Laplace Transform of a Triangle Wave

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Triangle wave with period T=2 and amplitude A=2

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Introduction

This article explains how to transform a periodic function (in this case a triangle wave). This is especially useful for analyzing circuits which contain triangle wave voltage sources.

Define F(t)

$m1=\frac{2+2}{.5+.5}=4$

$m2=\frac{-2-2}{1.5-.5}=-4$ So,

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Using the theorem for the transform of a periodic function,

$L\left\{F\left(t\right)\right\}=\frac{1}{1-e^{-2s}}[\underset{-.5}{\overset{.5}{\int }}4t{e}^{-st}dt+\underset{.5}{\overset{1.5}{\int }}(-4t+4.5)e^{-st}dt]$

$L\left\{F\left(t\right)\right\}=\frac{1}{1-e^{-2s}}[\underset{-.5}{\overset{.5}{\int }}4t{e}^{-st}dt+\underset{.5}{\overset{1.5}{\int }}-4t{e}^{-st}dt+\underset{.5}{\overset{1.5}{\int }}4.5e^{-st}dt]$

$\underset{}{\overset{}{\int }}4t{e}^{-st}=L\left\{4t\right\}=\frac{4}{s^2}$

$\underset{}{\overset{}{\int }}-4t{e}^{-st}=L\{-4t\}=-\frac{4}{s^2}$

$\underset{}{\overset{}{\int }}4.5e^{-st}=L\{4.5\}=\frac{4.5}{s}$

Author

• Michael Vier