The errors that arrive before the gain

Four rungs of this argument have measured the same three-amplifier instrument, and each of them found a different owner of the one number on its data sheet.

The rejection four resistors decide found the four resistors around the difference stage, whose matching gives $(1+G)/4t$ and contains nothing of the amplifiers at all. The four resistors that decide found that the two resistors setting the gain are not among them, and may be ten per cent apart without moving the answer a hundredth of a decibel. The rejection the parts have put each amplifier’s own finite rejection into the netlist and found that two matched mediocre input amplifiers cost nothing, while the difference between two good ones sets a ceiling with no gain in it. And the corner the instrument has no part in connected the thing to a source, and found a corner that belongs to the cable.

Each of those was measured with the other three switched off, which is the right way to isolate a mechanism and is not the circuit anybody builds. This rung puts all four into one netlist and asks which of them binds.

What is being solved #

One netlist, at one frequency, driven twice. Two source resistances and two input capacitances in front of it; two input amplifiers with a resistive bridge between them; four resistors and a third amplifier behind that. Each amplifier’s own finite rejection is in the netlist as a dependent source of $v_\mathrm{cm}/\mathrm{CMRR}$ in series with its non-inverting input, which is where the quantity is defined and keeps the network linear. The two drives are a differential one and a common-mode one, and what is read out is the ratio of the two output voltages.

The quantity kept is not the ratio, though, and that is the change that makes this rung possible. What is kept is the complex common-mode gain — the output, in volts, for one volt applied to both inputs together. A rejection in decibels is a magnitude of a ratio, and magnitudes of ratios do not add. Common-mode gains do, in the sense that a linear network’s response is the sum of the per-source solves, and that is the property the whole of this essay turns on.

The amplifiers are nullors here rather than the one-pole macro-model the rungs below used. That is deliberate: a finite open-loop gain is a fifth mechanism worth 146 decibels at unity gain, two orders below the smallest of the four, and its only effect on the arithmetic below would be to blur it.

The reproduction, which comes before anything else #

Each mechanism switched on alone must return the number the rung that found it reported, or nothing after this is worth reading.

It does. The four difference resistors at a tolerance of one part in a thousand give a common-mode gain of 1.998 mV/V, which is 53.98 decibels of rejection at unity gain and 93.99 at a gain of a hundred — the closed form’s $(1+G)/4t$, in the same place it was. Two input amplifiers of 100 and 98 decibels give 111.74 decibels, which is $1/|1/\mathrm{CMRR}_1 - 1/\mathrm{CMRR}_2|$ to the digit the rung below quoted. A kilohm of source imbalance against ten picofarads gives 110.06 decibels at 50 Hz, and the corner and the twenty-decibels-a-decade slope above it are the ones already drawn.

The source’s contribution deserves a stronger statement than agreement, because it is exact. The solved common-mode gain per unit of first-stage gain is $3.141593\times10^{-6}$, and $\omega\Delta\tau$ for a kilohm against ten picofarads at 50 Hz is $3.141593\times10^{-6}$. Eight figures, over three decades of source resistance and three of frequency. The netlist and the time-constant argument are not two models that agree; they are one statement reached two ways.

Two of the four are divided by the gain, and it is a question of position #

Now the thing that only appears when all four are in one netlist.

The difference resistors’ common-mode gain is 1.998 mV/V at every gain from one to ten thousand — the same number to six figures. So is the difference amplifier’s own, at 0.0100 mV/V. The input pair’s mismatch is $2.589\times10^{-6}$ per unit of gain, which means it is 2.589 µV/V at a gain of one and 25.89 mV/V at ten thousand. So is the source’s, at 3.142 µV/V per unit of gain.

Two constants and two proportionalities, and the reason is not arithmetic. It is where in the circuit each error is injected.

The four resistors and the difference amplifier are at the end of the chain. Whatever they do wrong, they do to a signal that has already been amplified, so their contribution to the output is fixed while the differential gain the rejection is measured against rises with $G$. Every decibel of first-stage gain is a decibel of rejection against them, which is precisely the mechanism the first rung on this ladder described and priced.

The input pair’s mismatch and the source’s imbalance are at the beginning. The pair’s error is referred to the input by construction and is amplified by exactly the gain the signal is. The source’s is worse than that: by the time the instrument sees it, the common-mode voltage has already become a differential one, so it is not an error the instrument makes at all — it is signal, and it is amplified as signal. Both are flat, and they are flat for the same structural reason.

That is the design statement the four separate measurements could not make, and it is short. Gain repairs the errors made after it and does nothing whatever to the errors made before it.

Why four separate measurements could not have found it #

The omission looks careless in retrospect and it is not, and saying why is worth a paragraph because the same shape recurs.

Each of the four rungs below measures one mechanism with the other three removed, and each is right to. A measurement with two mechanisms in it cannot attribute what it finds, so isolating them is not a simplification but the only way any of those four numbers means anything. What isolation cannot produce is a comparison of laws: with one term present the rejection is whatever that term allows, and whether it rises with gain or does not is a property of the curve rather than a difference between curves. The statement “two of these are divided by the gain and two are not” needs at least two of them at once, and there is no measurement any single rung could have made that would have said it.

The stronger version of the same point is the cancellation. It has no meaning at all in a single-mechanism measurement, because a term cannot cancel itself. It is not an interaction either — the network is linear and the four contributions superpose to a part in eight thousand — so it is not that the mechanisms affect each other. It is that they are signed, and a sign is only visible against another sign. Four magnitudes, each correct, contain no information about which way any of them points.

The four add as complex numbers, and two of them have opposite signs #

Superposition is the reason a budget is possible at all, so it is measured rather than assumed. At a gain of a hundred, with a kilohm of imbalance at 50 Hz, the four common-mode gains are

and their sum is $1.7771\times10^{-3}$ against the $1.7773\times10^{-3}$ the whole netlist solves in one go — 1.25 parts in ten thousand. The residual is first order in the resistor tolerance, which is what a perturbed resistance rather than an added source should cost.

Two things in that table are worth more than the agreement. The input pair’s term is negative: its error subtracts from the difference resistors’ rather than adding to it. And the source’s term is imaginary, because it is $j\omega\Delta\tau$ and a time constant is a phase.

So the total is not the worst of the four, and it is not their sum in decibels either. Adding the magnitudes gives $2.5811\times10^{-3}$ — a rejection of 91.8 decibels — where the network gives 95.00. A budget built by adding decibels would have been wrong by three of them in the safe direction, which is the direction that gets a design signed off.

The departure: the instrument has a best gain #

If two of the four terms are constants and two are proportional to $G$, and if the largest of each kind have opposite signs, then there is a gain at which they cancel. There is, it is sharp, and it is not near anything a designer would choose by instinct.

The real part of the total common-mode gain is $a - bG$, with $a$ the two terms injected after the gain and $b$ the two before it. Bisected on the solved network it crosses zero at a first-stage gain of 776.3. The three constants, measured separately, predict $a/b = 775.5$. That is a tenth of a per cent, from two computations that share the netlist and nothing else.

At that gain the whole instrument’s rejection is 110.066 decibels. The source alone, at that same gain, allows 110.057. The gap is nine thousandths of a decibel.

Which is the finding, and it is worth stating without hedging: at its best gain, the instrument is exactly as good as the cable in front of it, because everything that belonged to the instrument has cancelled. Below that gain the difference resistors dominate; above it the input pair’s mismatch does; at it, the two are equal and opposite and the only thing left is the one term that is at right angles to both and cannot be cancelled by either.

Matching buys the gain, not the ceiling #

The peak is a cancellation between a constant and something proportional to $G$, so improving the proportional term moves the gain at which they meet and leaves the height of the meeting alone. That is a testable prediction with an obvious way to fail, and five instruments differing only in how well their input pair is matched give it a test.

Half a decibel of matching between the two input amplifiers is worth a factor of two in the gain at which the instrument is at its best, and is worth nothing at all in how good that best is. A data sheet’s single rejection figure carries neither of those numbers, and the one it is usually read as promising — a ceiling that better parts raise — is the one that does not exist.

What the best gain costs #

A gain of 776 in the input stage is not free, and the cost is in a quantity this ladder has so far kept out of the frame.

So the peak is real and it is narrow in a second sense: it is available at direct current and at mains frequency, and it is gone by a few kilohertz because the amplifiers have run out. An instrument built for a strain gauge, reading at a few hertz through a hundred metres of cable, can have it. One reading a kilohertz of bridge excitation cannot, and its best gain is whatever bandwidth allows.

What it does not say #

It does not say the cancellation is a design. It is a coincidence of signs between a resistor tolerance nobody controls to better than a factor of two and a pair mismatch nobody controls at all, and the rung below already said as much about the same feature seen without a source. What is new here is what sits at the bottom of it: not an arbitrarily large rejection, but the source’s own — so the cancellation’s value is bounded by something knowable even though its position is not.

It does not say the source term is always imaginary. It is $-\omega\Delta\tau/(1+j\omega\Delta\tau)$, whose real part is $\omega\Delta\tau$ times its imaginary one, so at a kilohm and 50 Hz the real part is nine parts in a billion of the total and at a hundred kilohms it is three parts in ten thousand. That second-order term is small and it is not nothing: including it moves the predicted cancellation gain from 775.5 to 747.0 at a hundred kilohms, and the bisection on the network says 747.8. Leaving it out was the one thing that made the two routes disagree, which is the useful kind of failure — a two per cent gap that names its own cause.

And it does not say the four mechanisms are all there is. A real instrument has a fifth: the amplifiers’ finite open-loop gain, deliberately removed above, which behaves like the difference amplifier’s own rejection and is worth 146 decibels at unity gain and 186 at a hundred. It has a sixth in the leakage that the current that never reaches the input measures, and a seventh in the input currents that the current the instrument draws does. None of those is a common-mode error, which is why they are elsewhere.

In the units a measurement is made in #

Decibels of rejection are a ratio, and a ratio is not an error. One volt of mains interference on a bridge with a kilohm of source imbalance appears, referred to the input, as

• 2005 µV at unity gain,
• 17.8 µV at a gain of a hundred,
• 3.14 µV at the best gain of 776.

Against a two-millivolt bridge output those are 100 per cent, 0.9 per cent and 0.16 per cent. And with a hundred kilohms of imbalance instead, the same three become 314 µV at a gain of a hundred and 314 µV at the best gain — because there the source is the whole of the answer at every gain, and the instrument’s own cancellation has nothing left to cancel.

That last pair is the practical form of the finding. Choosing the input stage’s gain is worth doing when the cable is good and is worth nothing when it is not, and which of those a design is in can be read off one number: whether $\omega\Delta\tau$ at the frequency of the interference is above or below the difference resistors’ own $4t/(1+G)$.

What one number on a data sheet is a measurement of #

A monolithic instrumentation amplifier quotes a common-mode rejection at a stated gain, usually with a second figure at unity gain, and both are measured with the two inputs driven from a laboratory source through matched resistors — which is to say with two of the four mechanisms here deliberately absent.

That test is the right one for the part. It is a measurement of the thing the manufacturer controls: the laser trim on four resistors and the matching of two input transistors. What it is not is a prediction of the instrument, and the gap between the two is not a tolerance to be added. Two of the four terms above are outside the package entirely, one of them is not even an error the part makes, and the quantity that decides which of them binds — the first stage’s gain — is chosen by whoever buys it.

Read structurally instead, the data sheet’s pair of numbers is more informative than either alone. The unity-gain figure is the four difference resistors, since nothing else is visible there; the high-gain figure, minus twenty log of that gain, is what the input pair’s matching leaves. A part quoting 90 decibels at unity and 116 at a gain of a hundred is telling a reader that its resistors are matched to about a part in ten thousand and its pair to about 116 decibels — and therefore that its two mechanisms meet somewhere near a gain of two hundred, which is the only number a design actually needs and is the one number not printed.

The number worth carrying #

776, and 110.066 decibels.

The first is where two errors cancel and is a property of a resistor tolerance and a pair mismatch. The second is what is left when they have, and is a property of a kilohm of cable imbalance and ten picofarads.

The habit that goes with it is the one this collection keeps arriving at from different directions, and every model has an edge is its general form. Four mechanisms each measured alone are four correct measurements and are not a budget, because what a budget needs is the quantity that adds — and the quantity that adds here is a complex gain, not a decibel. Isolating a mechanism is how it is understood; putting them back is how the instrument is. The ranges those two activities are right over are different, and the second one is a region rather than a line: below a gain of a few hundred the arrangement decides, above it the parts do, and in one narrow place neither does and the cable is left holding the answer alone.