This blog contain about the Mesh analysis and Mesh current analysis.
Mesh analysis
Figure 1: Essential meshes of the planar circuit labeled 1, 2, and 3. R1, R2, R3, 1/sc, and Ls represent the impedance of the resistors, capacitor, and inductor values in the s-domain. Vs and is are the values of the voltage source and current source, respectively.
Mesh analysis (or the mesh current method) is a method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in the circuit. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other. A more general technique, called loop analysis (with the corresponding network variables calledloop currents) can be applied to any circuit, planar or not. Mesh analysis and loop analysis both make use of Kirchhoff’s voltage law to arrive at a set of equations guaranteed to be solvable if the circuit has a solution. Mesh analysis is usually easier to use when the circuit is planar, compared to loop analysis.
Mesh Current Analysis Circuit
One simple method of reducing the amount of math’s involved is to analyse the circuit using Kirchoff’s Current Law equations to determine the currents, I1 and I2 flowing in the two resistors. Then there is no need to calculate the current I3 as its just the sum of I1 and I2. So Kirchoff’s second voltage law simply becomes:
- Equation No 1 : 10 = 50I1 + 40I2
- Equation No 2 : 20 = 40I1 + 60I2
therefore, one line of math’s calculation have been saved.
Mesh Current Analysis
A more easier method of solving the above circuit is by using Mesh Current Analysis or Loop Analysis which is also sometimes called Maxwell´s Circulating Currents method. Instead of labelling the branch currents we need to label each “closed loop” with a circulating current.
As a general rule of thumb, only label inside loops in a clockwise direction with circulating currents as the aim is to cover all the elements of the circuit at least once. Any required branch current may be found from the appropriate loop or mesh currents as before using Kirchoff´s method.
Another way of simplifying the complete set of Kirchhoff’s equations is the mesh or loop current method. Using this method, Kirchhoff’s current law is satisfied automatically, and the loop equations that we write also satisfy Kirchhoff’s voltage law. Satisfying Kirchhoff’s current law is achieved by assigning closed current loops called mesh or loop currents to each independent loop of the circuit and using these currents to express all the other quantities of the circuit. Since the loop currents are closed, the current that flows into a node must also flow out of the node; so writing node equations with these currents leads to identity.
Let us first consider the method of mesh currents.
We first note that the mesh current method is only applicable for “planar” circuits. Planar circuits have no crossing wires when drawn on a plane. Often, by redrawing a circuit which appears to be non-planar, you can determine that it is, in fact, planar. For non-planar circuits, use the loop current method described later in this chapter.
To explain the idea of mesh currents, imagine the branches of the circuit as “fishing net” and assign a mesh current to each mesh of the net. (Sometimes it is also said that a closed current loop is assigned in each “window” of the circuit.)
 The schematic diagram
 The “fishing net” or the graph of the circuit
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The technique of representing the circuit by a simple drawing, called a graph, is quite powerful. Since Kirchhoff’s laws do not depend on the nature of the components, you can disregard the concrete components and substitute for them simple line segments, called the branches of the graph. Representing circuits by graphs allows us to use the techniques of mathematical graph theory. This helps us explore the topological nature of a circuit and determine the independent loops. Come back later to this site to read more about this topic.
The steps of mesh current analysis:
Assign a mesh current to each mesh. Although the direction is arbitrary, it is customary to use the clockwise direction.
Apply Kirchhoff’s voltage law (KVL) around each mesh, in the same direction as the mesh currents. If a resistor has two or more mesh currents through it, the total current through the resistor is calculated as the algebraic sum of the mesh currents. In other words, if a current flowing through the resistor has the same direction as the mesh current of the loop, it has a positive sign, otherwise a negative sign in the sum. Voltage sources are taken into account as usual, If their direction is the same as the mesh current, their voltage is taken to be positive, otherwise negative, in the KVL equations. Usually, for current sources, only one mesh current flows through the source, and that current has the same direction as the current of the source. If this is not the case, use the more general loop current method, described later in this paragraph. There is no need to write KVL equations for loops containing mesh currents assigned to current sources.
Solve the resulting loop equations for the mesh currents.
Determine any requested current or voltage in the circuit using the mesh currents.
For example: : i1 = I1 , i2 = -I2 and I3 = I1 – I2
We now write Kirchoff’s voltage law equation in the same way as before to solve them but the advantage of this method is that it ensures that the information obtained from the circuit equations is the minimum required to solve the circuit as the information is more general and can easily be put into a matrix form.
For example, consider the circuit from the previous section.
These equations can be solved quite quickly by using a single mesh impedance matrix Z. Each element ON the principal diagonal will be “positive” and is the total impedance of each mesh. Where as, each element OFF the principal diagonal will either be “zero” or “negative” and represents the circuit element connecting all the appropriate meshes. This then gives us a matrix of:
Where:
- [ V ] gives the total battery voltage for loop 1 and then loop 2.
- [ I ] states the names of the loop currents which we are trying to find.
- [ R ] is called the resistance matrix.
and this gives I1 as -0.143 Amps and I2 as -0.429 Amps
As : I3 = I1 – I2
The combined current of I3 is therefore given as : -0.143 – (-0.429) = 0.286 Amps.
LEARNING
Our topic was all about mesh analysis. I learned was that you need to follow the three steps to determine the mesh analysis such as Assign a mesh current to each mesh. Although the direction is arbitrary, it is customary to use the clockwise direction. Apply Kirchhoff’s voltage law (KVL) around each mesh, in the same direction as the mesh currents. If a resistor has two or more mesh currents through it, the total current through the resistor is calculated as the algebraic sum of the mesh currents. In other words, if a current flowing through the resistor has the same direction as the mesh current of the loop, it has a positive sign, otherwise a negative sign in the sum. Voltage sources are taken into account as usual, If their direction is the same as the mesh current, their voltage is taken to be positive, otherwise negative, in the KVL equations. Usually, for current sources, only one mesh current flows through the source, and that current has the same direction as the current of the source. If this is not the case, use the more general loop current method, described later in this paragraph. There is no need to write KVL equations for loops containing mesh currents assigned to current sources. Solve the resulting loop equations for the mesh currents. Determine any requested current or voltage in the circuit using the mesh currents. in order to easy for you to answer the problem.
