Feedback, and the margin

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.

16 min read 6 figures Two routes to a number The schematic is a label Every model has an edge

Assumes: The gain the loop closes against · What is left at crossover

The gain the loop closes against separated two gains that every inverting amplifier has. The signal gain is what a schematic is drawn to deliver: two equal resistors, a gain of −1. The noise gain is what the loop sees: one plus the ratio of the feedback impedance to everything at the summing junction, which for the same two resistors is two. Bandwidth, stability and the amplification of the amplifier’s own errors all follow the second. That essay met the noise gain as a hazard — nine picofarads at the junction, less than a scope probe, quietly raising it at high frequency and spending the margin — and repaired it with a capacitor across the feedback resistor.

The same distinction has a constructive use, and it explains a product line. Most amplifier families include a “decompensated” member: the same die with a smaller internal compensation capacitor, a gain–bandwidth product several times higher, a slew rate higher in proportion, and a specification that reads stable for gains of ten or more. The obvious reading is that the part is for gains of ten. The precise reading is that it is for noise gains of ten, and a noise gain can be raised without changing the signal gain at all — so the part can serve as a gain of −1, or as a follower, if the loop is given something to close against. This essay measures what that costs and what it buys.

A gain of −1 on a part rated for ten

The two parts here are one design. The unity-gain-stable member has 10 MHz of gain–bandwidth and a second pole at 20 MHz, from the output stage and mirrors that no compensation capacitor touches. The decompensated member has a tenth of the compensation capacitance, 100 MHz of gain–bandwidth, and the same second pole. Both have a hundred thousand of gain at direct current and fifty ohms of output resistance, and both are solved as netlists — an amplifier is three controlled sources, two internal poles and a resistor — rather than as transfer functions written down.

Fig. 1 The 100 MHz part as an inverter with two 10 kΩ resistors. With nothing across its inputs the noise gain is two (dashed): 35.0° of margin and 4.4 dB of peaking. With 1.25 kΩ from the summing junction to ground the noise gain is ten, the margin 65.6°, the response flat to 14.1 MHz, and the signal gain at a kilohertz still one. The slider is the noise gain the resistor sets.

Wired as a gain of −1 with two ten-kilohm resistors, the fast part closes its loop against a noise gain of two. Its loop crosses unity at 28.6 MHz, past its own second pole, and the cut loop measures 35.0 degrees of margin. The closed-loop response peaks by 4.4 decibels before it falls, and its −3 dB point at 45.6 MHz is mostly that peak.

Now add one resistor, from the summing junction to ground. It carries no signal current, because the loop holds the junction at ground and a resistor with zero volts across it conducts nothing — to the extent, the node that is at ground for a while measured, that the loop is strong at the frequency in question. It does appear in the noise gain: the feedback resistor now works against $R_\text{in} \parallel R_x$ rather than $R_\text{in}$ alone, and the noise gain becomes $1 + R_f/(R_\text{in} \parallel R_x)$ . At 1.25 kΩ that is ten. The solve says the signal gain at a kilohertz is unchanged to two parts in ten thousand — exactly the loop’s finite-gain error $n/A_0$ at a noise gain of ten — while the margin rises to 65.6 degrees and the peaking vanishes. The response is flat to 14.1 MHz.

The slider takes the noise gain through two, four, six, ten and twenty, and the margin rises at every step. Nothing a reader of the schematic would call the circuit’s gain moves at all.

The margin answers to one number

Draw the margin against the noise gain for both parts and the circuit drops out of the picture.

Phase margin against the gain the loop closes against, for two parts cut from one die. computed by solving, not by drawing at 15 noise gains for each part. The unity-gain-stable part (10 MHz, dashed) has 76.4° at a noise gain of two and more above it. The decompensated part (100 MHz, solid), with the same second pole at 20 MHz, has 35.0° at two, crosses 45° at a noise gain of about 3.5, and has 65.6° at the ten it is rated for. The margin depends on the noise gain alone; nothing in the signal path appears on this axis. — Fig. 2 Phase margin against noise gain, from 1.25 to 30, for the unity-gain-stable 10 MHz part (dashed) and the decompensated 100 MHz part (solid), both with their second pole at 20 MHz. The slow part has 76.4° at a noise gain of two; the fast part has 35.0° there, crosses 45° near 3.5, and has 65.6° at the ten it is rated for.

The slow part has 76.4 degrees at a noise gain of two and only more above it. The fast part is marginal at two, crosses forty-five degrees at a noise gain of about 3.5, and reaches the 65.6 degrees it was rated for at ten. Below 3.5 it rings badly enough that most applications would call it unstable, and as a follower, closing against one, it has 25.2 degrees left.

None of those numbers depends on what the signal gain is. A non-inverting stage with a gain of ten, an inverting stage with a gain of nine and an inverter with a gain of one and 1.25 kΩ across its inputs all close the loop against ten and all have the same 65.6 degrees, because the loop gain is the amplifier’s open-loop gain divided by the noise gain and nothing else. What is left at crossover defined the margin on exactly that quantity, cut and measured, and two measurements of one margin read the same margin back off a step. The datasheet’s “stable for gains of ten” is therefore a statement about the loop and should be read as one: a minimum noise gain, which any of several circuits can supply.

What speed is left

The obvious objection is that raising the noise gain to ten divides the gain–bandwidth product by ten, and a hundred megahertz divided by ten is the ten-megahertz part again. The solve disagrees by a factor of two.

The bandwidth of a gain of −1, against the noise gain, on the two parts. computed by solving, not by drawing. The −3 dB point of the inverter's signal response as a resistor across the inputs raises its noise gain. The unity-gain-stable part (dashed) gives 6.43 MHz at its natural noise gain of two and falls as one over the noise gain. The decompensated part (solid) gives 45.6 MHz at two, much of it peaking, and 14.1 MHz at the ten it is rated for — 2.19 times the slower part's best, with 0.00 dB of peak. — Fig. 3 The −3 dB point of the gain of −1 against the noise gain, on the 10 MHz part (dashed) and the 100 MHz part (solid). The slow part gives 6.43 MHz at its natural noise gain of two. The fast part gives 45.6 MHz at two, much of it peaking, and 14.1 MHz at a noise gain of ten — 2.19 times the slow part’s best.

The slow part as an inverter has a noise gain of two, so its closed loop extends to about half its gain–bandwidth: 6.43 MHz, a little more than five because its second pole adds a trace of peaking. The fast part held at a noise gain of ten extends to about a tenth of its gain–bandwidth: 14.1 MHz. The comparison that matters is not 100 against 10 but each part at the noise gain it needs, and there the fast part is 2.19 times faster, flat, with more margin than the slow part has as a follower.

The reason is in the ratio. The fast part’s gain–bandwidth is ten times the slow part’s, and it is being asked to close against five times the noise gain, so a factor of two survives — slightly more, because its lower margin lifts the closed-loop corner further above crossover: 1.55 times its 9.07 MHz crossover, against 1.33 times the slow part’s 4.85. If the decompensated part were only five times faster the comparison would be even, and a designer would be buying the slew rate alone.

What the resistor costs

The resistor is free for the signal and expensive for everything the amplifier adds. The amplifier’s voltage noise, its offset, its drift and its input-referred interference all reach the output multiplied by the noise gain, which is now ten instead of two.

Fig. 4 Output noise density of the slow part as a gain of −1 at noise gain two (dashed) and the fast part held at ten (solid), with 5

\text{nV}/\sqrt{\text{Hz}}

of amplifier noise and every resistor’s Johnson noise each solved as its own source. At a kilohertz, 20.5 against 64.0

\text{nV}/\sqrt{\text{Hz}}

; the resistors’ share (lower solid) rises too, because the new resistor has noise of its own. Integrated to a gigahertz, 57.4 against 253.3 µV rms.

At a kilohertz the slow part’s output carries 20.5 nanovolts per root hertz: the amplifier’s five multiplied by two, and 17.9 from the two resistors. The fast part held at ten carries 64.0: the amplifier’s five multiplied by ten is 50, and the resistors’ share has risen from 17.9 to 40.0 because the new 1.25 kΩ resistor has Johnson noise of its own and reaches the output at the ratio $R_f/R_x$ , which is eight. The resistor that raises the noise gain is itself a noise source the noise gain multiplies. Integrated to a gigahertz, the outputs are 57.4 and 253.3 microvolts rms — a factor of 4.4, of which the noise gain is a factor of five at the amplifier and the wider band is the rest.

The same factor of five applies to the offset. An amplifier with a millivolt of input offset puts two millivolts on the output of a plain inverter and ten on the output of this one, and the drift follows. The bandwidth noise sees is the reminder that the rms numbers here are integrals of the curves, not densities times a −3 dB bandwidth, and they include the hump the second pole leaves near crossover.

How much gain to add

Ten is the number on the datasheet, and it is not the only choice. The resistor across the inputs is a continuous knob, and every setting trades margin against noise and bandwidth at once.

At a noise gain of four — 5 kΩ across the inputs — the fast inverter has 47.5 degrees of margin, 1.93 decibels of peaking and a −3 dB point at 29.7 MHz, with the amplifier’s noise multiplied by four. At six — 2.5 kΩ — it has 55.6 degrees, 0.75 decibels of peaking and 22.1 MHz. At ten it has the 65.6 degrees and 14.1 MHz already measured, and at twenty, with 556 Ω, it has 76.4 degrees — the slow part’s own figure as an inverter — and 6.40 MHz, which is the slow part’s bandwidth too. At a noise gain of twenty the fast part has become the slow part, with ten times its noise and its slew rate intact.

That last case is worth noticing because it is the limit of the knob: raising the noise gain to the ratio of the two parts’ gain–bandwidth products, times the slow part’s own noise gain, reproduces the slow part’s loop exactly, so every small-signal number matches and the only differences left are the large-signal one the fast part was bought for and the noise it pays for it. Between two and twenty the designer is choosing a point on a line whose ends are “fast and ringing” and “the slow part, louder”.

The useful settings are in the middle. A converter driver that must settle a step to a sixteenth of a least-significant bit wants the overshoot small and the settling short, and the step figure below says that at ten the overshoot is 4.3 per cent and the settling barely slower than the bare part’s; a noise-sensitive stage wants the lowest noise gain whose margin it can live with, which on this model is about four. The datasheet’s ten is a choice of margin made on the designer’s behalf. Where the behaviour is written down puts each of those margins on the plane as a pair of closed-loop poles, which is where the overshoot and the settling actually come from.

The gain raised only where the loop needs it

Stability is decided at crossover, near 14 MHz. The noise gain at a kilohertz has nothing to do with it. A capacitor in series with the resistor across the inputs removes the resistor at low frequency and leaves it in place above the corner the pair sets.

The resistor across the inputs with 470 pF in series: noise gain two at low frequency, ten where the loop crosses. computed by solving, not by drawing. With 1.25 kΩ alone across the inputs the noise gain is ten at every frequency (dashed), so the offset and the low-frequency noise are multiplied by ten. With 470 pF in series it is two below 271 kHz and rises to ten above it (solid), and the margin is 64.2° against 65.6° with the resistor alone. At a kilohertz the output noise is 20.5 nV/√Hz against 64.0 nV/√Hz; integrated to a gigahertz, 254.3 µV against 253.3 µV. Margins for 47 pF to 2.2 nF: 52.1°, 59.1°, 62.7°, 64.2°, 65.0°, 65.3°. — Fig. 5 The noise gain measured with 1.25 kΩ alone across the inputs (dashed, ten at every frequency) and with 470 pF in series (solid): two below 271 kHz, ten above. The margin is 64.2° against 65.6°. At a kilohertz the output noise falls from 64.0 to 20.5 $\text{nV}/\sqrt{\text{Hz}}$ ; integrated to a gigahertz it is 254.3 against 253.3 µV.

With 470 picofarads in series the corner is 271 kHz. Below it the noise gain is the inverter’s two, so the offset and the low-frequency noise are back to a plain inverter’s — 20.5 nanovolts per root hertz at a kilohertz, exactly the slow part’s figure. Above it the noise gain is ten, and at crossover the margin is 64.2 degrees, within 1.4 of the resistor alone. Other values make the trade visible: 47 picofarads puts the corner too close to crossover and leaves 52.1 degrees; 2.2 nanofarads leaves 65.3; the margin approaches the resistor’s own as the corner moves down.

The wideband total does not move: 254.3 microvolts against 253.3. Almost all of the integrated noise lives between a few megahertz and the loop’s own band edge, where the noise gain is ten either way, so a capacitor that removes the resistor at audio frequencies has moved noise out of a band nobody was listening to it in. The series capacitor buys the low-frequency half of the cost and none of the wideband half, which makes it the right repair for a precision stage that happens to need speed and the wrong one for a wideband stage whose noise is integrated by whatever follows it.

In time, and what the linear model leaves out

The step response is where a reader of an oscilloscope would meet all of this.

A one-volt step through the three gains of −1. computed by solving, not by drawing: the residues of each closed loop's poles, summed at 601 instants. The slow part overshoots 0.0% and is within 1% by 107 ns. The fast part with nothing across its inputs overshoots 35.1% and settles by 72 ns. Held at a noise gain of ten it overshoots 4.3% and settles by 75 ns. This is the small-signal step; the reason to buy the fast part is its slew rate, which a linear model does not contain. — Fig. 6 A one-volt step through the three gains of −1, from the residues of each closed loop’s poles. The slow part: no overshoot, within 1% by 107 ns. The fast part bare: 35.1% overshoot, settled by 72 ns. The fast part at noise gain ten: 4.3% overshoot, settled by 75 ns.

The slow part does not overshoot and is within one per cent by 107 nanoseconds. The fast part with nothing across its inputs overshoots 35.1 per cent — a waveform that would fail any review — and yet settles to one per cent by 72 nanoseconds, because it rings near its 28.6 MHz crossover and its ringing dies within a few cycles. Held at a noise gain of ten it overshoots 4.3 per cent and settles by 75. So for a small step the resistor removes the ringing and costs three nanoseconds of settling, and both versions of the fast part settle 30 per cent sooner than the slow one.

The comparison understates the fast part, and the reason is not in any netlist here. A small step is linear; a large one is limited by the rate at which the amplifier’s internal current can charge its compensation capacitor, and that capacitor is ten times smaller in the decompensated part. The slew rate is ten times higher, so for a large step — the full-scale transition of a converter driver, say — the decompensated part settles several times sooner. The step that is too big measured where a linear model of slewing stops being true, and it is above that amplitude that the decompensated part earns its place. The resistor across the inputs is the price of admission to a slew rate.

Three other ways to raise the same number

The resistor across the inputs is one of several circuits that supply a noise gain without a signal gain.

A follower with a resistor to ground at its inverting input and one in the feedback. That is simply a non-inverting gain of two; to make a follower from a decompensated part the feedback resistor is kept and the ground resistor is replaced by an RC to ground, which is the series-capacitor arrangement above moved to the other input. The follower’s gain at direct current stays one.

A capacitor across the feedback resistor, the repair of the gain the loop closes against, lowers the high-frequency noise gain. It is exactly the wrong move on a decompensated part, which needs the noise gain high at crossover. The two repairs are mirror images, and confusing them makes a marginal circuit unstable.

A summing amplifier with several inputs has a noise gain of one more than its number of inputs whether the designer wants one or not. That is the subject of the input that costs every other input, and on a decompensated part it is a free source of stability: a ten-input mixer on this die needs no extra resistor at all.

And the resistor that buys the margin back is the other resistor that stabilises an amplifier without touching its low-frequency gain — in series with a capacitive load rather than across the inputs. Both work by changing what the loop sees near crossover and nothing else.

Where the numbers come from, and their limits

Every figure is a netlist solved at each frequency: the amplifier as a gain stage, two buffered internal poles and an output resistance; the resistors as resistors. The margin is read from the loop cut at the amplifier’s inverting input and driven from a source, with every resistor still loading the feedback node, and bisected for unity magnitude. Noise is the transfer from a source in series with each element to the output, one solve per source per frequency, combined as independent powers. Steps are the residues of the closed loop’s poles, from its transfer function computed on the same netlist.

The model has two poles where a real amplifier has a dozen, so the margins are those of a two-pole part and the 45-degree point at a noise gain of 3.5 belongs to this model; a datasheet’s own “stable at ten” includes margin for the poles this one omits. It has no slew limit, so every time-domain number is a small-signal one. Its voltage noise is white, with no flicker corner, so the low-frequency noise figures are white-noise figures. And it has no input capacitance, which in a real part adds to the junction and raises the noise gain at high frequency on its own — harmless on a decompensated part, the very hazard on a unity-stable one.

Still open: the follower’s lead network, the slew it buys, and the noise it lets through

The decompensated follower. A follower has no inverting-input resistor to put anything across, and its stabilisation needs an RC network from the inverting input to ground with a feedback resistor to work against. The values that give a stated margin while keeping the input impedance and the direct current gain of a follower are a solve on this netlist with the non-inverting input driven, and the result would say how much of the fast part’s speed survives as a buffer.

The slew rate as a number. The reason to buy the decompensated part is a slew rate ten times higher, and nothing here contains it. Adding a current limit on the internal node would make the large-step comparison measurable: at what step size does the fast part’s settling advantage over the slow part grow from 30 per cent to a factor of several, and does the resistor across the inputs cost any of it.

Noise the series capacitor lets through. Between the capacitor’s corner and crossover the noise gain climbs from two to ten, and the amplifier’s noise is amplified across that region however quiet the band below it is. A stage followed by a filter whose band ends somewhere in that region pays for part of it, and the capacitor that minimises the integrated noise in a stated band, for a stated margin, is an optimum those three values only bracket.