The probe is part of the circuit

The divider and its load makes an argument about a divider: its ratio is a property of the divider and of what is across its output, so two dividers of identical ratio give 5.970 V and 1.000 V into the same load. This field is that argument taken seriously about the one thing usually exempted from it.

An instrument is not an observer. It is an element, it goes in the netlist, and every reading it gives is a reading of the circuit that includes it.

What is actually on the end of the cable #

A one-to-one probe is a wire, an input resistance of a megohm, and capacitance. The capacitance is the part that matters and it is larger than most readers expect: the instrument’s own input is about fifteen picofarads and the cable adds a hundred, so the tip presents about a hundred and fifteen picofarads across whatever it touches.

A hundred and fifteen picofarads is not a small component. Across a two-kilohm source it is a pole at 692 kHz, and the reading is one per cent low well before that — at 6.8 kHz, which is bisected on the two solved networks rather than estimated.

The number to hold on to is that one: 6.8 kHz. It is the frequency above which the picture on the screen is not the circuit’s response but the response of the circuit plus the probe, for a source impedance that nobody would consider high.

Where the one per cent is measured #

The edges in the figure are bisections rather than corner frequencies, and the difference between those two things is larger than it looks.

A corner frequency is where a response has fallen by 3 dB — thirty per cent. What is bisected here is where the displayed value departs from the true one by one per cent, which happens a long way below the corner: for a single pole, one per cent of magnitude error arrives at about a seventh of the corner frequency. The 692 kHz pole and the 6.8 kHz edge differ by almost exactly that factor.

Two things follow. The first is that a probe’s stated bandwidth, which is a −3 dB figure, is not the frequency up to which it is accurate; it is about a hundred times the frequency at which it is one per cent accurate on a given source. The second is that the accuracy edge depends on what accuracy is being asked for, so the figure’s slider could as easily have been the tolerance as the source resistance — a tenth of a per cent lands at a twentieth of the corner and a tenth of a per cent is what a measurement often needs.

The site’s habit of drawing an edge where a model fails rather than where a response falls is doing real work here. “Where the reading is one per cent wrong” is a statement about a measurement. “Where the response is 3 dB down” is a statement about a filter, and the probe is not being used as a filter.

What the ten-to-one probe actually does #

The obvious reading of a ten-to-one probe is that it is an attenuator, and that its purpose is to let a large signal fit on the screen. That is a side effect. Its purpose is capacitance.

The probe contains a nine-megohm resistor and, in parallel with it, a small trimmer capacitor. The instrument’s megohm and the cable’s hundred picofarads are on the far side of them. So the tip does not see the hundred and fifteen picofarads directly; it sees the trimmer in series with them, and a series pair is dominated by the smaller.

The trimmer is 12.78 pF, so the series combination is 11.5 pF — exactly a tenth of what the one-to-one probe presents, and not by coincidence. The compensating value of the trimmer is $C_\mathrm{load}/(\text{ratio} - 1)$, so the series pair works out to $C_\mathrm{load}$ over the ratio for any ratio at all. A hundred-to-one probe presents 1.15 pF.

The consequence is measurable and the figure asserts it: the two probes’ one-per-cent bandwidths stand in the ratio of their tip capacitances, 10.000 to within a twentieth of a per cent, over the part of the slider where the resistance is not itself the problem.

That assertion is conditional, and the condition is checked rather than assumed — which is worth a sentence because the first version of it was not. At five kilohms the one-to-one probe is already 0.50% wrong at direct current, so only half the one-per-cent budget is left for its capacitance and its edge arrives early; the ratio then comes out at 11.46 rather than 10.00. Nothing is wrong with either measurement. What is wrong is the claim, which is about capacitance and was being tested on a case where resistance had taken over. The figure now asserts the clean statement only where the resistive error is under a tenth of a per cent, and asserts the weaker one — that the ten-to-one probe reaches at least three times further — everywhere.

A conditional assertion with the condition measured is a better object than an unconditional one with a wider tolerance. The second passes everywhere and says less.

The error that is there at direct current #

There is a second error in every probe and it is present at zero frequency, where the capacitance does nothing at all.

A ten-to-one probe’s total resistance is ten megohms, and across a two-kilohm source that is a divider: the reading is 0.02% low before any frequency is applied. On a ten-kilohm source it is 0.10%; on a two-hundred-kilohm source it is 2%, and there is then no frequency at which the reading is within one per cent, so the probe has no bandwidth in the sense this figure measures.

That cliff is why the slider stops at five kilohms. Above about ten kilohms the one-to-one probe’s megohm is already eating the whole error budget, and a “bandwidth” is not a meaningful number for a measurement that is wrong at zero.

Splitting the two effects is worth doing explicitly because they behave differently. Resistive loading is flat: it is the same at every frequency, it is easy to correct for, and it depends only on the ratio of the probe’s resistance to the source’s. Capacitive loading rises: it is negligible at low frequency and total at high, it cannot be corrected for by scaling, and it depends on the product of the source resistance and the probe’s capacitance.

An instrument’s specification gives both numbers, and they answer different questions.

The same fact in the time domain #

An edge, rather than a sinusoid, makes the cost of a probe more visceral, and the arithmetic is different in a way worth knowing. Rise times combine in quadrature. A probe with its own rise time $t_p$ watching a signal with rise time $t_s$ shows an edge of $\sqrt{t_s^2 + t_p^2}$, so the cost is not linear in the probe’s speed and it is not negligible until the probe is much faster than the signal.

The quadrature is what makes the specification “the probe must be N times faster” rather than “the probe must be fast”. At equal speeds the measurement is 41% wrong. At three times it is 5%. At ten times it is 0.5%. Each factor of three in probe speed buys roughly a factor of ten in accuracy, which is the sort of relation that makes a fast probe worth what it costs and makes a slow one useless rather than merely imperfect.

The connection back to the frequency-domain figure is direct. A probe’s own rise time is $2.2\,R_s C_p$, so it is a property of the source and the probe together and not of the probe alone — the same statement as “the pole is at $1/2\pi R_s C_p$”, written in the other domain.

What a probe cannot fix, and what an active one does #

The ten-to-one probe’s trick is to divide before the capacitance rather than after. That gets a factor of ten and it costs a factor of ten in signal, which for most measurements is a good trade and for a small signal is not one at all.

The alternative is to put an amplifier at the tip, so that what the circuit sees is the input capacitance of one transistor — a picofarad or less — rather than a cable. That is an active probe, it costs a great deal more, and the model here does not include one. What it would change is only the number: the tip capacitance falls by another order and every edge in these figures moves up by the same factor, because the mechanism is unchanged.

What no probe fixes is that something is attached. An active probe’s picofarad on a hundred-kilohm node is a pole at 1.6 MHz, and the reading is one per cent wrong well below that. The measurement always includes the instrument, and the only question is by how much.

The number worth carrying away #

For a two-kilohm source and a one-to-one probe: 6.8 kHz.

That is not a criticism of any instrument. Every number in it comes from components that are exactly what they are advertised to be — a megohm of input resistance, fifteen picofarads of input capacitance, a hundred picofarads of cable — and the arithmetic is a first-order low-pass filter.

What it is, is the answer to a question that is very rarely asked. An oscilloscope with a bandwidth of a hundred megahertz, used with the probe it came with, on a source impedance of a few kilohms, gives a reading that is one per cent wrong at a few kilohertz. The instrument’s specification is about the instrument. The measurement’s accuracy is about the instrument and the circuit, and the second number is four decades below the first.

What to do about it, since something can be #

It would be a poor essay that measured a boundary and offered nothing, so here is what the same arithmetic says about avoiding it. None of it is novel; the point is that all of it is derivable from the two numbers already on the page. Probe at a low-impedance node. The edge is at $1/2\pi R_s C_p$, so it moves in inverse proportion to the source impedance. The output of an emitter follower and the input of the stage it drives may carry the same signal, and the first can be probed two decades higher than the second.

Use the highest ratio that leaves enough signal. Each factor of ten in ratio is a factor of ten in tip capacitance and therefore a factor of ten in edge frequency, traded against a factor of ten in amplitude. The trade is favourable whenever the instrument’s own noise floor is not the binding constraint, which for a signal of more than a few tens of millivolts it usually is not.

Keep the ground lead short, which this page has not modelled at all. The wire from the probe’s barrel back to the circuit’s ground is an inductance in series with the measurement, and a fifteen centimetre lead is of the order of a hundred and fifty nanohenries — enough to resonate with the tip capacitance somewhere in the tens of megahertz and produce ringing on an edge that the circuit does not have. That is a real and well-known effect, it is the single most common cause of a measurement that shows a fault which is not there, and the model in these figures contains no lead inductance, so the boundary it computes is optimistic by however much that resonance contributes.

Naming the last one as a shortfall rather than leaving it out seems better than a page that measures one loading mechanism carefully and implies it is the only one.

Why this field exists at all #

The three essays in this field are about three different instruments and they make one argument, which is worth stating once rather than three times.

Measurement is not observation. There is no arrangement in which a quantity is read without something being connected to the thing carrying it, and the something has properties. A probe has capacitance. A divider with capacitance in it has two ratios. A pair of leads has resistance. In every case the instrument’s own properties enter the answer, and in every case they enter in a way that is computable from the instrument’s specification and the circuit’s own impedance — which means the error is knowable in advance rather than discoverable after the fact.

That last clause is the part worth insisting on. Nothing in this field is a warning about uncertainty; every boundary on these pages is a number, bisected on two solved networks, and every one of them could have been computed before the measurement was made rather than blamed afterwards. The site’s rule — no model drawn without the frequency, amplitude or size at which it stops being true — applies to instruments exactly as it applies to circuits, and an instrument is the one place where it is almost never applied.

The nodes where a probe would be worst #

A hundred and fifteen picofarads across a two-kilohm source is one per cent at 6.8 kHz, and the same capacitance across other nodes in this collection would be very much worse or entirely harmless. Three are worth naming, because between them they say what makes a node hard to measure.

The node that is at ground for a while is the worst case in the collection. An inverting amplifier’s summing junction is a tenth of an ohm at direct current, ten ohms at a kilohertz and 909 ohms above a megahertz, so a probe there is negligible at low frequency and dominant at high — and worse than that, the capacitance itself changes the circuit: the gain the loop closes against finds nine picofarads at that junction, less than a probe, taking the phase margin from ninety degrees to forty-five. A probe on a summing junction does not merely load it; it destabilises the loop it is being used to investigate, while the measured gain at a kilohertz moves by three parts in a million.

Where the trouble is at the input is the node where the same capacitance is already the design. A photodiode amplifier’s difficulty is the diode’s own junction capacitance, which is the price of its area, so a probe is one more of the thing the whole design is arranged around — and, unusually, an amount that can be reasoned about rather than a surprise.

And a source below a frequency is where a probe is nearly free and the measurement is still hard. A regulator’s output impedance is 0.43 milliohms at direct current, so no probe capacitance matters — but the quantity being measured is a few hundred microvolts on top of five volts, through leads that carry the drive current and the sense signal at once, which is the problem two terminals measure the leads as well solves with four wires rather than with a better probe.

Which is the field’s general rule stated as a question. Before measuring a node, ask what its impedance is at the frequency of interest and what the instrument’s is — and if the second is comparable with the first, the trace is a picture of a circuit that exists only while the probe is attached.

The ten-to-one probe that removes most of this is a divider with two ratios, and it is worth knowing that the repair is not free of the same problem — it is the same problem with a component added to cancel it. Putting capacitance into a resistive divider makes it divide by resistance at direct current and by capacitance at high frequency, which are two different numbers unless one equation holds, and the trimmer on every probe exists for that single equation. So a ten-to-one probe reduces the loading by ten and introduces a condition that has to be adjusted on the day, against the instrument’s own input capacitance rather than against a published number.

The four errors an instrument brings #

A probe’s loading is the first of four, and the field measures the others separately. A divider with two ratios is the compensation, exact at one trimmer setting. The instrument’s own rise time is the bandwidth, and the root-sum rule that is usually quoted for it is optimistic against every real shape. The current the instrument draws is the direct-current error, and Two terminals measure the leads as well is the only arrangement in the field that removes one of the four rather than bounding it.