Linear Circuit Analysis

1. Introduction
2. Basic Concepts
3. Simple Circuits
4. Nodal and Mesh Analysis (DC)
5. Additional Analysis Techniques (DC)
6. AC Analysis
7. Magnetically Coupled Circuits
8. Polyphase Systems
9. Operational Amplifiers
10. Laplace Transforms
11. Time-Dependent Circuits
12. Two-Port Networks
Appendix

Summary of DC Analysis Methods

DC analysis is the process of determining the steady-state voltages and currents in an electric circuit driven by direct-current (DC) sources. DC analysis assumes that all sources are constant in time and that the circuit has reached equilibrium, meaning voltages and currents no longer change with time.

DC analysis is important because it establishes the operating point of a circuit, which is essential for understanding power consumption, verifying design constraints, and serving as the starting point for more advanced analyses such as transient or AC (frequency-domain) analysis. So far, we have discussed general methods for analyzing DC circuits containing only resistors and independent sources. However, many practical circuits also include energy-storage elements such as inductors and capacitors. This naturally raises the question: what happens when a circuit driven by DC sources contains inductors and capacitors?

It turns out that DC analysis of such circuits is relatively straightforward. In steady-state DC conditions, an inductor behaves as a short circuit, since a constant current produces no induced voltage, while a capacitor behaves as an open circuit, because a constant voltage prevents further current flow. As a result, after sufficient time has passed, a circuit with inductors and capacitors can be reduced to an equivalent purely resistive network by replacing inductors with wires and capacitors with open gaps. The resulting simplified circuit can then be analyzed using the same DC techniques introduced earlier, allowing voltages, currents, and power to be determined efficiently.

For instance, let us consider the circuit in Fig. 1. After a long time, after steady-state equilibrium is reached, inductors can be replaces with short circuits and capacitors with open circuits. Therefore, the two circuits are equivalent under steady-state condition and current $I_0 = \frac{2\,\class{mjunit}{V}}{10\,\class{mjunit}{\Omega}} = 0.2\,\class{mjunit}{A}$.

2V L I0 10Ω C 2V I0 10Ω
Fig. 1. Under steady-state conditions, inductors behave like short circuits and capacitors like open circuits. Therefore, currents $I_0$ in the circuits above are equal.

Below is a short overview of the principal methods used in DC circuit analysis. At the end of this section, you will find practice problems that allow you to analyze DC circuits using any method you prefer.

Nodal Analysis

Nodal analysis determines circuit behavior by solving for node voltages relative to a reference (ground). Kirchhoff’s Current Law (KCL) is applied at each non-reference node, expressing the sum of currents leaving or entering the node as zero. The resulting system of linear equations is solved for node voltages, from which branch currents and powers are easily obtained. Nodal analysis is particularly efficient for circuits with many current sources and is well suited for systematic, computer-based solutions.

Mesh (Loop) Analysis

Mesh analysis focuses on solving for mesh currents in planar circuits by applying Kirchhoff’s Voltage Law (KVL) around each independent loop. Each mesh equation relates voltage drops across resistive elements to the applied voltage sources. Once mesh currents are known, branch currents and voltages can be calculated. Mesh analysis is most effective for planar circuits with relatively few meshes and primarily voltage sources.

Superposition

The superposition principle states that, in a linear circuit with multiple independent sources, the total response is the algebraic sum of the responses due to each source acting alone. When analyzing the effect of one source, all other independent voltage sources are replaced by short circuits and independent current sources by open circuits. Superposition is conceptually powerful for understanding how individual sources contribute to circuit behavior, though it can be computationally inefficient for large circuits.

Source Transformation

Source transformation simplifies circuit analysis by converting a voltage source in series with a resistor into an equivalent current source in parallel with the same resistor, and vice versa. This equivalence preserves the external terminal behavior of the circuit. Source transformations are often used iteratively to reduce complex networks into simpler forms that are easier to analyze using nodal or mesh techniques.

Circuit Simplification

Circuit simplification reduces a network to a smaller equivalent circuit by combining resistors in series or parallel and eliminating unnecessary elements. The goal is to decrease the number of unknowns before applying formal analysis methods. While limited to networks where such combinations are clearly identifiable, simplification can greatly reduce computational effort and clarify the dominant electrical behavior.

Thévenin and Norton Equivalents

Thévenin’s and Norton’s theorems replace any linear two-terminal network with an equivalent voltage source in series with a resistance (Thévenin) or a current source in parallel with a resistance (Norton). These equivalents produce the same terminal voltage–current relationship as the original circuit. They are especially useful for analyzing load variations, comparing circuit designs, and modularizing complex systems into simpler, reusable blocks.

See also