AC Circuit Analysis

AC circuits are more complex than DC because the voltage and current are constantly changing, and capacitors and inductors respond to that change in ways resistors don't. The key insight that makes AC analysis tractable is impedance — a generalisation of resistance that works for capacitors and inductors too, using complex numbers to track the phase relationship between voltage and current.

The AC Waveform

V(t) = V_max × sin(ωt + φ)

• V_max — peak voltage
• ω = 2πf — angular frequency in radians/second
• f — frequency in Hz
• φ — phase angle in radians

For calculations involving power, use RMS values: V_RMS = V_max / √2 ≈ 0.707 × V_max

Reactance

Reactance is the opposition to current flow in capacitors and inductors. Unlike resistance, it depends on frequency:

Capacitive reactance:  X_C = 1 / (2πfC)    ohms
Inductive reactance:   X_L = 2πfL           ohms

At higher frequencies: X_C decreases (capacitor passes more current), X_L increases (inductor blocks more). This frequency-dependent behaviour is what makes filters possible.

Impedance

Impedance (Z) combines resistance and reactance into one complex quantity that obeys Ohm's Law for AC circuits:

Z = R + jX          (j = imaginary unit, √-1)

For a series RC circuit:   Z = R - jX_C
For a series RL circuit:   Z = R + jX_L
For a series RLC:          Z = R + j(X_L - X_C)

Magnitude: |Z| = √(R² + X²)
Phase angle: θ = arctan(X/R)

Ohm's Law for AC: V = I × Z — where V, I, and Z are all complex numbers (phasors).

Resonance

In an RLC circuit, X_L and X_C cancel when they're equal. The frequency where this happens is the resonant frequency:

At resonance: X_L = X_C
2πf₀L = 1/(2πf₀C)
f₀ = 1 / (2π√LC)

At resonance, the impedance is purely resistive (Z = R), and current is at maximum. This is how radio tuners work — adjust L or C until f₀ matches the station frequency, and that frequency gets maximum response.

Power in AC Circuits

Real power (P):     P = V_RMS × I_RMS × cos(θ)    watts (W)
Reactive power (Q): Q = V_RMS × I_RMS × sin(θ)    volt-amperes reactive (VAR)
Apparent power (S): S = V_RMS × I_RMS              volt-amperes (VA)

S² = P² + Q²

Power factor: PF = cos(θ) = P/S

Real power is what actually does work (heats, moves, computes). Reactive power is stored and returned each cycle by inductors and capacitors — it doesn't do useful work but it does flow in the wires and causes losses. Power factor close to 1.0 means the load is mostly resistive and efficient. Large motors and fluorescent lights have low power factors; this is why industrial customers pay power factor penalties.

RC Low-Pass Filter Example

        R
Vin ──[  ]──┬── Vout
            |
           [C]
            |
           GND

Cutoff frequency (f_c): where X_C = R
f_c = 1 / (2πRC)

At f << f_c: capacitor is high impedance, Vout ≈ Vin  (passes signal)
At f >> f_c: capacitor is low impedance, Vout ≈ 0      (attenuates signal)

Above the cutoff frequency, the output drops at -20dB/decade. This is a first-order filter. Stack two of them (or use an active filter with an op-amp) for steeper rolloff.