Circuit Analysis

Introduction

Circuit analysis is the systematic process of determining voltages, currents, power, energy, and dynamic behavior in an electrical network. At university level, the subject connects Electrical engineering, Physics, Linear algebra, differential equations, and complex numbers. You learn not only how to obtain numerical answers, but also how to choose an efficient model, justify assumptions, check signs and units, and interpret what a result means physically.

This aiMOOC focuses on linear lumped circuits built from ideal and practical models of sources, resistors, capacitors, and inductors. It progresses from DC resistive networks to transient analysis, sinusoidal steady state, AC power, resonance, transfer functions, and the Laplace-domain viewpoint. The methods also form part of the conceptual basis of circuit simulation tools such as SPICE.

Learning outcomes: By the end of the course, you should be able to formulate and solve nodal and mesh equations, apply network theorems correctly, predict first- and second-order transients, use phasors and impedance for sinusoidal steady-state analysis, calculate AC power, interpret resonance and Bode plots, and assess whether a mathematical answer is physically plausible.

Prerequisites: You should be comfortable with algebra, simultaneous linear equations, elementary calculus, complex numbers, and basic ideas about electric charge and energy.

Foundations of Circuit Analysis

Lumped Models and Circuit Variables

A circuit model replaces a physical system with interconnected idealized elements whose terminal behavior is described mathematically. The lumped-element assumption treats voltage and current as well-defined variables associated with circuit elements and nodes. This is highly effective when the physical dimensions of the circuit are small compared with the electromagnetic wavelength of interest and when distributed effects can be neglected.

The central variables are charge q, current i, voltage v, power p, and energy w. Current is the rate of flow of charge, so i = dq/dt. Voltage is energy transferred per unit charge. Under the passive sign convention, current enters the terminal labeled positive; then p = vi. Positive p means the element absorbs power, while negative p means it delivers power.

Always check dimensional consistency. Voltage is measured in volts, current in amperes, resistance in ohms, capacitance in farads, inductance in henries, power in watts, and energy in joules. Prefixes matter: 1 kΩ = 1000 Ω, 1 mA = 0.001 A, and 1 µF = 10^-6 F.

Ideal Elements and Constitutive Relations

An ideal resistor follows Ohm's law: v = Ri. Its instantaneous power under the passive sign convention is p = Ri² = v²/R, so an ideal positive resistance dissipates energy.

An ideal capacitor stores electric-field energy. Its constitutive relation is i = C dv/dt, and its stored energy is w = Cv²/2. Because an impulse current would be required to create an instantaneous finite jump in capacitor voltage, capacitor voltage is continuous in ordinary switching problems.

An ideal inductor stores magnetic-field energy. Its constitutive relation is v = L di/dt, and its stored energy is w = Li²/2. Inductor current is continuous in ordinary switching problems because an instantaneous finite jump in current would require an impulse voltage.

Independent voltage and current sources prescribe a voltage or current regardless of the rest of the ideal circuit. Dependent sources are controlled by another circuit voltage or current. They are essential in transistor and amplifier models and must remain active when a network is analyzed because their value depends on the circuit itself.

Nodes, Branches, Loops, and Reference Directions

A node is a set of points connected by ideal wire and therefore at the same potential. A branch is a single two-terminal element or a series group treated as one connection between nodes. A loop is a closed path. A mesh is a loop that contains no other loop inside it in a planar drawing.

Before writing equations, assign a reference node, voltage polarities, and current directions. These choices are conventions rather than predictions. If the solved value of an assumed current is negative, the actual current flows opposite to the assumed arrow.

Kirchhoff's Laws

Kirchhoff's circuit laws provide the topological foundation of classical lumped circuit analysis.

Kirchhoff's Current Law, KCL: the algebraic sum of currents at a node is zero. Equivalently, total current entering a node equals total current leaving it. KCL expresses conservation of electric charge in the lumped model.

Kirchhoff's Voltage Law, KVL: the algebraic sum of voltage rises and drops around a closed loop is zero. In the standard lumped approximation, KVL expresses energy consistency around the loop.

A robust sign convention is more important than memorizing a particular arrow pattern. State your convention, write every term consistently, and then let the algebra determine signs.

DC Resistive Networks

Series, Parallel, and Divider Relations

Resistors in series carry the same current and have equivalent resistance R_eq = R_1 + R_2 + ... . Resistors in parallel share the same voltage and satisfy 1/R_eq = 1/R_1 + 1/R_2 + ... .

For two series resistors driven by voltage V, the voltage divider gives V_2 = V R_2/(R_1 + R_2). For two parallel resistors carrying total current I, the current through R_1 is I_1 = I R_2/(R_1 + R_2). These divider formulas are shortcuts derived from Ohm's law plus Kirchhoff's laws, not independent physical laws.

Nodal Analysis

Nodal analysis chooses node voltages as the main unknowns and applies KCL. It is often the most efficient method when a circuit has many current sources or relatively few essential nodes.

A systematic procedure is:

Choose one reference node and define it as zero volts.
Label every remaining unknown node voltage.
Express each branch current in terms of node voltages.
Apply KCL at each unknown node or supernode.
Solve the simultaneous equations and check signs, units, and power balance.

If an ideal voltage source lies between two unknown nonreference nodes, treat those nodes as a supernode. Write one KCL equation for the supernode and one constraint equation from the voltage source.

Worked example: Suppose a node v is connected through 2 kΩ to a fixed 10 V node, through 3 kΩ to ground, and receives 1 mA from a current source directed from ground into the node. Taking currents leaving v as positive gives (v - 10)/2 kΩ + v/3 kΩ - 1 mA = 0. Multiplying by 6 kΩ gives 3v - 30 + 2v - 6 = 0, so v = 7.2 V. The result can be checked by calculating all three branch currents and verifying KCL.

Mesh Analysis

Mesh analysis chooses mesh currents as unknowns and applies KVL to each independent mesh of a planar circuit. When two mesh currents share a resistor, the branch current is the algebraic difference of the mesh currents, and the resistor voltage must be written from that difference.

An ideal current source on a branch shared by two meshes creates a supermesh. Write KVL around the perimeter that excludes the current-source branch, then add the current-source constraint relating the mesh currents.

Nodal and mesh methods are dual viewpoints. Nodal analysis naturally uses admittances and KCL; mesh analysis naturally uses impedances and KVL. The better choice is usually the one that produces fewer or simpler equations.

Matrix Form and Computational Thinking

For a linear resistive network, nodal equations can often be written as Gv = i, where G is a conductance matrix, v is the vector of unknown node voltages, and i represents source injections. This form connects circuit analysis with Linear algebra and numerical computation.

The matrix viewpoint matters because large circuits are not solved by repeatedly applying ad hoc reductions. Circuit simulators assemble sparse systems of equations from element models. Modified nodal analysis extends the nodal method so that voltage sources and other elements can be incorporated systematically.

Network Theorems and Equivalent Circuits

Superposition

In a linear circuit, the response caused by several independent sources equals the sum of the responses caused by each independent source acting alone. To suppress an ideal independent voltage source, replace it by a short circuit. To suppress an ideal independent current source, replace it by an open circuit. Dependent sources are not suppressed because they are controlled by circuit variables.

Superposition applies to voltages and currents, not directly to nonlinear quantities such as power. If you need total power, first find the total voltage or current and then calculate power from the combined response.

Thévenin and Norton Equivalents

Any linear two-terminal network can be represented at its terminals by a Thévenin equivalent: a voltage source V_th in series with an equivalent resistance or impedance Z_th. The same terminal behavior can be represented by a Norton equivalent: a current source I_N in parallel with the same equivalent impedance, with I_N = V_th/Z_th.

To find V_th, compute the open-circuit terminal voltage. To find the equivalent resistance of a resistive network containing only independent sources, deactivate the independent sources and calculate the resistance seen into the terminals. If dependent sources remain, apply a test source at the terminals and use R_th = V_test/I_test.

Source Transformation and Maximum Power Transfer

A voltage source V_s in series with resistance R can be transformed into an equivalent current source I_s = V_s/R in parallel with R, provided the transformation preserves the same two-terminal behavior.

For a purely resistive Thévenin source, maximum average power is transferred to a variable load when R_L = R_th. For sinusoidal AC with complex impedances, maximum average power transfer occurs when the load impedance is the complex conjugate of the Thévenin impedance, Z_L = Z_th*. Maximum power transfer is not the same as maximum efficiency, because at the resistive maximum-power condition the source resistance dissipates power too.

Bridge Circuits and Measurement

The Wheatstone bridge illustrates how network structure can make a measurement especially sensitive. In a balanced bridge, the detector branch carries zero current because its two terminals are at equal potential. The balance relation allows an unknown resistance to be inferred from known resistances without needing the detector's absolute calibration.

Transient Analysis

Initial and Final Conditions

A transient occurs when a switching event or changing source causes stored energy to redistribute. The key state variables in basic circuits are capacitor voltage and inductor current.

Immediately before switching, determine v_C(0-) and i_L(0-). Under ordinary finite-source conditions, continuity gives v_C(0+) = v_C(0-) and i_L(0+) = i_L(0-). Then analyze the circuit for t > 0 using the new topology. For a stable DC circuit after a long time, an ideal capacitor behaves as an open circuit and an ideal inductor behaves as a short circuit.

First-Order RC and RL Circuits

A first-order natural response has the form x(t) = x(0+)e^(-t/τ), while a first-order step response can be written x(t) = x(∞) + [x(0+) - x(∞)]e^(-t/τ).

For an RC circuit, τ = R_eq C, where R_eq is the resistance seen by the capacitor in the relevant source-deactivated network. For an RL circuit, τ = L/R_eq. After one time constant, about 63.2 percent of a step change has occurred; after five time constants, the response is usually within about one percent of its final value.

Second-Order RLC Circuits

A circuit containing two independent energy-storage elements can produce a second-order differential equation. For a standard series RLC natural response, the characteristic form is s² + 2ζω_0 s + ω_0² = 0, where ω_0 = 1/sqrt(LC). The damping ratio ζ determines the qualitative behavior.

If ζ > 1, the response is overdamped and returns without oscillation. If ζ = 1, it is critically damped. If 0 < ζ < 1, it is underdamped and oscillates with a decaying envelope. The circuit's initial capacitor voltage and inductor current determine the constants multiplying the natural modes.

Sinusoidal Steady-State Analysis

Phasors, Impedance, and Admittance

For a linear time-invariant circuit driven at a single angular frequency ω, sinusoidal steady-state analysis replaces differentiation and integration with algebra using complex numbers. A sinusoid is represented by a phasor containing magnitude and phase.

The element impedances are Z_R = R, Z_L = jωL, and Z_C = 1/(jωC). Ohm's law generalizes to V = ZI in phasor form. Admittance is Y = 1/Z. Kirchhoff's laws continue to apply to phasor voltages and currents because the circuit equations are linear.

AC Power

For RMS phasors, complex power is S = VI*, where I* is the complex conjugate of current. Complex power separates into S = P + jQ. Real power P is measured in watts, reactive power Q in volt-amperes reactive, and apparent power |S| in volt-amperes.

For sinusoidal voltage and current with phase difference φ, P = VI cosφ and Q = VI sinφ using RMS magnitudes. The power factor is cosφ. Inductive loads typically have lagging current and positive reactive power under the passive sign convention; capacitive loads typically have leading current and negative reactive power.

Resonance

In a series RLC circuit, resonance occurs when inductive and capacitive reactances cancel: ωL = 1/(ωC). The ideal resonant angular frequency is therefore ω_0 = 1/sqrt(LC). At resonance, the series impedance is purely resistive, the current magnitude is maximal for a fixed source voltage, and the source voltage and current are in phase.

Resonance is important in filters, communication systems, sensors, oscillators, and power networks. Real inductors and capacitors have parasitic resistance and other nonidealities, so measured resonance differs from an ideal model.

Frequency Response and Transfer Functions

Transfer Functions

For a linear time-invariant circuit with zero initial conditions, a transfer function is H(s) = Y(s)/X(s), the ratio of an output transform to an input transform. Poles are roots of the denominator, and zeros are roots of the numerator. Their positions influence stability, transient behavior, and frequency response.

For a simple RC low-pass filter with the output across the capacitor, H(s) = 1/(1 + sRC). Setting s = jω gives the sinusoidal frequency response H(jω). Its cutoff frequency is f_c = 1/(2πRC), where the magnitude has fallen to 1/sqrt(2) of the low-frequency value, corresponding to approximately -3 dB.

Bode Plots

A Bode plot displays magnitude, usually in decibels, and phase versus logarithmic frequency. It makes poles, zeros, bandwidth, cutoff behavior, and asymptotic slopes easier to interpret over many decades of frequency.

For a first-order low-pass pole, the high-frequency magnitude asymptote falls at about 20 dB per decade, and the phase transitions from approximately 0 degrees toward -90 degrees. Use exact calculations near a corner frequency; asymptotes are approximations.

Laplace-Domain Circuit Analysis

The Laplace transform generalizes circuit analysis beyond sinusoidal steady state. With suitable treatment of initial conditions, capacitors and inductors can be represented by algebraic s-domain relations. This converts linear differential equations into algebraic equations, making switching problems, impulse responses, step responses, poles, and transfer functions part of one framework.

The Laplace-domain method does not eliminate physical reasoning. You still need correct initial conditions, circuit topology, sign conventions, and a meaningful inverse transform. A useful check is to compare the initial-value and final-value behavior with what the physical circuit should do.

Modeling, Simulation, and Measurement

From Ideal Circuit to Real Experiment

Every circuit analysis rests on assumptions. Real resistors have tolerance and temperature dependence. Capacitors have leakage, equivalent series resistance, and parasitic inductance. Inductors have winding resistance, core loss, and saturation limits. Sources have finite output resistance, and measuring instruments load the circuit.

A strong engineering workflow is:

Create the simplest model that can answer the question.
Solve analytically to predict important values and trends.
Simulate the same model and compare results.
Measure a safe physical circuit if appropriate.
Explain discrepancies by refining the model rather than merely fitting numbers.

When working in a teaching laboratory, use isolated low-voltage supplies and follow local laboratory procedures. Do not connect student experiments directly to mains electricity unless a qualified laboratory supervisor and approved equipment explicitly provide a safe procedure.

Using SPICE Critically

SPICE-based tools are powerful for operating-point, transient, AC, and other analyses, but simulation output is only as meaningful as the model and settings. Before trusting a plot, ask whether component values are realistic, ground is defined, source waveforms are correct, initial conditions make sense, and the requested analysis matches the question.

Use hand analysis for limiting cases and order-of-magnitude checks. If a simulation contradicts conservation laws or gives an implausible scale, investigate the model before accepting the graph.

Problem-Solving Strategy

A repeatable strategy reduces algebraic errors and makes your work easier to review:

Identify the analysis regime: DC steady state, transient, sinusoidal steady state, or general s-domain behavior.
Redraw the circuit if necessary and mark nodes, polarities, reference directions, and known values.
Choose an efficient method such as reduction, nodal analysis, mesh analysis, or an equivalent-circuit theorem.
Write equations symbolically before inserting numbers when practical.
Solve the equations with units and retain signs.
Check KCL, KVL, power balance, limiting cases, and physical plausibility.
Interpret the result in words rather than stopping at a number.

Interactive Tasks

Quiz: Test Your Knowledge

Which statement best expresses Kirchhoff's Current Law? (The algebraic sum of currents at a node is zero) (!The algebraic sum of resistances in a loop is zero) (!Voltage is always equal at every node) (!Current is always equal in parallel branches)

Under the passive sign convention, what does positive instantaneous power mean? (The element absorbs power) (!The element must be a source) (!The circuit has zero energy) (!The current direction is undefined)

What is the main unknown in standard nodal analysis? (Node voltage) (!Mesh resistance) (!Branch energy) (!Source frequency)

How does an ideal capacitor behave in DC steady state after a long time? (As an open circuit) (!As a short circuit) (!As an ideal voltage source) (!As a negative resistor)

What two-element form represents a Thévenin equivalent? (A voltage source in series with an equivalent impedance) (!A current source in series with an equivalent impedance) (!A voltage source in parallel with an ideal short) (!Two current sources in parallel)

What is the time constant of a simple RC circuit? (Resistance multiplied by capacitance) (!Resistance divided by capacitance) (!Capacitance divided by resistance) (!Resistance added to capacitance)

Which quantity is normally continuous through an ordinary switching event in an ideal inductor? (Inductor current) (!Inductor voltage) (!Resistor voltage) (!Source power)

What is the impedance of an ideal inductor in sinusoidal steady state? (j omega L) (!one over j omega C) (!R divided by j omega) (!zero at every frequency)

For maximum average power transfer in AC, how should the load impedance relate to the Thévenin impedance? (The load is the complex conjugate of the Thévenin impedance) (!The load is always zero) (!The load is always infinite) (!The load equals only the Thévenin reactance)

At the cutoff frequency of an ideal first-order RC low-pass filter, what is the magnitude change relative to the low-frequency value? (Approximately minus three decibels) (!Exactly plus twenty decibels) (!Exactly zero decibels at all frequencies) (!Approximately plus three decibels)

Memory Game

Kirchhoff current law	Charge conservation at a node
Kirchhoff voltage law	Voltage consistency around a closed loop
Thevenin equivalent	Voltage source in series with equivalent impedance
Norton equivalent	Current source in parallel with equivalent impedance
Time constant	Characteristic time scale of a first order transient
Phasor	Complex representation of a sinusoid at one frequency
Resonance	Condition in which inductive and capacitive reactances cancel

Drag and Drop

Match the correct terms.	Topic
Node-voltage method	Apply KCL to unknown node potentials
Mesh-current method	Apply KVL to independent planar loops
Superposition	Sum responses from one independent source at a time
Thevenin reduction	Replace a linear two-terminal network by a voltage source and series impedance
Source transformation	Exchange equivalent voltage-series and current-parallel source forms

Match each method with the description that states its defining analytical idea.

Crossword Puzzle

Current	What circuit quantity is the rate of flow of electric charge?
Voltage	What quantity represents electric potential difference?
Impedance	What complex quantity generalizes resistance in sinusoidal steady state?
Thevenin	Which equivalent theorem uses a voltage source in series with an equivalent impedance?
Resonance	What condition occurs when inductive and capacitive reactances cancel?
Transient	What do you call the temporary response following a switching event or disturbance?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Circuit diagram audit: Easy — Choose a small resistor network, redraw it neatly, label every node and reference direction, and write a short explanation of how your labels prevent sign errors.
Low-voltage resistor experiment: Easy — Build or simulate a safe low-voltage series-parallel resistor circuit, predict three voltages and currents before measurement, and compare prediction with data.
Kirchhoff visual guide: Easy — Create a one-page image or infographic that explains KCL and KVL using one original circuit example and clear sign conventions.
Engineering interview: Easy — Interview a laboratory technician, electronics student, or practicing engineer about how circuit analysis is used before measurement or troubleshooting, then summarize three insights in your own words.

Standard

Nodal analysis notebook: Standard — Solve one three-node circuit symbolically and numerically, present the conductance-matrix form, and verify the solution with KCL and a simulator.
RC transient investigation: Standard — Use an isolated low-voltage source, resistor, and capacitor or a simulator to record a charging or discharging curve, estimate the time constant from data, and explain deviations from the ideal model.
Thevenin laboratory study: Standard — Determine a two-terminal network's Thévenin voltage and resistance analytically, verify them by measurement or simulation, and test at least three load values.
Frequency response project: Standard — Design an RC low-pass filter for a chosen cutoff frequency, predict its Bode magnitude and phase, obtain simulated or measured data over at least two frequency decades, and discuss the cutoff region.

Advanced

RLC model comparison: Advanced — Derive a second-order model for an RLC circuit, classify its damping, and compare analytical, numerical, and simulated transient responses for at least two parameter sets.
Nonideal component study: Advanced — Create an ideal and a nonideal SPICE model of the same circuit, include at least two parasitic effects, and write a technical analysis of when the ideal model stops being adequate.
Sensitivity and uncertainty analysis: Advanced — Select a circuit whose output depends on component tolerances, derive or estimate sensitivity, run a parameter sweep or Monte Carlo study, and recommend tolerances for a stated performance target.
Circuit analysis mini-lecture: Advanced — Visit a university electronics laboratory or analyze its publicly documented equipment, then produce a short teaching video that connects one real measurement setup with its equivalent circuit, uncertainty sources, and safety assumptions.

Learning Assessment

Method selection assessment: Given three different linear circuits, justify whether nodal analysis, mesh analysis, or an equivalent-circuit theorem is the most efficient method for each and explain what would make another method less convenient.
Error diagnosis assessment: Analyze a worked solution containing at least three planted errors in signs, source deactivation, or units; identify each error, correct the calculation, and explain the physical check that reveals it.
Transient transfer assessment: Predict how doubling a resistance or capacitance changes a first-order transient, then transfer the same reasoning to an RL circuit and explain which quantity changes analogously.
AC power assessment: For a complex load, calculate P, Q, apparent power, and power factor, then explain how adding a compensating reactive element could change source current without changing the load's real power demand.
Model validation assessment: Compare hand calculations, simulation, and measured or synthetic experimental data for one circuit; quantify disagreement and argue whether component tolerance, instrument loading, parasitics, or modeling assumptions best explain it.
Design transfer assessment: Starting from a required cutoff frequency and load constraint, design a passive first-order filter, justify component values, predict sensitivity to tolerance, and state a practical limitation of the design.

Evidence of Learning

Evidence of learning should show more than correct final numbers. A strong portfolio should include:

Knowledge evidence: Accurate explanations of element models, Kirchhoff's laws, network theorems, transients, phasors, impedance, AC power, resonance, and transfer functions.
Analytical skill evidence: Correct nodal, mesh, equivalent-circuit, transient, and sinusoidal steady-state solutions with consistent reference directions and units.
Mathematical skill evidence: Clear use of simultaneous equations, complex arithmetic, differential equations, and Laplace-domain reasoning where each mathematical step is connected to a circuit assumption.
Product evidence: At least one reproducible technical artifact such as an annotated schematic, calculation notebook, simulation file, measured response plot, design report, infographic, or teaching video.
Validation evidence: A comparison among analytical prediction, simulation, and measurement or synthetic experimental data, including an explanation of meaningful discrepancies.
Communication evidence: A solution that another learner can reproduce, with stated assumptions, labeled variables, unit checks, and a concise physical interpretation.
Transfer evidence: Application of circuit-analysis methods to a new design, troubleshooting, sensing, filtering, energy, or electronics problem that was not used as a worked example in the course.
Reflection evidence: A reasoned account of which model or method was efficient, what limitations remained, and what additional measurement or model refinement would improve confidence.

OERs on the Topic

The English Wikipedia article on circuit analysis provides a useful starting point for linked definitions and related methods. Use it as a reference overview, then return to the worked reasoning and tasks in this aiMOOC for active practice.

You can also use the open undergraduate materials in MIT OpenCourseWare: Circuits and Electronics for lecture notes, problem sets, laboratory ideas, and additional examples.

Linked Learning Areas

Circuit Analysis

Circuit analysis connects mathematical modeling with physical measurement. In electrical and electronic engineering, it supports the design and diagnosis of power supplies, filters, sensors, amplifiers, communication hardware, and control systems. In physics, it provides tractable models of energy storage and dissipation. In mathematics and computation, it gives concrete applications of linear systems, differential equations, complex arithmetic, and sparse matrix methods.