MESH AND NODAL ANALYSIS
This page is on mesh and nodal analysis of electrical circuits. It is possible to apply mesh and nodal analysis to ac circuits also. For mesh analysis, we form loop equations based on Kirchoff's voltage law. For
nodal analysis, we form equations at nodes based on Kirchoff's current law. Essentially, we do what has been explained for dc circuits. The difference with ac circuits is that we are dealing with
phasors and complex values. Description of the techniques of mesh and nodal analysis of ac circuits is brief, since there is nothing essentially new. The examples presented later illustrate the application of mesh and nodal analysis to ac circuits. However, a general example is presented below.
To explain
mesh analysis, the circuit in Fig. 28 is used. We apply mesh analysis to this circuit as follows.
Divide the circuit into cells, such that they are adjacent to each other and ensure that cells should not overlap each other. Then assign loop currents. The number of independent loop currents equals the number of unknown loop currents. Let the loop currents flow in a clockwise direction. All the loop currents should be either in the clockwise direction or in the anticlockwise direction. Ensure that the current through any element is either a
loop current or the difference of two loop currents.
For this circuit in Fig. 28, the matrix equation that contains the loop
impedance matrix is shown by equation (95). The values of cells in equation (95) are expressed by equation (96), shown below.
In equation (96), it is seen that the self-impedances of the loops, expressed by
Z11,
Z22 and
Z33, are positive, whereas the mutual impedances are negative. If all the loop currents have the same direction and if they do not overlap, the self impedances are positive and the mutual impedances that form part of two loops are negative. Numerical examples are presented later.
A simple circuit is presented in Fig. 29, to explain
nodal analysis. The circuit has only one node at which the voltage is not known. The equations relevant to these circuits are shown below.
When we form nodal equations, currents flowing away from a node are considered to be positive, whereas those currents flowing towards the node are negative. If no branch currents have already been assigned, you can treat all currents at a node to be flowing away from the node. The current through an element is the product of the admittance of the element and the voltage across it. In circuit in Fig. 29, the voltage across admittance
Y1 is the difference of voltage
VC minus voltage
VA. This is so, since we are forming the equation at node
C, and then the current flowing through
Y1 is away from the node
C. You can see that passive
sign convention has been followed in forming an expression for current through admittance
Y1 . Follow the same technique to get currents through admittances,
Y2 and
Y3 . The sum of three currents incident at node
C is zero, as shown by equation (97). From this equation, we can get an expression for the node voltage
VC, as shown by equation (98).