Analysis Techniques

1. Linearity

Linear circuits have these properties:

Homogeneity Property: Also known as the scaling property, this property states that if the current is increased through a resistive element, the voltage is increased by the same factor.
Superposition Principle: If we have more than one source, we can calculate the response of each individually and add them. With voltage sources, we must short the terminals; with current sources, we leave them floating.

Common circuit elements that are linear include resistors, capacitors, inductors, transformers, and dependent sources. When a circuit is composed of only linear elements, they can described as a system of linear equations, which lend themselves to more straightforward solutions.

Moreover, in a linear system, the total response to multiple inputs is the sum of the responses to each individual input acting alone (superposition).

2. Node Voltage Analysis

Node voltage analysis (NVA), also known as nodal analysis, is a systematic and efficient procedure for determining all the voltages and currents in a circuit. The procedure is:

Identify all extraordinary nodes, select one of them as a reference node (ground), and then assign node voltages to the remaining \(n-1\) extraordinary nodes.
At each of the \(n-1\) remaining extraordinary nodes, apply KCL.
Solve the \(n-1\) independent simultaneous equations to determine the unknown node voltages.

Example: Nodal analysis

We can write the following system of equations:

\begin{align} \frac{V_1-V_0}{R_1+R_2} + \frac{V_1}{R_3} + \frac{V_1-V_2}{R_4} &= 0 \notag \\ \frac{V_2-V_1}{R_4}+\frac{V_2}{R_5} - I_0 &= 0 \notag \end{align}

\(V_1\) and \(V_2\) are our two unknowns, and we have two independent equations that we can solve for.

3. Mesh Current Analysis

Mesh current analysis (MVA), also known as the mesh current method, is a systematic procedure for determining all the voltages and currents in a circuit. The procedure is:

Identify all meshes and assign each mesh and unknown (clockwise) current.
For each mesh, apply KVL.
Solve the resulting simultaneous equations.

Example: Mesh current analysis

Applying KVL to each mesh gives:

\begin{align} V_0-I_1R_{12} - (I_1-I_2)R_3 &= 0 \notag \\ -(I_2-I_1)R_3 - I_2R_4-(I_2-I_3)R_5 &= 0 \notag \end{align}

But from the last mesh, we know that \(I_3 = -I_0\), so we have:

\begin{align} V_0-I_1R_{12} - (I_1-I_2)R_3 &= 0 \notag \\ -(I_2-I_1)R_3-I_2R_4-(I_2+I_0)R_5 &= 0 \notag \end{align}

We now have two equations with two unknowns, so we can find \(I_1\) and \(I_2\).

4. Analysis by Inspection

Node voltage and mesh current methods can be used to analyze any planar circuit. However, for circuits with only independent sources, standard patterns allow for analysis by inspection. In these cases, all sources must be exclusively either current or voltage sources.

4.1. Nodal Analysis by Inspection

The key here is that from Ohm’s Law, we have:

\begin{align} \boxed{GV=I} \end{align}

where \(G\) is a matrix of inductances. The procedure is as follows:

Convert all sources to current sources.
Assign ground and label all unknown node voltages.
Compute the conductance matrix \(G\) as:
- \(G_{ii}\) is the sum of all conductances connected to node \(i\).
- \(G_{ij}=G_{ji}\), where \(i \neq j\) is the negative of conductances connecting nodes \(i\) and \(j\).
Determine the total current entering node \(i\).
Solve the matrix equation \(GV=I\).

Example: Nodal analysis by inspection

Consider the following circuit:

Converting the voltage source to a current source yields the following equivalent circuit:

Computing the conductance matrix:

\begin{align} G = \begin{bmatrix} G_1 + G_2 & -G_2 \\ -G_2 & G_2+G_3 \end{bmatrix} \notag \end{align}

Then, computing all of the currents entering each node, we get:

\begin{align} I = \begin{bmatrix} -I_a \\ I_a + I_b \end{bmatrix} \notag \end{align}

Now, we can solve the following matrix equation to get the voltages:

\begin{align} \begin{bmatrix} G_1 + G_2 & -G_2 \\ -G_2 & G_2+G_3 \end{bmatrix}\begin{bmatrix}V_1 \\ V_2\end{bmatrix} = \begin{bmatrix} -I_a \\ I_a + I_b \end{bmatrix} \notag \end{align}

4.2. Mesh Analysis by Inspection

Here, from Ohm’s Law, we have:

\begin{align} \boxed{RI=V} \end{align}

where \(R\) is a matrix of resistances. The procedure is as follows:

Convert all sources to voltage sources.
Label all meshes and assign clockwise mesh currents.
Compute the resistance matrix \(R\) as:
- \(R_{ii}\) is the sum of all resistances in mesh \(i\).
- \(R_{ij}=R_{ji}\), where \(i \neq j\) is the negative of the common resistance between mesh \(i\) and \(j\).
Determine the total sum of all voltage sources in mesh \(i\), going clockwise.
Solve the matrix equation \(RI=V\).

Example: Mesh analysis by inspection

Consider the following circuits:

Since all of the sources are voltage sources, we can directly build the resistance matrix:

\begin{align} R = \begin{bmatrix} 8 & -2 & -5 \\ -2 & 9 & -3 \\ -5 & -3 & 14 \end{bmatrix} \notag \end{align}

And the total voltages are:

\begin{align} V = \begin{bmatrix} 5 \\ 0 \\ 5 \end{bmatrix} \notag \end{align}

Now we can solve the matrix equation:

\begin{align} \begin{bmatrix} 8 & -2 & -5 \\ -2 & 9 & -3 \\ -5 & -3 & 14 \end{bmatrix} \begin{bmatrix}I_1 \\ I_2 \\ I_3 \end{bmatrix} = \begin{bmatrix} 5 \\ 0 \\ 5 \end{bmatrix} \notag \end{align}