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Phasor Transformation Visualizer

How to Use

  • Amplitude Vₘ slider — set the peak voltage of the sinusoid (1–10 V).
  • Phase φ slider — set the initial phase angle of the frozen phasor (−180° to 180°).
  • Speed slider — control the animation speed of the rotating phasor.
  • Animate / Pause button — toggle the rotating phasor animation on or off.

What to Observe

  • The green arrow on the right panel is the frozen phasor V = Vₘ∠φ — it captures amplitude and phase at t = 0.
  • When animating, the blue arrow rotates counterclockwise, representing V·e^(jωt).
  • The red dot on the real axis traces the real part of the rotating phasor — exactly matching the time-domain waveform on the left.
  • At t = 0, the rotating phasor coincides with the frozen phasor at angle φ.
  • Changing φ shifts the waveform left (positive) or right (negative) in time.

Key Equations

\[v(t) = V_m \cos(\omega t + \phi)\]
\[\mathbf{V} = V_m \angle \phi = V_m e^{j\phi} = V_m\cos\phi + jV_m\sin\phi\]
\[v(t) = \text{Re}\left\{\mathbf{V} e^{j\omega t}\right\}\]

The phasor V contains all the information about the sinusoid — amplitude and phase — without carrying the time variable. Since all signals in an AC circuit share the same frequency ω, the phasor representation is complete.

Key Concepts

  • Phasor: A complex number representing amplitude and phase of a sinusoid
  • Magnitude |V|: Peak amplitude of the waveform
  • Phase angle φ: Starting angle at t = 0 (positive = leads, negative = lags)
  • Rectangular form: V = a + jb, where a = Vₘcos(φ), b = Vₘsin(φ)
  • Polar form: V = Vₘ∠φ