The four resistors that decide, and the two that do not

The rejection four resistors decide ended on a number and a complaint. One-tenth-per-cent resistors around a difference amplifier give 53.99 decibels of rejection — a factor of five hundred, which turns somebody else’s five hundred microvolts into nine hundred and ninety-nine nanovolts rather than removing it — and the amplifier itself contributes nothing to that number at all.

Five hundred is not enough for a measurement of a ten-microvolt thermocouple beside a hundred amps of somebody else’s return current, and the standard repair is to put two more amplifiers in front. This is what those two amplifiers actually do, and it is not what the arrangement’s name suggests.

The arrangement, and the thing to notice about it #

Two amplifiers, each configured as a follower with feedback to its own inverting input, and one resistor $R_g$ tied between those two inverting inputs. Each amplifier’s output goes through its own $R_f$ back to its own inverting node. Then an ordinary difference amplifier reads the two outputs.

The differential gain of the front stage is $1 + 2R_f/R_g$: a differential input puts a voltage across $R_g$, a current flows through it and through both feedback resistors, and the outputs move apart by the amplified amount.

Now do the same with a common-mode input. Both inverting nodes are driven to the same voltage by their own feedback, so the voltage across $R_g$ is zero, so no current flows in it. No current in $R_g$ means no current in either $R_f$, which means no drop across either, which means both outputs sit at exactly the input common-mode voltage.

The front stage’s common-mode gain is therefore exactly one, at every differential gain, and that is the whole mechanism.

Which makes the arithmetic trivial and the conclusion surprising #

If the front stage multiplies the difference by $G_1$ and leaves the common mode alone, then the difference amplifier behind it sees a signal $G_1$ times larger and an interference exactly the same size. Its own rejection is unchanged; the ratio has improved by $G_1$.

$$\text{CMRR}_{\text{total}} = \text{CMRR}_{\text{diff stage}} + 20\log_{10} G_1$$

Measured on the solve at four gains, against the same difference stage alone:

and the common-mode gain of the whole instrument comes back as 1.998 mV/V at every one of them — the difference stage’s own, unmoved. The figure asserts both: the sum, and the constancy of the common-mode gain that makes the sum true.

The surprising half is the reading. The two extra amplifiers are not rejecting anything. They are not improving the matching of the four resistors that do the rejecting. They are making the signal bigger before it reaches the place where the interference gets in, which is the oldest trick in instrumentation and is usually described as a noise argument rather than a rejection one.

The two resistors that do not matter #

The consequence that follows from “no current flows in $R_g$” is the one worth the essay, and it contradicts the instinct that everything in a matched circuit must match.

If no current flows in either feedback resistor under common-mode drive, then their values are irrelevant to the common-mode gain. Both outputs sit at the input common mode whether $R_f$ on one side is twenty-five kilohms and on the other twenty-two.

The figure mismatches them by nothing, one per cent, and ten per cent, and the rejection reads

$$93.9800,\quad 93.9800,\quad 93.9800\ \text{dB}$$

— unmoved in the fourth decimal place, which is the assertion. A hundred times the tolerance the four resistors that do matter are bought to, on the two that do not, changes nothing.

What a mismatch in $R_f$ does change is the differential gain, which becomes $1 + (R_{f1}+R_{f2})/R_g$ rather than $1 + 2R_f/R_g$ — a gain error, not a rejection error. Gain errors are calibratable and rejection errors are not, which is the whole reason the distinction is worth making. Six resistors in the arrangement, four of them in the answer, and the two that are not can be bought loose.

This is also why an integrated instrumentation amplifier has one external pin pair. The four that decide the rejection are trimmed on the die, where matching is a fraction of a per cent and tracks with temperature; the one that sets the gain is brought out, because it is the one whose absolute value the user needs and whose accuracy only costs gain.

What the number is worth in the measurement it was bought for #

Ninety-four decibels is a factor of fifty thousand, and it is worth turning that back into the volts the previous essay was arguing about.

That essay’s case: a ten-millivolt sensor measured against a ground that somebody else’s hundred milliamps is also using, putting five hundred microvolts of common-mode interference across the conductor between the two ends. Four resistors at one-tenth per cent leave 999 nanovolts of it in the reading — a hundredth of a per cent of the signal, which is fine for a ten-millivolt sensor and not fine for a hundred-microvolt one.

With a first stage of a hundred the same five hundred microvolts leaves 10.0 nanovolts referred to the input. Against a hundred-microvolt thermocouple signal that is a hundredth of a per cent again, which is the point: the extra forty decibels have bought exactly two orders of sensor smallness, and nothing else has changed.

What has not improved is the interference itself. It is still five hundred microvolts arriving at the input, and it is still there in every other respect — it eats input range, it is amplified by any subsequent imbalance, and if it is large enough to take an input outside its own range no amount of rejection applies at all. Rejection is a ratio and an input range is an absolute, and the second is the one that fails abruptly.

What ends it, and why the corner is the gain #

Everything above is at direct current. The rejection does not stay there, and what takes it away is the same quantity that bought it.

The common-mode gain is flat with frequency: it is a resistor ratio, and resistors do not have corners. The differential gain is not: the front stage is a feedback amplifier of closed-loop gain $G_1$ built from parts with a gain–bandwidth product, so it holds $G_1$ only to

$$f_c = \frac{\text{GBW}}{G_1}$$

and falls at twenty decibels a decade after. Rejection is the ratio of the two, so it is flat to $f_c$ and falls at twenty decibels a decade after that.

Measured with one-megahertz parts and a gain of a hundred: 93.97 dB flat to about ten kilohertz, 90.94 dB at ten kilohertz, and 73.68 dB at a hundred kilohertz — a decade past the corner and twenty decibels down, as promised.

So the trade is exact and has no slack in it. The gain that buys the rejection is the same gain that divides the bandwidth over which the rejection exists, and their product is the amplifier’s gain–bandwidth. Choosing $G_1$ is choosing a rectangle of fixed area.

That is worth putting beside what the interference actually does with frequency. Mains interference is at fifty or sixty hertz and its harmonics, where the rejection is flat and the whole 94 decibels is available. A switching converter’s common-mode injection is at tens or hundreds of kilohertz, where the rejection has already left. So the arrangement is superb against exactly the interference it was invented for and progressively useless against the interference that has arrived since.

The measurement, which is two solves and a ratio #

The number this essay is about is not a property the circuit has; it is a ratio between two experiments, and saying which two is most of the discipline.

The differential experiment. Drive the two inputs to $+\tfrac12$ and $-\tfrac12$, solve, read the output. That gives $A_d$.

The common experiment. Drive both inputs to $+1$, solve, read the output. That gives $A_c$.

Rejection is $20\log_{10}(A_d/A_c)$. Nothing else is involved, no small perturbation is taken, and no expression is evaluated: two netlists differing only in their two source values, solved at the same frequency.

That matters because the closed form usually quoted — $(1+G)/4t$ for a difference stage with fractional resistor error $t$ — is a first-order result in $t$, and the site’s habit is to check first-order results rather than assume them. Solved against it at four tolerances, the previous essay found the worst disagreement to be 0.000% and the fitted exponent in the tolerance to be $-1.0000$. Here the same two solves are simply repeated with a stage in front, so whatever was true of the closed form remains true and the new factor is measured on top of it.

It also means the frequency dependence costs nothing extra. The two experiments are solved at every frequency on the axis rather than at direct current, so the curve in the figure is the same measurement repeated, not a magnitude response with a rejection quoted beside it.

Where the amplifiers’ own rejection comes in #

There is a term this essay has been quietly leaving out, and it is the one that decides a real part.

Each amplifier has a common-mode rejection of its own — its output moves a little when both inputs move together, independently of any resistor. In the front stage, whose common-mode gain is exactly one by the argument above, an imperfection in each amplifier’s own rejection appears as a small differential output for a common-mode input, and that is indistinguishable from a signal.

Two things about it are worth knowing. It is not improved by $G_1$, because it happens at the input where the signal has not been amplified yet; it is referred to the input directly. And it is the reason a real instrumentation amplifier’s data sheet shows rejection rising with gain and then flattening: at low gain the four resistors dominate, at high gain the input amplifiers’ own rejection does, and the corner between them is a property of the part.

The figure’s amplifiers are modelled with an infinite common-mode rejection of their own, so what it measures is the resistor term alone. That is the honest scope of it: the arrangement’s architecture is what is being measured, and the reason the architecture is worth measuring separately is that it is the part a designer chooses.

The last of the four settings is the one where the four resistors stop being the whole story, and it is worth reaching because it is where the argument’s own limit is.

What this does not repair #

Two things sit outside the rejection this essay measures, and both are worth naming because a reader who has just bought 94 decibels may believe they have been bought too.

The source impedance imbalance. Rejection is defined for a common-mode voltage arriving at two inputs through equal impedances. If a sensor’s two leads have different impedances — different lengths of cable, one leg through a connector — a common-mode current through the input impedance produces different voltages at the two inputs, and that is a differential input the instrument cannot tell from a signal. The number that decides it is the imbalance divided by the input impedance, and it is routinely the dominant term in a real installation. It is why the arrangement’s very high input impedance is not a luxury: it is what keeps a lead imbalance from becoming a differential error.

Anything that is not common. The whole argument is about a voltage that arrives equally at both inputs. Interference that is picked up by a loop between the two leads is differential when it arrives, and no rejection acts on it at all. That is a shielding and a twisting problem, not an amplifier problem, and this collection’s crosstalk essay measures the mechanism.

Neither is a criticism of the arrangement. They are the reason the arrangement’s specification is quoted with a stated source imbalance, and the reason a rejection figure taken with both inputs tied together is an upper bound on what an installation will see.

What the rejection is spent on #

Rejection is bought here and spent elsewhere in the instruments field. The rejection four resistors decide is where it is spent: a shared return puts a common-mode error in front of the instrument, and the four resistors decide how much of it survives. The millivolts in the wire is where that error comes from, in milliohms and nanohenries of ordinary copper. The ideal amplifier, and where it stops being one is the ceiling on all of it — the corner in this page’s rejection is the amplifier’s gain-bandwidth divided by the gain that bought the rejection. The current the instrument draws and The current that does not reach the input are the two errors this arrangement does not address at all.

What is checked #

Four assertions, and the second is the one that makes the first mean anything.

That the rejection is the difference stage’s plus twenty log of the first-stage gain, to two parts in a thousand, at four gains from one to a thousand. Both quantities are solved on the same netlist, differentially and in common, rather than read from an expression.

That the common-mode gain of the whole instrument is the difference stage’s own, to half a per cent, at every one of those gains. That is the mechanism rather than the result: without it the sum above would be a fit.

That ten per cent between the two feedback resistors moves the rejection by less than a hundredth of a decibel, which is the two-resistors-that-do-not-matter claim asserted at a hundred times the tolerance the ones that do are held to.

And that the whole of it expires above the gain–bandwidth divided by the gain, which is the edge this collection insists every model is drawn with — here, unusually, an edge that the design’s own main parameter chooses.

Two ceilings this improvement runs into #

Twenty times the log of the gain is an unbounded expression and the architecture is not, so it is worth naming the two things that stop it — one inside the instrument and one outside.

Inside, the rejection the parts have gives each of the three amplifiers a finite rejection of its own and finds the result agreeably strange: two matched but individually mediocre parts cost nothing at all, because their error appears as a common-mode signal at the difference stage and is rejected there along with everything else. What costs is the difference between them, and it sets a ceiling with no gain in it — so the expression above is exact until the architecture’s contribution reaches the parts’ mismatch, and flat afterwards.

Outside, the corner the instrument has no part in finds a mechanism that has nothing to do with the instrument at all. A source with a kilohm of imbalance and ten picofarads at each input puts a corner at 290 Hz on the rejection of a part that is still doing 95 decibels at direct current, falling twenty decibels a decade after it. What converts common mode into differential there is the difference of two time constants, so the gain this essay measures multiplies a differential signal that already contains the interference — and no amount of it helps.

The practical ordering that follows is the useful part. Below a few hundred hertz and from a balanced source, the four resistors decide and this essay’s expression is the whole answer. From a real source at any frequency worth measuring, the source’s own imbalance decides, and the repair is a capacitor rather than a better part. And at the top, where both have been dealt with, what is left is the mismatch between three amplifiers, which is the one term nothing outside the package can improve.

The most useful part of the result is the negative half, and it is worth restating on its own. The two resistors that set the gain do not matter, at ten per cent, to a hundredth of a decibel — while four others are held to a tenth of a per cent and decide the whole specification. That is a statement about which components a manufacturer has to trim and which a user may fit externally, and it is why the gain-setting resistor of a three-amplifier instrumentation amplifier is brought out to a pin while the other four never are. A reader who took “precision instrumentation amplifier” to mean that all six resistors are precise would have the economics of the part exactly backwards.

It also says what a user can and cannot spoil. Fitting a poor gain resistor costs gain accuracy and nothing else; fitting anything at all across the four inside the package is impossible, which is the point of the package. The one way a user can destroy the rejection without touching a resistor is to present an unbalanced source, which is the corner the instrument has no part in and which costs eleven decibels at a kilohertz for an ordinary kilohm of imbalance.