The current a capacitor adds up

The current the instrument draws measured an amplifier’s input current against the resistance it flows in, and found it negligible until the resistance is large: fifty nanoamps is nothing until it flows in a megohm, and then it is fifty millivolts. The cure that becomes a different circuit then solved the classical cure — making the resistances the two inputs look into equal — and found it worth the ratio of the bias current to the offset current, about ten.

Both measured a static error: a current times a resistance, a fixed number of millivolts. There is a large class of instruments in which the input current does not flow into a resistance at all, but into a capacitor — an integrator, a sample-and-hold, a charge amplifier, the input of a slow electrometer. In each of them the same fifty nanoamps produces not an offset but a rate: the capacitor adds the current up, and the error grows for as long as the circuit runs. Time takes the place of resistance as the multiplier, and every design choice in these circuits turns out to be a trade against time.

This essay measures that in four instruments, each marched or solved with the bias currents present as the elements they are.

The integrator, marched with its own currents #

The integrator is the textbook inverting form: a 100-kilohm input resistor from a grounded source to the inverting input, a one-microfarad capacitor from the output back to it, and an ideal amplifier — a nullor — forcing the two inputs together. The amplifier’s imperfections are added as the elements they are defined as: a current source flowing out of each input, and where needed a voltage source in series with the non-inverting input for the offset voltage. The network is marched forward from rest by the trapezoidal rule, four hundred steps over ten seconds.

With 50 nanoamps out of each input and the non-inverting input grounded, the inverting input’s current has nowhere to go but through the capacitor, since the amplifier holds that node at zero and the input resistor then carries no current. The output ramps at $I_B/C$: 50.0 millivolts per second, to six figures, and after ten seconds it is half a volt away from where it started. Nothing in the circuit stops it short of the rail.

The classical cure works the same way it does in a resistive stage, and the measurement confirms it. With 100 kilohms in the non-inverting input — the same resistance the inverting input sees through the input resistor — the non-inverting input’s current develops a voltage that the amplifier copies across the input resistor, and that cancels the common part of the current. What is left is the offset current, five nanoamps, and a ramp of 5.00 millivolts per second. A factor of ten, as in the resistive stage.

But a factor of ten on a rate is still a rate. The balanced integrator drifts by 18 volts an hour, and an integrator meant to hold a reading for minutes cannot be fixed by balancing at all. The resistive stage’s error had a size; the integrator’s has a speed, and the only way to give it a size is to bound the time.

A leak turns the rate back into an offset #

The usual bound is a resistor across the capacitor — a leak — so that the capacitor cannot charge forever.

With a leak resistor the output ramps until the leak’s current equals the bias current, and then it stops at $I_B R_{\text{leak}}$: 50 millivolts for a megohm, half a volt for ten. The integrator has become, at DC, an inverting amplifier with a gain of $R_{\text{leak}}/R$, and its bias-current error is a resistive one again — the case the two earlier measurements were about, with the leak as the feedback resistor.

The price is that the circuit is no longer an integrator at low frequencies. Below $1/(2\pi R_{\text{leak}}C)$ it is an amplifier, so a leak of ten megohms across a microfarad stops integrating below 15.9 millihertz, and anything slower than about a minute is amplified rather than accumulated. So the leak is a trade, and an exact one: a lower offset needs a smaller leak, which raises the lowest frequency, and the two move in proportion. The average an integrator can hold meets the same trade from the noise side, where an integrator’s leak decides how long it can average before its own drift dominates.

The input current or the offset voltage #

The integrator has a second input error, its offset voltage, and that error is also a rate: the amplifier holds its inverting input at the offset voltage rather than at zero, so the input resistor carries a current $V_{os}/R$, and that current also flows into the capacitor.

So the two drifts compete, and the input resistor decides which wins. The offset voltage’s drift falls as the resistor grows, because a larger resistor converts the same offset voltage into a smaller current; the bias current’s drift does not depend on the resistor at all. They cross where the two currents are equal, at $V_{os}/I_B$ — one kilohm for 50 microvolts and 50 nanoamps.

That is the same crossing the current the instrument draws found for a resistive stage, and it has the same meaning: below it a better offset voltage is what to buy, above it a smaller input current. What is new is what does not decide it. An integrator’s time constant $RC$ sets its scale, and the same time constant can be built with a small resistor and a large capacitor or the reverse. The choice between those is a choice of which input error dominates: a one-kilohm, one-millifarad integrator is limited by its offset voltage, and a one-megohm, one-microfarad integrator with the same time constant is limited by its input current. Most practical integrators use large resistors, to keep the capacitor reasonable, and are therefore bias-current limited.

A hold capacitor’s droop and its acquisition #

A sample-and-hold is an integrator whose input is switched off: during the hold the capacitor is meant to keep its voltage while the buffer reading it draws its input current from it. The voltage falls — droops — at $I_B/C_h$, and a bigger capacitor droops less.

A bigger capacitor also charges more slowly through the sampling switch. To acquire a new value to within half of a 16-bit step, the capacitor must settle for about $\ln(2^{17})$, nearly twelve, time constants of the switch’s resistance and its own capacitance. Both the acquisition time and the inverse of the droop are proportional to the capacitance, so their product does not depend on it at all: every capacitor lies on one hyperbola, here 386 step-microseconds.

That product is set by the input current and the switch’s resistance, and it is unforgiving. Holding for a millisecond with half a step of droop needs 655 nanofarads, and through a 100-ohm switch that capacitor takes 0.77 milliseconds to acquire — most of the hold it is protecting. A faster sample needs a smaller input current or a smaller switch resistance; no choice of capacitor helps. The hold capacitor’s size is also what sets its sampled noise, $kT/C$, and the amplifier inside the sample measured how the buffer in the same loop changes that; the input current pushes the capacitor up from the side the noise does not, so the two limits agree on a large capacitor and only the acquisition time objects. The leak no switch can hold adds the switch’s own off-state leakage to the same budget, and in many designs it is larger than the buffer’s input current.

A charge amplifier’s offset and its corner #

A charge amplifier — the front end of a piezoelectric sensor, a photodiode in charge mode, a particle detector — is an integrator whose input is a source of charge rather than of voltage. Its feedback capacitor converts charge into voltage, and a bleed resistor across it keeps the output from drifting away on the input current, exactly as the leak did.

The offset is the input current times the bleed resistor, and the low-frequency corner is one over $2\pi$ times the bleed and the feedback capacitance. Their product contains no bleed resistor:

$$V_{\text{offset}} \cdot f_{\text{low}} = I_B R_b \cdot \frac{1}{2\pi R_b C_f} = \frac{I_B}{2\pi C_f}.$$

For a picoamp into a nanofarad it is 0.159 millivolt-hertz. A charge amplifier that must pass a sensor’s signal down to a hundredth of a hertz must accept 16 millivolts of offset from a single picoamp; one that must reach a ten-thousandth of a hertz must accept 1.6 volts. The only ways out are a smaller input current or a larger feedback capacitor, and the larger capacitor lowers the gain — volts per picocoulomb — in exactly the proportion it helps.

This is the reason charge amplifiers for slow sensors use inputs with femtoamps of leakage, and why their specification gives a low-frequency limit as well as a sensitivity: the two are tied together by the input current, and a quoted pair that violates the product is a sign that one of them is measured under conditions the other is not.

The rate as a measurement #

The same arithmetic runs the other way, and it is how very small currents are measured. A current too small to develop a readable voltage in any practical resistor — femtoamps, the leakage of a capacitor’s dielectric, the dark current of a photodiode — can be integrated on a known capacitor and read as a rate. A femtoamp into a picofarad is a millivolt a second, which a slow voltmeter resolves easily, where a femtoamp through a gigohm is a microvolt buried in that resistor’s own noise.

So the electrometer’s measuring mode is the integrator above with its input current as the signal rather than the error, and it inherits the same two limits. The amplifier’s own input current adds to the current being measured, which is why electrometer inputs are specified in femtoamps; and anything else that delivers charge to the input node — leakage across the board from a nearby rail, charge released by a cable flexing — is integrated as faithfully as the signal. The current that does not reach the input measured how a guard ring held at the input’s own potential removes the board leakage, and in an integrating electrometer that guard is not an option: an unguarded teraohm to a fifteen-volt rail is fifteen picoamps, a thousand times the current being measured.

The rate also gives the measurement a natural self-check. A constant current produces a straight ramp; a ramp that curves is reporting something that is not a constant current — dielectric absorption relaxing, a leakage that depends on the voltage the ramp has reached, or a temperature drifting during the measurement. The integrator turns the question “is the error constant” into the question “is the line straight”, which can be asked of the record itself.

A sensor’s capacitance is part of the integrator #

In a charge amplifier the sensor’s own capacitance sits across the amplifier’s input, and it changes the picture in one respect that matters for the input current. The bias current still flows into the feedback capacitor, because the amplifier holds the input node still and the sensor’s capacitance therefore carries no current at DC; the offset and the corner are unchanged. But the sensor’s capacitance sets the noise gain of the stage — the ratio of total input capacitance to feedback capacitance — and so multiplies the amplifier’s voltage noise at the output. A designer who enlarges the feedback capacitor to buy a lower corner at a given offset, as the product above invites, lowers that noise gain too, and the input current’s trade and the noise’s trade turn out to point the same way. The sensor an integrator does not see measured what that same capacitance does to the stage’s stability, which is the third side of the same choice.

The rule in all four #

The four circuits have one structure. A current flows into a capacitor; the capacitor converts it into a rate $I/C$; and the design has to bound the time over which the rate acts. An integrator bounds it by a reset or a leak, a sample-and-hold by its hold time, a charge amplifier by its bleed. In each case the bound converts the rate into a fixed error and charges for the conversion in the circuit’s time behaviour: the lowest frequency it integrates, the acquisition it can afford, the slowest signal it passes.

So the useful number for an input current in a capacitive circuit is not millivolts but a rate, and the useful comparison is between that rate and the time the circuit is asked to work over. Fifty nanoamps into a microfarad is 50 millivolts a second: negligible for a circuit that measures for ten milliseconds, catastrophic for one that holds for an hour. The balancing cure buys its factor of ten here too, but a factor of ten on a rate lengthens the usable time tenfold and does not remove the limit.

There is a quieter consequence for the temperature behaviour, which the current the instrument draws measured: a field-effect input’s leakage doubles every ten kelvin. In a resistive stage that doubles an offset. In an integrator or a hold it doubles a rate, and so halves the time the circuit can work for at a given accuracy — a hold that is good for a second on the bench is good for a quarter of a second twenty kelvin warmer.

What the measurement leaves out #

The capacitors are ideal. A real integrating capacitor has dielectric absorption, which makes it partly recover a voltage after a reset, and leakage of its own, which acts as an unintended leak resistor; the capacitor that remembers measures the first, and for film capacitors the second is often smaller than the amplifier’s input current but not always.

The input currents are constants. A real amplifier’s input current depends on its common-mode voltage, and in an integrator the non-inverting input sits at a fixed voltage while the inverting input follows it, so that matters less here than in a follower; but a field-effect input’s gate leakage rises steeply near its rails, and a hold buffer whose input swings across a large range sees a different droop at each end.

And the amplifier is ideal in every other respect: its offset voltage is a constant, its gain is infinite, and it has no output limit. An integrator that ramps to its rail stops integrating there, and recovering from that saturation takes time a real circuit has to budget for. The march here never reached a rail, because ten seconds at 50 millivolts a second is half a volt.

The number to keep #

An input current in a capacitor is a rate, $I_B/C$: 50 millivolts a second from 50 nanoamps into a microfarad, and balancing reduces it to the offset current’s rate without bounding it. Bounding it trades the error against time: a leak resistor gives an offset of $I_B R_{\text{leak}}$ and an integrator that stops below $1/(2\pi R_{\text{leak}}C)$; a hold capacitor’s droop times its acquisition time is fixed, so 50 nanoamps through 100 ohms needs 655 nanofarads and 0.77 milliseconds for half a 16-bit step; and a charge amplifier’s offset times its low corner is $I_B/2\pi C_f$ whatever its bleed. An integrator’s offset voltage and its input current cross at $V_{os}/I_B$ in input resistance, one kilohm here.

Still open: the two currents that are not one, the reset, and the current that depends on the signal #

The cure assumes one current at two pins. Balancing removes what the two input currents share. A part whose input current has been cancelled internally has two small residues that need not share anything, and for it the cure may remove nothing and add a resistor’s noise. How much the two currents have in common decides it, and that is measurable over a population of parts.

The reset. An integrator bounded by a periodic reset rather than a leak has an error that is a sawtooth, and a reset switch’s own charge injection adds a step at every reset. Whether a reset or a leak gives the smaller error for a given measurement time is a comparison these figures do not make.

A current that depends on the signal. In a sample-and-hold the buffer’s input current varies with the held voltage, so the droop is a function of the sample — a nonlinearity rather than an offset — and it is the distortion a precision hold actually specifies.