Phasor Playground

Two phasors on a rotating dial, and the sine waves they trace out beside them. Change magnitude and phase and watch both views move together — the whole point being that a spinning arrow and a sine wave are the same information drawn two ways.

Frequency (cycles across window)1.0

v₁ magnitude5.0v₁ phase (°)0°v₂ magnitude5.0v₂ phase (°)-60°

Phase difference: v₁ − v₂ = 60°(v₁ leads by 60°)

A rotating arrow and a sine wave are the same thing

Take an arrow of fixed length, spin it at a steady rate, and plot its vertical projection against time. What you get is a sine wave. The arrow is a phasor: its length is the amplitude and its angle is the phase. Every sinusoid in a circuit at the same frequency can be represented this way, which is what makes the whole method work.

Why bother, when you already have the waveform

Because adding sine waves algebraically is painful and adding arrows is not. Two sinusoids of the same frequency, whatever their phases, sum to another sinusoid of that frequency — so you can add them head-to-tail like vectors and read the answer off. Every derivative and integral becomes a multiplication. That is the entire reason AC analysis is tractable.

The frequency has to be common

A phasor diagram freezes the rotation, which only makes sense if everything rotates together. That is why the widget above has one frequency slider driving both phasors: two sinusoids at different frequencies have a phase difference that changes constantly, so there is no fixed angle to draw. Mixed-frequency problems need superposition — solve at each frequency separately, then add the results in the time domain.

Leading and lagging

A phasor at a greater angle reaches its peak earlier in time, so it leads. In a capacitor the current leads the voltage by 90°; in an inductor it lags by 90°. Those two facts, drawn as arrows at right angles to the resistive component, are where the impedance triangle comes from.

Learn more → Sine Wave Generation — Learn

Quick experiments

Watch phase become a time shift. Leave v₁ at 0° and drag v₂ to −60°. On the dial it is an angle; on the waveform it is a horizontal shift. At 50 Hz, 60° is one sixth of 20 ms, so about 3.3 ms of delay — the same fact in two languages.
Find the quadrature point. Set the phase difference to exactly 90°. One phasor is at its peak precisely when the other crosses zero. This is the capacitor and inductor relationship, and the reason reactive elements consume no average power.
Make two waves cancel. Give both phasors equal magnitude and set them 180° apart. The arrows point opposite ways and the waves are exact mirrors — add them and you get nothing. This is destructive interference, and how noise cancellation works.
See that magnitude does not move the peaks. Change v₂'s magnitude while leaving its phase alone. The wave grows and shrinks but its zero crossings stay put. Amplitude and phase are genuinely independent, which is why a phasor needs both numbers.
Speed up the rotation and watch the phase hold. Drag the frequency slider. Both waves compress, but the angle between the phasors never changes — because they share one frequency. That invariance is exactly the condition that makes phasor analysis legal.

Formula reference

Sinusoid to phasor: $v(t) = V_m\cos(\omega t + \phi) \;\Longleftrightarrow\; \mathbf{V} = V_m \angle \phi$
Amplitude and phase; the frequency is carried separately.
Rectangular and polar forms: $\mathbf{V} = a + jb = |V| \angle \theta, \quad |V| = \sqrt{a^2+b^2}$
Add in rectangular, multiply in polar.
Adding two phasors: $|V| = \sqrt{V_1^2 + V_2^2 + 2V_1V_2\cos\Delta\phi}$
The cosine rule. At 90° it reduces to Pythagoras.
Phase difference as time: $\Delta t = \frac{\Delta\phi}{360°} \times T$
60° at 50 Hz is about 3.3 ms.
Angular frequency: $\omega = 2\pi f$
Radians per second — the rotation rate of the phasor.
Reactive phase shifts: $\mathbf{Z}_L = j\omega L \qquad \mathbf{Z}_C = \frac{1}{j\omega C}$
The j is the 90° rotation: current lags in L, leads in C.

Symbol	Meaning	Unit
$V_m$	Peak amplitude — the phasor's length	V
$\phi$	Phase angle — the phasor's direction	°
$\omega$	Angular frequency	rad/s
$\Delta\phi$	Phase difference between two phasors	°

Common mistakes

Drawing phasors for signals at different frequencies on one diagram.
The angle between them would change continuously, so there is no diagram to draw. Phasor analysis requires a single common frequency. For mixed-frequency circuits use superposition: solve at each frequency, then add in the time domain.
Adding the magnitudes of two phasors arithmetically.
Only valid when they are exactly in phase. Two 5 V phasors 90° apart give 7.07 V, not 10 V, and 180° apart they give zero. Add them as vectors, or use the cosine rule.
Mixing peak and RMS values in the same diagram.
Both conventions are used, and they differ by a factor of √2 for a sine. Pick one and label it. A phasor diagram in RMS gives power directly; one in peak matches the waveform you see on a scope.
Confusing leading with 'bigger'.
Leading is about angle, not magnitude. A small phasor at +90° leads a large one at 0°, meaning it peaks earlier in time. Length and angle carry completely separate information.
Forgetting that the phase reference is arbitrary.
Only differences between phasors are physical. Rotating the whole diagram changes nothing, which is why you are free to pick any signal as the 0° reference — usually the source, or the current in a series circuit.

Frequently asked questions

What is a phasor?

A rotating arrow whose length is the amplitude of a sinusoid and whose angle is its phase. Plot the arrow's vertical projection against time and you get the sine wave back, so the two are the same information drawn differently.

Why use phasors instead of the waveforms?

Because adding sinusoids algebraically is awkward while adding arrows is not, and calculus becomes arithmetic: differentiation turns into multiplication. That is what makes AC circuit analysis tractable.

Can I draw phasors for signals at different frequencies?

No. The angle between them would change continuously, so there is no fixed diagram. Phasor analysis needs one common frequency; for mixed-frequency circuits solve at each frequency separately and add the results in the time domain.

How do I add two phasors?

As vectors, not as numbers. Use the cosine rule: the magnitude is the square root of V1 squared plus V2 squared plus twice V1 V2 times the cosine of the phase difference. Two 5 volt phasors 90 degrees apart give 7.07 volts, not 10.

What does leading and lagging mean?

A phasor at a greater angle reaches its peak earlier in time, so it leads. In a capacitor the current leads the voltage by 90 degrees; in an inductor it lags by 90 degrees. Leading is about angle, not about magnitude.