RC filter on the bench

// A function generator into an RC low pass, with a scope on both ends of
// it and a meter across the output.
//
// R = 16 k, C = 100 nF. The corner is where the capacitor's reactance equals
// the resistance:
//
//   f0 = 1 / (2 pi R C) = 1 / (2 pi x 16000 x 100e-9) = 99.5 Hz
//
// The generator is set to 100 Hz, which is as good as on the corner, so the
// filtered trace (blue, CH2) comes out at 0.707 of the yellow one and a whole
// eighth of a cycle behind it: 2 V peak in, 1.41 V peak out. The meter reads
// the same thing the other way around: true RMS, so 0.99 V against the
// 1.41 V RMS going in.
//
// Sweep it by hand. The - and + under the frequency on the generator's face
// step 1, 2, 5 per decade without opening a panel:
//
//   20 Hz   the traces sit almost on top of each other, lag about 11 degrees
//   100 Hz  0.707 and 45 degrees, the corner
//   500 Hz  down to about a fifth, lag nearly 80 degrees
//   2 kHz   the blue trace has all but gone
//
// Count the lag in divisions. At 100 Hz one cycle is 10 ms, the time base is
// 2 ms a division, so a whole cycle is five divisions and 45 degrees is
// five eighths of one.
//
// Two things the instruments will not do, and both are marked on the parts.
// The generator drives through 50 ohms, which is nothing against 16 k here but
// halves the amplitude into a 50 ohm load. The meter samples at 4 kHz, so
// above about a kilohertz its AC reading is under-sampled and reads low;
// the scope is the instrument for that, which is the real lesson.
//
// Sweep it all at once: the Bode plot.
//
// Analysis... in the more menu, then Frequency response. Drive gen1, leave the
// input net on the generator's own node, watch the filtered node, and sweep
// 1 Hz to 100 kHz at 20 points a decade. What the engine draws:
//
//   below the corner  flat, at -0.15 dB
//   about 101 Hz      the marked -3 dB point, and -45 degrees of lag
//   1 kHz             -20.1 dB, -84 degrees
//   10 kHz            -40.0 dB, -89 degrees
//
// Ten times the frequency is twenty decibels less: that is what one capacitor
// does, and it is a straight line on a log axis.
//
// Now the interesting part. The formula says the corner is at 99.5 Hz and the
// pass band should be 0 dB, and the plot says 101 Hz and -0.15 dB. Both of
// those are the instruments, and both are real. The scope's probe is 1 Mohm
// and the meter on volts is 10 Mohm; the two of them in parallel are 909 k,
// sitting across the capacitor. That divider is the 0.15 dB, and the R in
// 1 / (2 pi R C) is no longer 16 k but 16 k in parallel with 909 k, which is
// 15.7 k, so the corner moves up to 101 Hz. Take both instruments off the
// node and it goes back to 99.5 Hz. A real 1 Mohm probe on a real 16 k source
// does exactly this, which is why bench probes are 10 Mohm on the x10 range.
//
// What the plot will not show you, because the engine does not model it: no
// noise, no distortion, and no second pole in an op-amp.