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All DC analysis techniques — node voltage, mesh current, superposition, Thevenin/Norton — apply directly to AC circuits by replacing element resistances with complex impedances and sources with their phasor representations. The result is a system of complex algebraic equations whose solution gives phasor voltages and currents. The transfer function H(jω) = Y(jω)/X(jω) describes the ratio of output to input phasors and captures all frequency-domain behavior. When multiple source frequencies are present, superposition must be applied separately at each frequency.
Solve the same RLC circuit first in the time domain (differential equations) and then with phasors to appreciate the efficiency gain. Draw phasor diagrams to visualize phase relationships between voltages and currents. Practice finding Thevenin equivalents in the frequency domain with complex Z_th.
When you first learned node voltage or mesh current analysis, you solved DC circuits where every quantity was a real number. Then AC circuits introduced differential equations governing inductors (v = L di/dt) and capacitors (i = C dv/dt), which seemed to demand entirely different techniques. Phasor analysis makes a striking claim: no new techniques are needed. Every DC analysis method — node voltage, mesh current, superposition, Thevenin/Norton — applies directly to AC circuits once you replace resistance with complex impedance and represent sinusoidal sources as phasors.
The foundation is the phasor transform. A sinusoidal signal v(t) = V_m cos(ωt + φ) is represented by the phasor V = V_m∠φ — a complex number that encodes amplitude and phase but discards the common factor ejωt that all quantities share in steady state. When the circuit operates at a single frequency ω, the differential relationships for reactive elements collapse into algebraic ones: the capacitor's i = C dv/dt becomes I = jωC·V, and the inductor's v = L di/dt becomes V = jωL·I. Combined with the resistor's V = IR, these define complex impedances: Z_R = R, Z_C = 1/(jωC), Z_L = jωL. KVL and KCL hold for phasors exactly as they hold for DC quantities, because both are linear superposition relations.
With impedances replacing resistances, you can write node voltage or mesh current equations in the phasor domain by direct inspection — the same procedure as DC analysis, but with complex arithmetic. The solution gives complex phasor voltages; their magnitudes are peak amplitudes and their angles are phase shifts relative to the reference. Thevenin and Norton equivalents generalize to a complex Thevenin impedance Z_th and a frequency-dependent phasor V_th. The entire DC analysis toolkit transfers intact.
The transfer function H(jω) = V_out/V_in (or any output-to-input phasor ratio) is the principal payoff of this approach. Because H(jω) is a function of frequency, it encodes the circuit's behavior for all sinusoidal inputs — not just the one you happen to be analyzing. |H(jω)| gives the magnitude response (does the circuit amplify or attenuate at this frequency?) and ∠H(jω) gives the phase response. A low-pass filter has |H| ≈ 1 at low frequencies and |H| → 0 at high frequencies, which is immediately visible in the transfer function expression but obscured in any particular time-domain solution.
The hard constraint to remember is that phasor analysis assumes a single excitation frequency throughout. Inductor and capacitor impedances (jωL and 1/jωC) depend on ω; they take different numerical values at different frequencies. If a circuit has sources at two different frequencies, you cannot combine them in one phasor analysis — the impedances cannot simultaneously be correct at both frequencies. The correct procedure is superposition: analyze the circuit at each frequency separately using the appropriate impedances, convert each phasor result back to a time-domain sinusoid, then add the time-domain waveforms. This is not an approximation; it is exact for linear circuits.