Circuit Response
every model has an edge, and the figure carries it
Field

Measurement, which is a circuit on a circuit

An instrument is not an observer; it is an element, it goes in the netlist, and every reading is a reading of the circuit that includes it. A probe is a capacitance with a bandwidth. A divider with capacitance in it has two ratios and one equation that makes them agree. Two terminals measure the leads as well, and below ten ohms that is most of the answer.
-6-4-201101001k10k100k1M10M100Mfrequency (hertz)log₁₀ of the error in the reading1× probe, 115.0 pF at the tip10× probe, 11.5 pFone per cent1× is 1% out at 6.79 kHz10× at 69.2 kHzsolved, then checked — the node with and without the probe1% wrong at 6.79 kHz with a 1× probe

The probe is part of the circuit

A one-to-one oscilloscope probe on a two-kilohm source gives a reading that is one per cent wrong at 6.8 kHz. Not because the instrument is inaccurate — it is reading correctly — but because the hundred and fifteen picofarads at the end of the cable are across the node, and above that frequency the trace on the screen is a picture of a circuit that only exists while the probe is attached.

 00.0500.10002.5057.5010time, in network time constantsoutput ÷ inputboth ratios: 0.1000the divider9M12.8 pF1Msolved, then checked — two ratios, one network0.0% out at 12.8 pF

A divider with two ratios

Put capacitance in a resistive divider and it divides by resistance at direct current and by capacitance at high frequency, and those are two different numbers unless one equation holds. The adjustable trimmer on every oscilloscope probe exists for that single equation, and the square wave on the instrument's front panel is a display of which of the two ratios is currently winning.

 -6-4-201m10m100m1101001kresistance being measured (ohms)log₁₀ of the reading's errortwo wiresfour wiresone per centtwo wires are 1% out at 10 Ωsolved, then checked — two arrangements, not two formulastwo wires fail below 10 Ω

Two terminals measure the leads as well

Fifty milliohms in each lead makes a two-wire measurement one per cent high at ten ohms, ten per cent high at one ohm, and a hundred per cent high at a tenth. Not approximately — the reading is the resistance plus the leads, and below about ten ohms most of what is being reported is the wire between the instrument and the thing.

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