Linear Circuit Analysis

1. Introduction
2. Basic Concepts
3. Simple Circuits
4. Nodal and Mesh Analysis
5. Additional Analysis Techniques
6. AC Analysis
7. Operational Amplifiers
8. Laplace Transforms
9. Time-Dependent Circuits
10. Two-port networks

Mesh Analysis

Mesh analysis is a method based Kirchhoff's voltage law that can be used to compute the currents flowing in each branch of a circuit. The technique is based on the fact that, if a circuit has $n$ minor meshes, the $n$ mesh currents can be found by solving a system of $n$ independent equations that can be obtained by applying KVL, in which the voltage across each component is computed based on its voltage-current characteristic. If the circuit contains $m$ current sources (notice that in the case of current sources the voltage vs. current is a multivalue function), $m$ equations will be the current constrained equations of each current source, as described below.

Algorithm

Assume we have a circuit with $n$ meshes (excluding the outside mesh), $m$ current sources and $c$ control variables.

Step 1. Identify the meshes in the circuit. The outside mesh is selected as a reference mesh.

Step 2. Label the currents at each of the $n$ meshes with $i_1$, $i_2$, ..., $i_n$.

Step 3. Write the system of mesh analysis equations, which will have $n+c$ equations ($m$ current constrained equations, $n-m$ KVL equations, and $c$ equations for the control variables). It is a good practice two write the $n+c$ equations in the order specified below:

A. Write $m$ current constrained equations. Write one current constrained equations for each source $$\begin{equation}I_i=i_{i1}+i_{i2}\end{equation}$$ where $i=1,...,m$ and $i_{i1}+i_{i2}$ is the algebraic sum of the two mesh currents adjacent to source $m$ going in reference with the direction of the current source. When all mesh currents are considered to be in the same direction (usually clockwise), one mesh current will usually enter with a plus sign, while the other with a minus sign in this formula.

B. Write $c$ equations for control variables. Express each control variable (that usually appear in the definition of dependent sources) in terms of the $n$ mesh currents.

C. Write KVL equations for regular meshes. Write KVL for each of regular mesh that does not connect a current source. When writing KVL, the current through each resistor (or impedance in the case of AC circuits) should be computed using Ohm's law.

D. Write KVL equations for supermeshes, if any. Write KVL for each supermesh. As before, when writing KVL, the current through each resistor (or impedance) should be computed using Ohm's law.

Step 4. Solve the system on mesh analysis equations to compute the $n$ mesh currents and $c$ control variables.

Step 5. Compute the sought variables.

Notes
  • A supermesh is a generalized mesh containing only current sources inside but is not connected to any current source on the outside. Usually, we will need to write equations for supermeshes whenever we have current sources that are not part of the outside mesh or near the outside mesh through any other current sources. Since a supermesh contains current sources, a supermesh does not have its own current (in fact, it will contain multiple regular meshes).
  • The number of KVL equations that we write at steps 3.C and 3.D should be $n-m$.
  • Sometimes, it is convenient to add the equations for the sought variables to the $n+c$ mesh analysis equations. In this case, the solution of the system will also contain the values of the sought variables.
  • It is not mandatory to chose the outside mesh as a reference mesh. In principle, one could chose any mesh as the reference mesh and include the outside mesh as a regular mesh. However, this is almost never done in practice when solving mesh analysis problems.
  • Mesh analysis can be used to analyze both linear and nonlinear circuits, DC and AC circuits, as well as time-dependent circuits. In the case of DC circuits we obtain a system of equations with real coefficients; in the case of AC circuits we obtain a system of equations with complex coefficients; in the case or time-dependent problems we obtain a system of integro-differential equations (which can usualy be reduced to a system of ordinary differential equations).
Sample Solved Problems
See also