Am I correct in thinking that the 50 ohm impedance relationship in the RG174 is created by the core vs the position of the braid?
Yes, or more precisely (as long as it's coaxial), the ratio of diameters.
Consider the facing surfaces of the wires. This gives capacitance. A thinner wire has less capacitance, as does a greater separation between wires.
Now consider the space between wires. This allows room for magnetic field around them. More separation means higher inductance, as does thinner wire.
A coax cable made with (literally) hair-fine core and regular sized shield could achieve an impedance of a few hundred ohms. Average stuff, obviously, achieves 50-75 ohms. A very thin insulator with a heavy core could go down into the 10 ohm range (which starts to look not so much like coax, as, were you to take two strips of copper, a thin dielectric, and rolled that up).
A pair of wires in space have impedance to each other, which for typical insulated wire at the typical diameter of the insulation, this is around 75-120 ohms. A wide separation increases inductance and reduces capacitance, so that common "twin lead" is 300 ohms.
If you have no companion wire at all, but just wire traveling through "free" space, it works against the impedance of free space alone, which is 377 ohms; actual impedances still depend on resonance and geometry, so that an antenna (a wire sticking up with no adjacent ground pair) still exhibits a fairly low impedance (50-100 ohms for common types like 1/4 wave whip or 1/2 wave dipole) despite that.
Getting impedances much over 300 ohms is kind of hard (for the odd case where you need it). The conductors need to be very thin and distant, while also not coupling to free space (i.e., radiating), while also not resonating (you can get quite high impedances if you reflect the radiation with a shield... but then it only works at certain resonant frequencies!).
So, that's kind of a very gross qualitative touchy-feely look at transmission line impedance.
When you see advice like "minimize inductance", what they generally mean is not actually minimizing the lumped equivalent inductance of the structure, but reducing the impedance (but not minimizing -- at least, not except for very rare cases) so that the transmission line / lumped line / whatever equivalent more or less sort-of matches the circuit's desired characteristics.
In this case, since we're talking high resistance, we should also be talking high inductance -- but not to the point of actually using an inductor, because that forms a low pass filter with parasitic capacitances. Suppose you laid it out so that the transmission line sections are still 50 ohms; they'll look like capacitors, and act to resonate or attenuate the highest frequencies, rather than transmitting them proportionately. So the more pressing minimization, if there should be one, would be capacitance -- but true absolute minimization is rarely the ideal goal, and going for an optimal combination of inductance and capacitance leads to the best result.
Are you suggesting that the same thing needs to happen with the resistor in the picture and the ground lead coming along side it? How would you adjust it to obtain a certain impedance? Would that impedance be relation to the size of the resistor? Is Mark's message above about using two resistors to cut parasitic capacitance related to this?
Sort of. The thing about resistors is, there's a semi-conductive layer on the thing, which therefore acts as a transmission line against nearby conductive surfaces. That's line to ground action. But you also have the fact that the resistor is built from two conductors, and you have coupling between the faces of those conductors. The resistor is, in effect, shunted by a very short transmission line in parallel with its (DC) resistance. Because it's very short, it doesn't play much role at most any frequency, but it becomes more noticeable for larger R and higher F.
Using resistors in series effectively connects those parasitic electrode-to-electrode transmission lines in series, increasing the total series impedance. Which is good. But, each of those nodes between the resistors (in addition to the resistors themselves) have impedance to ground, so ultimately you're still not dividing it evenly, and you'll get weird rises and dips in the frequency response.
The low frequency story is, capacitance from between-resistors-node to ground causes a droop at some intermediate frequency. If you connect capacitors in parallel with the resistors, you can swamp this effect, and extend the high frequency range, at least until the chain of capacitors itself causes too much loading (or has too much ESR, ESL, weird transmission line effects, etc.) to do its job anymore.
The ultimate case, then, one should quite reasonably suspect -- there IS no such thing as a true ideal lumped resistor and capacitor divider, but it's all just restated transmission lines. And in that light, one should suppose the best result will be a 450 ohm transmission line dividing into a 50 ohm transmission line, or whatever -- which looks odd, because that 450 ohm line is connected "line to ground" from source to load, not "line to line". You can't really implement this, because of common mode rejection. But if you can approximate this in some series-parallel equivalent, including resistive materials, you'll be doing very well.
What I'd really love to do is make a compact style probe like this - where the RG174 ends and there are a couple of wires/leads coming off of it that have the resistance embedded in them and then have a nice pin for connecting them to a through hole probe point, breadboard, etc. Can someone explain to me what happens at this point where the RG174 stops and these two leads begin that would be awesome. I am assuming the shorter the better, how would 2" for the signal and 2" for the ground affect the probes capability?
Two things:
1. You get an impedance discontinuity. Which, if it's short enough for the frequencies of interest*, won't matter, but you will want to deal with it so you can match it correctly. Of course, I couldn't tell you offhand just what arrangement of resistors and transmission-line-ey action would be necessary.. you'd have to faff around quite a lot to get it just so. And then it loses its usefulness as a probe, because everything has to be just so: the probe wire positions, the cable, the ground, where it is over the circuit (and whether the circuit is ground-planed or not), etc. (I'd be happy to give it a try figuring it out for you, but I'm not cheap...
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*Which is undoubtedly the case -- by which I mean, if you're using RG-174 which is just awful stuff even for short lengths, for frequencies over 100MHz -- you won't have enough bandwidth at the far end of the cable to even care!
2. Even if you get the impedances and dividers and stuff laid out correctly so that the frequency response at the feed point (i.e., even ignoring cable losses) is correct, other reasons may pull weight first -- in particular, in the presence of common mode noise (easily present in digital and switching circuits, and careless or poorly terminated RF amps, and ambient noises, and... anything else), any non-coaxial connection is just asking for interference. Try to do a sensitive measurement and you'll just get a mess! See Linear Tech AN-47 for good probe ideas and examples.
Tim
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