AC Waveform & Phasor Diagram Simulator

A phasor is a complex-vector representation of a sinusoid's magnitude and phase. In power engineering the magnitude is commonly represented as RMS, while the corresponding physical sine-wave peak is √2 times RMS. A rotating-vector picture is a useful visualization after choosing a scale, but the phasor itself is normally treated as a stationary complex quantity in steady-state analysis. Voltage and current phasors at the same frequency maintain a fixed phase difference θ.

In a pure resistor, voltage and current are always proportional and in step (θ=0°) because Ohm's law applies instantaneously at every point in the cycle. In a pure inductor, voltage leads current by 90° because an inductor opposes the rate of change of current, not the current itself — current peaks a quarter-cycle after voltage. In a pure capacitor, the relationship reverses: voltage lags current by 90° because a capacitor's voltage depends on accumulated charge, which peaks a quarter-cycle after the current that's been charging it.

Z = R + j(X_L−X_C), where X_L=2πfL is inductive reactance and X_C=1/(2πfC) is capacitive reactance. The magnitude |Z|=√(R²+(X_L−X_C)²) sets how much current flows for a given voltage (I=V/|Z|), and the phase angle θ=arctan((X_L−X_C)/R) sets how far current lags (positive θ, net inductive) or leads (negative θ, net capacitive) voltage. When X_L exactly equals X_C, they cancel — the circuit becomes purely resistive at that specific frequency, the condition known as resonance.

Real power P=VIcosθ is the power actually converted to useful work (heat, light, mechanical output) — only the in-phase component of current contributes. Reactive power Q=VIsinθ is the power that sloshes back and forth between source and reactive components (inductors and capacitors) without being consumed, but still requires current-carrying capacity from the supply and wiring. Apparent power S=VI is what the supply actually has to be rated for, and cosθ (the power factor) is the fraction of apparent power that's actually doing useful work — which is why utilities and facility managers care about correcting poor power factor even though it doesn't show up as 'wasted energy' on a simple kWh meter.

At a real 60 Hz supply frequency, the voltage and current phasors complete 60 full rotations every second — far too fast to see the rotation or read the phase angle by eye. This simulator deliberately slows the animation to a fixed, visually useful pace regardless of the frequency you enter, so you can actually watch the phasors rotate and see the waveforms trace out; the frequency value still correctly affects the calculated reactances, impedance and phase angle, just not the on-screen animation speed.

In a series circuit, V_R, V_L and V_C don't add up arithmetically to the supply voltage — they add vectorially, because each carries a different phase relative to the common current: V_R is in phase with I, V_L leads I by 90°, and V_C lags I by 90°. Their vector sum always equals the total supply voltage exactly (this is Kirchhoff's voltage law applied to phasors), even though V_R+V_L+V_C as plain numbers can add up to far more than the supply voltage — a genuinely counterintuitive result the first time you see it, and exactly why series resonant circuits can show component voltages well above the source voltage.

A parallel circuit flips the series picture: voltage is the same across every branch, and it's the branch currents that carry different phases — I_R in phase with V, I_C leading V by 90°, I_L lagging V by 90°. The total current is their vector sum, found using admittance Y=1/R+j(1/X_C−1/X_L) rather than impedance. One easy-to-miss consequence: increasing R in a series circuit reduces its Q factor, but increasing R in a parallel circuit increases its Q — the two topologies respond to resistance in opposite directions.

At the resonant frequency f₀=1/(2π√(LC)), inductive and capacitive reactance exactly cancel, and a series circuit's impedance drops to its minimum (just R), while a parallel circuit's impedance rises to its maximum. The quality factor Q describes how sharp that resonance peak is: Q=ω₀L/R for series, Q=R√(C/L) for parallel — a higher Q means a narrower, more sharply-tuned response, with bandwidth (the frequency range over which the response stays within about 70.7% of its peak) equal to f₀/Q.

A balanced three-phase system has three equal-magnitude voltages 120° apart in phase — phase B at −120° and phase C at +120° relative to phase A for the standard A-B-C sequence (equivalently C may be written as −240°). For a wye (star) connection, line-to-line voltage is √3 times phase voltage (roughly 1.732×), not simply double or triple — a direct consequence of subtracting two phasors 120° apart rather than two in-phase quantities. Balanced three-phase power is constant over time (unlike single-phase, whose instantaneous power pulses at twice the supply frequency), which is exactly why three-phase is preferred for motors and heavy industrial loads.

It comes down to what each component actually opposes. An inductor generates a back-EMF proportional to the rate of change of current, so current can't jump instantly — it builds up gradually after voltage is applied, putting current a quarter-cycle behind. A capacitor's voltage is proportional to the charge it has accumulated, so current has to flow first to deposit that charge before voltage can rise — putting current a quarter-cycle ahead of voltage.

It means the inductive and capacitive reactances exactly cancel at the frequency you're operating at (XL = XC), which happens at one specific frequency for any given L and C — the resonant frequency. At that frequency, the phase angle is exactly zero and the circuit draws current as if the inductor and capacitor weren't there at all, even though real energy is still oscillating back and forth between them internally.

Because the electrical supply, wiring, and transformers still have to be sized for the full apparent power (S = VI), including the reactive component — a low power factor means more current has to flow for the same useful real power delivered, which means larger conductors, more resistive (I²R) losses in the wiring itself, and less spare capacity on the same equipment. That's a real cost even though the reactive power itself returns to the source rather than being consumed.

Not exactly — real inductors have winding resistance and core losses, real capacitors have small leakage and dielectric losses, and both can show frequency-dependent behavior beyond the simple XL=2πfL and XC=1/(2πfC) formulas at high frequencies. This simulator uses ideal, lossless component models, which is the standard simplification for teaching phase relationships and is a good approximation for many real components well within their rated frequency range.

Because they add vectorially, not arithmetically — near resonance in a series RLC circuit, VL and VC can each become very large while mostly canceling each other (since they're 180° apart in phase), leaving a modest net voltage that satisfies Kirchhoff's law even though the individual component voltages are far higher than the source. This is a real, useful effect (it's how some voltage step-up resonant circuits work) but also a real hazard if component voltage ratings aren't checked against it.

In a series circuit, resistance is in the current's only path, so more R directly damps the resonance more (lower Q). In a parallel circuit, resistance is one of several parallel paths, so more R means less current diverted through that damping path relative to the reactive branches, sharpening the resonance instead (higher Q) — the same physical component playing structurally different roles depending on topology.

Because line voltage is the vector (phasor) difference between two phase voltages that are 120° apart, not a simple sum — subtracting two equal-length phasors 120° apart geometrically gives a result √3 times their length (a direct trigonometric consequence of the 120° angle between them), not double or triple as you'd get from either adding two in-phase quantities or naively multiplying.