The ring the ground lead adds

The probe is part of the circuit put a one-to-one probe’s 115 pF across a node and found the reading one per cent wrong at 6.8 kHz on a two-kilohm source. The probe that takes a tenth cut the tip capacitance by ten with a ten-to-one divider and found the bandwidth returned and the noise multiplied by 31. Both essays, at the end, named the same omission. The first: “Keep the ground lead short, which this page has not modelled at all.” The second: “the ground lead is still absent.”

The omission was the convenient one. A probe’s ground clip and its wire are not part of the divider and do not appear on any probe schematic; they connect the probe’s shield, which is the instrument’s reference, to the circuit’s ground. But every current the probe draws from the node returns that way, and a wire fifteen centimetres long with a loop of that size around it has an inductance of about 150 nH. That inductance is in series with the tip capacitance, across the node, and the two of them are a resonant circuit that the instrument is reading.

Where the inductance sits #

The probe here is the ten-to-one probe of the earlier essays: a nine-megohm resistor and its compensating capacitor in front of the instrument’s one megohm and its input and cable capacitance, which together present 11.5 pF at the tip. In those essays its low end was the circuit’s ground. Here it is a node of its own, the probe’s shield, and the shield reaches the circuit’s ground through the lead’s inductance. The instrument reads its input against the shield, because that is what its own ground is.

The node being measured is driven by a source — a step through 50 Ω in the hero figure — and the netlist is solved twice. Once by recovering the transfer function from the probe’s input to the screen and expanding its step response in residues; once by marching the netlist itself forward in time with the trapezoidal rule. The two reach the same peak, 1.4942, to the fourth decimal. With the shield at the circuit’s ground instead, the same probe on the same source shows a clean first-order rise to 1.000, its time constant the tip capacitance against 50 Ω.

The size of the effect is easy to see without any of the solving. A 1 V edge that rises in a nanosecond pushes $C\,dV/dt = 11.5$ mA into the tip capacitance. If that current builds over the same nanosecond, the lead’s 150 nH develops $L\,di/dt \approx 1.7$ V across a wire the instrument treats as zero volts — the same size as the signal, in series with it. The instrument’s reference is moving by as much as the thing it is referring to, and nothing on the screen says so: a scope displays the difference between its tip and its ground, and has no way of knowing that its ground has left the circuit’s.

With the lead in place the screen shows something the node does not do: a ring at 121 MHz and a first peak 49.4 per cent high. The step at the node itself is clean. The ring is the probe’s, and it is in every edge the probe looks at.

A resonance damped by the thing being measured #

The mechanism is a series resonance. Current that charges the tip capacitance flows back through the lead, so the tip capacitance $C$ and the lead inductance $L$ are in series, driven by the source through its resistance $R_s$, and the instrument reads the voltage across the capacitor — the standard second-order low-pass. Its natural frequency is $1/(2\pi\sqrt{LC})$, 121 MHz for 150 nH and 11.5 pF, and its damping ratio is

$$\zeta = \frac{R_s}{2}\sqrt{\frac{C}{L}},$$

0.219 for a 50 Ω source. The probe contributes no resistance of its own to the loop at these frequencies: its nine-megohm resistor is shunted by its compensating capacitor, and the one-megohm input is shunted by the cable. The only damping is the source’s.

That makes the error run the wrong way for intuition. A designer making a node better — lowering its output impedance, terminating a line in its characteristic impedance, adding a stronger driver — is removing the probe’s damping. At 10 Ω the screen shows 87.1 per cent of overshoot; at 50 Ω, 49.4; at 150 Ω, 6.48; and above $2\sqrt{L/C} = 228$ Ω none at all. The solved overshoot follows the second-order law at every source resistance on the figure, because the circuit is the second-order circuit; the law and the netlist differ only in that the netlist also carries the probe’s slow megohm-scale pole, which is irrelevant on the nanosecond scale of the ring.

The earlier essays found the opposite dependence for loading. A probe’s capacitance loads a soft source: the error in the probe is part of the circuit grew with the source resistance, because a capacitor against a large resistance makes a low corner. The ground lead’s error grows as the source resistance falls. A probe therefore has two failure modes on two sides of a source impedance: too soft and the tip capacitance slows the edge; too stiff and the lead rings on it. In between, near the 228 Ω where the resonance is critically damped, a ten-to-one probe with a clip lead shows an edge truthfully.

The probe that was made better, and rings for it #

The ten-to-one probe exists to put less capacitance on the node: 11.5 pF instead of the one-to-one probe’s 115. The probe that takes a tenth priced that in noise. The ground lead prices it a second way, because the damping ratio contains the square root of the tip capacitance. A one-to-one probe on the same fifteen-centimetre lead resonates at 38.3 MHz, with $\zeta = 0.692$ at a 50 Ω source, and by the same second-order law that the solved steps above follow, it overshoots by 4.91 per cent. The ten-to-one probe, with a tenth of the capacitance, resonates at 121 MHz with $\zeta = 0.219$ and overshoots by 49.4 per cent.

So the improvement that raised the probe’s bandwidth by ten raised its resonance by $\sqrt{10}$ and lowered its damping by $\sqrt{10}$, and the critical source resistance went from 72.2 Ω to 228. A one-to-one probe on a 50 Ω node with a clip lead shows a modest overshoot that a user might believe; a ten-to-one probe on the same node shows a ring a user will notice. That is not a reason to prefer the one-to-one probe, whose capacitance slows every edge it touches; it is a reason why the ten-to-one probe’s other half — the short ground — is not optional. A low-capacitance tip and a long ground are a high-Q resonator, and the tip capacitance a probe maker works to lower is the capacitance that keeps the lead’s resonance damped.

The lead’s length #

The lead’s inductance is roughly a nanohenry per millimetre of loop, so the length of the clip lead is the knob a user actually has.

Shortening the lead does two things at once. It raises the ring’s frequency, as $1/\sqrt L$: from 85.7 MHz with 300 nH of lead to 664 MHz with the 5 nH of a spring tip pressed against a ground pad beside the node. And it raises the damping, as $1/\sqrt L$ in the other direction, since $\zeta = (R_s/2)\sqrt{C/L}$: at 50 Ω the overshoot falls from 61.1 per cent with 300 nH to 9.5 with 20 nH to none at all with 5 nH. The spring tip works twice — it damps the ring with the source the probe was already loading, and it pushes whatever ring is left above the bandwidth of most of the instruments it is used with, so that the instrument’s own rise time filters it out before it reaches the screen.

That second effect is worth being careful with. A 200 MHz instrument looking at a 121 MHz ring shows most of it. The same instrument looking at a 664 MHz ring shows a little of it, smeared into its own step response; a 2 GHz instrument shows it all. A ground lead that looks harmless on a slow oscilloscope can become the dominant artefact on a fast one, which is the reverse of what a faster instrument is bought for.

A source resistance below which the probe rings #

The critical value is the number to remember, and it depends on the lead.

Bisected on the solved steps — the source resistance at which the overshoot first exceeds a part in a thousand — the boundary is 38 Ω with a spring tip, 76 Ω with 20 nH of lead, 208 Ω with a fifteen-centimetre clip and 294 Ω with thirty. The second-order law reaches a thousandth of overshoot at a damping of 0.83, so its prediction is $1.66\sqrt{L/C}$, and the bisected values lie on it to within five per cent.

The figure’s horizontal line is a 50 Ω node, which is the ordinary node in fast digital and radio work — a terminated transmission line, the output of a matched driver, the input of a matched receiver. Terminated at both ends and its neighbours spent several essays getting such nodes to show clean edges. Every lead of ten nanohenries or more — a centimetre of wire — rings on them. The clean edge that a termination buys cannot be seen with a clip lead, and an engineer who sees ringing on a properly terminated line and starts adjusting the termination is adjusting the wrong circuit.

Damping it at the tip #

When a short ground is not available, the remedy is to supply the damping the source lacks. A resistor between the node and the probe’s tip is in series with the resonant loop, and it adds to the source’s resistance as a plain sum.

The figure measures both sides of the trade. With no tip resistor the screen shows 49.4 per cent of overshoot and a 1.60 ns rise; with 100 Ω, 6.48 per cent and 2.60 ns; with 150 Ω, 0.334 per cent and 3.65 ns; with 200 Ω, none and 5.00 ns. The overshoot follows the second-order law with $R_s + R_{tip}$ as the damping resistance, because that is all the tip resistor is. The rise time grows because the tip capacitance now charges through the tip resistor as well as the source, so the resistor that removes the ring slows the edge it was meant to reveal.

Probes built for high-speed work carry such a resistor in the tip, chosen by the maker for a source impedance the maker had to guess. The figure says what that guess costs on a different source: a tip resistor right for 50 Ω over-damps a 10 Ω node only slightly, and on a 200 Ω node it adds two hundred ohms of resistance in series with the probe for no benefit, slowing the edge further. A divider with two ratios found that a probe’s compensation trimmer is set for one cable and one input; the tip resistor is the same kind of setting, made for one source.

The ring as a measurement #

There is a constructive reading of the artefact, and it follows from the same two formulas. A ring whose frequency and decay can be read off the screen carries two numbers, and the second-order model turns them into the lead’s inductance and the node’s resistance. The frequency gives $L = 1/((2\pi f)^2C)$: a ring at 121 MHz on an 11.5 pF tip is 150 nH of lead. The decay gives the damping — the ratio of successive peaks is $\exp(-2\pi\zeta/\sqrt{1 - \zeta^2})$ — and the damping gives the source: $R_s = 2\zeta\sqrt{L/C}$. A 50 Ω node shows successive peaks falling by a factor of 4.1 per cycle at $\zeta = 0.219$, and the factor read off a screen recovers the node’s resistance at the frequency of the ring.

That is a measurement of the node’s output impedance at a hundred megahertz made with nothing but a probe and its worst habit. It is crude — the model assumes the node’s impedance is a resistance at the ring’s frequency, which a driver’s output often is not — but it is a measurement the clean ten-to-one probe with a spring tip cannot make, because a probe that does not ring tells nothing about what it is not ringing against. The best damping is not the one to build found a second-order system’s settling to be exquisitely sensitive to its damping near the optimum; here the same sensitivity is the useful property, because a small change in the source’s resistance is a visible change in the ring.

What the ring is, and is not #

The ring is a property of the measurement, not of the node, and that has a practical test: change the ground connection and see whether the ring changes. If it moves in frequency with the lead’s length and in size with the source’s impedance, it is the probe. Where the current comes back found that a return current takes the path of least inductance, and a ground lead is the path the probe’s own return current is given — so the lead’s inductance is in the probe’s loop and nowhere else, which is why the node itself shows nothing.

Two simplifications are worth naming. The lead is treated as a pure inductance, with no resistance and no capacitance to anything; at a few hundred megahertz a fifteen-centimetre wire is a small fraction of a wavelength, so the lumped model is reasonable, but at the gigahertz frequencies of a spring tip’s ring it is not, and Kirchhoff’s own frequency is where that boundary is drawn. And the lead’s inductance is treated as fixed, where a real clip lead’s depends on how it is draped: a loop held flat against the board has less inductance than one held up in the air, and the ring moves as the lead does.

The numbers #

A ten-to-one probe with 11.5 pF at its tip and 150 nH of ground lead resonates at 121 MHz, damped only by the source: $\zeta = (R_s/2)\sqrt{C/L}$. On a clean step it shows 87.1 per cent of overshoot from 10 Ω, 49.4 from 50 Ω, 6.48 from 150 Ω and none above 228 Ω, and the solved steps follow the second-order law at every source. A spring tip’s 5 nH rings at 664 MHz and not at all from 50 Ω. The source resistance below which the probe shows its own ring is $1.66\sqrt{L/C}$ — 38 Ω for a spring tip, 208 Ω for a clip lead — so every 50 Ω node rings on a clip. A 150 Ω tip resistor brings a clip lead under 2 per cent at a 50 Ω source, and slows the displayed rise from 1.60 ns to 3.65.

Still open: the lead and the instrument together, a lead that couples, and the differential probe #

The lead and the instrument’s own bandwidth. The ring the screen shows is the probe’s resonance filtered by the instrument’s response. Composing the two — the second-order ring against the instrument’s own poles, which the instrument’s own rise time modelled — would give the displayed overshoot as a function of lead length and instrument bandwidth together, and say which instruments make a clip lead look harmless.

A lead that picks up as well as rings. A loop of lead is also an antenna for the magnetic field of whatever current is switching nearby, and the voltage it picks up appears in series with the probe’s reading. The pick-up scales with the loop’s area as the inductance does, so the two artefacts grow together; separating them on a bench needs the probe shorted to its own ground at the node, which reads the pick-up alone.

The differential probe. A probe that reads two tips against each other has no ground lead in its signal path, and its return current flows through the circuit under test rather than through a wire. Whether its input capacitance and its two tips’ lead inductances still make a resonance, and with what damping, is the comparison that would say when a differential probe is worth its price at a single-ended node.