Counting a 555

// Counting a 555, three ways.
//
// The astable is the textbook one: R1 = 10 k from the supply to DIS, R2 = 10 k
// from DIS to THR, and 100 nF from there to ground.
//
//   high   =  0.693 x (R1 + R2) x C  =  1.386 ms
//   low    =  0.693 x R2 x C         =  0.693 ms
//   period =  2.079 ms               ->  481 Hz, high for 2/3 of it
//
// The counter reads about 479 Hz, and under it the period, the duty and how
// long each cycle spends high. That is within half a percent of the 481 on
// paper: the formula is for an ideal 555, and the simulated one is very
// nearly, but not exactly, that. The multimeter, on its Hz position, says the same
// number, because underneath they are the same comparator and the same
// counter. What differs is the gate: the counter's is a whole second by
// default and the meter's is a quarter of one, which is why the meter follows
// a changing frequency sooner and the counter is steadier.
//
// Neither of them samples. Both are told the instant the net crosses the
// trigger level, so the reading does not step in units of anything and a
// frequency a thousand times higher would be counted just as exactly.
//
// Things to try:
//
//   Make C1 1 uF and the frequency drops by ten, to about 48 Hz. The arithmetic
//   above is linear in C, so check it.
//   Make R2 100 k and the duty goes from 67% towards 52%, because the charge
//   path (R1 + R2) and the discharge path (R2) become nearly equal.
//   Press RST on the counter and the pulse total starts again from zero. At
//   about 480 Hz it is up in the thousands within seconds, which is what a
//   counter's TOTAL function is for: counting things that happen, not
//   measuring a rate.
//
// What neither instrument can tell you: there is no noise in here, so there is
// no trigger sensitivity and no false counting on a slow edge, and no input
// bandwidth; a real counter's front end would have opinions about both.