I am not sure whether I have mentioned in this forum that I write a column for a hobby magazine. The magazine is Australian Model Engineering. My column is called “Sparks ‘n’ Arcs”, a rather uncomplimentary way of suggesting that the column focuses on electrical matters. The readers are mostly skilled artisans, but not people well versed in electronic matters. I started writing the column some years ago when I felt compelled to rescue the model engineering fraternity from some very strange ideas about how to provide variable speed drive for a traction motor.
I have just signed off the galley sheets for the third of a series of articles in which I attempt to explain why some control loops are stable and some are not – all for a readership with very little mathematics.
After considerable thought about how to tackle the problem, I came up with a sequence of concepts that ran like this:
1. Integer exponents. You know the sort of thing: x times x times x times x is equal to x to the power 4 because we multiplied together four instances of x.
2. Non-integer exponents. For this, I chose adiabatic expansion of a gas. You remember:
Pressure times Volume to the power Gamma is equal to a constant.
The value of gamma depends on the number of degrees of freedom of the gas molecules. It has the value of about 1.67 for a monotomic gas. I defined a very simple model in which a piston moved along in a cylinder to allow the gas inside to expand. the gas in the cylinder consisted of only one atom that bounced back and forth between the cylinder cover and the piston. Very simple equations of motion describe what is going on. If the atom velocity is such that it will bounce back and forth about a thousand times during the piston stroke, then a thousand points of data will show how the pressure varies with volume. Snap! It obeys the simple gas law! I was quite pleased with that. I have never seen anything like that before. Anyway, this showed the mechanically minded viewer how very simple equations could give rise to non-integer exponents, even though the notion of exponent was originally defined (in my work) by how many instances of x one multiplied together (a definition that seems to be restricted to integer exponents).
3. On to logarithms. Not the treatment that my readers would have got at school.
4. Very quickly moved on to graphs with logarithmic scales. Showed how a power law comes out to be a straight line on such axes. There is an example of the use of this in steam engines so that was not too far off the familiar ground for the reader.
5. From a power law is a straight line, I moved on to the way some equations come out to have two straight lines with a “bend where there is a transition between them.
6. Straight from there, we had a Bode plot representing a single pole.
7. It wasn’t hard to devise some Heath Robinsonesque mechanisms that would exhibit a single pole response.
8. Then multiple poles and their effect on amplitude and phase. It is pretty easy to explain how 180 degrees phase shift turns negative feedback into positive feedback, and the notion that if this occurs at a frequency at which the loop gain is at or more than unity, we have instability, does not stretch the imagination too far.
I gave them an example of a Watt governor on a steam engine.
It will be interesting to see what sort of reception these articles get. It has been a very interesting challenge.
Didactic Diversion Closer to Home – The role of this Blog.
I do get some interesting comments to this Blog. Of course there are all those who think that I will be wanting to have my wedding tackle modified. WordPress provides means to save me from bothering with those. Some comments are from those who understand what I have written but feel the need to straighten me out on some detail. Many of these people are my friends and acquaintances who tend to send me a personal email rather then using the Blog comment facility. Often I bring them out into the open by responding to their remarks in a later post. I do get a few, however that do not fall into either of the above category. Is the writer seeking a tutorial response? Sometimes this is not clear. If the real drift of a comment is not clear, but I suspect that there is a genuine query in there somewhere, then I respond directly to the comment author. Only if some point of more general interest emerges, would I then bring the matter up for us all to see.