High speed passive probe - contradiction between authors or different points of view?

In a document Hiscocks et al. describes some basics of Oscilloscope probe theory. The document is very understandable and seems coherent. Notice in particular that for him, the bad guy is the parallel capacitance of the coaxial cable and of the oscilloscope that should be compensated by adding a capacitance in parallel to the tip of the probe (so, the capacitance of the tip is increased).

Then comes d. smith with his method to build a 1 GHz passive probe. First, it is not entirely clear why he terminates his probe by a 50 ohm resistance: to avoid reflections, isn't it sufficient that one side of the probe (that is the oscilloscope side) be terminated by a 50 ohm resistance? I presume that this is to kill even more the reflections. So, let it be. But what is strange for me is that he does not take into account the capacitance of the cable, nor the capacitance of the oscilloscope. In particular, for him, the beast that has to be killed is the tip capacitance (so he increases the parallel capacitance of the cable), the exact converse of what says Hiscoks in the document above. If this man were a newbie, I would say that he does not understand why his probe works, and that he actually increases the capacitance of the tip with his copper foil. But hey! this man is a guru of probes that published several articles in different journals.

And now the best of the best, The Art of Electronics, 12.2 p. 808: to do a high speed passive probe? very simple:

... and make your own by hooking a series resistor (we like 950 ohm) onto a length of skinny 50 ohm coax (we like RG-178); you temporarily solder the coax shield to a nearby ground, plug the other end into the scope (set for 50 ohm input) and voila - a high speed 20 x probe!.

If my understanding is right, the 950 ohm resistor with the 50 ohm characteristic impedance of the cable make a 1:20 resistor divider (up to now OK), but what about probe compensation etc.? uh!

Can someone tell me what is going on?

For 100 MHz and slower probes, the wavelength of the signals in question is long enough that the cable doesn't really act like a transmission line and the probe tip pretty much directly 'sees' the input impedance of the scope. Also, the probe impedance and scope input impedance do not match the cable characteristic impedance. In this case, the capacitance is really the main thing that needs to be controlled and compensated for. This is described in the Hiscocks et al. document.

At high frequencies, the cable acts like a transmission line and the probe tip doesn't see the scope input impedance directly. Instead, the probe tip sees the cable's characteristic impedance. Usually for high frequency probes, standard 50 ohm RF design techniques are used. Everything just gets matched to 50 ohms - both the scope input and the probe tip.

As for the difference between d. smith and art of electronics, they're basically trying to do more or less the same thing. d. smith adds a parallel resistance to ground to form one side of a voltage divider to produce a ~40:1 probe. That 50 ohm resistance appears in parallel with the 50 ohm cable for an equivalent 25 ohm resistance. This then forms a voltage divider with the 976 ohm series resistor. Apparently the tip capacitance of his probe is high enough that extra compensation was required to get a flat frequency response. Note that this resistor isn't really necessary as a termination resistor--presuming the other end of the line (at the scope) is properly terminated into 50 ohms, then there should be no reflections coming back up the cable that could reflect off of an impedance mismatch at the probe head.

The art of electronics design does the same thing, but only uses the cable's characteristic impedance as one side of the voltage divider. In combination with a 950 ohm series resistor, this produces a 20:1 probe. This probably works 'well enough' up to reasonably high frequencies without additional compensation if the right resistor is used, but I presume you could do a little better if you add a properly-sized capacitor to ground between the 950 ohm resistor and the coax cable. The attenuation of the art of electronics design is also lower than the d. smith design, which likely makes the mismatch in capacitance less of a problem. In general, I think the art of electronics design is really intended to be a quick-and-dirty technique that works well enough for debugging but could be improved upon if more accuracy is required.

• The best answer, but the reader is invited to have a look at the answer of Jasen (and comments) below to understand the question in deep and complete this answer. Commented Jan 22, 2018 at 10:09

Indeed the Hiscocks document is quite clear, 9 M series resistance in probe, 1 M to ground in scope. Add capacitors in parallel so that for high frequencies the 10:1 ratio is maintained. That all makes sense.

A good 10:1 probe made like this can achieve up to 300 MHz of bandwidth I believe.

The other solutions try to achieve a higher BW (bandwidth). Then the first limitation we need to get rid of (compared to the standard 10:1 probe) is the probe cable. The cable used for 10:1 probes is the limiting factor for the BW. We need to use a high BW cable and those have almost always a 50 ohms characteristic impedance, like the RG-178. To be able to use that BW that lenght of cable must be terminated on both sides with 50 ohms. That makes the cable a transmission line.

Both D. Smith and the Arts of Electronics use this transmission line as their basis. Note that the 50 ohm termination resistor usually sits inside the oscilloscope (you have to change a setting on the scope), if it does not have such a setting you have to add the 50 ohm yourself some way.

To couple into that 50 ohm transmission line both use a resistor with an optional capacitor. The Arts of Electronics are apparently already happy with the BW they get. Note how they mainly talk about those digital signals having a nice shape!

Also, since the transmission line behaves as a 50 ohms impedance without much capacitance you would not "see" all of the RG-178's capacitance at the input. So you would only need a very small capacitance across the 950 ohm resistor to get proper frequency compensation.

• +1 for the answer. At the end, you entirely rely upon the theory exposed in Hiscocks; but Jasen in his answer above says that the capacitance of the cable is cancelled by its inductance. Who is right? Commented Jan 22, 2018 at 9:30
• You can cancel a capacitance with an inductance but that only works at a certain frequency where the L and C resonate. A transmission line can be seen as a distributed LC network, now that I think about that, since the T-line is 50 ohms in, you would not "see" the full capacitance of the cable so the last paragraph in my answer needs an edit. Commented Jan 22, 2018 at 10:18

Probe compensation is needed when you have a scope with 1 megaohm impedance

When the scope and cable's impedance match there's nothing to compensate. The cable is a transmission line and the inductance of the cable cancels the effect of its capacitance.

The reason why most scopes dont have have 50 ohm probles is that it puts a signifcant load on the circuit being measured, and care would be needed to not cause undesired operation just by connecting the probe. with a high impedance probe you can probe the circuit with less disturbance.

Smith terminates both ends of his coaxial cable I'm not sure what he's getting from that, and then needs to compensate the capacitance of his termination, I'm not sure that he's gaining anything.

The Art of Electronics, has been proof-read by many experts and is well regarded

• So, what about the probe of d. Smith? Also, can you explain mathematically why this is true? Commented Jan 22, 2018 at 8:24
• maybe he wants a 2:1 probe? Commented Jan 22, 2018 at 8:25
• I think this is a 1:40 probe. Commented Jan 22, 2018 at 8:28
• probe compenation is needed when you have a scope with 1 megaohm impedance I would add to that: ...and are using a 10:1 probe That excludes the 1M ohm 1:1 probes which have crap bandwidth anyway! Commented Jan 22, 2018 at 8:29
• @MikeTeX look up "characteristic impedance" Commented Jan 22, 2018 at 9:29