In fact I try to avoid doign maths if I can find
a neat solution to
the problem. Recently I've been looking at a resisotr network
circuit [1].
...dangling pointer here...
Sorry ...
[1] Part of the DAC board in an HP9125A plotter, so it's on-topic-ish.
I can (and did) write down the equations givign
the output voltage as
a function of the input voltage (using the superprosition principle
to solve for one input at a time) and solve them.
I'd say that counts as doing math(s)!
So would I. My point is that while I _can_ do this, and while I can
therefore show the circuit has the effect I would think it needs to, I
don't realyl feel I _understand_ the circuit.
This shows me it does indeed behave as I'd
esxpect, but I still don't
feel satisfied. I don't feel I really understnad it. I am still
looking for an intuitive way to figure it out.
Depending on the resistor network, there may not be one.
That may be true.
I wanted to do something of the sort myself, once (prompted by the
resistor network in xkcd #730). I treated it as a case of generalizing
an uninteresting problem enough to make it interesting and then solving
it; I built a program that takes a resistor network, including
resistances and the ability to specify voltages at some nodes, and sets
up the equations and solves them, giving current through each
resistance and voltage at each node. (Of course, anyone who wants a
copy is welcome to it; if you have git set up, you can ge tit by
cloning
git://git.rodents-montreal.org/Mouse/resistor-networks; if you
don't, I can mail out copies. You'll also need lcs-cvt, as in
ftp.rodents-montreal.org:/mouse/local/src/lcs-cvt/, or a little hand
tweaking.)
There are a couple of tricks that might be useful. both only work for
circuits contianing linear impedances, like resisotrs, not non-linear
things like semiconductors :
The first is very useful if you are solving crertian problems by hand.
It's called the cuperpostion principle. It goes like this. Pick a
resistor in the circuit (whichever one you are interested in). Now, short
out all the power sources in the circuit aaprt from one. Calcualte the
current through the resistor in the modified circuit. Now repeat
'unshorting' each power source in turn. Add up all the currents (take
account of direction, of course). The resulting current is the current
thriough the resistor in the complete circuit (all power sources on).
this is a very quick way to solve the typical 2-battery, 3-reisostr 'T'
circuit
The second method is much more general. Firstly, you work in terms of
admittance, which is 1/impedance. The point being that admitances in
parallel add.
Now, if you haev an addmitance Y(i,j) between points i and j in the
circuit, then the currnet throuigh it is, of course, Y(i,j)*(v(i)-V(j)).
That is equivalent to taking that current from node i and feeding the
same current into node j. You can now neatly represent that equitaion as
part of an n*n matrix equation (where n is the total number of nodes) --
there's a matrix (n*n) containing the admitances suitableed added and
subtracted multiplying a vector of the voltages at the nodes. This, of
course equals the resdultant current flowing into or otu of each node,
again a vector. Noramlly this will eb zero, but power sources are
replaced by current sources, which contribute a current into or out of
the notdes to which they are conencted.
Now, if that matrix is non-singular, you simply invert it, multiply the
cvector of courrent sources by it, and you get the voltages at all the
nodes of the circuit. The problem is the matrix is _always_ singular.
This represents the physical fact that there are no absolute voltages,
only differneces. No matter, simply take a note (say node 0) to be
ground, and delete the appropriate row an column from the matrix, the
element from the currnet and voltage vectors. The resulting matrix is
non-singular, invert, etc...
On the other hand, I am a lot more mathmeatically
inclined than 90%
(or more) of the populatiion. It worries me -- a lot -- that none of
the neighbours here had any idea what 'e' was (base of natural
logarithms, not the elctronic chaerge), for example.
I once was at work when someone walked in wearing a T-shirt which said
"SPEED LIMIT" with a formula, 4 pi h-bar / 137 mu_0 e^2 is perhaps
I asusme that evaluates to c.
clear enough as a text form of it. I said "h-bar
is Planck's constant,
Hmm.. I thought h-bar was conventionally plancks constant _divided by 2pi_
137 is the fine structure constant, mu-zero is the
permittivity of free
space, but what's e?" - because I was pretty sure e didn't represent
antilog(1) there.
My first guess would have bene the electronic charge. On second thoughts,
permitivity?
He said, "You're the first person to have gotten as far as `h-bar is
Planck's constant'.". (e turned out to be a constant which depends on
the material in the vicinity - the expression gives the local speed of
light, as opposed to the vacuum speed.)
That does sound like perimitivity. But why mu_0, and not mu, then? OK,
mu_r (relative permiablitly) is very close to 1 for non-magnetic
materials, the variation in the speed of light is due to the perimitvity
change mostly, but...
-tony