The corner the instrument has no part in

Three rungs of this argument have measured the same instrument and each found a different mechanism limiting it. The four resistors around the difference stage decide one term, which rises with the input stage’s gain. The input amplifiers’ own rejection decides a second, which does not rise with anything — and the surprise there was that two matched mediocre amplifiers cost nothing at all, while the difference between two good ones sets a ceiling of 111.7 dB with no gain in it.

All three measurements are at direct current, and all three are measurements of the part.

The instrument is connected to something. This essay puts a source in front of it, and the rejection acquires a corner.

Two low-pass filters that are not the same filter #

The mechanism is one line and it does not involve the instrument at all.

Each input has a source resistance in front of it and a capacitance to ground behind it — the amplifier’s own input capacitance, the connector, the cable, a few picofarads of board. Together they make a low-pass filter of time constant $R_sC_\mathrm{in}$, and there are two of them.

A common-mode signal arrives at both inputs equally. It comes out of the two filters unequally, because the filters are not the same filter, and the difference is a differential signal at the instrument’s inputs. At that point it is signal. No rejection repairs it, because the instrument is doing exactly what it was asked to do: amplifying the difference between its inputs.

To first order the differential signal is $v_\mathrm{cm}\,\omega\,\Delta\tau$, where $\Delta\tau$ is the difference of the two time constants, so the rejection is

$$\mathrm{CMRR} \approx \frac{1}{\omega\,\Delta\tau}$$

which falls at twenty decibels a decade and has no instrument in it.

Measured on the netlist with the instrument’s own imperfections removed — perfect resistors, ideal amplifiers, so that nothing else can contribute — the rejection is 144.04, 124.04, 104.04, 84.04 and 64.04 dB at 1, 10, 100, 1,000 and 10,000 Hz for a ten-nanosecond time-constant gap. Twenty decibels a decade to two decimal places, and the value at a kilohertz is $1/(2\pi\cdot 10^3\cdot 10^{-8})$ to five figures.

Where the corner falls, which is lower than anyone expects #

With the instrument’s own 95 dB back in place, the two mechanisms add and the corner is where they are equal. For a kilohm of imbalance and ten picofarads it is at 290 Hz.

That number is the essay. A kilohm is not a large source impedance — it is a thermocouple with some cable, a strain-gauge bridge, a divider on a sensor board. Ten picofarads is not a large capacitance; it is roughly what an amplifier’s own input pins have before anything is connected to them. And the two of them together take an instrument specified at 95 dB and give it 84 at a kilohertz.

Swept across the imbalance, the corner moves exactly as the expression says it should:

The mains row is the one to read twice. At a kilohm of imbalance the rejection at 50 Hz is untouched, so an instrument measured on a bench with short leads looks perfect. At ten kilohms it has lost six decibels of mains rejection; at a hundred, twenty-five. The failure arrives at the frequency the interference is at, and the specification that was checked is at direct current.

The cure is a capacitor, and it goes on the wrong input #

If what matters is the difference of two time constants, then a resistance imbalance can be cancelled by a capacitance imbalance in the other direction. That is a design statement the three rungs below have no way to make, because none of them has a source.

The prediction is exact and it is checkable: the cancelling capacitance is $C_2 = R_1C_1/R_2$, the value that makes the two products equal. Sweeping the second input’s capacitance and golden-sectioning the maximum, with the instrument’s own imperfections removed:

100.0000 pF measured against 100.0000 pF predicted, and the rejection there is 327 dB — which is to say, limited by nothing but the arithmetic.

With real parts the same sweep finds 96.7 pF, three per cent away, and the reason is worth having: the peak is broad. Once the source’s contribution falls below the instrument’s own 95 dB floor, improving it further changes nothing, so there is a plateau rather than a maximum and its centre is poorly determined. That is not a defect in the measurement; it is the correct statement of how precisely the capacitor needs to be chosen, which is not very.

The practical form of this is a small trimmer, or a fixed capacitor, on the input with the lower source impedance — which is the opposite of where anybody would put it by instinct, since that is the input that is already fine. Balancing means making both inputs equally bad, and there is no arrangement that makes both good, because the resistances are the source’s and cannot be changed from here.

What it is worth in the units of the measurement #

Decibels of rejection are a ratio and a ratio is not an error. What a designer has is a common-mode voltage of some size at some frequency, and what matters is how many microvolts of it reach the output divided by the gain — that is, how much it looks like signal.

Take the ordinary case this instrument is built for: a bridge or a thermocouple with a differential signal of a few millivolts, and a common-mode interference of one volt at mains frequency, which is what a metre of unscreened cable near a wall picks up.

• With a hundred ohms of imbalance, the rejection at 50 Hz is 95.1 dB and one volt of interference appears as 17.5 µV referred to the input.
• With ten kilohms, it is 88.9 dB and the same volt appears as 35.9 µV.
• With a hundred kilohms, 70.0 dB and 316 µV.

Against a signal of two millivolts those are 0.9%, 1.8% and 16%. The last one is not a measurement of anything, and the instrument is still meeting its 95 dB specification exactly.

The same numbers at a kilohertz — which is where a switching supply’s interference is, rather than the mains — are 94.8, 64.1 and 44.0 dB, so the hundred-kilohm source lets 6.3 mV of a one-volt interference through. That is three times the signal.

Why the three rungs below could not have found this #

It is worth saying plainly, because the omission looks careless in retrospect and is not.

Each of the three rungs below measures the instrument as a two-port driven by ideal sources: a differential source and a common-mode source, connected directly to the two inputs. That is the right model for the questions they ask, and it is what a data sheet’s own test circuit is. Under it the input capacitance does nothing at all — an ideal voltage source drives any capacitance without a phase shift — so the mechanism this essay is about has a gain of exactly zero in all three of them.

There is no measurement any of the three could have made that would have shown it. The instrument’s model was not wrong; its boundary was in the wrong place, and moving the boundary out by two components is the whole of this rung.

That is the same shape as the fault the transients field found when it priced a diode’s recovery and discovered ninety per cent of the energy in the transistor: the arithmetic was right and the system it was arithmetic about was a component too small.

Where a bridge puts its own imbalance #

One source deserves its own paragraph because it is the one this instrument is most often connected to, and because the imbalance is not an accident of wiring but the signal itself.

A resistive bridge presents each of its outputs through the parallel combination of its two arms — about $R/2$ at each, and nominally equal, which is why a bridge is a good source for this instrument. What makes the two unequal is the arm that has changed, which is the quantity being measured. At a strain of one part in a thousand on a 350 Ω bridge the two source impedances differ by about 0.09 Ω, so the time-constant gap is under a picosecond-ohm and the corner is at several megahertz. That is why bridges are the easy case.

The hard case is what is in front of the bridge: a hundred metres of cable, one core of which has a bad crimp. A single degraded connection puts kilohms in one input and nothing in the other, and the symptom is a mains hum that appears at one channel of a multi-channel instrument and not at the others — with every channel’s amplifier measuring the same 95 dB on a bench.

What this does and does not explain about a real instrument #

Two boundaries on the result, and each is a place where something else takes over.

Above the instrument’s own bandwidth this is not the mechanism. The amplifiers’ rejection falls with frequency too, because their gain does, and by ten kilohertz the instrument alone is down to 91 dB from 95. The two effects are both present in the measurement and the source’s is much the larger of them at every imbalance in the table — but at 100 Ω of imbalance they are comparable, which is why that row’s corner is the least sharp.

Below the corner nothing here is happening at all. The three rungs below remain the whole story there, which is why they were right to measure at direct current and why this rung is a fourth and not a correction. The direct-current rejection of the instrument is unchanged by anything in this essay: 95.08 dB with a kilohm in front of it and 95.08 dB with nothing.

Why this belongs to the netlist rather than to an expression #

Everything above could be written as an expression, and the expression is in every application note that mentions the subject. Two things are different about having it in the solve.

The first is that the two mechanisms add in the netlist. The resistors’ mismatch, the amplifiers’ own rejection and the source’s time-constant gap are three separate error sources at three separate places, and the total is what the difference stage sees after all three have travelled through the arrangement. Writing them as three expressions and adding the results in quadrature is a guess about their correlation; solving the network is not.

The second is the balance measurement. The cancelling capacitance is a maximum of the solved rejection, found by golden-section search on the network, and the prediction $R_1C_1/R_2$ is checked against it rather than assumed. The two agree to a part in a thousand with an ideal instrument and to three per cent with a real one, and the difference between those two numbers is itself the finding — it says how much of the balance matters and where the plateau starts.

What the instrument does and does not decide #

A corner that belongs to the source rather than to the instrument is the sharpest form of this field’s standing complaint. The probe is part of the circuit is the ordinary case, where the instrument does decide it. The corner that says nothing about an edge is the corner an instrument advertises and a pulse defeats. The rejection four resistors decide and The millivolts in the wire are the errors the wiring contributes rather than either end. Two terminals measure the leads as well is the one arrangement in the field that removes such an error rather than bounding it, and The current the instrument draws is the error that is the instrument’s alone.

What is checked #

The low-frequency end of every sweep is asserted to sit within half a decibel of the instrument’s own rejection, which is the claim that this rung adds nothing to the three below rather than replacing them. A corner is required to exist inside the swept range at every source imbalance on the slider, which is what fixed the sweep’s lower limit at a hundredth of a hertz — at the largest imbalance the corner is 0.28 Hz and a sweep starting at one would have shown a falling line with no corner on it.

The slope above the corner is asserted to be between seventeen and twenty-three decibels a decade, fitted over the points above it rather than taken between two of them, because the claim is that one mechanism is responsible and a fitted slope is what distinguishes that from a collection of them.

The balancing capacitance is asserted against $R_1C_1/R_2$ to a part in a thousand with an ideal instrument and to a tenth with a real one — two assertions rather than one, because the difference is the plateau and asserting only the loose bound would have hidden it. And balancing is asserted to be worth at least two decibels against leaving the capacitances equal, so that a figure drawn where there is nothing to gain would fail rather than be quietly drawn.

What is not modelled: the amplifiers’ own rejection falling with frequency, which is a real second corner and would need a frequency-dependent error source rather than the constant one this arrangement uses; the input bias currents, which flow in exactly these source resistances and produce a differential offset by the same imbalance; the cable, whose capacitance is distributed rather than lumped at the input; and the resistors’ own tolerance drifting with temperature, which is what actually limits a good instrument over a year.

What the three rungs below it were measuring instead #

The corner on this page is a property of the source, and the three measurements it stands on are all properties of the instrument — which is the whole point, and is worth setting out because the four numbers are easy to mistake for four estimates of one thing.

The rejection four resistors decide measures the difference amplifier alone and finds 53.99 decibels from tenth-per-cent parts against a closed form’s 53.98, with the number being one plus the gain over four times the resistor tolerance and the amplifier having nothing to do with it. The four resistors that decide, and the two that do not adds the input stage and finds it worth exactly twenty times the log of its gain — because the stage passes common mode at unity and the improvement is entirely the differential signal arriving larger. The rejection the parts have gives the amplifiers rejections of their own and finds matched mediocrity free and mismatch expensive, with a ceiling containing no gain.

All three are measured at direct current from a balanced source, and all three are correct. What this rung adds is that a real measurement is made at a frequency, from a source that is not balanced, and that under those conditions none of the three numbers is the one that binds: a 95 dB instrument delivers 84 dB at a kilohertz into an ordinary source imbalance, and the eleven decibels lost are not attributable to any component in the instrument.

Which changes what a specification is for rather than making it wrong. A rejection figure is a property of a part and this essay’s corner is a property of a connection, so the figure is the right thing to print and the wrong thing to design against — the same distinction the probe is part of the circuit draws about an oscilloscope probe, where a reading is two solves and the quantity drawn is the difference, and no property of the instrument alone states the error.

What the corner is worth in the application the whole field is built around is the last thing to say. The millivolts in the wire puts the interference at half a millivolt on a ten-millivolt sensor from a hundred-milliamp load sharing ten millimetres of copper, and finds the shared impedance turning inductive above 79.6 kilohertz so that the error rises a decade per decade with no ceiling. Set against the corner measured here, the two curves cross badly: the interference climbs above eighty kilohertz and the rejection falls above two hundred and ninety hertz, so the difference between them worsens at forty decibels a decade over most of the band where a switching load actually operates. A calculation made at direct current — five hundred microvolts divided by five hundred — is optimistic by that whole slope. Which is the practical form of this essay’s result: the rejection figure and the interference figure are both quoted at direct current, and they diverge above a few hundred hertz at forty decibels a decade.