The source that rings against its guard

The sign of what the guard gives back found that a guarded input presents a negative conductance: the follower that holds the ring at the input’s potential lags, and a capacitance driven by a lagging copy of its own voltage delivers power rather than absorbing it. The resistor that moves the lag put a resistor in the ring’s drive and found that it weakens the negative resistance and makes it its own, without removing it. Both essays measured the input alone, with the source taken away, and both ended on the same caution: a negative conductance at a capacitive node is not yet a fault. It becomes one when the node has something to resonate with.

The ordinary thing to resonate with is the source’s inductance. A long lead, a coil, a transducer with windings — every real source has some, and an inductance in series with a resistance, feeding a capacitance with a negative loss, is an oscillator in outline. This page solves the whole netlist, source included, and finds the inductance at which it becomes one in fact.

The guard as a negative resistance in series #

The earlier essay drew the guarded input as a parallel conductance, which is the natural way to read an admittance. For a question about a series source there is a more useful way. The cable’s capacitance $C$ runs from the input to the ring, the ring follows the input by a follower of gain $A = 1/(1 + j\omega/\omega_t)$, and the cable therefore takes $j\omega C(1 - A)$. Since $1 - A = (j\omega/\omega_t)/(1 + j\omega/\omega_t)$, the impedance of that path is

$$Z = \frac{1 + j\omega/\omega_t}{j\omega C \cdot j\omega/\omega_t} = \frac{1}{j\omega C} - \frac{\omega_t}{\omega^2 C}.$$

That is a capacitor $C$ in series with a resistance of $-\omega_t/(\omega^2 C)$: negative, and falling as the square of frequency. An element whose impedance is a negative real number proportional to $1/\omega^2$ is sometimes called a frequency-dependent negative resistance, and filter designers build them on purpose. A guard builds one by accident, out of a cable and a follower’s lag.

Put a source with resistance $R$ and inductance $L$ in series with it and the loop’s impedance is $R + j\omega L + 1/(j\omega C) - \omega_t/(\omega^2C)$. Its imaginary part vanishes at $\omega^2 = 1/(LC)$, the resonance of the source’s inductance with the cable. At that frequency the negative resistance is $-\omega_t/(\omega^2 C) = -\omega_t L$, and the loop’s real part is $R - \omega_t L$. So the loop has net negative resistance at its own resonance — it oscillates — once

$$\frac{L}{R} > \frac{1}{\omega_t}.$$

The source’s own time constant against the follower’s. The cable’s capacitance has cancelled out of the condition entirely; it only sets the frequency, $1/(2\pi\sqrt{LC})$.

The poles of the whole netlist #

That argument used a one-pole follower and left out everything else at the node. The netlist has more in it: the amplifier’s own input capacitance to ground, which the guard does not bootstrap, the board’s leakage, the follower’s direct-current gain. So the argument is tested by computing the rightmost natural frequency of the complete circuit — every pole of the network with the source’s inductance in it — and asking where its real part crosses zero.

Each source’s curve crosses zero and the input starts to oscillate: at 17.6 µH for 100 Ω, against 15.9 from $R/\omega_t$, and in proportion for the others — 1.75 µH at 10 Ω and 187 µH at a kilohm. The solved values sit about ten per cent above the expression, which is the amplifier’s own input capacitance adding a little positive loss to the node. The frequencies of oscillation, 3.59 MHz for the 100 Ω source, sit within about five per cent of $1/(2\pi\sqrt{LC})$, 3.79 MHz.

It helps to put numbers on the balance at the resonance. With 17.6 µH against 100 pF the loop resonates near 3.8 MHz, where the guard’s negative resistance, $-\omega_t/(\omega^2 C)$, is about −110 Ω: just enough to cancel the source’s 100 Ω and the small positive loss the input capacitance adds. Halve the inductance and the resonance rises by $\sqrt 2$, the negative resistance halves to about −55 Ω, and the source’s 100 Ω wins comfortably; double it and the negative resistance doubles to −220 Ω and wins. The limit is a balance between two resistances at one frequency, and the inductance moves the frequency so that one of them changes.

The unguarded input, with the same source and the same cable, is stable at every inductance. The cable is then an ordinary capacitance to ground with a positive loss, a series RLC with nothing negative in it.

The curves also fall back towards zero at large inductance. A very large inductance puts the resonance at a very low frequency, where the negative resistance $-\omega_t L$ is large but the loop’s rate of growth — a real part per second — is small, because everything happens slowly. The input still oscillates there; it simply takes longer to build.

A time constant against a time constant #

Swept against the source’s resistance, the critical inductance lies on a line of slope one for each follower: it is proportional to the resistance, and the constant of proportionality is $1/\omega_t$ — 159 ns for a one-megahertz follower, 1.59 µs for a hundred kilohertz, 15.9 ns for ten megahertz. A faster follower tolerates ten times less inductance. That is the same inversion the earlier essay found for the negative resistance itself: the guard’s defect is proportional to how good the amplifier is, because a faster follower’s lag reaches further into the band before it is corrected.

The cable is not in the line. At 100 Ω behind a one-megahertz follower the critical inductance is 32.2 µH with 10 pF of cable, 17.6 with 100 and 16.2 with a nanofarad: a factor of two across a hundredfold range of length, and the factor comes from the input capacitance, which matters more beside a short cable than a long one. For every cable long enough to dominate the node, the length does not decide whether the input oscillates, only at what frequency.

Ten times the cable gives nearly the same critical inductances — 1.61, 16.2, 171 µH — and frequencies lower by $\sqrt{10}$: 1.24 MHz for the 100 Ω source against 3.59 with the shorter cable. That is the expression’s prediction exactly. The condition depends on the source and the follower; the cable chooses where on the spectrum the oscillation lands.

What 159 nanoseconds means for a real sensor #

The condition is a time constant, and time constants of sources are easy to estimate. A metre of ordinary wire has about a microhenry of inductance. A thermocouple or a strain gauge on a few metres of lead is a few ohms and a few microhenries: an L/R of a few hundred nanoseconds to a microsecond, which is above the 159 ns of a one-megahertz follower. A pickup coil of a millihenry and fifty ohms has an L/R of twenty microseconds, a hundred times over. Even a short lead from a low-resistance source can be over the line.

The sources a guard is built for are usually the opposite: a gigohm photodiode or an ion chamber, whose resistance is so large that no plausible inductance brings L/R anywhere near a microsecond. For them the condition is comfortably met and the guard is safe, which is why the effect is rarely seen in the application guards were designed for. It appears when a guarded input built for a high-impedance source is reused with a low-impedance, inductive one — a general-purpose electrometer front end connected to a coil, or a guarded amplifier input left driving a cable when the source is swapped. The same amplifier that was stable yesterday rings today, and the only thing that changed is a time constant nobody thought was the amplifier’s business.

Marched, either side of the limit #

A pole’s real part is a statement about a linear network, and the thing a reader of a bench would see is a waveform. So the same source is marched in time from a small step, just below and just above its critical inductance.

Below the limit, the step sets the input ringing and the ringing dies away — to 0.152 of its early amplitude by the end of the trace. Above it, the same step starts a ringing that grows by a factor of 7.54 over the same span and would go on growing until something in a real amplifier limits it. The march shares nothing with the pole calculation except the netlist, and it agrees with it on everything a waveform can show: the side of the limit each source is on, the frequency it rings at, and a growth of the size the pole’s real part, 0.737 per microsecond, gives over the two and a half microseconds between the trace’s first third and its last.

On a bench this looks like an input that is fine with one sensor and oscillates with another that has a longer lead. Nothing about the guard has changed, and nothing about the amplifier; the source’s inductance has crossed a line that depends on the amplifier’s bandwidth, and the guard is the component that made the line exist.

The resistor in the drive, as a remedy #

The resistor the previous essay studied does raise the limit, and steadily: 33.7 µH through 100 Ω, 164 µH through a kilohm, 1.32 mH through thirty. It does so for the reason that essay found. Once the resistor’s own lag dominates, the negative resistance is no longer the follower’s $-\omega_t/(\omega^2C)$ but a weaker one the resistor sets, and a weaker negative resistance needs a larger source inductance to overcome the source’s resistance.

And it charges for it in the same currency. The bootstrap ends near $1/(2\pi R_g(C + C_s))$, which for 1 kΩ is about 800 kHz and for 30 kΩ about 27 kHz. A resistor large enough to make a millihenry source stable leaves the guard bootstrapping the cable only up to the low tens of kilohertz. For a slow sensor that may be fine. For a guarded input whose point was bandwidth, it is the guard partly given up.

The comparison between the two remedies is the comparison the previous essay drew, now with a stability limit attached. Through 1 kΩ the limit rises ninefold and the bootstrap still reaches most of a megahertz; through 30 kΩ the limit rises seventyfold and the bootstrap stops in the tens of kilohertz. So the resistor is a good remedy for a source that is a little over the line and a poor one for a source that is far over it, and the point where it stops being worth it is where the bootstrap bandwidth it leaves falls below the band the measurement needs.

What the other remedies are #

The condition $L/R > 1/\omega_t$ lists its own remedies, and each has a cost that can now be stated.

A slower follower raises $1/\omega_t$ and so the limit, in proportion — a hundred-kilohertz follower tolerates ten times the inductance of a one-megahertz one. It also bootstraps the cable only over a tenth of the band, which is the same trade as the resistor in a different form.

More resistance in series with the source raises $R$ and the limit in proportion, and adds its own Johnson noise and its own attenuation of the signal; for a source whose resistance is the thing being measured, it is not available at all.

Damping the resonance — a resistor across the source’s inductance, or a capacitor with a resistor across the input — adds positive loss at the resonant frequency, which is what the negative resistance has to overcome. The node that is an inductance found a virtual earth behaving as an inductance and resonating with stray capacitance; the remedy for that node and for this one is the same kind of thing, a loss placed where the resonance is.

And the guard can simply not be driven at those frequencies, which is what the resistor in the drive does. Every remedy on the list works by moving something — the follower’s bandwidth, the source’s resistance, a loss, the guard’s bandwidth — and none of them removes the negative resistance a lagging bootstrap makes. That is the guard’s nature, and the design question is only where to put it.

Where this sits among the input’s other problems #

The current that does not reach the input measured the guard’s purpose, and it is at direct current: leakage, which the guard removes by five orders of magnitude. The probe is part of the circuit measured how any instrument attached to a node becomes part of it. This page finds the guard doing both at once: removing an error at direct current and adding a mechanism for oscillation at megahertz, through the same wire. A designer who installs a guard for its direct-current benefit and never looks above a kilohertz will not see the second effect until a sensor with a long lead makes the input ring.

How the numbers were obtained #

The input is the same netlist as in the earlier essays — a one-pole follower built from a transconductance and a compensation capacitance, 100 pF of cable from the input to the ring, the board’s leakage split between the rail and the ring — with 10 pF of input capacitance to ground and a source of the named resistance and inductance in series. The natural frequencies are the roots of the determinant of the network’s nodal matrix, recovered by sampling it on a circle after equilibrating its rows and columns — the route where the behaviour is written down uses for a network’s poles. The critical inductance is bisected on the sign of the rightmost root’s real part. The time traces march the same netlist by the trapezoidal rule, as one step, computed twice checks it against residues, from a 1 mV step, 24,000 steps over twelve periods of the oscillation, and the growth factor is the ratio of the envelope over the last third of the trace to that over the first.

What it leaves out #

The amplifier’s large-signal behaviour. An oscillation that grows will be limited by the follower’s slew rate and output swing — the regime the step too large to have an impedance measures for a follower driving a capacitance — and what it settles to — its amplitude and its effect on the reading — is a question for a nonlinear march, not for the poles.

A source with distributed inductance and capacitance. A long lead is a transmission line, not a lumped inductance, and above its own quarter-wave frequency it presents an impedance that the lumped source here does not capture; whether the guard can make a line resonate at one of its own modes is a different calculation.

And a guard driven by more than a one-pole follower. A real amplifier has a second pole, which adds lag near its crossover, and the negative resistance there is larger than the one-pole expression says.

Still open: the amplitude it settles to, a lead that is a line, and a second pole in the follower #

The amplitude of the oscillation. Marching the guarded input with a follower that slews and saturates would say what amplitude the oscillation settles to and what it does to the direct-current reading — whether a ringing guard shows up as noise, as an offset, or not at all at a slow instrument’s output.

A lead that is a transmission line. Replacing the lumped inductance with a line of stated length and impedance would give the input a set of resonances rather than one, and the question is whether the guard’s negative resistance, which grows towards low frequency, can make the lowest of them oscillate for a lead of a few metres.

A follower with a second pole. A second pole adds phase near the follower’s crossover and makes $1 - A$ larger there, which should lower the critical inductance for sources whose resonance falls near that frequency. Solving it would say whether the one-pole condition $L/R > 1/\omega_t$ is safe or optimistic for a real amplifier.