A divider with two ratios

A resistive divider has one ratio. Put a capacitor across each resistor and it has two, and they are different numbers.

At direct current the capacitors do nothing and the ratio is R₂/(R₁ + R₂). At high frequency the resistors do nothing and the ratio is C₁/(C₁ + C₂) — the top capacitance over the sum, which is the reciprocal-looking arrangement that follows from a capacitor’s impedance being inverse in its value. Those two expressions are equal only when R₁C₁ = R₂C₂.

What the step response is showing #

The shape is the whole diagnostic and it reads directly.

The edge lands at the capacitive ratio, because at the instant of a step the capacitors are the only components that matter — a resistor’s current is proportional to voltage and a capacitor’s to rate of change, and at an edge the rate of change is unbounded.

It then relaxes to the resistive ratio over the network’s time constant, which is (R₁ ∥ R₂)(C₁ + C₂) — 115 µs for the values drawn.

So an edge that lands above where it settles means the capacitive ratio exceeds the resistive one, which means C₁ is too large: over-compensated. An edge that climbs from below means C₁ is too small: under-compensated. The direction of the error is legible at a glance, which is why the adjustment is made against a square wave rather than against a meter.

The measured numbers make it concrete. At the balancing value of 12.78 pF, the edge lands at 0.10000 and settles at 0.10000, and the overshoot is 0.000%. At 19 pF the balance ratio is 1.487, the edge lands at 0.14179 and the overshoot is 41.3%. At 9 pF the balance is 0.704, the edge lands at 0.07258 and the response climbs, 27.1% short.

The settled value is the same in every case, and that is the first thing to check on any of these traces. Whatever the trimmer is set to, the divider settles at 0.10000 — because at direct current the capacitors are not there and the resistors decide, and no capacitor value changes a resistive ratio. A probe that is badly compensated reads correctly on a steady voltage and incorrectly on everything else, which is exactly the kind of error that survives a calibration check.

The time constant does move a little with the setting: 108.9 µs at 6 pF, 115.0 µs at balance, 126.0 µs at 25 pF. That is (R₁ ∥ R₂)(C₁ + C₂) with C₁ varying, and it is a reminder that the trimmer is a real component in the network rather than a correction applied to the answer.

The ramp, and why a true step would have been wrong #

There is a detail in how the figure is computed that started as a bug and ended as a correctness condition, and it is worth setting out because the symptom was so unhelpful.

The drive is a ramp with a rise time of a three-hundredth of the network’s time constant, not a discontinuous step. The reason is that C₁ couples the input directly to the output, so a true step makes the output jump at t = 0 — while the integrator, like every initial-value solver, starts every state at zero. Those two statements contradict each other, and the trapezoidal rule resolves the contradiction by ringing.

The first version of this figure reported a settled value of −1.5×10⁻⁸ for a divider that settles at 0.1, and an “overshoot” of −1.3×10⁷ per cent. Nothing about that was subtle once seen; what made it worth recording is that the shape of the trace looked like a plausible oscillation, and a figure that had merely been drawn rather than asserted on would have shipped.

A ramp three hundred times faster than the network is a step as far as the network is concerned, and a real generator’s edge is finite anyway. The residual cost is that the measured landing value is about one per cent past the capacitive ratio — the network relaxes measurably during the ramp — which is why the figure asserts that agreement to three per cent rather than to the arithmetic.

How precisely the trimmer has to be set #

The balancing value needs no search: it is R₂C₂/R₁, one division, and for the values here it is 12.7778 pF exactly.

What is worth measuring is the window — how far from that value the setting can be before the edge is visibly wrong — because that is what decides whether an adjustment is needed at all.

Four and a half per cent, for two per cent of flatness. One and a bit per cent, for half a per cent. At ten per cent of flatness — which is a badly adjusted probe, visibly wrong on a square wave — the window is 22.6% wide.

The scaling in those three numbers is worth reading. The window widens roughly in proportion to the flatness accepted: 1.13%, 4.52% and 22.6% for 0.5%, 2% and 10%. That is the linear relation from the paragraph below stated as a window rather than as a slope, and it says something practical about the adjustment — halving the acceptable overshoot halves the range of settings that achieve it, so each further factor of two in flatness costs the same effort as the last. There is no point at which the adjustment becomes suddenly harder or suddenly easier.

Standard capacitor values step by about ten per cent, so no fixed component reliably lands in either window — and that, rather than any argument about manufacturing tolerance, is why the adjustment exists. It is not there to compensate for drift or for temperature; it is there because the target is narrower than the grid of available values.

The second thing the shape says is that the error is linear through zero rather than quadratic. A five per cent error in the trimmer gives about five per cent of overshoot. There is no flat region around the optimum, which is why the adjustment is fiddly and why it can be made accurately: a quantity with a sharp minimum is easy to find and a quantity with a broad one is not.

Why the response is flat when the two agree, not merely equal at the ends #

The equation R₁C₁ = R₂C₂ makes the ratio at zero frequency equal the ratio at infinite frequency. That is what has been shown so far, and it does not by itself say the ratio is the same at every frequency in between. It could equally describe a response that departs and returns.

It does not, and the reason is worth a paragraph because it is the algebraic heart of the arrangement.

Each arm of the divider is a resistor in parallel with a capacitor, so each has an impedance of R/(1 + sRC). The divider’s ratio is the lower impedance over the sum, and when the two time constants are equal the factor (1 + sRC) is the same in both arms and cancels completely. What is left is R₂/(R₁ + R₂), a constant with no s in it anywhere.

So the response is not approximately flat or flat over a band. The pole and the zero of the divider land on top of each other and annihilate, and the network has no frequency dependence at all. That is a much stronger statement than “the two ends agree”, and it is why the step response goes immediately to its final value rather than reaching it.

It is also why the failure is a single time constant rather than something more complicated. Away from balance the pole and the zero separate but there is still only one of each, so the departure is always a single exponential relaxation from the capacitive ratio to the resistive one — which is exactly what every trace in the figures does, at every setting.

What the divider is for #

It is worth stepping back to say why anybody would put capacitance into a divider deliberately, since the whole essay is about the trouble it causes.

They would not. The capacitance in a probe’s lower arm is not a design choice; it is the instrument’s input capacitance and the cable’s, and it is there whether or not anyone wants it. The trimmer in the upper arm is the fix: given that C₂ exists and cannot be removed, adding a C₁ that satisfies R₁C₁ = R₂C₂ makes the divider’s two ratios agree and gives it a flat response.

That framing changes what the equation means. It is not a constraint the design must respect; it is a repair, and what it repairs is a frequency response that would otherwise fall away above 1/2πR₂C₂ — about 1.4 kHz for the values here. A ten-to-one probe with no trimmer at all would be a low-pass filter with a corner in the audio band.

What the adjustment does not fix #

Three things, and they are worth separating from the one it does.

It does not change the tip capacitance. The trimmer’s value is what makes the tip present $C_\mathrm{load}$ over the ratio, so compensating the divider and minimising the loading are the same adjustment — which is a genuinely fortunate coincidence and is why the previous essay and this one share a number. But minimising is not eliminating: 11.5 pF is what a correctly compensated ten-to-one probe presents, and the probe is part of the circuit is about what that costs.

It does not survive a different instrument. The balancing value depends on C₂, which is the instrument’s input capacitance plus the cable’s. Move a compensated probe to a different oscilloscope and C₂ changes, the equation no longer holds, and the probe needs re-adjusting. That is why the adjustment is on the probe rather than inside the instrument, and why the ritual is performed on connection rather than once at the factory.

It does not extend to high frequency. Everything here is a two-element model of each arm, and a real probe has inductance in its leads, distributed capacitance along the resistor, and a cable that is a transmission line rather than a lumped capacitance. Above a few tens of megahertz the flat response this adjustment produces stops being flat for reasons no trimmer addresses, which is why a fast probe has more than one adjustment in it.

The same arrangement, in three other places #

The pole-zero cancellation above is not a probe technique. It is a general one, and recognising it elsewhere is most of what makes it worth understanding rather than memorising.

Every attenuator in a wideband instrument. A switched attenuator is a chain of dividers, and each of them has stray capacitance across each arm. Each therefore carries a trimmer, and the alignment of an instrument’s attenuator is the same adjustment repeated once per range. A step response that changes shape when the range is switched is a divider somewhere in the chain whose time constants no longer match.

High-voltage dividers. A hundred-to-one divider for a kilovolt supply has a large resistance in its upper arm, which makes even a picofarad of stray capacitance across it consequential. The same equation applies, and the physical construction of the upper arm — the spacing, the shielding, the grading rings — is largely an exercise in controlling C₁ so that it can be matched.

Lead compensation in a feedback loop. A network that places a zero on top of a pole to cancel it is doing algebraically what the trimmer does: making a factor appear in both numerator and denominator so that it disappears. The feedback field meets it as a way of buying phase margin, and it has the same fragility — a cancellation that is exact only when two independently drifting quantities stay equal.

That last one is the useful warning. A cancellation is a coincidence between two components, and it holds only as well as the coincidence does. A trimmer set to a twentieth of a per cent by a careful adjustment is set to that precision on the day; a resistor and a capacitor with different temperature coefficients will not keep it, and nothing about the flat response gives any warning as it drifts.

Two ratios, one equation #

The result is unusually clean and worth restating in its shortest form.

A divider with capacitance has a ratio at zero frequency and a ratio at infinite frequency. They are equal if and only if the two arms have the same time constant. The step response shows which is larger. The setting window for two per cent of flatness is 4.5% wide, so the adjustment is a trimmer rather than a part.

Every one of those sentences is a measurement on a solved network, and the last one is the reason the component exists.

What makes this one of the more satisfying corners of the subject is how completely the picture and the arithmetic agree. The equation is one line. The failure is one exponential. The diagnostic is a shape on a screen that says both how far off the setting is and in which direction. And the window — the quantity that decides whether the adjustment is needed at all — is a number that falls out of the same solve, and it turns out to be narrower than the spacing of the parts anybody could buy.

Very little in this collection lines up that neatly. Most models here are excellent inside a range and wrong outside it, with the boundary somewhere inconvenient; this one is exact inside a window and the window is measurable. The difference is that the failure has one degree of freedom, and a one-dimensional problem can be adjusted rather than merely characterised.

A window that has to be trimmed, and three that cannot be #

The compensation trimmer exists because one equation has to hold between four components, two of which step in values ten per cent apart. That is a narrow window bought with an adjustment, and it is worth setting beside the collection’s other narrow windows, because in most of them adjustment is not available.

Two requirements pulling one capacitor has a regulator’s stability floor at 939 milliohms and its best transient at 850 — ten per cent apart, on a parasitic that is not specified tightly, doubles between room temperature and −20 °C, and cannot be trimmed because it is a property of the capacitor rather than a component in its own right.

The two resistors a ladder was designed between has a passband window of 0.886 to 1.137 times the design resistance — tighter than a resistor is usually specified to — and the repair there is to fit a real resistor of the design value rather than to adjust anything, which is this essay’s solution without the trimmer.

And the gain that is exactly one is the limiting case with no window at all: an oscillator’s design condition is an equality met on a set of measure zero, so no tolerance on any manufacturer’s list makes the circuit work and the repair is a nonlinearity that enforces the condition continuously rather than a trimmer that sets it once.

Read together, those four are a small taxonomy. A window wide enough to hit with a component is met by buying the component; one too narrow for a component and wide enough for a screwdriver is met by adjustment, which is this essay; one too narrow for a screwdriver is met by a feedback mechanism that holds it; and one of zero width is met only by the last of those. Which of the four a design is in is decided by two numbers — the width of the window and the tolerance of the parts — and both are usually computable before anything is built.

What the trimmer here is adjusting against is the last thing worth naming, because it is not a component in the probe. The equation involves the instrument’s own input capacitance, which varies between instruments and with the cable, so the adjustment cannot be made at the factory and cannot be transferred between instruments — which is why every oscilloscope has a square wave on its front panel and why the first thing done with a probe is to adjust it against the instrument it will be used with. The probe is part of the circuit is the same observation applied to the far end of the same cable: an instrument is an element in the netlist, and a probe is an element in two netlists at once.

What the trimmer does not reach #

A compensated divider is exact at one trimmer setting and the instrument still has three errors it cannot touch. The probe is part of the circuit is the loading, which is a change to the circuit rather than to the reading. The instrument’s own rise time is the bandwidth, which the compensation makes flat and does not make wide. The corner that says nothing about an edge is the low-frequency end of the same network, where the trimmer is not involved at all. The divider, and the thing it does not know about is the resistive argument underneath, and One step, computed twice is the machinery the edge is computed on.