# Triangular Waveform

## Triangular waves are a form of electronic waveform where the voltage level ramps up linearly and falls away linearly at the same rate for both ramps.

Electronic & Electrical Waveforms Includes:
Waveform types & basics     Sine wave     Square & rectangular waves     Triangular wave

Triangular waves or waveforms are often found within electronics and are used for a variety of purposes.

A triangular waveform consists of a pair of consecutive straight ramps moving in opposite directions, where the voltage of variable first moves up and then down. The inclination of the ramp is generally the same, although obviously one is the negative or mirror version of the other.

Triangular waveforms can be easily formed using an integrator to give first a ramp in one direction and then once it has reached a required voltage, it reverses to give a ramp in the other direction.

## Triangular waveform basics

For a triangular waveform, the rise and fall slopes are generally the same. If no specification is given, then this is generally the case. This gives the waveform a 50% duty cycle.

However it is possible to have a non-symmetrical triangular waveform where the rise and fall slopes differ.

In the extreme case the non-symmetrical sawtooth can have a ramp and a very steep or vertical fall or rise to give a sawtooth waveform.

The triangular waveform is often used within musical instruments as it is rich in harmonics and this gives it an interesting sound.

The sound of a triangular wave signal

In terms of the sound, the triangular wave has a less harsh sound than a square wave which is very rich in harmonics. The triangular waveform has harmonic levels, which are odd harmonics, but the levels fall away far more quickly than that of a square wave.

## Mathematical elements of the triangular wave

There are some interesting aspects to the triangular waveform and it has some relationships to other waveforms.

The triangular wave consists of only odd harmonics of the fundamental waveform repetition frequency, where the amplitude or each harmonic is equal to 1/ (harmonic number)2

The triangular wave can be expressed mathematically by the equation below. The equation may seem rather complicated, but it provides a complete representation for the waveform.

$g\left(t\right)=\sum \left({\left(-1\right)}^{i}\left(\frac{A}{{n}^{2}}\right)\mathrm{sin}\left(2\pi fnt\right)\right)$

Where
n = 2i-1
A = amplitude of the overall waveform
the summation is from i = 1 to infinity.

The summation makes use of the fact that -sin(x) = sin(x = π) as this handles the odd harmonics that are 180° (&pi radians) out of phase with each other.

Another interesting aspect of the triangular waveform is that it is the integral of a square wave.

As a result of this, triangular waveforms can easily be created by passing a square wave through an integrator, such as an op-amp integrator circuit. As the square wave switches between the two states, so the integrator first linearly ramps up and then down.

Triangular waveforms are one of the basic waveforms that are often seen and sued as examples for waveform types. They can be generated by most function generators and they can be created by integrating a square waveform. They can also be converted into a reasonable representation of a sine wave by passing the triangular waveform into an electronic circuit that includes a pair of back to back diodes. Additional resistors are required to ensure the elvels are correct for this.

More Basic Electronics Concepts & Tutorials:
Voltage     Current     Power     Resistance     Capacitance     Inductance     Transformers     Decibel, dB     Kirchoff's Laws     Q, quality factor     RF noise     Waveforms