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

Power supply rejection — where it appears

How much of a disturbance on a circuit's supply rail reaches its output, quoted as a ratio in decibels. It is not one mechanism: a fixed offset at the input stage is corrected wherever the loop has gain, while a compensation capacitor returning to the rail is a feedforward path the loop never sees, and the two behave quite differently with frequency.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Rejection is not a property of the loop. The same regulator with the same loop gain and the same 46.5° of phase margin, drawn twice. With the amplifier's output referred to ground the rail reaches the output at -67.3 dB at the bottom and 5.79 dB at 10.0 kHz — where it is amplified. Referred to the rail instead, every point is 60.00 dB lower, which is 20 log(gm·ro) for the pass device and not a design choice.

What gets through from the rail

A regulator's job is to hold its output still while its input moves, and the measurement says it does that well at ten hertz, badly at a kilohertz, and not at all at ten kilohertz — where this one puts out 1.9 times what arrives. Then the same netlist with one node moved, the same loop gain and the same 46.5 degrees of margin, rejects 60.009 decibels better at every frequency in six decades. The sixty decibels is the pass device's own intrinsic gain and it is not a design choice.

applied · Regulator One channel's load step reaches another through the supply, and the compensation decides by 4497×. computed by solving, not by drawing. Two identical amplifiers on one rail — sharing no signal node — with an ampere of load step pulled from the first and the second's output read. With the wiring left out the coupling is exactly zero, which is what the seven rungs below this one computed. With 30 nanohenries and fifty milliohms of rail and 10 microfarads of decoupling it is not: 3.84 microvolts per ampere at 271 kHz if the compensation capacitor returns to ground, and 17.29 millivolts per ampere at 2.33 MHz if it returns to the rail. That is a factor of 4497 decided by a modelling choice, which is why both are drawn. The channel that caused the step is unaffected: its own loop corrects the disturbance along with everything else, and the crosstalk is entirely a problem for the channel that did not.

The rail the load moves

Seven rungs of this ladder end by saying the same thing: the supply is an ideal voltage source, so a load step is drawn from a node that cannot be disturbed. Giving the rail an impedance turns out to change the disturbing channel's own output impedance by three parts in ten million — its loop corrects the supply along with everything else — and to open a path to a second amplifier that shares nothing with it but a wire. How large that path is is a modelling choice: 3.84 microvolts per ampere with the compensation capacitor returned to ground, 17.3 millivolts with it returned to the rail, a factor of four and a half thousand.

feedback · Capacitive load The ripple 1000 µF leaves, through the regulator: 60.62 mV rather than 21.11 mV. One settled cycle of the reservoir's output — 1.331 V peak to peak across 1000 µF — split into 1000 Fourier lines, each passed through the solved regulator's rail-to-output response, and summed. The output is 60.62 mV peak to peak. The ripple times the rejection at 100 Hz, -36.00 dB, gives 21.11 mV, which is 2.872 times too little: the rejection worsens at twenty decibels a decade, so each harmonic of the sawtooth arrives at nearly the size of the first. The 100 Hz line carries 23.4% of the output's mean square and the lines above 1 kHz 10.4%; the first three arrive at 7.731 mV at 100 Hz, 7.200 mV at 200 Hz, 6.388 mV at 300 Hz. The dashed curve is the 100 Hz line alone.

The ripple that arrives as a comb

The rejection essay multiplied two numbers: 1.331 volts of reservoir ripple and the regulator's 36.0 decibels of rejection at a hundred hertz, for 21 millivolts at the output. The ripple is a sawtooth, a comb of lines at every multiple of a hundred hertz, and the rejection worsens at twenty decibels a decade — so the second line arrives at nearly the size of the first, and the third too. Summed with their phases, the output is 60.62 millivolts, 2.87 times the estimate, and the hundred-hertz line carries only 23.4 per cent of it. A reservoir twenty-one times larger cuts the rail's ripple 14.8 times and the regulated ripple 5.95.

applied · Regulator The capacitor across the upper resistor: 90.9° of margin at 836.5 pF, and less ripple past it. The regulator's phase margin against a capacitor across the upper divider resistor, with the output ripple the 1000 µF reservoir leaves beside it. With no capacitor the margin is 46.47° at a crossover of 9.73 kHz, the output impedance at 10 kHz is 1.95 Ω, the worst rail rejection is 5.79 dB and the ripple 60.62 mV. The margin is greatest, 90.929°, at 836.5 pF — a zero at 6.34 kHz and a pole at 25.4 kHz around a crossover moved to 16.9 kHz — where the output impedance at 10 kHz is 828 mΩ, the worst rail rejection -1.50 dB and the ripple 52.33 mV. At 10 nF the margin has fallen back to 66.34° and the ripple is 35.40 mV.

The capacitor across the upper resistor

The rejection essay said a regulator reproduces its reference times its divider's four, and that a capacitor across the lower divider resistor brings that down to one at high frequency. Measured, the loop peaks the reference's gain to 5.68 near its crossover before any capacitor is added; a nanofarad across the lower resistor raises the peak to 11.2; and the capacitor that brings it down belongs across the upper resistor, where a nanofarad keeps the gain from ever exceeding four. The same capacitor is a lead pair in the loop: 836.5 picofarads takes the phase margin from 46.5 to 90.9 degrees, and ten nanofarads, past that optimum, still holds 66 while cutting the output ripple from 60.6 millivolts to 35.4.

applied · Regulator

Named alongside it

The objects these essays reach for when they reach for this one.

Loop gainSeries-pass regulatorDesign tradeoffOutput impedanceCapacitive loadCommon-impedanceConduction angleCrosstalkDecouplingFeedforwardIntrinsic gainLead compensation

All concepts