Circuit Theory

Circuit analysis replaces a spatial electromagnetic problem with terminal voltages and branch currents. Reference arrows and polarity marks are bookkeeping choices; the passive sign convention determines whether positive power means absorption or delivery.

The basic one-port elements each impose one voltage-current constraint. Resistors dissipate energy, while capacitors and inductors store it and therefore carry state across time.

KCL and KVL are the topology-level conservation laws of lumped circuits. Their independent equation counts come from the graph of nodes and branches, not from the component types.

Real sources change terminal voltage or current when loaded. Thevenin and Norton source models make that interaction explicit and show why measuring open-circuit output is not enough to predict delivered power.

A circuit experiment validates a model only when measurement loading, instrument uncertainty, component tolerance, and residual definitions are explicit. Agreement is a quantitative comparison, not two rounded numbers that look close.

Resistive reduction removes internal nodes while preserving terminal behavior. Series and parallel rules are local; delta-wye conversion handles three-terminal resistor networks that contain neither pattern.

Direct branch-current analysis assigns an unknown to every branch and closes the system with independent KCL, KVL, and element equations. It is general and transparent, but usually larger than nodal or mesh formulations.

Nodal analysis solves for node potentials and obtains branch currents from element laws. It naturally handles current sources and produces a sparse conductance matrix; voltage sources require direct constraints or supernodes.

Mesh analysis assigns circulating currents to the elementary loops of a planar circuit. Shared-branch currents become differences of mesh currents, and current sources are handled by direct constraints or supermeshes.

Superposition decomposes a linear circuit response into contributions from independent sources. Source transformations replace a voltage-source/series-impedance pair with its Norton dual while preserving the same terminal relation.

Any linear one-port can be reduced at a chosen pair of terminals to one source and one impedance. The equivalent is terminal-specific and must retain dependent-source behavior through a test-source calculation.

Under one consistent sign convention, the algebraic sum of instantaneous powers in a lumped network is zero. The balance is both an energy statement and one of the best diagnostics for polarity and source-sign mistakes.

A potential diagram plots cumulative node potential along a chosen circuit path. It makes voltage polarities, source rises, resistor drops, and KVL closure visible while preserving the arbitrariness of the reference node.

A sinusoid is determined by amplitude, angular frequency, and phase. RMS converts its heating effect to an equivalent DC value, while phase differences determine timing and power exchange.

Phasors encode the magnitude and phase of same-frequency sinusoids as complex numbers. Differentiation becomes multiplication by $j\omega$, turning RLC differential equations into algebraic impedance relations.

AC steady-state analysis reuses the DC topology methods with complex impedances and RMS phasors. The algebra is unchanged, but frequency, conjugation, and reference phase must remain explicit.

Complex power packages average energy transfer and reactive energy exchange into one phasor product. Power factor is the fraction of apparent power that becomes average real power, with sign and leading/lagging behavior set by phase.

A phasor diagram shows complex vector relationships at one frequency; a topographic voltage diagram arranges those phasors in circuit-path order. Both are geometric audits of equations, not decorative plots.

Coupled inductors share magnetic flux, so changing current in either winding induces voltage in both. The dot convention records the sign of the mutual terms; coupling coefficient separates ideal linkage from leakage.

Driving-point functions relate voltage and current at the same port; transfer functions relate variables at different ports. Their numerical values depend on source and load conditions unless the termination is part of the definition.

A linear two-port compresses an internal network into relations among two terminal voltage-current pairs. ABCD parameters are arranged so cascaded sections combine by matrix multiplication.

T and Pi networks realize two-port behavior with three impedances or admittances. Their chain matrices can be built from elementary series and shunt sections and fitted to a target reciprocal two-port when the required parameters are realizable.

Image parameters describe a reciprocal two-port in the termination conditions that make cascades repeatable. A matched image termination reproduces the corresponding input impedance and lets the propagation constant add across identical sections.

A one-port model reproduces the voltage-current behavior seen at two terminals while hiding internal structure. Model topology determines what can be identified and which fitted parameters retain physical meaning.

Instantaneous power $p(t)=u(t)i(t)$ remains exact for arbitrary waveforms, while the familiar sinusoidal power triangle does not. Average power and RMS define a useful orthogonal current projection, but reactive-power decompositions are model-dependent.

Waveform-based identification fits an equivalent circuit directly to terminal voltage and current. A valid result requires excitation that separates parameters, noise-aware estimation, physical constraints, and validation on data not used for fitting.

A fitted resistance, inductance, or capacitance may depend on operating point or time, and incremental values can differ from static ratios. Signed or rapidly varying parameters demand an explicit energy and causality interpretation before they can be treated as physical elements.