The rejection the parts have

Two essays in this field have measured how well an instrumentation amplifier rejects what is common to both its inputs, and both were about the arrangement.

The first found that a one-amplifier difference stage’s rejection is $(1+G)/4t$ and contains nothing of the amplifier at all: four resistors decide it, and one-tenth-per-cent parts give 54 decibels — a factor of five hundred rather than a removal. The second found that putting a two-amplifier stage in front works for a reason that is not the obvious one. The input stage passes a common-mode voltage at exactly unity and a differential one at $1 + 2R_f/R_g$, so the difference stage’s common-mode gain is untouched — 2.00 millivolts per volt at every gain — and the whole improvement is the differential signal being larger when it arrives. Rejection is therefore the one-amplifier figure plus $20\log G_1$, exactly, and the two resistors that set the gain need not match at all.

Every number in both was computed with amplifiers whose own rejection is infinite. That is a statement about an arrangement. This essay puts the parts in.

Where an amplifier’s own rejection goes in a netlist #

A real amplifier’s output is not $A(v_+ - v_-)$. It is $A(v_+ - v_-) + A_\mathrm{cm}\,v_\mathrm{cm}$, and the ratio $A/A_\mathrm{cm}$ is what a data sheet calls its common-mode rejection.

Written that way it is a property of the amplifier’s transfer function, which is awkward to put in a netlist. Written the other way it is an input quantity: the same behaviour is produced exactly by a perfect amplifier with an error voltage of $v_\mathrm{cm}/\mathrm{CMRR}$ in series with its non-inverting input. That is where it goes here — a dependent source whose control is the very input it is in series with, which keeps the netlist linear, keeps the amplifier model untouched, and means the error is solved along with everything else rather than added to the answer afterwards.

It is the same move the instruments field makes with every other instrument it measures: the thing that does the measuring goes into the circuit as an element, and every reading is a reading of the circuit that includes it. A probe is a capacitance with a bandwidth; an ammeter is a resistance; an amplifier’s imperfect rejection is a source in series with its input.

Matched parts cost nothing at all #

The first result is the one that is not obvious from the data sheet, and it is worth stating before any numbers.

Give both input amplifiers the same finite rejection and the instrument’s rejection does not change. Not “changes by a negligible amount”: the measured figure with two 98-decibel parts is 93.971 decibels, and with two perfect ones it is 93.971 decibels.

The reason is in the arrangement rather than in the parts. An error of $v_\mathrm{cm}/\mathrm{CMRR}$ at the input of the upper amplifier appears at its output; the identical error at the input of the lower one appears at its output; the two outputs have therefore both moved by the same amount, which is a common-mode signal at the input of the difference stage — and rejecting common-mode signals is what the difference stage is for. It removes them with its own $(1+G)/4t$, exactly as it removes the original common-mode voltage.

So the specification a designer reads on the data sheet of the input amplifiers is not, by itself, in the answer at all. What is in the answer is the difference between the two, and a monolithic instrumentation amplifier has both input amplifiers on one piece of silicon a few hundred microns apart, made in the same process minutes apart, at the same temperature — which is the whole reason such a part exists rather than being assembled from three separate amplifiers.

The ceiling the mismatch sets #

What the difference between the two amplifiers leaves is a floor, and it is a clean expression:

$$\mathrm{CMRR}_\mathrm{floor} = \frac{1}{\left|\dfrac{1}{\mathrm{CMRR}_1} - \dfrac{1}{\mathrm{CMRR}_2}\right|}$$

with no gain in it, no resistor tolerance, and no frequency. Two parts specified at 100 and 98 decibels — a two-decibel difference, which is nothing at all on a data sheet — give 111.7 decibels and no arrangement of the six resistors around them will do better.

Measured against the netlist, the expression is exact to the ninth decimal place, which is what one expects of a statement that is really about superposition. And it has the shape that makes mismatch specifications so unforgiving: two parts at 80 decibels that differ by one per cent give 120 decibels, while two parts at 120 decibels that differ by a factor of two give 126. Matching matters more than magnitude, by a wide margin, and it is the quantity no discrete amplifier’s data sheet gives.

The consequence for the architecture is a corner. The four resistors’ term rises decibel for decibel with the input stage’s gain; the parts’ term does not rise at all; so the two cross, and above the crossing more gain buys nothing. For the parts here the instrument is 3 decibels short of what the arrangement alone would give at a gain of 1903, and by a gain of $10^5$ it is at 111.9 decibels where the arrangement alone would have been worth 148.

What a decibel of rejection is actually worth #

The numbers above are large and abstract, so it is worth converting one of them into the quantity a measurement actually loses.

An instrumentation amplifier reading a strain gauge bridge sees perhaps ten millivolts of differential signal sitting on two and a half volts of common mode, which is the bridge’s own excitation divided by two. At 111.7 decibels of rejection the common mode contributes $2.5/3.85\times10^{5} = 6.5$ microvolts of apparent differential signal — 650 parts per million of the ten-millivolt full scale, or about ten counts of a fourteen-bit reading.

At the 94 decibels the arrangement alone gives at a gain of a hundred, it contributes 50 microvolts: five thousand parts per million, and eighty counts. At the 54 decibels a bare difference stage manages, it contributes five millivolts — half the signal.

So the ladder this essay ends has taken a measurement from unusable, to usable, to limited by something else. That last part is the point of computing the floor: at 111.7 decibels the rejection has stopped being the thing that decides the reading, and the next term in the error budget — the amplifier’s own offset drift, the bridge’s own excitation stability, the reference — is where the next decibel has to come from.

A number that has stopped mattering is a useful thing to have established, and it cannot be established without knowing where the ceiling is.

The gain at which it beats its own floor, which is a coincidence #

There is one feature of the measured curve that is not in the algebra above, and it is worth pointing at because it would otherwise look like a result.

At one particular gain the instrument’s rejection goes above the floor — 125 decibels at a gain of a thousand, against a floor of 111.7 — and stays above it for about a decade of gain before settling back down onto it from above.

That is a cancellation, not a mechanism. The resistors’ error and the parts’ error are two independent contributions with independent signs, and at one gain they are equal and opposite. It is real, it is repeatable for that exact set of components, and it is worth nothing to a designer: the gain at which it happens depends on the sign and size of a resistor mismatch nobody controls to a decade, and building a production run on it would give a spread of rejections from the floor to a hundred and twenty-five decibels with no way to sort them but measurement.

This collection has met the same shape before, in the same field: the two-wire ohmmeter whose lead error and voltmeter loading cancel exactly at one point on a grid. The right response is to name it as a coincidence and to make the claim about the asymptote instead — which is what the figure’s own assertions do.

The two ways a common mode becomes a difference #

It is worth separating the two mechanisms that this figure holds in one number, because they fail differently and are fixed differently.

The first is a gain mismatch. The four resistors of the difference stage form two dividers, and if the two ratios are not identical, a voltage applied equally to both inputs arrives at the amplifier’s two inputs unequally. That error is proportional to the common-mode voltage, is flat with frequency until the amplifier runs out of loop gain, and is what $(1+G)/4t$ measures.

The second is an amplifier error, and it is not a divider at all: the part responds to the average of its two inputs as well as to their difference, by an amount that is a property of its input stage’s own symmetry — the tail current source’s output impedance, in a bipolar pair, which is exactly the quantity the semiconductor field’s cascode essay is about.

The two enter the answer at different points, which is why they scale differently with gain, and they are both called “common-mode rejection” in the literature — one of a network, one of a device. An instrumentation amplifier’s specification is the combination, measured on the assembled part, and the useful thing about separating them is that only one of the two can be improved without changing parts.

There is a third mechanism that this netlist does not contain and that is often the real one: the source impedances are not identical either, and a common-mode current flowing through unequal source impedances produces a differential voltage before the instrument has seen anything at all. That belongs to the circuit being measured rather than to the instrument, which is why it is not here — but it is the reason a real bridge is driven from a low impedance on both sides.

Which of the three amplifiers matters, and by how much #

There are three amplifiers in the arrangement and they do not enter equally.

The difference stage’s own rejection is divided by the first stage’s gain, exactly as the resistor mismatch is, because everything at that stage’s input is being compared with a differential signal that is $G_1$ times larger than it was. An 80-decibel difference amplifier behind a gain of a hundred contributes as an 120-decibel one would alone: measured, it takes the instrument from 93.97 to 93.55 decibels, four tenths of a decibel.

The input amplifiers’ rejection is not divided by anything, because their error is amplified by exactly the gain the signal is. Only their mismatch survives, and it survives in full.

So the design rule that falls out is short and is not what a parts list would suggest. Buy matching in the input stage and magnitude in the difference stage: the input pair must be matched and need not be individually excellent, and the difference amplifier can be an ordinary part because its own error is attenuated by all the gain in front of it. Which is exactly how monolithic instrumentation amplifiers are built, and this is the measurement that says why.

Why this is the rung that ends the ladder #

Three rungs on one instrument, and each measured a different owner of the same number.

The first belonged to the resistors: four of them decide the rejection of a difference stage, one part in a thousand of mismatch gives 54 decibels, and no amplifier appears in the expression. The second belonged to the architecture: putting gain in front multiplies the rejection by exactly that gain, and identifies which two resistors are in the answer and which two are not. This one belongs to the parts, and it is where the sequence has to stop, because what it finds is a term that nothing about the arrangement can reach.

The resistor mismatch can be improved by buying better resistors or by trimming one of them, and the site of the improvement is on the board. The architecture’s term can be improved by moving gain from the difference stage into the input stage, and the site of the improvement is in the schematic. The input amplifiers’ mismatch can be improved only by choosing a different pair of amplifiers, and there is nothing a designer can do at the board or the schematic level that touches it.

That is what makes it a floor in this collection’s sense rather than a term in a budget: it is the quantity that remains when every variable the design controls has been set to its best value, and it is the number a specification should be written against.

Where the floor is met, and what it is spent on #

A rejection floor set by the mismatch between two amplifiers is the ceiling on the arrangement measured next door. The four resistors that decide, and the two that do not is that arrangement, where the rejection rises with the first stage’s gain until it meets this floor. The rejection four resistors decide is where the rejection is spent, against a shared return’s common-mode error. The millivolts in the wire is where that error comes from. What a pair cancels, and what it only halves is the same symmetry argument on a single device pair, where the cancellation is exact rather than limited by matching, and The ammeter that is a resistor is the measurement that most often needs all of it.

What is checked #

The matched-parts result is asserted as an equality rather than as an inequality: the instrument with two identical finite amplifiers must give the same rejection as one with two perfect ones, to a part in a thousand. The floor is asserted against its closed form to the ninth decimal. The instrument is asserted to have settled onto that floor at the largest gain drawn, whatever the arrangement alone would have given there. The excursion above the floor is asserted to exist and is described as the coincidence it is. And at unity gain the parts are asserted to be invisible, which is the case the rung below measured and is the check that the new elements do nothing where they should do nothing.

What is not measured here: the frequency dependence of a real amplifier’s own rejection, which falls with frequency as its loop gain does and would put a corner on this essay’s floor as well as on the architecture’s; and the input bias currents, whose mismatch flowing in the source impedance is a second mechanism that turns a common-mode voltage into a differential one and which has nothing to do with any amplifier’s rejection at all.

The last of those is not a footnote #

The bias-current mechanism named at the end of that list is worth separating from the rest, because it is not a smaller version of what this essay measures — it is a second route from a common-mode voltage to a differential one, with no rejection in front of it at all.

The current the instrument draws is where its size is established, and the framing there is the one this essay needs: an amplifier’s input current is not an idealisation of a small quantity but an idealisation of one whose size is decided by something outside the part. Fifty nanoamps is nothing until it flows in a megohm, and then it is fifty millivolts. What matters here is the mismatch of two such currents flowing in two source impedances that differ, and that essay’s classical cure — balancing the two resistances — removes the bias current and leaves the offset current, worth a factor of ten rather than a thousand, at a cost of forty per cent of the noise density.

Read against the 111 decibels this essay reaches, the arithmetic is uncomfortable. A hundred and eleven decibels is a factor of 350 000, so on a one-volt common-mode signal it leaves under three microvolts. A nanoamp of offset current in a kilohm of source imbalance is a microvolt, arriving as a differential signal that the architecture has no view of — so on any source with an imbalance, the term that this ladder spent three rungs improving is comparable with a term none of the three rungs contains.

The corner the instrument has no part in is the same conclusion arrived at in the frequency domain rather than in the offset budget, and the pairing is worth noticing: the imbalance a source has is the quantity that limits the instrument at direct current through the bias currents and at frequency through the capacitances. It is the one number a data sheet cannot carry, and it decides both ends.

Which puts the three architecture results of this ladder in their place. The rejection four resistors decide establishes what four tenth-per-cent resistors are worth — 53.99 decibels against a closed form’s 53.98, a factor of five hundred and not a removal — and the four resistors that decide, and the two that do not establishes what an input stage adds, exactly twenty times the log of its gain. Both are exact, both are about the instrument, and both are ceilinged by the mismatch measured here and floored by a source imbalance neither of them contains.