Laplace Analysis
Laplace Analysis
This project was inspired by the phenomenon of “ringing” in electrical circuits, and how to analyse ringing using the mathematical technique of Laplace Transforms. For a description of what is meant by “ringing” please see this article
on Wikipedia.
These are the goals of the project:
For example, 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.
But hasn’t this been done before?
Having taken on the task, my initial approach was to Google the subject: 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 ringing 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 interested in. Also, for some of the configurations I knew I would want to consider, there appeared to be no existing analyses at all.
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 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 in this project, but only to verify results which have been first obtained from the analytical Laplace models. If you haven’t come across it before, LTSpice is a powerful, free SPICE simulation which is quite brilliant and available at this location. Well worth a look if you haven’t already.
Some words on formatting.
Inevitably this project is going to involve equations. Lots of equations. I’m not sure to what extent my web-hosting platform supports equations. LaTex maybe? I don’t know. So 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.
And one more thing…
These are the goals of the project:
- Identify a simple circuit example which has a tendency to exhibit ringing.
- Develop a mathematical model describing current flow during a ringing event in that circuit.
- Produce mathematical models of the voltages appearing across the various circuit elements during the event.
- Present the models in clear, concise language which directly addresses the problem at hand and avoids over-complicated maths whenever possible.
For example, 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.
But hasn’t this been done before?
Having taken on the task, my initial approach was to Google the subject: 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 ringing 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 interested in. Also, for some of the configurations I knew I would want to consider, there appeared to be no existing analyses at all.
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 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 in this project, but only to verify results which have been first obtained from the analytical Laplace models. If you haven’t come across it before, LTSpice is a powerful, free SPICE simulation which is quite brilliant and available at this location. Well worth a look if you haven’t already.
Some words on formatting.
Inevitably this project is going to involve equations. Lots of equations. I’m not sure to what extent my web-hosting platform supports equations. LaTex maybe? I don’t know. So 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.
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 via the Contacts form. 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.
You can access my work on Laplace Transforms using the navigation bar at the top of the page or via the buttons below: