The rejection two capacitors lose at their corner

The corner that says nothing about an edge measured one AC-coupled input with a pulse and found that its specification — a low-frequency corner — says nothing about the sag a pulse suffers or where a train of pulses settles. It ended on a case it had not drawn. The numbers transfer to a differential input side by side, it said; what they do not cover is the common-mode path, where two coupling capacitors that are not matched turn a common-mode step into a differential one, and the mismatch that matters there is a tolerance rather than a value.

This essay draws that case. It solves the two coupling networks as one circuit driven by one common-mode source, and asks how much of that source arrives as a difference. The answer is small at direct current, small far above the corner, and largest at the corner itself, where it is set by nothing but the capacitors’ tolerance. At that frequency the best differential amplifier available is irrelevant: the rejection is the pair’s, and for ordinary parts it is worse than a resistor divider’s.

Common mode through two filters #

The circuit is two copies of the one the earlier essay measured, joined at the source: one common-mode generator drives both inputs, each through its own capacitor into its own megohm to ground, and what the differential amplifier sees is the difference between the two coupled outputs. With both capacitors at 1 µF the two outputs are identical at every frequency and the difference is zero. Make them 1 µF plus and minus half a per cent and the time constants differ by one per cent, and the difference is the gap between two first-order high-passes whose corners are one per cent apart.

That gap is a band-pass. At direct current both capacitors block, both outputs are zero, and nothing converts. Far above the corner both capacitors are short circuits, both outputs equal the input, and again nothing converts. In between, the two responses have the same shape at slightly different frequencies, and their difference per volt of common mode is

$$\left|\frac{j\omega\tau_a}{1 + j\omega\tau_a} - \frac{j\omega\tau_b}{1 + j\omega\tau_b}\right| \approx \frac{\omega\tau\,\delta}{1 + \omega^2\tau^2},$$

for a fractional mismatch $\delta$ between the time constants. It peaks at $\omega\tau = 1$ — the corner — at exactly δ/2.

The netlist agrees. Solved at 281 frequencies, the difference between the two outputs matches the difference of the two closed-form high-passes to $7\times10^{-14}$, and it peaks at the corner, 0.159 Hz, at $5.000\times10^{-3}$ per volt: half the one per cent. That is 46.0 dB of common-mode rejection, at the one frequency the input’s specification names. Move the slider to 0.1 per cent and the dip is 66.0 dB; to 20 per cent, a pair of ordinary electrolytics, and it is 20.0 dB.

Why the corner, and why half #

The peak’s position and height have a reason that is worth having, because it says the result does not depend on the particular values chosen. A first-order high-pass is $H = j\omega\tau/(1 + j\omega\tau)$, and its change with its own time constant is

$$\frac{\partial H}{\partial \ln\tau} = H\,(1 - H),$$

the high-pass multiplied by its complementary low-pass. The conversion per unit of mismatch is that product’s magnitude. Far below the corner the high-pass is small; far above it the low-pass is; at the corner both are $1/\sqrt2$ and their product is one half. So any pair of first-order networks — coupling capacitors, input filters, anti-alias sections — converts common mode at their corner at half their fractional mismatch, and at no other frequency as much.

The same product appeared in half to the amplifier, half to the resistors, where a feedback loop’s sensitivity and its complementary sensitivity summed to one and both mattered most at the loop’s crossover. A coupling network is a loop of the simplest kind — a capacitor charging towards its own input — and its corner is where the part of the signal it passes and the part it holds back are equal, which is where it is most sensitive to anything that moves the boundary between them.

Where the pair decides #

A differential instrument has its own rejection, and the question that matters is which of the two a measurement meets. The dashed curve adds the pair’s conversion to a 90 dB amplifier’s in the worst phase. Far from the corner the amplifier’s 90 dB is what the instrument delivers. Near the corner the pair’s conversion is larger, and the instrument delivers the pair’s.

The band is wide. Behind an 80 dB amplifier a pair one per cent apart decides the instrument’s rejection from 0.00159 Hz to 15.9 Hz, four decades; behind 90 dB, from 0.0005 Hz to 50.3 Hz, five. The band’s edges sit where the pair’s conversion, which falls one-for-one with frequency away from the corner on either side, has fallen to the amplifier’s, so its width in decades is twice the logarithm of the ratio between the pair’s peak conversion and the amplifier’s. A better amplifier widens it: 100 dB behind a 20 per cent pair hands the rejection to the capacitors from 8 µHz to 3.22 kHz — mains frequency and its harmonics included.

That last case is the practical one. An instrument AC-coupled to remove a DC offset from a bridge or a biopotential, specified for 100 dB of common-mode rejection and fitted with electrolytic coupling capacitors, delivers the capacitors’ rejection at 50 Hz, not the amplifier’s. The pair’s conversion at 50 Hz is its mismatch divided by $\omega\tau$, 0.2 over 314, so the instrument’s rejection there is about 64 dB — set by how far 50 Hz is above the 0.159 Hz corner — and the data sheet’s 100 dB plays no part in it. The pattern is the one the four resistors that decide found at direct current for a difference amplifier: the rejection belongs to whichever matched pair is worst, and here the worst pair is the one added to fix something else.

A common-mode edge, received as a pulse #

The frequency picture has a counterpart in time that is easier to recognise on a bench.

Step the common-mode voltage by 10 V — a supply switching on, a cable plugged in, a patient moving — and both coupled outputs jump by 10 V and decay back to zero with their own time constants. They start together and end together, and in between the faster one is always a little ahead. The difference is $V(e^{-t/\tau_a} - e^{-t/\tau_b})$, a pulse that rises from nothing, peaks one time constant after the edge at the step times $\delta/e$ for a small mismatch, and dies away over the next few.

Marched on the netlist, a one per cent pair turns the 10 V step into a 36.8 mV differential pulse peaking at 1.05 s, five per cent into 184 mV and twenty per cent into 737 mV. The marched traces match the closed-form difference of exponentials to four parts in ten thousand of the mismatch times the step. The pulse’s area is the step times the difference of the time constants, 100 mV·s at one per cent — which is what an integrating instrument downstream would accumulate from a single common-mode edge.

A differential instrument sees that pulse as signal. Its shape is not an artefact of any amplifier: it has the time constant of the coupling, it follows every common-mode edge, and its polarity is set by which capacitor is larger. On an electrocardiograph with electrolytic coupling, a patient’s movement shifting the electrodes’ common potential by three hundred millivolts produces, through a 20 per cent pair, a differential transient of 22 mV lasting seconds — twenty times the millivolt signal being measured.

A tolerance, not a design #

The mismatch in all of this is not a designer’s choice. It is the difference between two parts that were both bought to the same value, and its size is a draw from their tolerance.

Two parts drawn uniformly within a tolerance $t$ differ by an amount whose distribution is triangular, from zero to $2t$, and whose median is $(2 - \sqrt2)t$ — 0.586 of the tolerance. The median pair’s rejection at the corner is therefore $2/(0.586t)$. Over four hundred seeded pairs from each class, 1 per cent film parts give a median of 50.3 dB and a worst of 40.2; 5 per cent film, 36.5 and 26.7; and 20 per cent electrolytics, 25.0 and 14.1. The worst pair of four hundred comes within a couple of decibels of $1/t$, two parts at opposite ends of the band.

Twenty-five decibels is sixty-five below the ninety an ordinary instrumentation amplifier is bought for. It is what a large coupling capacitor — the only kind that gives a corner low enough to pass a slow physiological or mechanical signal — delivers in the common-mode path of a differential input, at the frequency where the input is most exposed. The rejection the parts have found an instrumentation amplifier indifferent to two matched but individually poor amplifiers; the coupling pair is the opposite case, where two individually good parts are poorly matched, and the mismatch is all that counts.

The mirror of a source imbalance #

The corner the instrument has no part in met the same conversion in the other direction. There a source-resistance imbalance and the input capacitance formed two low-passes whose corners differed, and the converted common mode grew with frequency — as $\omega\Delta\tau$ — from nothing at direct current to a plateau set by the imbalance, so the instrument’s rejection had a corner at 290 Hz and fell twenty decibels a decade after it.

The coupling pair is the same arithmetic on a high-pass, and its shape follows. A low-pass mismatch converts at high frequency and never recovers; a high-pass mismatch converts only near its corner and recovers on both sides. The first costs rejection over everything above a corner; the second costs it over a band centred on one. An instrument with both — AC-coupled inputs, and a source with imbalanced resistance feeding its input capacitance — has two such regions, one at a fraction of a hertz and one at hundreds, and between them the amplifier’s own rejection. The errors that arrive before the gain found four such contributions adding as complex numbers; this is a fifth, with its own frequency and its own sign.

Couple the difference instead #

The conversion exists because the coupling is done twice, once on each side, with two parts that cannot be identical. Do it once, after the subtraction, and there is nothing to mismatch.

With the coupling on each side and the capacitors five per cent apart, the instrument’s rejection falls to 32.0 dB at the corner. With a single capacitor after the differential stage, its rejection is the amplifier’s 90 dB at every frequency — 92.9 dB at the corner and more below it, where the capacitor attenuates whatever the amplifier let through. The dip has gone, and it has gone because the common mode was never presented to a pair of parts in the first place.

The price is the reason the coupling was put on each side. A capacitor in front of each input removes each input’s DC level, so the amplifier sees only the signal and the common-mode AC; a capacitor after the difference removes only the difference’s DC, and the amplifier must accept the whole common-mode level — the bridge’s half-supply, the electrode’s offset — within its input range. Where that level is volts and the amplifier’s range is millivolts, the coupling cannot move. There the remedy is matching: selected or trimmed pairs, a single dual capacitor, or time constants matched with a trimmed resistor in exactly the way a divider with two ratios matches a probe’s two arms, because what converts is the difference of the two time constants and either element can close it.

What a specification would have to say #

A differential instrument with AC-coupled inputs is usually specified by its amplifier’s rejection and its coupling corner, separately. The two numbers cannot be combined into what the instrument does without a third, and the third is the capacitors’ matching. At the corner the rejection is $2/\delta$; so 60 dB at the corner needs the two time constants within 0.2 per cent of each other, 80 dB within 0.02 per cent, and 100 dB within twenty parts per million. No capacitor large enough to give a corner below a hertz is bought to any of those, and the first is at the edge of what selected film parts hold over temperature.

So a specification that is honest about this has two forms. It can state the common-mode rejection at the coupling corner, which is a number the instrument meets whatever parts it contains; or it can state the rejection away from the corner, with the band around the corner excluded and its width given. What it cannot do is quote the amplifier’s figure unqualified, because over four or five decades of frequency the amplifier is not what the measurement meets.

The band is also a clue to diagnosis. An instrument whose common-mode rejection measures well at mains frequency and poorly at a tenth of a hertz — or whose baseline jumps when the common-mode level is disturbed and then recovers over a few seconds — is showing its coupling pair, and the recovery’s time constant identifies it. The fix is then in the coupling network rather than in the amplifier, and no better amplifier will find it.

How the numbers were obtained #

Every figure solves one netlist: a common-mode voltage source driving two series capacitors, each into its own 1 MΩ resistor to ground, with the capacitors set to $1\,\mu\text{F}\cdot(1 \pm \delta/2)$. The frequency response is the complex solve at each frequency, and the conversion is the magnitude of the difference between the two output nodes; the closed-form difference of two first-order high-passes is computed alongside and compared, never drawn. The band edges are bisected on the solve for the frequency at which the pair’s conversion equals the amplifier’s, which is modelled as a common-mode gain of $10^{-A/20}$ adding in the worst phase. The step is marched by the trapezoidal rule at 6,000 steps over five time constants. The tolerance populations draw each capacitor uniformly within its tolerance from a seeded generator, 400 pairs a class.

What it leaves out #

The source’s own impedance. Each coupling capacitor sees the source’s output resistance in series, and an imbalance there moves one time constant and not the other. That is a second mismatch added to the capacitors’, and it is where the source-imbalance essay’s low-pass conversion and this one’s high-pass conversion meet in one circuit.

Leakage and dielectric absorption. An electrolytic’s leakage current flows through the megohm and appears as an offset on each side, differently on each; its absorption adds a slow tail to each side’s step response. Both differ between the two parts of a pair.

The resistors. Every megohm here is exact. The resistors’ tolerance changes the time constants exactly as the capacitors’ does, so a pair of one per cent resistors behind a pair of one per cent capacitors has a combined mismatch of up to two per cent.

Still open: two stages in a row, the resistor that trims, and the cable’s capacitance #

Two coupling stages in cascade. Instruments are often AC-coupled twice — once at the input, once between stages — and a second stage changes the single input’s pulse response rather than adding to it. The tail after a pulse, which one stage leaves on one side of zero, has to cross zero with two, and the sag doubles. That is a question about a single channel, and it is the next thing to march.

Trimming the time constant. One resistor trimmed against the other can null the conversion at the corner exactly, and the nulled pair’s residual is then set by how the two capacitors’ values drift apart with temperature and age — a tolerance on tracking rather than on value, the same shift the tolerance that is not on any part describes for a resistive divider’s ratio.

A cable in the common-mode path. A screened cable’s capacitance to its screen is part of each side’s coupling network when the screen is driven with the common mode, and two conductors in one cable are closer matched than two discrete capacitors. Whether a guarded cable improves the pair or merely moves the corner is a solve on the combined network.