Circuit Response
every model has an edge, and the figure carries it
Measurement, which is a circuit on a circuit

The current that does not reach the input

A teraohm across a board from a fifteen-volt rail is fourteen millivolts of error through a gigohm source, and a humid morning takes that resistance down two decades. A ring held at the input's own potential leaves a nanovolt — proportional to the signal rather than to the rail, so an offset has become a gain error of one part in a billion — and the same wire multiplies the input resistance by the loop gain, which makes it 10¹⁸ Ω at direct current and 10¹² Ω at a megahertz.
14 min read 8 figures The schematic is a labelEvery model has an edgeMeasured, not tabulated

Assumes: The current the instrument draws · The millivolts in the wire

The current the instrument draws put the amplifier’s own input current into the netlist and found that the difference between the two inputs is what binds: the bias current can be cancelled by making the two source resistances equal, and the offset current cannot, so the classical cure is worth a factor of ten rather than a factor of a thousand.

Every number in that essay is a property of the amplifier. This one is about the other current at the same node, which is not the amplifier’s at all. It comes across the surface of the board, from whatever is nearby, through a resistance that is a property of the laminate, the flux residue and the weather — and on a high-impedance input it is larger than anything the device does.

A clean board is 14.0 mV of error, and a guard makes it 1.0 nV. computed by solving, not by drawing. Two boards differing by one wire, solved with the leakage present and again with it removed, so the number plotted is the leakage's own contribution and nothing else. Unguarded, a teraohm across the laminate from a 15 V rail through a 1 GΩ source is 13.99 millivolts — and a humid morning takes that resistance down two decades, which is the left-hand end of this axis. Guarded, the ring is held at the input's own potential by the amplifier, so what is across the leakage is the amplifier's own error and the result is 1.00 nanovolts. The guarded line is flat in the rail and proportional to the signal: the offset has become a gain error of 1.00 parts per billion, and the reading is low by it rather than high: the ring sits a little below the input, so the last of the leakage pulls the input down.
Fig. 1 The error at the input from a fifteen-volt rail leaking across the board, against how good the board is, with and without a guard ring. Two solves per board: with the leakage present and with it removed, so what is plotted is the leakage’s own contribution.

What a board is worth, and what it is worth in the rain

A clean board with a solder mask and no residue is somewhere around 10¹² ohms between adjacent features. That is not a small number and it is not large enough. Fifteen volts across a teraohm is fifteen picoamps; through a one-gigohm source resistance that is fourteen millivolts of error, which is four orders of magnitude above the amplifier’s own contribution.

The resistance is also not a constant. Relative humidity above about sixty per cent puts a monolayer of water on the surface and ionic residue dissolves into it; two decades is an ordinary morning’s variation, and the left-hand end of the figure above is what that looks like. An instrument that reads correctly in a dry laboratory and drifts by a volt on a wet one has not developed a fault.

Where an amplifier's reading comes from, against the source it is reading. computed by solving, not by drawing. Three errors with three different dependences on the source, each measured by a solve with the other two set to zero. The offset voltage is flat — 50 microvolts wherever the source is. The bias current times the imbalance is linear in the source and is what balancing removes. The offset current times the source is linear too and is what balancing leaves. Unbalanced, the current overtakes the voltage at 953.20 kΩ; balanced, at 10000.0 kΩ, which is the offset voltage divided by the OFFSET current and is the ratio of the two currents further along. Below about a kilohm, balancing makes the reading worse — the feedback network is already the larger resistance, and equalising means adding to the source.
Fig. 2 The amplifier’s own two input currents, from the rung this one sits beside, drawn for a fifty-picoamp part rather than the fifty-nanoamp one. Its bias current does not overtake its fifty microvolts of offset voltage until 953 kΩ of source, and balanced it does not overtake until 10 MΩ. That is the number a device is chosen for, and it is three orders below what the board above contributes.

The comparison is the whole reason the technique exists. Choosing an amplifier whose input current is fifty femtoamps rather than fifty picoamps is a real decision with real cost, and on the board described above it changes the answer by nothing at all, because the laminate is contributing three orders of magnitude more.

Which error is worth choosing an amplifier for. computed by solving, not by drawing. The board's leakage error, guarded and not, against the error the amplifier's own 50 femtoamp input current makes through the same 1 GΩ source — 50.00 µV, and a horizontal line because it does not know what the board is made of. Unguarded, the laminate is above it by three orders of magnitude at every resistance drawn here, so the choice of amplifier changes nothing. Guarded, it is below it everywhere, and the amplifier is the limit again. That is the whole argument for the ring: not that it removes an error, but that it moves the binding one back to the part that was chosen for it.
Fig. 3 The board’s leakage against the device’s own input current. Unguarded, the laminate is above the device everywhere; guarded, it is below it everywhere. That is the argument for the ring.

One wire, and what is across the leakage

A guard is a conductor that surrounds the input trace — a ring on both faces of the board, or a driven shield on the cable — and is held at the input’s own potential by the amplifier. The leakage path from the rail then lands on the ring instead of the input, and the amplifier supplies whatever current that takes from its output, where it does no harm.

What is left is the leakage from the ring to the input, and across that resistance there is not fifteen volts. There is the difference between the input and the guard, which is the amplifier’s own error: the signal divided by its open-loop gain. With a gain of a million and a signal of a volt, the leakage current is a millionth of what it was.

The netlist says 1.00 nanovolts where the bare board said 13.99 millivolts. And it says something about the character of what is left as well as its size. The bare error is proportional to the rail, which makes it an offset, drifting with whatever the rail does. The guarded error is proportional to the signal, which makes it a gain error of one part in a billion — a quantity that is calibrated out by the same operation that calibrates out the divider ratio, and that does not drift when the rail does.

Measuring with 50 mΩ of lead in each wire. computed by solving, not by drawing at 61 resistances, twice each. The two-wire arrangement measures the leads too, so its error is 2×50 mΩ over whatever is being measured: one per cent at 10 Ω, and 10000% at 1 mΩ. The four-wire arrangement senses on a separate pair that carries almost no current, and its error stays under 1.0e-2% across the whole range.
Fig. 4 The same move in a different field: a measurement arranged so that the error current flows somewhere it does not develop a voltage across the thing being measured. A guard is a Kelvin connection for leakage.

That is the same structural move two terminals measure the leads makes with a four-terminal resistance measurement, and it is worth naming the shared idea: neither technique removes an error current. Both arrange for it to flow somewhere that does not matter.

What is actually in the netlist

The board is three elements and the amplifier is four, and it is worth writing them out because the whole argument is which node each one connects to.

The leakage is a resistor from a rail node at fifteen volts. On the bare board it goes to the input node; on the guarded board it is split into two, one from the rail to the ring and one from the ring to the input, with the split set by how much of the path the ring intercepts. The cable is a capacitance, from the input to ground on the bare board and from the input to the ring on the guarded one — the same component, moved to a different node, which is the entirety of what a driven shield is.

The amplifier is the one-pole macro-model the feedback field already uses: a transconductance into a capacitance, then a follower, so its direct-current gain and its gain-bandwidth are two element values rather than two fitted parameters. The ideal amplifier and its bandwidth is where that model’s own range was measured, and this essay is entirely inside it.

The same wire, and the same loop gain, is 11160× the bandwidth. computed by solving, not by drawing. The −3 dB frequency from the source to the output, with 100 pF of cable on a 100 GΩ source. Unguarded the cable and the source are a low-pass filter at 0.02 Hz and there is nothing to be done about it. Guarded, the shield follows the input, so there is no voltage across the capacitance and no current into it — the amplifier drives it from its output, where it is a load rather than a pole, and the corner moves to 196 Hz. It is not the full loop gain, because the guard's own accuracy runs out with frequency: the answer is where those two curves cross, and the netlist finds it.
Fig. 5 What is actually in the netlist, at a hundred gigohms of source rather than one: the cable’s own leakage and capacitance as elements, with the guard driven from the follower’s output. Nothing here is a correction applied to an answer — the guard is a node in the matrix, and what it does to the error is whatever the solve says it does.

The error is then two solves rather than one: the same board with the leakage present and with it raised to 10¹⁸ ohms. The difference is the leakage’s contribution and nothing else, so the amplifier’s own offsets and the divider’s own gain cannot leak into the number being reported. That is the same two-solve habit two solves that add argues for, applied to a subtraction rather than a superposition.

An input resistance that is a function of frequency

The quantity a data sheet calls the input resistance is the signal divided by the current the input draws, and on a guarded board that current is the residual leakage. So the resistance is the physical leakage multiplied by the loop gain — and a loop gain is a function of frequency.

The input resistance is 10¹⁸ Ω or 10¹² Ω, depending on when you ask. computed by solving, not by drawing. The leakage current the signal itself drives, divided into the signal, which is the resistance the source sees. Guarded, it starts at 1.00e+18 ohms — the teraohm of laminate multiplied by the amplifier's open-loop gain — and falls twenty decibels a decade above the amplifier's own corner, arriving back at the bare leakage by 1.00 MHz. Half of what the guard bought is gone by 1.73 Hz. The dashed line is the unguarded board, which has no loop gain in it and is therefore flat. A data sheet's input resistance is a direct-current statement about a circuit nobody uses at direct current.
Fig. 6 The input resistance against the frequency the question is asked at. Ten to the eighteen at direct current, ten to the twelve above the gain-bandwidth product, twenty decibels a decade in between.

It starts at 10¹⁸ Ω, falls twenty decibels a decade above the amplifier’s own corner at one hertz, and arrives back at the bare 10¹² Ω by a megahertz. Half of what the guard bought is gone by 1.73 Hz.

There is nothing wrong with the guard here and nothing wrong with the amplifier. The quantity itself is not a resistance: it is a resistance multiplied by a frequency-dependent number, and calling the result the input resistance is the mistake. A data sheet’s figure is a direct-current statement about a circuit that nobody uses at direct current, and the frequency at which it stops being true is the amplifier’s own dominant pole rather than anything about the board.

The dashed line in that figure is the unguarded board, which has no loop gain anywhere in it and is therefore flat. The unguarded number is smaller and it is honest.

The same loop gain, spent on the cable

A guard does something else that is usually described as a second benefit and is the same mechanism counted twice. The shield around the input conductor is held at the input’s potential, so the capacitance between them has no voltage across it and draws no charging current. The capacitance is still there and the amplifier still drives it — from its output, where it is a load rather than a pole.

The same wire, and the same loop gain, is 1230× the bandwidth. computed by solving, not by drawing. The −3 dB frequency from the source to the output, with 100 pF of cable on a 1 GΩ source. Unguarded the cable and the source are a low-pass filter at 1.59 Hz and there is nothing to be done about it. Guarded, the shield follows the input, so there is no voltage across the capacitance and no current into it — the amplifier drives it from its output, where it is a load rather than a pole, and the corner moves to 1960 Hz. It is not the full loop gain, because the guard's own accuracy runs out with frequency: the answer is where those two curves cross, and the netlist finds it.
Fig. 7 The −3 dB frequency from source to output, with a hundred picofarads of cable on a gigohm source. The same wire is worth 510 times the bandwidth.

A gigohm and a hundred picofarads is a corner at 1.59 Hz, and there is nothing to be done about it: it is a low-pass filter made of the source and the cable, and no amplifier improves it. Guarded, the corner moves to 812 Hz — a factor of 510.

It is not the full loop gain, and the reason it is not is instructive. The guard’s own accuracy falls with frequency exactly as the input resistance above does, so the bootstrapping stops working before the loop gain has run out. Where the answer lands is the crossing of two falling curves, and the netlist finds it; no expression in the shape C/(1 + T) gives 812 Hz, because T is not one number.

Where the guard is driven from, and what that costs

For a follower the guard is the output, and there is nothing to choose. For a non-inverting amplifier of gain G the guard has to be driven from the feedback divider, at the potential the inverting input sits at — which is the input’s potential only to the extent that the divider is accurate and the amplifier’s common-mode rejection is good.

That puts a second, quite different error in series with the first: a resistor tolerance. A divider mismatched by a tenth of a per cent holds the guard a thousandth of the signal away from the input, which caps the improvement at a thousand however large the loop gain is. The mechanism is the same one the rejection the parts have measures in an instrumentation amplifier: an active circuit’s performance limited by the passive components’ matching rather than by anything the device does.

A clean board is 1272.7 mV of error, and a guard makes it 99.9 nV. computed by solving, not by drawing. Two boards differing by one wire, solved with the leakage present and again with it removed, so the number plotted is the leakage's own contribution and nothing else. Unguarded, a teraohm across the laminate from a 15 V rail through a 100 GΩ source is 1272.73 millivolts — and a humid morning takes that resistance down two decades, which is the left-hand end of this axis. Guarded, the ring is held at the input's own potential by the amplifier, so what is across the leakage is the amplifier's own error and the result is 99.90 nanovolts. The guarded line is flat in the rail and proportional to the signal: the offset has become a gain error of 99.90 parts per billion, and the reading is low by it rather than high: the ring sits a little below the input, so the last of the leakage pulls the input down.
Fig. 8 The same board with a hundred-gigohm source. Everything scales: the unguarded error is a hundred times larger and the guarded one is too, because both are a current through the source resistance.

Both errors scale with the source resistance, which is worth stating because it says what the technique is for. At a source impedance of a kilohm the bare leakage is fourteen nanovolts and nobody guards anything. The whole subject exists above about a hundred megohms, which is electrometers, pH probes, photodiodes at low light and insulation testers — and in every one of them the source is a real physical thing whose own resistance is the measurement.

A guard that is not driven, and why it is worse than none

There is a third arrangement, and it is common, and it is a trap: a ring connected to ground rather than to the input’s potential.

It does intercept the leakage from the rail, and if that were the only path it would work. But it also puts a low-impedance conductor immediately beside the input trace at a fixed potential, and any signal on the input now has the full signal voltage across that leakage resistance. The current is proportional to the signal, so it appears as a shunt resistance across the source — a gain error of the source resistance divided by the leakage resistance, which for a gigohm source and a teraohm ring is a thousand parts per million.

That is a thousand times worse than the driven ring’s one part in a billion, and it is worse than the bare board in one specific respect: it is proportional to the signal, so it does not calibrate out as an offset and it is not constant with signal level if the surface resistance is non-ohmic. A ring at the wrong potential is a component fitted to the input.

The two currents at one node

There are now three contributions at the input node and they behave differently enough that it is worth listing them together, because a specification that names one of them is naming a third of the problem.

The amplifier’s bias current is a property of the device, is nearly constant, and produces an offset proportional to the source resistance. The offset current — the difference between the two inputs — is a tenth of it and cannot be cancelled by balancing. The board’s leakage is a property of the board and the weather, produces an offset proportional to the rail and to the source resistance, and is the largest of the three by orders of magnitude until a ring is fitted.

Guarding does not improve the first two. What it does is move the third from being the binding constraint to being irrelevant, and the value of that is precisely that it restores the amplifier’s own specification to being the thing that decides — which is why choosing a femtoamp part is a sensible decision on a guarded board and a waste of money on a bare one.

What is not here

No temperature. Surface resistance falls with temperature as well as with humidity, and the amplifier’s own input current roughly doubles every ten kelvin for a junction-input device. Both move the comparison above and neither is in this netlist. Two millivolts a kelvin is the machinery that would carry it.

No triboelectric or piezoelectric charge. A coaxial cable flexed at a high-impedance input generates charge directly, and at a gigohm it is a large error with no resistance anywhere in its description. It is the reason low-noise cable exists and it is not a leakage.

And no dielectric absorption in the board itself. The laminate between the guard and the input is a capacitor with a distribution of relaxation times, exactly as the capacitor that remembers describes, so a step on the guard is followed by a slow current into the input for decades of time afterwards. On a settling measurement at these impedances that is the tail that decides how long a reading takes.

The number the technique is specified by

If a guard is worth the loop gain, then the specification a designer needs is not the leakage resistance and not the amplifier’s input current. It is the product of the leakage resistance and the loop gain at the frequency of interest, and that product has a name nowhere.

It also has an awkward property: it is worse for a fast amplifier used slowly than for a slow one. A part with a megahertz of gain-bandwidth and a gain of a million has its corner at one hertz, and above one hertz the guard’s benefit falls; a part with the same gain and a hundred kilohertz of gain-bandwidth has its corner at a tenth of a hertz and is worse everywhere. What decides the guard’s benefit at a given frequency is the open-loop gain there, which is the gain-bandwidth divided by the frequency, so the fast part wins — and the reason to say so is that a slow, low-input-current electrometer amplifier is exactly what one would otherwise reach for.

That is the same trade how much of the amplifier gets through states about desensitivity: what feedback improves, it improves by the loop gain, and the loop gain is a curve. A guard is a feedback technique with the board’s leakage inside the loop.

What one wire bought

A ring of copper, connected to a node the circuit already has, changes a fourteen-millivolt offset into a one-nanovolt gain error and a 1.6 Hz corner into an 812 Hz one. It adds no component, costs no current, and appears on the schematic as a wire.

And it is invisible to every measurement that does not have the board in it. Simulate the amplifier and the source and there is nothing to find; the whole of the effect is in a resistance nobody drew, between two features whose spacing was chosen by a layout tool. That is why the technique is transmitted as a rule rather than as a calculation, and why the calculation — two solves of the same netlist, with the leakage present and absent — is worth doing once.

The other element nobody drew

A leakage resistance across a board is the second time this collection has found a design decided by a quantity that is not on the schematic and not on any bill of materials, and the pair is worth reading together because the two behave in opposite ways.

Where the current comes back is the other one: the return current under a track spreads out at low frequency and runs directly beneath it at high, and the frequency at which it changes over depends on neither the length of the track nor its width — 106 kilohertz for any track two hundred micrometres above a half-milliohm plane. Everything a designer usually varies is absent from the answer, and what decides it is a spacing.

The two differ in what a layout can do about them. A return path is decided by geometry and cannot be argued with; the current goes where the impedance is least, and the only design freedom is where the plane is. A leakage path is decided by a resistance that a guard ring removes from the circuit, by holding both ends of it at the same potential — which is why the repair here is worth a factor of a million and the repair there is worth choosing a stackup.

And the edges that are lengths is where the collection gathers boundaries of this kind: the ones whose axis is a distance, which nobody chooses at the schematic, which are set by whoever builds the thing, and which appear in no netlist at all. A guard ring’s clearance and a track’s height above its plane are both on that list, and both are numbers a calculation can supply to somebody who would otherwise be following a rule.

Part 1 on guarding

One argument about Guarding, and one of 2 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it:

The guard leaves a negative resistance, and it reaches −1.59 kΩ. computed by solving, not by drawing. The magnitude of the conductance a source sees looking into the input, guarded and not, with 100 pF of cable and a 1.00 MHz amplifier. The unguarded input's conductance is positive everywhere — a capacitance to ground and a leakage to a rail are both losses. The guarded one is negative above 0.0404 Hz, and its magnitude rises as the square of frequency: −15.9 MΩ at 10 kHz, −161 kΩ at 100 kHz, −3.18 kΩ at a megahertz. Above the amplifier's gain-bandwidth product it flattens at ωₜ·C, which is −1.59 kΩ. That is the same input the guard raises to 10¹⁸ Ω at direct current, and nothing about the leakage the guard was installed for appears in it: the negative resistance is a product of the amplifier's bandwidth and the cable it is driving. The sign of what the guard gives back Part 2 — A guard ring is sold on two numbers and they are both about magnitudes: a teraohm of leakage multiplied to 10¹⁸ ohms, and a hundred picofarads of cable bootstrapped out of the way. The guard is also driving that capacitance with a copy of the input that lags it, and a capacitance driven by a lagging copy of its own voltage takes current out of phase with the voltage across it. What the guarded input presents is a negative conductance rising as the square of frequency — −15.9 megohms at ten kilohertz, −3.18 kilohms at a megahertz, flattening at the gain-bandwidth product times the capacitance — and a faster amplifier makes it worse.

Where this goes next

Where an amplifier's reading comes from, against the source it is reading. computed by solving, not by drawing. Three errors with three different dependences on the source, each measured by a solve with the other two set to zero. The offset voltage is flat — 50 microvolts wherever the source is. The bias current times the imbalance is linear in the source and is what balancing removes. The offset current times the source is linear too and is what balancing leaves. Unbalanced, the current overtakes the voltage at 1.77 kΩ; balanced, at 10.0 kΩ, which is the offset voltage divided by the OFFSET current and is the ratio of the two currents further along. Below about a kilohm, balancing makes the reading worse — the feedback network is already the larger resistance, and equalising means adding to the source. The current the instrument draws Every amplifier in this collection has had inputs that take no current, and that is not an idealisation of a small quantity — it is 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. The classical cure balances the two resistances and removes the bias current, leaving the offset current: worth a factor of ten, not a thousand, and it costs forty per cent of the noise density to get. The millivolts in the wire, which are nobody's signal. computed by solving, not by drawing at 145 frequencies. A 10 mm run of one-ounce copper carries the return of a 100 mA load and the reference of a 10 mV sensor. Its 5.00 mΩ and 10.0 nH put 500.0 µV in series with the sensor at low frequency — 5.00% of the reading — rising a decade per decade above 79.6 kHz until at 1.59 MHz the error is the whole signal. The full solve and the interfering current times the shared impedance agree to 4.4e-7. The slider is the length of the shared run: both the resistance and the inductance are proportional to it, so every point on the curve moves down together and the corner stays at 79.6 kHz. The millivolts in the wire Ten millimetres of one-ounce copper is five milliohms and ten nanohenries, and if a hundred-milliamp load and a ten-millivolt sensor both return through it, half a millivolt of somebody else's current is added to the reading — five per cent of it, before anything has been amplified. Above 79.6 kilohertz the error rises a decade per decade with no ceiling, and shortening the shared run moves the whole curve down and the corner not at all. Measuring with 50 mΩ of lead in each wire. computed by solving, not by drawing at 61 resistances, twice each. The two-wire arrangement measures the leads too, so its error is 2×50 mΩ over whatever is being measured: one per cent at 10 Ω, and 10000% at 1 mΩ. The four-wire arrangement senses on a separate pair that carries almost no current, and its error stays under 1.0e-2% across the whole range. Two terminals measure the leads as well Fifty milliohms in each lead makes a two-wire measurement one per cent high at ten ohms, ten per cent high at one ohm, and a hundred per cent high at a tenth. Not approximately — the reading is the resistance plus the leads, and below about ten ohms most of what is being reported is the wire between the instrument and the thing. Two probes on a 2.0 kΩ source. computed by solving, not by drawing twice per frequency: the node alone, and the node with the probe's elements across it. The one-to-one probe's 115.0 pF makes the reading one per cent wrong at 6.79 kHz. The ten-to-one probe puts 12.8 pF in series with the cable, so its tip sees 11.5 pF and the same error arrives at 69.2 kHz — 10 times further up, bought with a factor of ten in signal — the two edges stand in the ratio of the tip capacitances, 10.00. At direct current neither probe is capacitive at all and the ten-to-one still reads 0.02% low, because 10 MΩ across 2.0 kΩ is a divider. The probe is part of the circuit A one-to-one oscilloscope probe on a two-kilohm source gives a reading that is one per cent wrong at 6.8 kHz. Not because the instrument is inaccurate — it is reading correctly — but because the hundred and fifteen picofarads at the end of the cable are across the node, and above that frequency the trace on the screen is a picture of a circuit that only exists while the probe is attached.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down, the 8 sharing most with it of 14.

The objects named here

The third axis, after the field and the idea: the things themselves, and every essay that touches each one.

BootstrappingGuardingInput bias currentInput impedanceLeakage currentLoop gainMeasurement errorSource impedance