AC circuits and phasors
Learn AC by playing with it — sine waves and rms; rotating phasors projected onto sines, a complex-number calculator and Euler's formula; R, L and C (ELI the ICE man), reactance and live impedance triangles; resonance, Q and the energy sloshing between L and C; power, power factor and correction; three-phase and the neutral current; transformers and the conjugate match — each with a randomised exercise to solve.
Added
Works with
- Timing, filters, and analog design Electronicsquick RC, RL and LC time constants, filter cutoffs and Bode plots for the same parts
- Complex numbers, visually Mathematicsthe general maths of complex numbers behind phasors — roots, maps and the Euler series
- Unit circle and waves Mathematicsthe unit circle that a phasor's tip runs round, and sinusoids fitted to data
- Smith chart Radio and RFimpedance at radio frequencies — move a load with series and shunt parts and lines
- Circuit simulator Electronicsbuild any of these circuits and run an AC sweep or a transient on a simulated scope
How to use it
AC circuits are easiest to understand by watching them. Each tab takes one idea, shows it moving,
explains it, and ends with a Solve this exercise whose values change every time — type your
answers (engineering notation such as 4.7k, 100n or 416u works), Check, and Show working
to see the method.
The animations run in slow motion — a quarter of a turn per second, whatever the frequency — while the numbers are the real ones. Pause animation (above the tabs) freezes them all; Speed changes the pace.
How to use it
-
1 · Sine waves — choose a sine, square or triangle and set its peak, frequency and phase; the plot marks the peak, peak-to-peak, rms and rectified average, and the table gives the form and crest factors. 230 V 50 Hz and 120 V 60 Hz load the mains. Below, add two sines: at the same frequency the sum is another sine (the phasors add head to tail, right); at different frequencies it beats.
-
2 · Phasors — an arrow spinning at ω, with its height traced out as a sine. Tick second phasor B and drag its phase to see lead and lag; show A + B adds them. The calculator below takes
3+4j,3-j4,5∠53.13or2←30and adds, subtracts, multiplies or divides, with the working in both forms. Euler’s formula shows ejθ = cos θ + j sin θ on the unit circle. -
3 · Impedance — ELI the ICE man: the voltage and current of a lone R, L or C. Then pick series or parallel RL, RC or RLC, type R, L and C, and drag the frequency: the reactance plot, the impedance (or admittance) triangle and the phasor diagram all follow. Go to f₀ jumps to resonance.
-
4 · Resonance — |Z| and the current against frequency for a series or parallel RLC, with f₀, Q, the bandwidth and the magnification (VC = Q·V in series), and the safety note that goes with it. Below, a charged capacitor rings into the inductor: watch the energy move between C and L and leak into R; raise R for critical and over-damping.
-
5 · Power — v, i and p = v·i for a load R + jX: the shaded areas are energy delivered and returned. The power triangle shows P, Q and S, and with a target power factor the capacitor that corrects it (dashed). Below: a three-phase capacitor bank in star and in delta, and what correction does — and does not — save on a bill.
-
6 · Three-phase — three phasors 120° apart with the line voltage √3 × phase at +30°. Load it in star with a neutral, star without one, or delta; set R and X per phase (or tick balanced). The current diagram lays Ia, Ib and Ic head to tail: balanced, they close and the neutral current is zero; unbalanced, the gap is IN — or, without a neutral, the star point shifts.
-
7 · Transformers — turns, primary voltage and a complex load give Vs, Is, Ip and the impedance the source sees, a²·ZL. Below, maximum power transfer: a map of load power over the RL–XL plane with the conjugate match marked, a slice through it, and the best pure resistor.
Settings and your exercise score are remembered in this browser.
Method and assumptions
-
Everything is linear and at a single frequency (sinusoidal steady state), except the ringing capacitor, which uses the exact solution of the series RLC for each damping regime. Phasors are rms unless shown as peak; the angle of an impedance is positive for inductive loads (current lagging).
-
Power factor is cos φ, the displacement power factor; distorted (non-sinusoidal) currents from rectifiers lower the true power factor further and put third-harmonic current in a neutral even when the load is balanced — outside this model.
-
The parallel RLC is ideal R ‖ L ‖ C, with Q = R√(C/L); a real inductor’s series resistance makes the formulas approximate at low Q. The ringdown for a parallel tank is drawn as the series loop with the same Q.
-
The transformer is ideal: no losses, no magnetising current, perfect coupling.
-
The bill model is illustrative: a kVA maximum-demand charge, a reactive-energy charge beyond a power factor of 0.95 (common in UK and EU business tariffs), the energy charge and the I²R loss in a supply cable. Real tariffs vary.
-
Three-phase uses sequence a–b–c with Va at 0°. Nominal voltages from IEC 60038 (230/400 V) and ANSI C84.1 (120/208 V, 277/480 V).
Sources: Hughes, Electrical and Electronic Technology (AC chapters, three-phase and transformers); Boylestad, Introductory Circuit Analysis (phasors, resonance, power); Horowitz and Hill, The Art of Electronics (reactance, resonance and Q).
Feedback is a gift. Something to improve, a feature to add, or a bug? Tell me.
Runs entirely in your browser; nothing you type leaves this page.