Circuit Response
every model has an edge, and the figure carries it
Series

Loop gain — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Loop gain of a three-pole amplifier closed for a gain of 100. Unity loop gain at 5.73 kHz, where 34.9° of phase remains before −180°. The phase reaches −180° at 89.6 kHz, where the loop gain is 46.1 dB below unity.

    What is left at crossover

    A feedback loop is stable or not according to one number read at one frequency — how much phase remains before −180° at the point where the loop gain passes unity. The loop gain here is obtained the way it is obtained on a bench: cut the loop, drive one side of the cut, and measure what comes back to the other.

    part 1 · feedback
  2. A loop whose phase comes back, at a gain of 1.0e+7. computed by solving, not by drawing. Three identical poles take the phase to −270°, two lead sections bring it back to −158°, and their own poles take it down again — so ∠L = −180° at 189 Hz, 3.46 kHz and 23.5 kHz. The gain moves the magnitude curve and not those three frequencies, so it decides only which side of each the unity-gain point falls. At 1.00e+7 the crossover is 10.6 kHz, in the recovered band, and the loop is stable.

    Stable, and unstable with less gain

    A three-pole loop with two lead sections is stable for gains between 1.81 × 10⁶ and 3.36 × 10⁷ and unstable on both sides of that window. Turning its gain *down* is what breaks it. Neither margin can see this, because a phase margin describes the loop at one frequency and a gain margin at one other, and this loop's phase crosses −180° at three: 189 Hz, 3.46 kHz and 23.5 kHz.

    part 2 · feedback
  3. Loop gain of a two-pole amplifier closed for a gain of 100. Unity loop gain at 5.73 kHz, where 35.0° of phase remains before −180°. The phase never reaches −180° at any frequency, so there is no gain margin to quote: 2 poles contribute at most 180° and the last of it arrives only at infinity.

    The loop that never crosses

    Every loop this field draws carries two margins, and one of them is not always a number. Take the third pole out of the standard loop and its crossover moves by 2.45 parts per million and its phase margin by 0.164 degrees — the third pole's own arctangent there, to five decimal places — while its gain margin goes from 46.06 decibels to no number at all. The phase reaches −180° only where the magnitude has already reached −62 decibels, and the two instruments that are supposed to notice report 3.19 × 10⁻⁶ either way.

    part 3 · feedback
  4. One loop, cut at its output wire with 160 nF on it: the voltage injection crosses unity twice, the current injection reads 57 dB too much at low frequency, and together they are the loop gain. computed by solving, not by drawing. A three-pole amplifier closed for a gain of a hundred, with 100 Ω of output resistance and 160 nF of load. The loop gain read at the controlled source's own input, which draws no current (solid); read by a voltage injected in series with the output wire (dashed); and by a current injected into the same wire (dotted). The true loop crosses unity at 5.31 kHz with 8.9° of margin. The voltage reading first crosses at 5.04 kHz with 15.1°, and rises through it again at 10.6 kHz, because above 9.95 kHz the load is a lower impedance than the source. The current reading crosses at 5.82 kHz with 19.8°. Combined through 1/(1 + T) = 1/(1 + Tv) + 1/(1 + Ti) (dots), the two give the controlled source's reading to 3e-9 at every frequency.

    Two wrong readings and one loop gain

    A loop gain measured on a bench is measured at a wire, and a wire has an impedance on each side. Cut an amplifier's output with 100 Ω behind it and 160 nF ahead, inject a voltage in series, and the reading is the loop gain plus (1 + T)·Zs/Zl — 15.1° of margin where the loop has 8.9°, and a second unity crossing at 10.6 kHz that the loop does not have. Inject a current into the same wire and the error is (1 + T)·Zl/Zs: 19.8°, and 57 dB too much loop gain at low frequency. The two errors are reciprocal, so the readings combine exactly: 1/(1 + T) = 1/(1 + Tv) + 1/(1 + Ti) returns the loop gain at the controlled source's own input to a few parts in a billion, at every frequency and every load.

    part 4 · feedback

All series