AC Sources and Phasor Representation

Core Idea

AC sources produce sinusoidal voltage and current: v(t) = V_m·sin(ωt + φ). Phasor representation converts sinusoidal signals to complex numbers in the frequency domain: V = V_m∠φ = V_m·e^jφ. This transformation converts differential equations to algebraic equations, making AC circuit analysis practical. Phasors assume all signals operate at the same frequency, a reasonable assumption for circuits with one source frequency.

Explainer

You've worked with DC circuits where voltages and currents are constant. AC circuits introduce a new complication: voltages and currents vary sinusoidally with time. A sinusoidal voltage v(t) = V_m·sin(ωt + φ) has three properties you need to track simultaneously: its amplitude V_m (peak value), its angular frequency ω (how fast it oscillates, in radians per second), and its phase φ (where in its cycle it starts at t = 0). Analyzing circuits with reactive elements under sinusoidal excitation means solving differential equations — the current through a capacitor is C·dv/dt, the voltage across an inductor is L·di/dt. Doing this from scratch for every circuit would be prohibitively tedious.

Phasor representation is the key simplification. If you know that every signal in a circuit oscillates at the same frequency ω — a reasonable assumption when there is one AC source — then the only information that distinguishes one signal from another is its amplitude and phase. The phasor for v(t) = V_m·sin(ωt + φ) is simply the complex number V = V_m∠φ = V_m·e^jφ. You have encoded all the relevant information into a single complex number, stripping away the time-varying factor e^jωt that is identical for every signal in the circuit.

The power of this representation becomes apparent when you differentiate. In the time domain, differentiating a sinusoid gives another sinusoid at the same frequency but with shifted phase and scaled amplitude. In the phasor domain, this corresponds to multiplying by jω — a purely algebraic operation. This is the central insight: differential equations in the time domain become algebraic equations in the phasor domain. Kirchhoff's voltage and current laws still hold, but now applied to complex numbers rather than time-varying functions. Series and parallel combinations, voltage dividers, node voltage analysis — all of the techniques you know from resistive circuits carry over to AC circuits once you replace resistances with complex impedances: R for resistors, 1/(jωC) for capacitors, and jωL for inductors.

The connection to complex exponentials (your prerequisite) is what makes this rigorous rather than a computational trick. Euler's formula gives e^jθ = cos(θ) + j·sin(θ), so a sinusoid is the real part of a rotating complex exponential: V_m·cos(ωt + φ) = Re{V_m·e^jφ·e^jωt}. The phasor V = V_m·e^jφ is the complex amplitude — the "snapshot" of the rotating vector at t = 0. When you solve a circuit in the phasor domain and want the time-domain answer, you attach e^jωt back and take the real part. This formal grounding ensures that phasor analysis gives exactly the same answers as solving the differential equations directly — just far more efficiently.