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Single-phase AC analysis — phasors, complex impedance, resonance, max power

Phasor algebra, Z_L = jωL, Z_C = -j/(ωC), series + parallel RLC, resonance with Q & bandwidth, and conjugate-matched maximum power transfer.

Sophomore ~11 min

Step 1 — Phasor representation: sinusoid v(t) ↔ complex amplitude V∠φ

Animation speed 0.55×

f — |Z| — φ —

Reference notes

Use Next → to walk through single-phase AC analysis: phasors, complex impedance, RLC circuits, series and parallel resonance, and maximum-power-transfer in AC.

Phasor representation

A sinusoidal voltage v(t) = V_m·cos(ωt + φ) is fully described by amplitude V_m and phase φ. Represent it as a phasor V = V_m∠φ = V_m·e^(jφ). The ωt rotation drops out because every element in a linear circuit oscillates at the same ω in steady state.

Complex impedance

Each linear element has a complex impedance Z = V/I measured in Ω (ohms):

Resistor: Z_R = R (real, current in phase with voltage)
Inductor: Z_L = jωL (imaginary positive, current lags voltage by 90°)
Capacitor: Z_C = 1/(jωC) = −j/(ωC) (imaginary negative, current leads voltage by 90°)

Series elements add Z's; parallel elements add admittances Y = 1/Z.

RLC circuits — frequency response

Series R-L-C:

Z(ω) = R + jωL + 1/(jωC) = R + j(ωL − 1/(ωC))

|Z| vs ω: a V-shape, minimum = R at the resonant frequency.

Parallel R-L-C (fed from a current source): admittance Y has the V-shape, so |Z| has an inverted V (maximum at resonance).

Series resonance

ω₀ = 1 / √(LC) f₀ = 1 / (2π · √(LC))

Q = ω₀·L / R = 1 / (ω₀·R·C) = (1/R)·√(L/C)

Bandwidth Δω = ω₀ / Q

At resonance, X_L = X_C → net reactance zero → |Z| = R (minimum), current is maximum I = V_s/R. Voltage magnification: V_L = V_C = Q·V_source.

Used: filters, oscillators, antenna matching, harmonic-injection circuits. Q = 1 is critically damped (broadband); Q > 10 is high-Q narrow-band.

Parallel resonance — the "rejection" circuit

Parallel L-C at ω_0: L and C currents are equal magnitude and 180° out of phase → they CIRCULATE within the LC tank without drawing from the source. Source sees |Z| = MAXIMUM at f_0.

Used: notch filters, IF rejection in radio receivers, harmonic-trap filters in power systems (e.g., a 5th-harmonic trap is a series LC tuned to 5·f_1 placed in parallel with the bus — shorts out the 5th but presents high impedance to other frequencies).

Maximum power transfer

For a Thevenin source with internal impedance Z_s = R_s + jX_s feeding a load Z_load = R_L + jX_L:

Max P transferred when Z_load = Z_s* = R_s − jX_s

P_max = |V_s|² / (4·R_s)

The load's reactive component cancels the source's reactive component (conjugate); then R_s and R_load divide the source voltage equally. Applications: antenna matching to transmission lines, audio amplifier output stages, PV MPPT (PV cell is mostly resistive at the operating point, so conjugate ≈ resistive match), RF networks at every wireless interface.

Take-away. Phasors replace time-varying sinusoids with complex amplitudes. Z_R = R, Z_L = jωL, Z_C = −j/(ωC). Series RLC resonates at ω_0 = 1/√(LC) with Q = ωL/R determining sharpness. Parallel LC at resonance rejects f_0. Maximum power transfer in AC requires conjugate-matched load Z_load = Z_s*. P_max = |V_s|²/(4·R_s).

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