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 in Time-Dependent Circuits

The mesh analysis method that we introduced for DC and AC circuits can be generalized to time-dependent circuits. In this case, the method results into a a system of integro-differential equations which can be reduced to a system of ordinary differential equations (ODE) that needs to be solved to compute the time-dependent mesh currents. The system of nodal analysis equations contains:

  • Current constrained equations (one equation for each current source)
  • Kirchhoff's voltage law equations (one equation per mesh or supermesh)
When we solve the system of mesh analysis equations in time-domain, we should always impose initial conditions for the current going throught the inductors and voltage across the capacitors (otherwise, this system would not have unique solution). These initial conditions are usually computed in advance.

Mesh analysis

The current constrained equations are written in the same way as in the case of DC circuits, the only difference being that the currents of the sources are time-dependent: $$I_i(t)=i_{i1}(t)+i_{i2}(t)$$ where $i=1,...,m$ and $i_{i1}(t)+i_{i2}(t)$ is the algebraic sum of the two mesh currents adjacent to current source $m$.

Voltages across components

When we write KVL, the voltage across each device depends on the type of the device, as described in Table 1, where $I(t)$ is the current going through the device, which can be written in terms of the adjacent mesh currents. 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 the expression of $I(t)$.

Table 1. Voltages across various components as a function of the current going through the component. The current going through each device, $I(t)$, can be written in terms of the adjacent mesh currents.
Name Symbol Voltage
Independent voltage source − + V(t) V $$\begin{equation}V(t)\end{equation}$$
Resistor − + V(t) I(t) R $$\begin{equation}V(t)=I(t){R}\end{equation}$$
Capacitor − + V(t) I(t) C $$\begin{equation}V(t)=V(t_0)+\frac{1}{C}\int_{t_0}^{t} I(t) {dt}\end{equation}$$
Inductor − + V(t) I(t) L $$\begin{equation}V(t)=L\dfrac{dI(t)}{dt}\end{equation}$$

For instance, the mesh equation in time-domain for mesh current $i_{2}$ in the circuit represented in Fig. 1 is

$$\begin{equation}i_{2}R_{2}+V_{C0}+\frac{1}{C_1}\int_{0}^{t}(i_2-i_3){dt} - V_{1} + L\dfrac{d(i_{2}-i_{1})}{dt}=0\end{equation}$$

where $V_{C0}$ is the voltage across the capacitor at $t=0$. Notice that the previous equation is valid for $t>0$. To simplify notations, we have not denoted explicitly that $i_1$, $i_2$, and $i_3$ are functions of $t$, however, we should consider it as $i_1(t)$, $i_2(t)$, and $i_3(t)$.

R1 R2 − + VC0 C1 L1 V1 I1 i1 i2 i3
Fig. 1. Mesh analysis in time-domain.
Algorithm
(Notice the similarity with DC circuits)

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 (ordinary differential) 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 $$\begin{equation}I_i(t)=i_{i1}(i)+i_{i2}(t)\end{equation}$$ where $i=1,...,m$ and $i_{i1}(t)+i_{i2}(t)$ 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. 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.

Sample Solved Problems
See also