Waveforms

Chief Noter

July 31, 2021
3 min read
Electronics Modulation Waves

In electronics, signals are constantly being recorded, processed and given out. These can be analogue or digital with sinusoidal & geometric waveforms respectively. Either way, it is important to understand the waves and be familiar with reading them.

Sinusoidal Waveforms
Geometric Waveforms
Summary

Sinusoidal Waveforms

Mains electricity is delivered as an alternating current at 50 Hz. This can be represented on sine
and cosine waves:

$v=V\sin (\omega t)=V\sin (\frac{2\pi t}{T})$

$v=V\cos (\omega t)=V\cos (\frac{2\pi t}{T})$

$v$ is the quantity being plotted against time
$V$ is the maximum value of $v$
$\omega$ is the frequency in radians per second
$t$ is time
$T$ is the time period

$\omega=\frac{2\pi}{T}$

To convert between rad/s and rpm:

$rad s^{-1} = \frac{\pi}{30}(rpm)$

Phase

Two sinusoidal waves can be in phase if they overlap, or out of phase if they are the same, but
slightly offset in time. This time is called the phase lead or phase difference, $\phi$ :

$v=V\cos(\omega t+\phi)$

Root Mean Square Amplitude

The actual average of a sine wave is zero, but this is not helpful. Therefore, the root mean square
is used to find an effective average of sinusoidal graphs:

$V_{eff}=\sqrt{\frac{1}{T}\int^T_0\frac{v^2}{R}dr}$

The effective voltage for a sine wave is given by:

$V_{eff}=\frac{V_m}{\sqrt{2}}=0.707 V_m$

Geometric Waveforms

An analogue signal can be approximated digitally using square waveforms.

square wave form, digital wave, digital signal, electronics signal

Alternatively, sawtooth and triangular waveforms might be used:

sawtooth wave, triangular wave, geometric waveformsl

The average value of a periodic waveform like the ones above can be found by integrating and dividing by the time period:

$\bar X=\frac{1}{T}\int^T_0x(t)dt$

The amplitude of the waveform is the distance between the maximum and average values:

$A=\max(x(t))-\bar X$

This is not necessarily the distance from the x-axis.

Pulse Width Modulation

Often, we want to regulate the power output of a circuit. This can be done by varying the current, voltage or resistance:

$P=IV=I^2R=\frac{V^2}{R}$

However, for a square waveform, it can also be done by varying the time the wave is on for in each cycle:

pulse width modulation, signal modulation, digital modulation

$\alpha=\frac{on \quad time}{off \quad time}$

The greater the value of α, the greater the output power.

We can calculate the root mean square voltage and current for a square wave form:

$V_{rms}=V_{max}\sqrt{\alpha}$

$I_{rms}=I_{max}\sqrt{\alpha}$

Piecewise Waveforms

To better approximate and analogue signal, a piecewise waveform can be used instead of a square waveform:

This is a square wave with varying amplitudes for $N$ fixed intervals. The root mean square voltage is found using the sum of all the intermediate voltages squared:

$V_{rms}=\sqrt{\frac{1}{N}(V_1^2+V_2^2+V_3^2+...)}$

$V_{rms}=\sqrt{\frac{1}{N}\sum{V_n^2}}$

Analogue signals generally have a sinusoidal wave
Digital signals generally have geometric waveforms
Pulse width modulation regulates power, where the power ratio is given as $\alpha=\frac{on}{off}$
A piecewise waveform is a better approximation of a sine wave than a square waveform: $V_{rms}=\sqrt{\frac{1}{N}\sum{V_n^2}}$

Waveforms

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