Various Waveforms: Sine, Square, Triangle and Transients

What is a waveform?

A waveform is a graph. Time goes along the bottom. Voltage (or current) goes up the side. When the line is above the middle, the push is in one direction. When it is below, the push is reversed.

The time for one full pattern is the period (T). How many patterns fit in one second is the frequency (f). They are linked: f = 1 / T. India's mains: 50 Hz, so T = 0.02 s.

The highest point is the peak (V_m). The RMS value is the steady DC voltage that would heat a lamp equally. For a sine wave, V_rms = V_m / √2 (about 0.707 V_m).

Sine, square, triangle and saw waves

Sine wave: smooth. It is what a spinning generator coil makes, so mains power is a sine wave. A plain sine wave is called sinusoidal.

Any wave that is not a sine is non-sinusoidal. Common ones:

• Square wave: jumps between two levels. Made by switches and digital circuits. V_rms = V_m.
• Triangle wave: straight slopes up and down. V_rms = V_m / √3.
• Saw (ramp) wave: slow rise, quick drop. Used in old TV scanning and in tone makers.

A sine wave is "clean". The others carry extra faster wiggles, and these can heat motors and disturb radios.

Non-sinusoidal waves are a sum of sines

A French scientist, Fourier, showed that any repeating wave is a sum of sine waves. The first sine is the fundamental. The others are harmonics, with 2, 3, 4 ... times the frequency.

A square wave needs only odd harmonics: fundamental + 1/3 of the 3rd + 1/5 of the 5th + ... The more you add, the sharper the corners. A triangle wave also uses only odd harmonics, but they shrink much faster, so it stays smooth.

Harmonics matter in real life: an inverter, a charger or a dimmer pushes harmonic currents back into the wiring.

Transient: what happens just after a switch

Close a switch on a resistor R in series with a capacitor C. The capacitor voltage does not jump. It climbs in a curve and then settles. This short settling stage is the transient. After it, the steady state begins.

The speed is set by the time constant τ = R × C (in seconds, with R in ohms and C in farads).

• After 1 τ: voltage is about 63% of final.
• After 3 τ: about 95%.
• After 5 τ: about 99%, so we call it "done".

Charging: v = V (1 − e^−t/τ). Discharging: v = V e^−t/τ. An R-L circuit does the same with τ = L / R.

Try it: build a wave yourself

In the 3D scene press Build. Slide the "sine waves" slider from 1 to 9. Predict first: will the top become flatter or rounder? Then check. Next press Switch-on. Move τ to 3. Predict: does the curve rise faster or slower? Check where the green bar meets the curve: it is always at 63%.

At home: watch a small LED with a big capacitor and resistor in a kit. A larger R or C makes the LED fade more slowly.

Key formulas and definitions

• f = 1 / T
• Sine wave: V_rms = V_m / √2 ≈ 0.707 V_m
• Square wave (±V_m): V_rms = V_m
• Triangle wave: V_rms = V_m / √3 ≈ 0.577 V_m
• Square wave = sin x + (1/3) sin 3x + (1/5) sin 5x + ... (times 4/π)
• RC time constant: τ = R × C; RL time constant: τ = L / R
• Charging capacitor: v = V (1 − e^(−t/τ))

Worked examples

Common mistakes

• Using V_m / √2 for every wave. That formula is only for a sine wave. Square gives V_m and triangle gives V_m / √3.
• Thinking the capacitor voltage jumps instantly. It rises along a curve; only after about 5 τ is it nearly full.
• Mixing up period and frequency. Period is seconds per cycle; frequency is cycles per second. f = 1 / T.
• Using microfarads directly in τ = RC. Convert first: 100 µF = 0.0001 F.