Math-nerds, Machinists

See Saw Margery Daw

I remember a time when everything seemed possible. I could perhaps have gone anywhere and been anything. This was almost scared out of me by truisms, such as nursery rhymes, that cautioned me to specialize in one thing – to pick a trade and get good at it. I tried, but I couldn’t do it. I just couldn’t find one thing and be the best at it.

So now, I am the master of nothing…

But is that so bad, really? I like hanging out with math-nerds and machinists. I can’t do what the best of them can do, but I can see their complementarity. Hanging out in the space between them is pretty cool (to me). I get to see things this way…

Basis Functions

Basis functions are the building blocks of electrical circuits, not, as one might suppose, resistors capacitors and coils. Many, many years ago I began to learn and repeat this fascinating story (see here) that I recently revisited. The grand sense of wonderment I get from electrical engineering is like the feeling one might get from other adventure stories like Alice in WonderlandPeter Pan and so many more. (Lewis Carroll, the person who first met adventurous Alice was also Charles Dodgson, the Oxford mathematics lecturer, a coincidence of significance.) The electrical engineering story goes something like this: We use mathematics to create machines to create mathematics to create …

 Thus electrical engineering is an example of a self-referential system
 

We use mathematics to create machines to create mathematics to create…

 
The story came back to mind as I was surveying possible gain equalizers for a potential customer of a friend of mine. The desired performance is detailed in the scatterplot below right. There are many ways (all interrelated, of course) to view how to design a circuit with these characteristics, but I prefer to view it as a ‘lossy’ band stop filter. (NOTE: For a brief introduction to filter synthesis, go here.)

I explored the land of equalizers by messing around with four different types of designs:
  1. Lumped element lossy bandstop (5, 7, 9 poles)
  2. Lumped element lossy LPF || lossy HPF combo, with and w/o Wilkinsons
  3. PI inverter coupled shunt resonators
  4. TL coupled shunt resonators
By far the best performance resulted from option #4 (below).

This structure also turns out to be the simplest to construct, since the lumped elements need to be connected with transmission line elements anyway. The reason why this is the best design for the task is that the impedance equation  representing a transmission line is a better basis function for this particular response than the impedance equation for a reactive element, e.g. C or L. Transmission lines just naturally fit the requirement best this time.

Profiling Tools

A machinist or trim carpenter will sometimes create a special tool with a profile that mirrors a portion of the project she needs to produce.  Individual profiling operations can be performed in series, so that their sum creates the desired shape.

You can also view individual resonators or individual antenna elements as special tools. By combining a bunch of them together you can possibly create the filter shape or antenna pattern you desire.

Machinists and carpenters accrue many thousands of dollars worth of cutters of different sizes and profiles in the course of their careers. Understanding (and remembering) that basis functions – each represented by one or more physical structures – are closely analogous ‘tools’ for shaping signals is the math-nerd equivalent.

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