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

Noise gain — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. An inverting unity gain, and the 100 pF that only the loop can see. computed by solving, not by drawing. Two ten-kilohm resistors around a 10 MHz amplifier make a gain of 1.00, and a loop that closes against 2.01 — one plus the ratio, not the ratio. Adding 100 pF at the summing junction leaves the closed-loop gain at a kilohertz unchanged — 0.99998051 against 0.99997988, three parts in a million at the far end of the slider — and takes the phase margin from 90.0° to 14.4°, because the noise gain now rises a decade per decade and the loop closes at forty decibels per decade instead of twenty. Forty-five degrees is reached at 9.00 pF, bisected on the netlist. The capacitance is not part of the signal path and does not appear in any expression for the gain.

    The gain the loop closes against

    An inverting amplifier with two equal resistors has a gain of one and a loop that closes against two, so it has half the bandwidth of a follower built from the same part — 4.99 megahertz against ten. Nine picofarads at the summing junction, less than a scope probe, takes the phase margin from ninety degrees to forty-five, and a hundred picofarads puts twelve decibels of peaking on a response whose designed gain is nought decibels and whose measured gain at a kilohertz has not moved by three parts in a million.

    part 1 · feedback
  2. A gain of −1 on a part stable only at gains of ten, with 1.25 kΩ across its inputs. computed by solving, not by drawing. A 100 MHz amplifier whose second pole is at 20 MHz, wired as an inverter with two 10 kΩ resistors. Its signal gain is one at a kilohertz on every curve. Its noise gain is two with nothing across the inputs (dashed): 35.0° of margin and 4.4 dB of peaking. A resistor from the summing junction to ground raises the noise gain without touching the signal: at a noise gain of 10 the margin is 65.6°, the peak is 0.00 dB and the −3 dB point is 14.1 MHz.

    The gain added to buy a margin

    An amplifier sold as stable only at gains of ten — 100 MHz of gain–bandwidth, its second pole at 20 MHz — has 35.0° of margin and 4.4 dB of peaking as a gain of −1, because that gain closes its loop against two. A 1.25 kΩ resistor from the summing junction to ground raises the noise gain to ten without touching the signal gain, which stays one to a part in ten thousand, and the margin becomes 65.6°: still 14.1 MHz of bandwidth, 2.19 times what a unity-stable part from the same die gives at its natural gain. The price is everything the amplifier contributes, multiplied by ten instead of two — 64.0 nV/√Hz at the output against 20.5 — and a capacitor in series with the resistor buys back the low-frequency half of it and none of the wideband total.

    part 2 · feedback
  3. One channel of a summing amplifier whose junction is shared by 8 inputs. computed by solving, not by drawing. Every input has its own 10 kΩ resistor into one junction and a gain of −1 at a kilohertz; the 10 MHz part closes its loop against a noise gain of 9. Input 1's −3 dB point is 1.17 MHz with 8 inputs, against 6.43 MHz with one: 1: 6.43 MHz, 2: 3.95 MHz, 4: 2.21 MHz, 8: 1.17 MHz, 16: 603 kHz. No other input has to be driven for this to happen; it is enough that its resistor is there.

    The input that costs every other input

    A summing amplifier gives each of its inputs a gain of −1 and closes its loop against one more than the number of inputs, so every input added divides every channel's bandwidth and multiplies the amplifier's own noise — 6.43 MHz with one input, 1.17 MHz with eight, 603 kHz with sixteen, on a 10 MHz part. Above 5.4 inputs the amplifier's 5 nV/√Hz, multiplied by n + 1, outgrows the resistors' Johnson noise, which rises only as its square root. A muted channel whose source is grounded still costs: a sixteen-input mixer is 11.5 dB louder with two channels in use than one whose unused resistors are disconnected. A channel of gain 0.1 beside one of gain 10 is 13.7 dB noisier than alone. A tree of smaller summers doubles the bandwidth and does not make the sum any quieter.

    part 3 · feedback

All series