Re: mathematica doesn't find periodic response with laplace transform

• To: mathgroup at smc.vnet.net
• Subject: [mg106977] Re: mathematica doesn't find periodic response with laplace transform
• From: Noqsi <jpd at noqsi.com>
• Date: Sat, 30 Jan 2010 07:12:15 -0500 (EST)
• References: <hjp7ot\$d69\$1@smc.vnet.net>

```On Jan 27, 4:24 am, nukeymusic <nukeymu... at gmail.com> wrote:
> In every course on Laplace transforms you'll find that the Laplace
> transform of a periodic signal with period tee is found by multiplying
> the Laplace transform of one period by 1/(1-Exp[-tee*s]). That factor
> replaces the series expansion  1 + Exp[-tee*s] + Exp[-2*tee*s] + Exp
> [-3*tee*s]+ ...
> Unfortunately Mathematica doesn't seem to work according to this.
> As an example I have taken a square wave with amplitude 3V and period
> tee which is applied to a low pass filter with time constant tau.
> You'll see that Mathematica doesn't calculate the correct response
> using the above mentioned factor.
> As a workaround and also to show what the correct result should be
> (for three periods) I also calculated the response by multiplying by
> (1 + Exp[-tee*s] + Exp[-2*tee*s] + Exp[-3*tee*s])
> Here is my code:
> hs = 1/(1 + tau*s)
> hps = 1/(1 - Exp[-tee*s])
> h1Ts = Exp[-tee*s]
> tau = 33.0*^-6
> tee = 330.0*^-6
> ui1t = 3*HeavisideTheta[t]
> ui2t = -6*HeavisideTheta[t - tee/2]
> ui3t = 1 - HeavisideTheta[t - tee]
> uit1 = (ui1t + ui2t)*ui3t
> Plot[uit1, {t, 0, 1500*^-6}]
> uis = LaplaceTransform[uit1, t, s]
> uisT = uis*(1 + Exp[-tee*s] + Exp[-2*tee*s] + Exp[-3*tee*s])
> uisTbis = uis*hps
> uitT = InverseLaplaceTransform[uisT, s, t]
> uitTbis = InverseLaplaceTransform[uisTbis, s, t]
> Plot[uitT, {t, 0, 1500*^-6}]
> Plot[uitTbis, {t, 0, 1500*^-6}]
> uos = uisT*hs
> uot = InverseLaplaceTransform[uos, s, t]
> Plot[uot, {t, 0, 1500*^-6}]
>
> does anybody here a way to make Mathematica act correctly for this
> case?

So, your complaint seems to be that Mathematica returned
InverseLaplaceTransform[uisTbis, s, t] unevaluated. It would have been
helpful if you'd said that.

What did you expect it to do?

Note that few functions have closed form inverse Laplace transforms.

```

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