topic Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation in Mathcad

Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

xyz123 — Tue, 16 Feb 2021 15:24:57 GMT

Hello Guys!

I want to analyse my circuit, using laplace transformation.

Step 1) First I have to generate a triangle waveform.

I have all informations given like peak-value, duty cycles, frequency, and so on.

I could obtain a triangle waveform by defining sections with the if-equation, according to the post from Werner:

https://community.ptc.com/t5/PTC-Mathcad/Plots-for-Conditional-Function/td-p/124048

My first idea was to multiply a sawtooth function with a rect function, but nevermind. I keep the solution from Werner I guess if the following Step 2) will work...

Another approach would be a fourier row to obtain the triangle shape. Maybe you have some older work / examples here for me?

Step 2-4) I want to make a Laplace Transformation from the triangle waveform, means: i(t) -> I(s)

--> is it posible with my function from Step 1) to do a laplace transformation, or do I have to go the approach with fourier rows?

(Step 3) Vout(s)=I(s)*G(s), with G(s) = Transfer-function of a circuit

(Step 4) Inverse Laplace Transformation Vout(s) -> vout(t) and plot

I am working with Prime 3.0

Best Regards

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

LucMeekes — Tue, 16 Feb 2021 16:09:10 GMT

Very first check: see if the Laplace transform likes an 'if'-statement. I know it deals with the Heaviside step function OK, but have my doubts about if...

Success!
Luc

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

Derbigdog — Tue, 16 Feb 2021 16:10:51 GMT

Here are a couple of handy documents showing how to generate and analyze waveforms with Mathcad.

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

Werner_E — Tue, 16 Feb 2021 17:53:08 GMT

Shouldn't you replace t.3 in your definition of f(t) by T. After all t.3 seems to be a duration, not a time stamp.

As Luc already wrote, the symbolic "laplace" does not work with conditionals (if's).

I am also not sure about trunc and/or mod (which you would use to make a periodic signal) and the used units. After all the symbolics does not know anything about units and treats them as unknown variables.

Using Fourier would require you to trunc the row and would just be an approximation.

EDIT: I just gave it a try and redefined f(t) using the Heaviside function as suggested by Luc.

We can laplace transform f(t), but as soon as we use "floor" to make it periodic, the symbolic is not able to transform it in a usable way - you see the calls to a (non-existent in Prime) laplace-function in the result.

I haven't tried, but I guess that different ways to make the signal periodic (using the "mod"-function e.g.) will have no success, too and you will have to resort to other methods to solve your task. Maybe the pdfs of @-MFra- , posted by @Derbigdog will be of help.

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

LucMeekes — Tue, 16 Feb 2021 22:24:54 GMT

And even if you succeed in finding the Laplace transform of a triangular waveform, which, according to Spiegel "Laplace transforms":

for a triangle with amplitude A and period T and no offset, will be something like:

Then on trying to apply the inverse Laplace, you (may) get:

which means that the inverse Laplace isn't known.

Success!

Luc

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

xyz123 — Tue, 16 Feb 2021 19:54:45 GMT

So this makes really fun, I must admit.

The pdf´s attached above are great, but unfortunately does not help...

See my attachment below, I have played and tried out a lot.

The solution is actually in the F1-mathcad-help-menu. Mathcad will not return any useful solution without the further command "assume, ALL>0"!

I also cleared all defintions, unfortunately clear(ALL) does not work ... Nevermind.

Completly symbolic it looks like this:

...

Tomorrow will follow more ... 🙂

Best Regards

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

Werner_E — Tue, 16 Feb 2021 21:11:03 GMT

You shouldn't "upgrade" to Prime 6 or any upcoming version.
Sooner or later the "legacy" symbolic (Mupad) will not be integrated anymore and the newly introduced symbolic isn't really ... competitive.

Remark: In Prime 6 you still have the option to switch between the two symbolic engines.

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

LucMeekes — Tue, 16 Feb 2021 22:08:16 GMT

In that case: upgrade now to Prime 6 while it still lasts (and you can still use the MuPad symbolics). Two weeks from now you are maybe stuck with Prime 7 for an upgrade.

Success
Luc

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

LucMeekes — Tue, 16 Feb 2021 22:27:28 GMT

Found another way to build a triangle. But it's neither symbolic-proof, nor fast:

Success!
Luc

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

Werner_E — Wed, 17 Feb 2021 03:15:15 GMT

In case you could not see the worksheet of the OP, as far as I understood it and depending on the sheet which was posted, the signal should look like this:

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

ttokoro — Wed, 17 Feb 2021 11:58:59 GMT

The next step is convolution of waveforms. i means 0.001s sampling,

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

xyz123 — Wed, 17 Feb 2021 12:14:43 GMT

Hi ttokoro,

can you attach a screenshot of the function x(t) and g(t) of your example?

With my waveform above I have a non periodical signal. Just a single pulse. But it seems to work quite nice. I must verify the solution with a simulation in Spice.

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

LucMeekes — Wed, 17 Feb 2021 14:27:37 GMT

Here's an example, first your triangle:

Don't try to use the repetitive signal, due to the mod() function, the Laplace transform will not carry through:

Now we need G(s), to calculate X(s). As an example we take a simple Low-pass filter:

From here, apply the inverse Laplace transform to obtain x(t):

(Yes, it's a big expression)

And plot f(t) and x(t):

Success!
Luc

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

LucMeekes — Wed, 17 Feb 2021 14:25:01 GMT

And with another filter, you may get this:

Success!
Luc

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

Werner_E — Wed, 17 Feb 2021 15:04:21 GMT

We might not be able to Laplace-transform the periodic made signal, but at least we can repeat it a couple of times using a simple summation which the laplace command accepts.

But I guess that less simple filters might be a problem for invlaplace

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

LucMeekes — Wed, 17 Feb 2021 16:01:52 GMT

Great idea Werner,

Luc

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

Werner_E — Wed, 17 Feb 2021 20:20:57 GMT

In case "invlaplace" fails it may help to "simplify" beforehand:

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

ttokoro — Wed, 17 Feb 2021 23:05:16 GMT

Number of input signal can change by as same as Werner_E posted.

If we have a impulse response, g(t), Mathcad can show the output signal waveform by using convolution even we can't write the equation of input function. (Just use the A/D signal data for input x(t).)

If we can write the both equations, convolution integral can show the output signal waveform.

can show the infinity input series response. It is well known as Z(jw) results.

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

xyz123 — Thu, 11 Mar 2021 21:24:23 GMT

Hello guys,

although it was solved I dont want to create a new topic.

With a simple RC-lowpass is was possible to obtain a solution.

Now I try to multiply Iout(s) * G(s) to obtain Vout(s), but with a far more complex filter.

When i do it on the symbolic way, then Mathcad is calculating for a long time with a result "too large to display" ...

Well, I tried it then with functions instead of symbolic math and maybe I have found a solution here, but:

- Please check if i did it correctly

- I cannot plot the final function

- My goal is to obtain a voltage ripple on the output of the filter (peak-to-peak) ...

I attached the sheet. The first part of the sheet is the older topic where the current by itself was plotted.

For the new issue please scroll below to page 13 ++

Best regards!

Re: Creation of Triangle Waveform and Circuit Analysis with Laplace Transformation

LucMeekes — Thu, 11 Mar 2021 22:07:59 GMT

You should still be able to use the symbolic result, even if it's 'too large to display'.

Assign your symbolic result to a function such as:

Vout(t, Vin, R, C) := H(s, R, C) * Ltriangle(s, Vin) invlaplace -> <symbolic expression too large to display>

Then you can (or at least, should be able to), plot Vout(t, Vin, R, C) versus time t.

Success!
Luc