Amplitude (signal)

Amplitude is the height of a waveform — its peak excursion away from zero. It is one of the three numbers that fully specify a single sinusoid, the other two being frequency and phase.

Write a sinusoidal voltage as

$v(t)=A\sin (\omega t+\phi )$

where $A$ is the amplitude in volts, $\omega$ is the angular frequency in rad/s, $t$ is time in seconds, and $\phi$ is the phase in radians. As $t$ runs forward, $\sin (\omega t+\phi )$ swings between $-1$ and $+1$, so $v(t)$ swings between $-A$ and $+A$. The constant $A$ is exactly how far the waveform reaches from the zero line at its highest point.

The three basic descriptors of a sinusoid $v(t)=A\sin (\omega t+\phi )$: amplitude $A$ (peak, peak-to-peak, RMS $=A/\sqrt{2}$), period $T$ and frequency $f=1/T$, phase $\phi$.

Three ways to quote amplitude

The same waveform gets described by three different “amplitude” numbers, and confusing them is a classic source of factor-of-2 and factor-of-$\sqrt{2}$ errors.

• Peak amplitude $A$ (also written $V_p$): the distance from zero to the maximum. This is the $A$ in the equation above.
• Peak-to-peak $V_{pp}$: the distance from the most negative point to the most positive point. For a symmetric sinusoid the waveform reaches $+A$ and $-A$, so $V_{pp}=2A$. Oscilloscopes typically display peak-to-peak because it is the easiest thing to read off a screen — you measure the total vertical span of the trace.
• RMS value $V_{rms}$: the Root mean square of the waveform. For a sinusoid, $V_{rms}=A/\sqrt{2}\approx 0.707A$. RMS is the amplitude that matters for power and heating.

For a $1\text{V}$ peak sine wave: $A=1\text{V}$, $V_{pp}=2\text{V}$, $V_{rms}=1/\sqrt{2}\approx 0.707\text{V}$. All three describe the same signal.

Why amplitude matters physically

A signal is the electrical stand-in for some real-world quantity, and amplitude is usually how the strength of that quantity gets encoded. For an audio signal, amplitude maps to loudness — a louder sound produces a microphone voltage with larger peak swing. For a radio wave, amplitude relates to transmitted power: the power a sinusoid delivers into a resistor goes as the square of its amplitude (this is why RMS is the right measure for power — see Signal power and Root mean square). Doubling the amplitude of a signal across a fixed resistor quadruples the power it dissipates.

Amplitude is also what an amplifier exists to change: the whole point of a voltage amplifier is to take a small-amplitude input and produce a larger-amplitude copy at the output, ideally without distorting its shape (i.e. without altering its frequency content or phase relationships).