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

Nodal Analysis

Nodal analysis is a method based Kirchhoff's current law that can be used to compute the potentials in a circuit. The technique is based on the fact that, if a circuit has $n+1$ nodes (including the ground node), the $n$ potentials (excluding the potential of the ground node, which is assumed to be $0$) can be found by solving a system of $n$ independent equations that can be obtained by applying KCL, in which the current going through each branch is computed based on the current-voltage characteristic of the element on that branch. If the circuit contains $m$ voltage sources (notice that in the case of voltage sources, the current vs. voltage is a multivalued function), $m$ equations will be the voltage constrained equations of each voltage source as described below.

Algorithm

Assume we have a circuit with $n$ nodes (excluding the ground node), $m$ voltage sources and $c$ control variables.

Step 1. Identify the nodes in the circuit and select one of them as a reference node. The reference node is treated as ground node.

Step 2. Label the potentials at each of the $n$ nodes with $v_1$, $v_2$, ..., $v_n$, where $n$ is the number of nodes (excluding the ground node).

Step 3. Write the system of nodal analysis equations, which will have $n+c$ equations ($m$ voltage constrained equations, $n-m$ KCL 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 $m$ voltage constrained equations. Write one voltage constrained equations for each source $$\begin{equation}V_i=v_{i+}-v_{i-}\end{equation}$$ where $i=1,...,m$ and $v_{i+}-v_{i-}$ is the difference between the potentials of the positive and the negative electrodes of source $m$.

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

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

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

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

Step 5. Compute the sought variables.

Notes
  • A supernode is a generalized node containing only voltage sources inside but is not connected to any voltage source on the outside. Usually, we will need to write equations for supernodes whenever we have voltage sources that are not connected to the ground node directly of through any other voltage sources. Since a supernode contains voltage sources, a supernode does not have its own potential (in fact, it will contain multiple regular nodes).
  • The number of KCL 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 $n+c$ nodal analysis equations. In this case, the solution of the system will also contain the values of the sought variables.
  • Nodal 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 system of integro-differential equations (which can usualy be reduced to a system of ordinary differential equations).
Sample Solved Problems
See also