Methods of Analysis
Introduction:
Having understood the fundamental laws of circuit theory (Ohm’s law and Kirchhoff’s laws), we are now prepared to apply these laws to develop two powerful techniques for circuit analysis: nodal analysis, which is based on a systematic application of Kirchhoff’s current law (KCL), and mesh analysis, which is based on a systematic application of Kirchhoff’s voltage law (KVL).
we can analyze almost any circuit by obtaining a set of simultaneous equations that are then solved to obtain the required values of current or voltage.One method of solving simultaneous equations involves Cramer’s rule,which allows us to calculate circuit variables as a quotient of determinants.
Nodal Analysis
Nodal analysis provides a general procedure for analyzing circuits using node voltages as the circuit variables. Choosing node voltages instead of element voltages as circuit variables is convenient and reduces the number of equations one must solve simultaneously.To simplify matters, we shall assume in this section that circuits do not contain voltage sources. Circuits that contain voltage sources will be analyzed in the next section.
Nodal analysis is also known as the node-voltage method.
In nodal analysis, we are interested in finding the node voltages.Given a circuit with n nodes without voltage sources, the nodal analysis of the circuit involves taking the following three steps
Step 1: Select a node as the reference node. Assign voltages v1, v2, . . . , vn−1 to the remaining n − 1 nodes. The voltages are referenced with respect to the reference node.
Step 2: Apply KCL to each of the n − 1 nonreference nodes. Use Ohm’s law to express the branch currents in terms of node voltages.
Step 3: Solve the resulting simultaneous equations to obtain the unknown node voltages.
Current flows from a higher potential to a lower potential in a resistor.
Super Node:
If the voltage source (dependent or independent) is connected between two nonreference nodes, the two nonreference nodes form a generalized node or supernode; we apply both KCL and KVL to determine the node voltages.
"A supernode is formed by enclosing a (dependent or independent) voltage source connected between two non-reference nodes and any elements connected in parallel with it."
Note the following properties of a supernode:
1. The voltage source inside the supernode provides a constraint
equation needed to solve for the node voltages.
2. A supernode has no voltage of its own.
3. A supernode requires the application of both KCL and KVL.
Mesh Analysis:
Mesh analysis provides another general procedure for analyzing circuits, using mesh currents as the circuit variables. Using mesh currents instead of element currents as circuit variables is convenient and reduces the number of equations that must be solved simultaneously. Recall that a loop is a closed path with no node passed more than once. A mesh is a loop that does not contain any other loop within it.
Mesh analysis is also known as loop analysis or the mesh-current method.
Nodal analysis applies KCL to find unknown voltages in a given circuit, while mesh analysis applies KVLto find unknown currents. Mesh analysis is not quite as general as nodal analysis because it is only applicable to a circuit that is planar. A planar circuit is one that can be drawn in a plane with no branches crossing one another; otherwise it is non-planar. A circuit may have crossing branches and still be planar if it can be redrawn such that it has no crossing branches.
A mesh is a loop which does not contain any other loops within it.
Steps to Determine Mesh Currents :
1. Assign mesh currents i1, i2, . . . , in to the n meshes.
2. Apply KVL to each of the n meshes. Use Ohm’s law to express
the voltages in terms of the mesh currents.
3. Solve the resulting n simultaneous equations to get the mesh
currents.
The directionof the mesh current is arbitrary—(clockwise or counterclockwise)—and does not
affect the validity of the solution
Super Mesh:
A supermesh results when two meshes have a (dependent or independent) current source in common
Note the following properties of a supermesh:
1. The current source in the supermesh is not completely ignored;
it provides the constraint equation necessary to solve for the
mesh currents.
2. A supermesh has no current of its own.
3. A supermesh requires the application of both KVL and KCL
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