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Sinusoidal Waveform Anatomy

Run the Sinusoidal Waveform Anatomy MicroSim Fullscreen

To experiment with the code, paste sinusoid-anatomy.js into the p5.js editor.

About This MicroSim

A sinusoidal voltage is completely described by three numbers: its amplitude \(V_m\), its frequency \(f\), and its phase \(\phi\):

\[v(t) = V_m \sin(2\pi f t + \phi)\]

This MicroSim draws that waveform on fixed axes (−50 ms to 200 ms, ±13 V) so that moving a slider changes the shape on the screen, not just the axis labels. Every part of the waveform's "anatomy" is labeled and updates as you move the sliders:

Label Color Meaning
Peak \(V_m\) red dashed lines and dots Maximum value measured from zero
RMS purple dotted line \(V_{rms} = V_m/\sqrt{2} \approx 0.707\,V_m\), the equivalent DC value for power
Peak-to-peak orange arrow \(V_{pp} = 2V_m\), the full swing from trough to peak
Period \(T\) teal arrow \(T = 1/f\), the time for one complete cycle
Zero crossings green circles Instants where \(v(t) = 0\), spaced \(T/2\) apart
Phase shift magenta bar How far the waveform is shifted from the dashed zero-phase reference

The formula at the top is rewritten with the current slider values, together with the angular frequency \(\omega = 2\pi f\) and the period \(T = 1/f\).

How to Use

  1. Move the Amplitude slider (1 V to 10 V). The peak, RMS and peak-to-peak labels all scale together.
  2. Move the Frequency slider (1 Hz to 100 Hz). More cycles fit in the same 250 ms window and the period arrow shrinks.
  3. Move the Phase slider (−180° to +180°). A positive phase slides the waveform to the left (it leads the dashed reference); a negative phase slides it to the right (it lags).
  4. Click RMS Level to show or hide the RMS line.
  5. Click Reference to switch between the sine form \(V_m \sin(\omega t + \phi)\) and the cosine form \(V_m \cos(\omega t + \phi)\). The dashed reference curve and the formula change with it.
  6. Click Reset to return to the defaults: 5 V, 10 Hz, 0°.

Check the numbers

With the defaults (\(V_m = 5\) V, \(f = 10\) Hz):

  • \(V_{rms} = 5/\sqrt{2} = 3.54\) V
  • \(V_{pp} = 2 \times 5 = 10\) V
  • \(\omega = 2\pi \times 10 = 62.8\) rad/s
  • \(T = 1/10 = 100\) ms

Setting the phase to +90° shifts the waveform left by a quarter period, \(T/4 = 25\) ms.

Iframe Embed Code

You can add this MicroSim to any web page by adding this to your HTML:

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<iframe src="https://dmccreary.github.io/circuits/sims/sinusoid-anatomy/main.html"
        height="577px"
        width="100%"
        scrolling="no"></iframe>

Lesson Plan

Learning Objective

Students will explain the relationship between the parameters of a sinusoid (amplitude, frequency, phase) and the resulting waveform shape.

Target Audience

College students in a first circuits course (electrical engineering or computer science majors).

Duration

10-15 minutes

Prerequisites

  • Sine and cosine functions and radian measure
  • Voltage as a quantity that can vary with time
  • Square roots

Activities

  1. Predict, then check (5 min): Before touching a slider, students predict what will happen to the picture when the amplitude doubles, when the frequency doubles, and when the phase changes from 0° to +90°. They then move each slider and compare.
  2. Guided measurement (5 min): Set \(V_m = 8\) V and \(f = 20\) Hz. Students compute \(V_{rms}\), \(V_{pp}\), \(\omega\) and \(T\) by hand and confirm each value against the labels (5.66 V, 16 V, 125.7 rad/s, 50 ms).
  3. Sine or cosine? (5 min): With the sine reference selected, set the phase to +90°. Switch to the cosine reference and set the phase to 0°. Students explain why the two solid curves are identical, using \(\sin(\omega t + 90°) = \cos(\omega t)\).

Assessment

  • Given a formula such as \(v(t) = 6\sin(2\pi \cdot 25 \cdot t - 45°)\) V, students state the peak, RMS, peak-to-peak, period, and whether the waveform leads or lags the zero-phase reference, then verify with the MicroSim.
  • Students explain in one or two sentences why the RMS line always sits at about 71% of the peak, no matter the frequency or phase.

References

  1. Wikipedia: Sine wave - Definition of a sinusoid and its amplitude, frequency and phase parameters
  2. Wikipedia: Root mean square - Why the RMS value of a sinusoid is the peak divided by the square root of two
  3. Wikipedia: Phase (waves) - Phase shift, leading and lagging
  4. p5.js Reference - Documentation for the p5.js library used to build this MicroSim