The input that costs every other input

An inverting amplifier with several inputs, each through its own resistor into one summing junction, is the most common way of adding signals in analogue electronics: an audio mixer, a sum of sensor channels, the output stage of a resistor-ladder converter. Each input sees a gain of $-R_f/R_i$ and nothing else. The junction is held at ground — as firmly, the node that is at ground for a while measured, as the loop is strong — so no input’s current affects any other’s; the superposition is exact; and a schematic of eight channels reads as eight independent gains of −1.

The gain the loop closes against gave the reason that reading is incomplete. Bandwidth, stability and the amplification of the amplifier’s own errors follow the noise gain — one plus the feedback impedance over everything at the junction — and the junction of a summing amplifier has every input’s resistor on it. With eight equal resistors the noise gain is nine, whichever inputs are driven. The gain added to buy a margin raised a noise gain on purpose, with a resistor across the inputs that carries no signal; a summing amplifier does the same thing by construction, once per input, and nobody chose it.

This essay measures what one input costs the others: in bandwidth, in noise, in a muted channel, in a quiet channel beside a loud one, and in the architecture that is supposed to avoid it.

Each input divides everyone’s bandwidth #

The amplifier is the unity-gain-stable 10 MHz part of the previous essay, with its second pole at 20 MHz. Each input has a ten-kilohm resistor into the junction and the feedback resistor is ten kilohms, so every channel has a gain of exactly −1.

Input 1’s gain at a kilohertz is one, to the loop’s finite-gain error, whether it shares the junction with nothing or with fifteen other inputs. Its bandwidth is another matter: 6.43 MHz alone, 3.95 with two inputs, 2.21 with four, 1.17 with eight and 603 kHz with sixteen. The noise gain is one more than the number of inputs, the loop crosses unity at the gain–bandwidth product divided by that — the arithmetic of the ideal amplifier, and where it stops being one applied to a gain nobody asked for — and at sixteen inputs the channel’s −3 dB point is the gain–bandwidth product over seventeen to within the loop’s own rounding. The first two values sit a little above $f_T/(n+1)$ because the second pole puts a trace of peaking into a loop that crosses close to it.

Nothing about the other inputs has to be active for this to happen. Their sources can be at zero volts; it is enough that their resistors connect the junction to something low-impedance. A designer specifying a mixer channel’s bandwidth from its own gain of −1 will specify it nine times too wide for an eight-channel mixer, and nothing in the schematic of one channel would say so.

The amplifier’s noise is multiplied, the resistors’ is not #

The same arithmetic applies to the amplifier’s own noise, and here the two kinds of noise source in the circuit behave differently enough to give a crossover.

The amplifier’s input voltage noise, five nanovolts per root hertz here, appears in series with the non-inverting input and reaches the output multiplied by the noise gain. Each resistor’s Johnson noise — 12.7 nanovolts per root hertz for ten kilohms at 290 K, the floor a resistor sets — reaches the output at that resistor’s own signal gain, which is one for every input resistor and one for the feedback resistor. So the amplifier’s contribution grows as $n + 1$ and the resistors’ together grow as $\sqrt{n + 1}$.

They are equal where $e_n(n+1) = \sqrt{4kTR\,(n+1)}$, that is at $n + 1 = 4kTR/e_n^2$: 5.4 inputs. Below that the resistors dominate the output noise and the choice of amplifier hardly matters; above it the amplifier dominates and the choice of amplifier is the noise specification. At thirty-two inputs the output carries 180.2 nanovolts per root hertz, of which 164.9 is the amplifier — 84 per cent of the noise power, from a component that a single channel considered alone would have contributed a quarter of.

The crossover moves with the resistor value: halving every resistor lowers the Johnson noise by $\sqrt2$ and moves the crossover to half as many inputs, which is the reverse of the usual advice that lower resistances are quieter. They are quieter per resistor, and they hand the noise budget to the amplifier sooner. The floor a circuit has found the same competition between a device’s voltage noise and a resistor’s for a single stage; a summing amplifier runs that competition with a multiplier on one side.

A channel that is muted still costs #

Mixers mute channels, and there are two ways to do it. One grounds the channel’s source, or switches its input to ground, leaving its resistor connected to the junction. The other opens a switch in series with the resistor and takes the resistor off the junction altogether.

Grounded, every muted channel keeps its resistor on the junction, so the noise gain stays at seventeen and the output carries 99.7 nanovolts per root hertz with one channel in use or with sixteen. Opened, the muted resistors leave, the noise gain falls to one more than the channels in use, and with two channels in use the output carries 26.6 — 11.5 decibels quieter for exactly the same two signals, and nearly six times the bandwidth.

The difference is a property of the switch’s position, not of anything a listener could hear in the signal path, and it is the reason a well-designed mixer disconnects rather than grounds. It also says how to read a mixer’s noise specification: a figure measured with all channels muted by grounding describes the loudest the mixer will ever be, not the quietest, and a figure measured with all channels assigned describes nothing about a session that uses two of them.

A quiet channel pays for a loud neighbour #

Not every input has the same gain. A mixer that sums a line-level source at a gain of 0.1 with a microphone pre-amplified at a gain of 10 has resistors of 100 kΩ and 1 kΩ on the same junction, and the noise gain is $1 + 0.1 + 10 = 11.1$.

Alone, the 0.1-gain channel has a noise gain of 1.1, and the noise at the output referred back to its input — divided by its own gain of 0.1 — is 143.7 nanovolts per root hertz, most of it the feedback resistor’s noise magnified by the division. Beside a channel of gain 10, the same input sees 697.0: 13.7 decibels worse, entirely because a neighbour it never shares a signal with has raised the noise gain to 11.1. Beside a gain of 100 it sees 5.2 microvolts per root hertz, a figure that would be unacceptable on any line input.

The rule is that every channel’s input-referred noise is set by the sum of all the channels’ gains, so the low-gain channels of a summing amplifier are the ones that pay for its high-gain channels. The remedy is to give the high-gain path its own stage — pre-amplify the microphone separately, then sum at a common low gain — which raises the parts count and lowers every quiet channel’s noise. The gain added to buy a margin used the same rule deliberately: a resistor that carries no signal raised everybody’s noise gain to stabilise a fast part.

Sixteen channels, worked #

Put the numbers into a mixer. Sixteen channels, each at a gain of −1 through ten kilohms, line-level signals of a volt rms, an audio band of twenty kilohertz, and the 10 MHz amplifier with its five nanovolts per root hertz. Two channels are in use and fourteen are muted.

With the fourteen muted by grounding, the output carries 99.7 nanovolts per root hertz across the band. White noise integrates as the root of the bandwidth, so over twenty kilohertz that is 14.1 microvolts rms, and against a one-volt signal the mixer’s signal-to-noise ratio is 97.0 decibels. With the fourteen muted by disconnecting their resistors the output carries 26.6 nanovolts per root hertz, 3.76 microvolts over the band, and the ratio is 108.5 decibels. The two mixers are the same parts on the same board, and one is 11.5 decibels better than the other because of where fourteen switches sit.

The arithmetic also says where the 97.0 comes from. Of the 99.7 nanovolts, 85.0 are the amplifier’s five multiplied by seventeen, and the resistors’ seventeen root-summed Johnson noises are 52.2: the mixer muted by grounding is amplifier-limited at nearly three times the crossover count, and a quieter amplifier would buy back most of the gap. The mixer muted by disconnecting, with three resistors on its junction, sits below the crossover — 15 nanovolts from the amplifier against 21.9 from the resistors — and a quieter amplifier would buy it almost nothing. Two designers measuring two configurations of one board would therefore give opposite advice about the same amplifier, and both would be right about their own measurement.

The bandwidth moves with the noise. Grounded, each channel’s −3 dB point is near 590 kHz, the gain–bandwidth product over seventeen; disconnected, with three resistors on the junction, it is the 3.95 MHz of the two-input curve in the first figure. Neither matters in an audio band. Both matter if the same summing stage is used for a wideband sum — a video mixer, or the output of a multi-channel converter — and a channel whose bandwidth depends on how many of its neighbours are switched in is a channel whose rise time does too.

That dependence is the summing amplifier’s whole character, and it is the opposite of what two solves that add found for the signals themselves. The signals superpose exactly, each channel’s output untouched by any other channel’s input; the noise gain is shared, and with it the bandwidth and every error the amplifier adds. A schematic of a summer shows the first property and hides the second.

A tree buys bandwidth, not quiet #

If one stage of eight inputs closes its loop against nine, the obvious repair is to sum in stages: two summers of four, then a summer of two; or a binary tree of two-input summers three deep. Each stage then closes against a smaller noise gain.

It works for bandwidth. One stage of eight gives each channel 1.17 MHz; four-then-two gives about 1.93, because the slower first stages now close against five and the second stage against three; the binary tree gives about 2.28.

It does not work for noise, and the reason is the other half of the noise budget. The amplifiers’ share does fall — 45.0 nanovolts per root hertz for one amplifier at a noise gain of nine, 38.4 for three amplifiers at five, five and three combined as independent sources — but every stage brings its own resistors, and each stage’s resistors reach the output at a gain of one — the resistor the noise comes from is the general rule that a resistor’s noise is filtered by everything after it, and here everything after it is a gain of one. The resistors’ share rises from 38.0 to 45.6 for the two-level tree and to 58.0 for the binary one. The totals are 58.9, 59.6 and 70.3: two levels is a wash, three is worse.

So the tree is a bandwidth architecture. It is the right answer for a wide-band sum whose amplifier is quiet compared with its resistors and the wrong answer for a quiet sum, and on a channel count below the 5.4-input crossover — where the resistors dominate anyway — it can only make the noise worse. The measurement that decides between them is the crossover count, not the number of inputs.

The capacitance the inputs bring #

Every input brings a track, a resistor body and a pad to the junction, and each adds a few picofarads. The gain the loop closes against showed what nine picofarads at one inverter’s junction does to its margin. A summing junction collects that capacitance from every input.

The result is the reverse of what the capacitance suggests. With four picofarads per input plus three of the amplifier’s own, the single-input inverter is the worst case at 39.6 degrees, and eight inputs with 35 picofarads have 72.3. Each input adds capacitance and resistance to the junction in the same proportion, so the resistive noise gain rises as fast as the capacitive one; the high-frequency noise gain still rises, but from a higher base, and the loop crosses unity lower, where the capacitance’s share of the junction admittance is smaller. More inputs mean more margin, with or without the capacitance.

That is a useful negative result. A designer who has made a single inverter stable with its junction capacitance has made every summer built from the same layout stable too; the hazard is concentrated at the smallest channel count, which is the one most likely to be tested on a bench.

Where the numbers come from, and what they leave out #

Each configuration is a netlist: one voltage source per input, its resistor into the junction, the feedback resistor, and the amplifier as a gain stage with two buffered internal poles and fifty ohms of output resistance. Gains and bandwidths are solved transfers; the −3 dB points are bisected on them. Noise is solved one source at a time — the amplifier’s as a source in series with its non-inverting input, each resistor’s as a source in series with that resistor — and combined as independent powers. The tree combines its stages as independent sources too, which is exact because no two stages share a noise source. The margins are read from the loop cut at the inverting input with every resistor and capacitor still on the feedback node, the measurement what is left at crossover defines.

The amplifier has white voltage noise and no current noise. A bipolar part’s current noise at the inverting input has nowhere to go but the feedback resistor, so it reaches the output as $i_n R_f$ whatever the channel count — ten nanovolts per root hertz at a picoamp — a constant floor under every curve above that moves none of the crossovers, since it neither grows nor shrinks with the inputs. The resistors are at 290 K and the amplifier’s noise is temperature-independent. And the sources driving each input are ideal: a real source’s own impedance adds to its input resistor, changing that channel’s gain and, through the junction, everyone’s noise gain by a small amount.

Still open: the converter that is a summer, the switch’s own resistance, and current noise #

A resistor-ladder converter is a summing amplifier. The ladder built for the rest of itself measured the divider chain such a converter is made from. A binary-weighted converter’s output stage sums currents whose resistors are switched between the junction and ground according to the digital word, so its noise gain is a function of the word being converted — every resistor that is switched to the junction raises it. Whether that makes the converter’s output noise, and its glitch, code-dependent in a way its datasheet does not quote, is the summing amplifier asked of a digital input.

The mute switch is a resistor too. An analogue switch in series with a channel’s resistor has an on-resistance that adds to the resistor and an off-capacitance that leaves a little of it connected at high frequency. A muted channel’s resistor is therefore not fully off the junction above the frequency where that capacitance’s reactance falls to the resistor’s value, and the noise gain at crossover — where the margin lives — may not fall as far as the noise gain at a kilohertz does.

The resistance a mixer should be built at. Raising every resistor raises the resistors’ Johnson noise as $\sqrt R$ and an amplifier’s current-noise term as $R$, while the voltage-noise term does not move; lowering them does the reverse and costs drive current. For a stated channel count and amplifier there is a resistance at which the sum of the three is least, and it moves the 5.4-input crossover in a direction that depends on whether the amplifier is bipolar or field-effect — a three-term optimum that this essay’s two-term figures only bracket.