Analysis of a Series RLC Circuit Using Laplace Transforms Part 1

Introduction.

This work was inspired by the phenomenon of “ringing” in electrical circuits, and how to analyse ringing using the mathematical technique of Laplace Transforms.

Several years ago, at the request of my boss at the company where I worked,  I took on the task of “sorting out ringing”. Ringing was increasingly becoming a problem in the circuits we were working on (essentially series RLC circuits, which I mention at this early stage only for the benefit of Googlebots) and at that time we had no mathematical model to describe what was happening. It became my job to develop that model, and because there was little time available at work to devote to the task, I was to spend significant amounts of my own time investigating what turned out to be a fascinating and absorbing subject. Those investigations were to take me into areas of mathematics that previously I had not even thought about, and well beyond the original remit of “sorting out ringing”.

Sadly, the large volume of work I produced during those investigations never really saw the light of day, for various reasons which I won’t bore you with here. I feel this is a great shame, as I know that ringing causes problems for many engineers and circuit designers and clear, concise explanations of the phenomenon seem to be very scarce. I am therefore publishing these pages in the hope that the reams of handwritten notes produced nearly a decade ago can be made available to a wider audience. And who knows? Someone out there might even find them useful.

 

What is ringing?

Ringing occurs when a system is subjected to a sudden stimulus and the system responds by oscillating in some way. For example if our “system” is a bell and we provide a stimulus by tapping the bell sharply with a hammer, then the bell will respond by making a noise. That’s ringing – quite literally! The amplitude (volume) of the ringing will depend on how hard we hit the bell (i.e. the magnitude of the stimulus) and eventually the ringing will decay away to nothing (a bell doesn’t ring forever, no matter how hard you hit it).

Similarly, certain types of electrical circuit can exhibit ringing when they are subject to sudden changes in voltage. Typically the ringing will appear as high frequency sinusoidal voltages across one or more parts of the circuit. As with the bell example above, the oscillations will eventually decay to zero, but they can still cause problems for the circuit designer: the amplitude of the ringing can cause components to be subject to higher voltages the they are intended to withstand and the high frequency component of the ringing waveforms can cause severe electromagnetic compatibility (EMC) problems. Needless to say, electronic engineers are keen to avoid ringing at all costs.

That’s my rough-and-ready description of ringing. For a another description plus informative diagram you might like to look here.

 

Goals of the project.

The goals of the “sort out ringing” project were as follows:

  1. From the various circuits we were working on, determine the simplest case which exhibits ringing.
  2. Produce a mathematical model describing current flow during a ringing event.
  3. Produce mathematical models of the voltages appearing across the various circuit elements during that event.
  4. Develop a mathematical model of how the ringing can be damped (eliminated) using a snubber. I’ll explain what I mean by “snubber” in the analyses.
  5. Present the models in clear, concise language which directly addresses the problem at hand and avoids over-complicated maths whenever possible.

To put it another way, if we have a circuit consisting of a series-connected  resistor, inductor and capacitor (R, L and C respectively), then we would like equations telling us how the current through the circuit (i), and the voltages across the components (VR, VL and VC) vary with time (t) as a function of those parameters i.e.

i(t) = F(R, L, C, t… )

VC(t) = F(R, L, C, t… )

and so on.

So, a nice clear project definition!

 

But hasn’t this been done before?

Having been set this task, my initial approach was to Google the subject (isn’t that always the case nowadays? And should Google as a verb be capitalised?) Surely a problem as common as this must have been analysed to death already? Well, I searched – a lot – and was surprised to find that this problem didn’t appear to be comprehensively analysed at all. Those analyses that I did find were either not well explained or examined the behaviour of circuits as filters rather than the case I was looking at. Also, for some of the configurations I knew I would want to consider, there appeared to be no existing analyses at all.

What was particularly interesting was that I could find no complete, rigorous description of how to design the snubber that I mentioned above. I was eventually to find out why this was, and the answer did take me down an unexpected but ultimately rewarding path, but I’ll save that particular surprise for later.

The lack of existing analyses of ringing added an extra frisson to the project: the chance to do something a bit original rather than just repeating things that have been done a million times already. Excellent – I live for stuff like that. Let me at it! And so began one of the most fulfilling project of my career so far.

 

Why Laplace Transforms?

When I started this project I did look around for clues as to the best way to approach the analyses. As I mentioned above, there wasn’t a great deal written about the exact scenario I was trying to tackle, and the majority of what there was seemed to focus on calculus-based approaches. Now, I’ve nothing against calculus, I might even say I like calculus, but niggling away at the back of my mind was a memory of something which might, just might, provide a simpler and more direct approach: Laplace Transforms.

Probably like many engineers, my exposure to the Laplace method came at college where I learnt the basics in order to pass an exam and then immediately forgot all about them. Fortunately, enough of a memory lingered to at least point me in the right direction, although with little useful knowledge except that the letter “s” was used a lot.

Now, open any textbook on Laplace Transforms and you’ll probably be confronted by the following:

Laplace definition

at which point you may well scream, slam the book shut and run from the room. This is the basic definition of the Laplace Transform, and I’m not in any way trying to say it’s wrong (it isn’t), what I am trying to say is that you don’t need to use it, or even be aware of it, in order to carry out perfectly valid and useful circuit analysis using the Laplace method. In fact, in all the analyses that I will demonstrate, I will not refer to this definition even once.

In terms of ease-of-use, the Laplace method has the following huge benefits:

  1. No need for calculus.
  2. Transforming circuit elements from the ordinary time-domain (t-domain) into their Laplace (s-domain) equivalents is extremely simple.
  3. Once transformed, the s-domain elements can be added, subtract, multiplied and divided just like ordinary resistances in the t-domain.
  4. Transforming back into the t-domain is usually only a matter of carrying out straightforward algebraic manipulations, and hence determining the final answer is also straightforward.

I hope to convince you of these advantages when I demonstrate the details of the Laplace method.

 

Why not just use simulation?

Well, yes, you could, and there is a very good (and free!) SPICE simulation package called LTSPICE from Linear Technology which would let you simulate circuits and observe ringing to your heart’s content. You could even make graphs of how, say, ringing frequency varies with L or amplitude varies with R. But – and this is the important “but” – this would not bring you any closer to having an analytical model of the phenomenon. At best, all you would have is a collection of empirical observations of the behaviour of ringing – if data collected from simulation can be called “empirical”. You would be nowhere nearer having a real understanding of what is actually going on. As it happens, I will be using LTSPICE quite a bit, but only to verify results which have been first obtained from the analytical Laplace models.

 

Some words on WordPress and equations.

Inevitably this project is going to involve equations. Lots of equations. WordPress (which hosts this blog) does support equations, but only by use of LaTeX. LaTeX is a typesetting language,  but  one with which I am not familiar and don’t have the time to learn. Instead, I have decided to write up the analyses using Word and make the pages available as PDFs via download links. I’ve been using Word for this sort of thing for years and for me it represents a zero-effort solution to the problem. Microsoft Equation 3.0 is probably the best thing that company has ever produced. Maybe one day I’ll learn LaTeX, but for now – Word it is. Incidentally, the equation I quoted above was a screenshot from a Word document blown up to 200% and converted to a graphic file via Pixelformer. I did briefly consider using this method for the documents as a whole, but discounted it on the basis of being too-long winded. Sticking with Word into PDF for now.

 

And one more thing…

One more thing I need to say: I’m a practical engineer by training and vocation, but I am not a mathematician even though I do love maths. I try hard to make sure that the mathematical terminology I use is correct, but it is possible – even likely – that the occasional error will creep in. I ask the reader’s forgiveness in advance and invite you to point out any errors in the Comments section.

So! I think that about covers it for the background information. The next logical place to go is into the first stage of analysis, and for that you’ll need to follow this link:

Analysis of a series RLC circuit using Laplace Transforms Part 1

Comments, as ever, are welcome!

PS if you are looking to learn more about Laplace Transforms, then I heartily recommend KA Stroud’s book “Laplace Transforms, Programmes and Problems” for a pain-free introduction to the subject. The Khan Academy also has an excellent series of videos on Laplace Transforms.