Can someone provide me with a link to a good explanation of how the Laplace transform is used, in practical terms, in EE?

I understand that it transforms certain differential equations in time domain into ordinary equations in frequency domain, and I have an understanding of what that means, and why it is useful (or, at least I think I do.)

I am more interested in the rules for how to apply it to circuits than in the theory explaining why it works. In other words, a practical, step by step guide to how one would use it for solving things like filter response curves. A little theory behind it is good too. I was a Physics major and my coursework in EM and circuit theory never got to it.

 

I am not in a positon to recommend a specific tutorial - however, according to my experience, most books about control systems contain a special chapter on Laplace transformation and its practical application.

 

Thanks, that is a place to start looking.

 

As far as my understanding goes, the Transformation is mostly used to design low pass / high pass / band pass filters, it is easier to determine the transfer curve of your filter in the mathematical domain (define zeros, poles, amplification, ect.), and then translate that into electronic domain (define values for R, L, C).
I have never used that in practice though, because this is sophisticated method used probably in design of audio, or mostly Ethernet/communication high frequency protocol devices.. not exactly hobby stuff..

 

not exactly hobby stuff

Yep, trying to bump up my skills.

 

I didnt mean to undermine you level of knowledge, with "hobby stuff" I meant DIY project, as opposite to designing a wireless router or graphic card of course there is no such thing as "not applicable knowledge", so more power to you on your new quest!

 

I didnt mean to undermine you level of knowledge, with "hobby stuff"

Didn’t take it that way, no problem. I am definitely a hobbyist.

 

I had to refresh my memory of these theories, I might have mislead you, designing filters has greater utility of the Fourier Transformation, which is like a special case of the Laplace Transformation, you can check that out too, I see some really educational videos on Youtube about this. Just have in mind that these are mathematical models... not everything you see about these transformations has a real life model, or implementation. Just like in mathematical theory you can have 5, 10, 15, infinite dimensions but in our dull perception of reality unfortunately we are stuck with 3 .

 

As with most tools, there are a number of ways that Laplace transforms are used in circuit design and analysis. Strictly speaking, Laplace transforms only apply to linear systems, but people have crafted techniques to apply them to non-linear systems under restricted circumstances. If you have ever used phasors or complex impedance, then you have used Laplace transforms, possibly without knowing it.

On the analysis side, Laplace transforms allow you to determine not only the frequency response, but also the transient response to an arbitrary input waveform. The frequency response is simply the output waveform when the input waveform is an impulse.

On the design side, if you want a particular frequency response or a particular output waveform in response to a particular input waveform, the ratio of the transforms of those waveforms gives you the Laplace transform of the circuit, which you can then pick apart to determine the circuit components needed and how to connect them.

In most EE curricula, transform methods are covered in Circuits II, so tracking down a text for such a course might be helpful. Here's some from my shelves:

Harrison, Transform Methods in Circuit Analysis
Nilsson and Riedel, Electric Circuits
Thomas and Rosa, The Analysis and Design of Linear Circuits
Kraniauskas, Transforms in Signals and Systems

The first one is the one I learned from. It's been a long time and my memories of it are that of a student learning the material for the first time. I recall it seeming to be very rigorous and math intense, but a solid text. The second is the text that was used when I was teaching it. The first part of the book is Circuits I and is pretty solid. The second part, which has benefited from less feedback, is Circuits II. It is pretty decent, but has some issues. The other two are ones I've picked up over the years and have not looked at them in any depth, but both seem pretty decent.

The rules for how to apply them to a circuit are reasonably straightforward -- the actual process of applying those rules is not always so.

Without transform methods, what you would normally do is analyze a circuit to determine the differential equation that describes the behavior of whichever voltage or current you are interested it. Then you would solve the differential equation in the presence of the forcing function that describes the input signal. For all but the simplest circuits, this quickly becomes a real nightmare -- even for a simply RC filter it isn't a lot of fun. And it is very error-prone.

A mathematician would analyze the circuit by first setting up the same differential equation and then using Laplace transforms as a tool for solving that equation. A lot better, but still a nightmare.

An electrical engineer ("engineer" being the usual polite euphemism for "lazy") side steps this by taking the Laplace transform of each component in the circuit first, which is almost trivial to do for most circuits, and then applies the wealth of techniques that have been developed to make the analysis of DC circuits very formulaic (again, we like "lazy"). The hardest part is usually transforming the input signal into the Laplace domain and transforming the output signal back to the time domain. But when you are interested in the frequency response, either from an analysis or a design perspective, there is usually not a need to do either of those, and we have developed lots of other formulaic ways that embrace our lazy desires.

 

As far as my understanding goes, the Transformation is mostly used to design low pass / high pass / band pass filters, it is easier to determine the transfer curve of your filter in the mathematical domain (define zeros, poles, amplification, ect.), and then translate that into electronic domain (define values for R, L, C).
I have never used that in practice though, because this is sophisticated method used probably in design of audio, or mostly Ethernet/communication high frequency protocol devices.. not exactly hobby stuff..

Sorry, but I have to disagree.
The transfer functions for frequency-dependent systems like filters, oscillators, control loops, etc. are derived directly in the s-domain using the classical impedances and rules for linear systems (remember the impedances sL, 1/sC...).
The Fourier as well as Laplace transformations are used only in case you want to switch to the time domain (diffential equations).

 

...
An electrical engineer ("engineer" being the usual polite euphemism for "lazy") side steps this by taking the Laplace transform of each component in the circuit first, which is almost trivial to do for most circuits, and then applies the wealth of techniques that have been developed to make the analysis of DC circuits very formulaic (again, we like "lazy"). The hardest part is usually transforming the input signal into the Laplace domain and transforming the output signal back to the time domain. But when you are interested in the frequency response, either from an analysis or a design perspective, there is usually not a need to do either of those, and we have developed lots of other formulaic ways that embrace our lazy desires.

 

The second is the text that was used when I was teaching it

Thanks, I found a used hardcover copy on Amazon for $25, while a new paperback is over $300. I had to order such a bargain.

 

Thanks, I found a used hardcover copy on Amazon for $25, while a new paperback is over $300. I had to order such a bargain.

Hope it helps. It really is a decent text, despite some weaknesses in the second half. The biggest gripe I have with it, and with nearly all EE texts, is that they give lip service to the importance of units, and then take the same sloppy approach in which they drop them entirely when doing the calculations and then tack on the units they want the answer to have at the end. I found a few errors in the 9th edition -- the 9th frickin edition! -- in which, had they tracked their units through the calculations, would have let them catch the error right where they made it. I reported them to the authors, so hopefully those were fixed going forward.

Oh... be sure that the one you buy is not an "international" or "global" edition. In my limited experience (four different texts) these are littered with errors because they are not written by the authors, but rather by folks contracted by the publishers to make stripped-down modifications for markets in other countries and the folks they contract with often have little clue about the material, and so they make changes, mostly to examples, exercises, and problems, that are totally wrong. The authors hate this practice, but they have zero control over it since the publisher owns the rights.

 

they give lip service to the importance of units, and then take the same sloppy approach

I was a physics major. I learned very early that units are my friend. Can’t remember whether the term of an equation is in the numerator or denominator? Look at the units!