ACcircuits

AC circuits are electrical networks in which currents and voltages vary with time, usually sinusoidally at a single frequency. They underpin power distribution, radio, and signal processing. Analysis commonly uses phasors and complex impedance to transform differential equations into algebraic ones.

In linear, time-invariant AC circuits, elements have frequency-dependent impedances: resistor Z_R = R, inductor Z_L = jωL, capacitor Z_C = 1/(jωC), with ω the angular frequency and j the imaginary unit. Ohm's law generalizes to V = IZ, where V and I are complex phasors.

Power in AC circuits is described by real power P, reactive power Q, and apparent power S. If V and I have a phase difference φ, P = VI cosφ, Q = VI sinφ, and S = VI. The power factor cosφ measures the efficiency of energy transfer.

Common configurations include RC, RL, and RLC networks. They can function as filters, resonators, or impedance matchers. For series circuits, resonance occurs when X_L = X_C, minimizing impedance; for parallel circuits, resonance maximizes impedance. Frequency response is typically depicted with Bode plots.

Applications range from power distribution and radio transmission to audio electronics and instrumentation. Real-world design must account for parasitics, nonlinearity, component tolerances, and frequency-dependent losses, which shape impedance magnitudes and phase relationships.