$-10+4{{i}_{2}}+6{{i}_{2}}+10\left( {{i}_{2}}-{{i}_{1}} \right)=0$
$2{{i}_{2}}-1={{i}_{1}}~~\text{ }\cdots ~~~\left( 2 \right)$
By solving equation (1) and (2), we have
${{i}_{1}}=1A$
${{i}_{2}}=1A$
Then
${{i}_{3}}={{i}_{1}}-{{i}_{2}}$
${{i}_{3}}=0$
Note that in above circuit, the voltage source of 10V is common to both loops, so, it must be included in both loop equations.
It is important to understand that Mesh or Loop analysis can be applied to either AC resistive circuit or DC resistive circuit. In any type of network, the numbers of linear equations are dependent on the number of loop currents.