💡 AC quantities and circuits

Alternating voltage and current vary sinusoidally: v = V₀ sin ωt, i = I₀ sin(ωt + φ). Effective (RMS) values are V_rms = V₀/√2 and I_rms = I₀/√2. Inductors and capacitors introduce reactances X_L = ωL and X_C = 1/(ωC), so the series RLC impedance is Z = √(R² + (X_L − X_C)²). At resonance ω₀ = 1/√(LC), current is maximum and the circuit is purely resistive. Average power is P_avg = V_rms I_rms cos φ, where cos φ is the power factor. Transformers use turns ratios to step voltage up or down while approximately conserving power.

AC calculators

RMS, pure R/L/C, series RLC, resonance, power & transformers

Peak voltage and frequency

Peak current and phase

V_rms = —, I_rms = —, V_avg(half) = —

Uses V_rms = V₀/√2, I_rms = I₀/√2, V_avg (half‑cycle) = 2V₀/π.

Choose element and parameters

Value (R, L or C)

|Z| = —, I_rms = —, φ = —, cos φ = —

Resistor: φ = 0°, cos φ = 1. Inductor: φ = +90°, cos φ = 0. Capacitor: φ = −90°, cos φ = 0.

Series RLC circuit (R, L, C, V_rms, f)

|Z| = —, I_rms = —, φ = —, cos φ = —, f₀ = —, Q = —

Uses Z = √(R² + (X_L − X_C)²), tan φ = (X_L − X_C)/R, f₀ = 1/(2π√(LC)), Q = ω₀L/R.

RMS values and phase (for power)

P = —, S = —, Q = —, cos φ = —

Average power P = V_rms I_rms cos φ, apparent power S = V_rms I_rms, reactive power Q = V_rms I_rms sin φ.

Ideal transformer (Vp, Np, Ns, η)

Secondary current (load)

V_s = —, I_p = —, P_p ≈ —, P_s ≈ —

Uses V_s/V_p = N_s/N_p, I_p/I_s = N_s/N_p (ideal), P_s ≈ η P_p/100.

📈 AC waveforms & phasors

🧮Step-by-Step Solution▼ Show

AC basics & pure elements

Concept	Formula	Notes
Instantaneous voltage	v = V₀ sin(ωt)	ω = 2πf
Instantaneous current	i = I₀ sin(ωt + φ)	φ = phase difference
RMS values	V_rms = V₀/√2, I_rms = I₀/√2	≈ 0.707 × peak
Half-cycle average	V_avg(half) = 2V₀/π	Used in rectifiers
Pure resistor	Z = R, φ = 0°	V and I in phase
Pure inductor	Z = jX_L = jωL	V leads I by 90°
Pure capacitor	Z = −jX_C = −j/(ωC)	I leads V by 90°

Series RLC, resonance, power & transformers

Concept	Formula	Notes
Series RLC impedance	Z = √(R² + (X_L − X_C)²)	tan φ = (X_L − X_C)/R
Resonance	ω₀ = 1/√(LC), f₀ = 1/(2π√(LC))	X_L = X_C, Z = R
Quality factor	Q = ω₀L/R = 1/(ω₀RC)	Sharpness of resonance
AC power	P = V_rms I_rms cos φ	Real power (watts)
Power triangle	S² = P² + Q²	S = V_rms I_rms, Q = V_rms I_rms sin φ
Transformer voltage	V_s/V_p = N_s/N_p	Turns ratio
Transformer currents	I_p/I_s = N_s/N_p	Power conservation (ideal)
Efficiency	η = (V_s I_s)/(V_p I_p) × 100%	< 100% in real transformers

About this AC circuits tool

This page summarizes alternating current formulas commonly used in exams: sinusoidal voltage and current, RMS and average values, pure R/L/C behavior, series RLC impedance and resonance, AC power and power factor, and ideal transformers. Each calculator is paired with a short step-by-step explanation to reinforce the concepts.

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