Oscilloscope probe loading and ground-lead ringing calculator

A probe's ground lead and its tip capacitance form a series resonance that rings on fast edges — an 11 pF probe with the standard 6-inch clip lead rings at about 140 MHz, hidden on a 100 MHz scope and plain on a 200 MHz one. The same tip capacitance loads the node it touches, and the scope and probe each add their own 0.35/BW rise time to whatever is displayed. Enter the probe, the lead, the node and the scope to get the ring frequency against the system bandwidth, the lead length that hides it, the loading on the node, and the rise time the screen will show for the edge you are actually measuring.

Fig 1 — Ground-lead ringing for 11.0 pF of tip capacitance at 0.77 nH/mm: a 152 mm lead rings at 140 MHz, inside the 166 MHz of the probe and scope together. Tek's rule is to put the ringing above the bandwidth, which on this chart means below the line — a shorter lead.

Probe tip capacitance (pF)

Probe input resistance (MΩ)

Ground lead length (mm)

Lead inductance (0 = Tek-derived 0.77) (nH/mm)

Source impedance of the node (Ω)

Signal rise time (ns)

Scope bandwidth (MHz)

Probe bandwidth (MHz)

Ground lead inductance · ring frequency: 117 nH · 140 MHz — inside the bandwidth, visible
Lead for the ring to clear the bandwidth: 108 mm
Rise time: scope · probe · together: 1.75 ns · 1.17 ns · 2.10 ns (166 MHz)
Capacitive loading on the node: corner · added rise: 289 MHz · 1.21 ns
Displayed rise for a 5.00 ns edge: 5.56 ns (+11.2 %)
Signal rise ÷ system rise (Tek: 3 to 5): 2.4× — too close
Resistive loading, DC: 100.000 % of the signal

A 152 mm lead rings at 140 MHz, which the 166 MHz system shows. Tek's rule is a lead short enough that the ringing sits above the bandwidth: under 108 mm here, which is a ground spring, not a clip. And Tek's test: if moving the cable or a hand near it changes the aberrations, they are the probe's, not the signal's.

The system's 2.10 ns is only 2.4× faster than the 5.00 ns edge; Tek asks for three to five. The displayed 5.56 ns is the instrument as much as the signal.

11.0 pF on a 50.0 Ω node adds 1.21 ns to the node's own edge — the probe is changing the circuit, not just the display. Keysight's check: touch a second probe to the point and watch the trace move.

How this is calculated

Standard: Tektronix ABCs of Probes; Keysight 5989-7894

$f_{ring} = \frac{1}{2\pi\sqrt{L_{lead} C_{tip}}}$: Tek §4: the ground lead and the tip capacitance as a series resonance. Ringing above the system bandwidth is not displayed.
$L_{lead} \approx 0.77\ \text{nH/mm} \times \ell$: Derived from Tek's "11 pF with 6-inch ground lead will ring at about 140 MHz": 117 nH over 152 mm. Editable.
$T_r = \frac{0.35}{BW}$: Tek §1, for the scope and for the probe.
$T_{r,sys} = \sqrt{T_{r,scope}^2 + T_{r,probe}^2}, \qquad T_{r,shown} = \sqrt{T_{r,sig}^2 + T_{r,sys}^2 + (2.2 R_s C_{tip})^2}$: Rise times in quadrature; Tek wants T_r,sys three to five times shorter than the pulse.
$f_{load} = \frac{1}{2\pi R_s C_{tip}}$: The RC the tip capacitance makes with the node's source impedance — Tek's "loading of greatest concern".
$\frac{V_{probed}}{V} = \frac{R_p}{R_s + R_p}$: Resistive loading, Tek's battery-divider example.

Assumptions

The ground lead is a straight wire of the entered length at the entered inductance per millimetre; loop shape changes the inductance by up to about a factor of two.
Bandwidths convert to rise times as 0.35/BW and combine in quadrature, which Tek notes is an approximation; a manufacturer's system figure "to the probe tip" is better where it exists.
The probe is compensated. Compensation errors are not modelled.
The node is a resistance in series with the signal; a transmission line or an inductive node is outside the model.
The resonance is treated as undamped for its frequency; the amplitude and decay of the ringing depend on the probe's input resistance and are not computed.

What sets what a probe does to the signal

A probe is not a wire. Tektronix's primer opens its physics with the point: a probe "must draw some signal current in order to develop a signal voltage at the oscilloscope input", so it loads the node, and it is "circuits composed of distributed resistance, inductance, and capacitance", so it has a bandwidth and a resonance of its own. Three things follow, and the calculator computes each.

Loading. The input resistance — 1 MΩ for a 1X probe, 10 MΩ for a 10X — divides against the source, which Tek's 100 kΩ example shows to be a 0.1 % matter. "Usually, the loading of greatest concern is that caused by the capacitance at the probe tip": at frequency the tip's 8–13 pF (for a 10X passive) is a reactance in parallel with the node, and with the node's source impedance it is an RC that slows the node's own edge. Keysight's hint 2 is the practical test — "connect your second probe to the same point. Ideally, you should see no change on your signal. If you see a change, it is caused by the probe loading."

Ringing. The ground lead "has some amount of distributed inductance", and "this inductance interacts with the probe capacitance to cause ringing at a certain frequency that is determined by the L and C values" — a series resonance "with only R_in for damping". Tek gives the one number everything else scales from: "an 11 pF passive probe with 6-inch ground lead will ring at about 140 MHz", which is 117 nH for the lead, about 0.77 nH per millimetre. The remedy is not to remove the resonance but to move it: "the effects of ringing can be reduced by designing probe grounding so that the ringing frequency occurs beyond the bandwidth limit of the probe/oscilloscope system." On a 100 MHz scope the 140 MHz ring "may not be seen at all"; on a 200 MHz scope it "will be apparent".

Rise time. "T_r = 0.35/BW", the probe and the scope each have one, and "when a probe is attached to an oscilloscope, you get a new set of system bandwidth and rise time limits". For a rise time measurement Tek wants the system "three to five times faster than that of the pulse being measured". The calculator adds the scope, the probe, the loading and the signal in quadrature to show what the screen will display, and how far from the truth that is.

Worked example: Tek's 11 pF probe, 6-inch lead, 200 MHz scope

The calculator's defaults: an 11 pF, 10 MΩ, 300 MHz passive probe with its standard 152 mm clip lead on a 200 MHz scope, probing a 5 ns edge from a 50 Ω source.

lead inductance   152 mm × 0.77 nH/mm                 = 117 nH
ring frequency    1 / (2π √(117 nH × 11 pF))          = 140 MHz   (Tek: "about 140 MHz")
system rise       √(0.35/200 MHz)² + (0.35/300 MHz)²  = √(1.75² + 1.17²) = 2.1 ns  → 166 MHz
                  140 MHz < 166 MHz                     → the ringing is on the screen
lead to hide it   L = 1/((2π × 166 MHz)² × 11 pF) = 84 nH → 108 mm; 20 mm spring: 490 MHz
loading           2.2 × 50 Ω × 11 pF                  = 1.2 ns added to the node's edge; corner 289 MHz
displayed rise    √(5² + 1.75² + 1.17² + 1.2²)        = 5.5 ns   (+11 %)
ratio             5 ns / 2.1 ns                       = 2.4×     (Tek wants 3–5×)

Three separate things are wrong with this measurement and none of them is the signal. The ringing on the edge is the ground lead, which Tek's test confirms in a second — "move the probe cable around. If placing your hand over the probe or moving the cable causes a change in the aberrations, the aberrations are being caused by the probe grounding system." The 5.5 ns is 11 % slow because the instrument is too close to the signal's speed. And the node itself is slower with the probe on it than off. A ground spring and a faster scope fix the first two; Keysight's answer to the third is an active probe, whose "significantly lower capacitive loading" left its 600 ps test signal at 630 ps where the 500 MHz passive probe with a 15 cm alligator lead loaded it to 740 ps and displayed 1.4 ns "with resonance".

Where the probe model stops being valid

The lead inductance is a loop, not a length. 0.77 nH per millimetre is what Tek's single example implies for a clip lead hanging in the usual shape. A lead dressed tight against the board, or a wide loop, differs by a factor of two either way; measure the ring on a known edge and enter the per-millimetre figure that matches.
Rise times add in quadrature only for Gaussian-ish responses. Tek notes "the relationship between system bandwidth and the individual oscilloscope and probe bandwidths is not a simple one" and that manufacturers therefore specify bandwidth "to the probe tip" for named probe models. The quadrature sum is the estimate; the specified system figure wins where it exists.
Compensation is assumed correct. An under- or over-compensated 10X probe misreports the edge and the flat top before any of this applies; Tek's advice is to "always compensate probes right after connecting them to the oscilloscope".
Even the tip wire counts. Tek's figure 1.12: a two-inch wire soldered to a test point "changed [the rise time] from 4.74 ns to 5.67 ns". The model has no entry for it; the lead figure can be used for it.
Ground loops are a different problem. Hum and noise from the scope's earth to the board's ground are not the resonance here; Tek's chapter 6 covers them separately, and a supply-ripple measurement is the everyday case where the clip lead is both the resonance and the antenna.

Common probing mistakes

The long clip lead on a fast edge. It is the default accessory and the default cause of ringing. Tek: "always use the shortest ground lead provided with the probe."
Extending the ground with a clip lead to reach a chassis. Tek's figure 1.13 — a 28-inch lead added to the 6.5-inch clip — is a textbook ringing waveform.
Believing the overshoot. The signal may be clean; the test is whether a hand near the cable changes the display.
A 1X probe on a fast or high-impedance node. 100 pF at the tip (Tek's table) is ten times the loading of a 10X probe; the 1X setting is for small, slow signals.
Measuring a rise time with a scope of the same speed. The displayed figure is the quadrature sum; at 1:1 it reads √2 times the truth, and Tek's 350 MHz scope on a 1 ns step is exactly that case.