The answer to most of these, is: dampen resonances of the inductor.
Play around with this, for example; read the supporting information:
http://hamwaves.com/antennas/inductance.htmlIt uses a helical waveguide model, and a bunch of corrections, to estimate the voltage and current through a single layer, wire solenoid. Apparently this geometry is remarkably difficult to analyze!
If the calculation succeeds (there is some root-finding that can fail, particularly at small pitch angles), you get the complex impedance of the element -- resonant modes accounted for!
The first (parallel) resonant mode is fine, that's what we want -- a high impedance between DC and EUT sides. What we don't want is the second (series) resonant mode, which acts in parallel with EUT, shunting signal and distorting the frequency response.
Knowing that standing waves are the culprit, we can employ resistors in strategic locations to dampen those modes. The series resonance has an antinode in the middle (voltage peak), and nodes (current peak) at the ends, which is why we measure a low impedance at the ends, at that frequency. A resistor from one end to the middle will act in parallel with that mode, dampening it. The resistance acts in parallel with the voltage peak, and gets 1/4-wave-transformed into a series equivalent resistance at the terminal. When R = Zo,* parallel R is transformed to series R, and that's simply the resistance you will measure at the port.
*For an ideal transmission line, Zo is simply Zo. Helical waveguide however is dispersive (velocity varies with frequency), meaning the Zo varies for each resonance; the resonances also aren't harmonically related. So it's not quite as simple as knowing the wire length of the coil (though that's close for the 1st mode, I think?), and it may be worth playing with the resonances in a calculator such as above, or testing real hardware.
[Note that helical waveguide can have quite high impedances, low kohms -- this makes them useful for bias tees for one, but also useful for delay lines in certain applications.
A vintage application was color TV sets: the impedance must be high (low kohms) to suit to the vacuum tubes used in early sets. The delay is applied to the luma signal (which is simply detected directly from the radio signal), to "catch it up" with the chroma signal, which ends up delayed due to additional filtering and processing stages. The required delay was about a microsecond. A typical delay line was a phenolic tube, wound with fine wire, and lined with a strip of foil -- not a complete wrap-around foil lining, that would make a shorted turn, defeating the helical mode; just a narrow strip to give some ground reference for the travelling wave.]
We can also employ loading materials; if we have a lossy ferrite or powdered iron material, we can use it to both increase the inductance (reducing the number of turns required, potentially raising the resonant frequency) and dampen the resonance (by magnetically coupling material losses to the resonances).
The loss has to be appropriate, of course; a high-mu powdered iron will be too conductive, and actually increase the capacitance more than dampen modes. A low-mu ferrite might have too high of a Q at these frequencies, and just not do much, or maybe make things worse. It may take some trial and error to perfect.
I haven't done this specifically for solenoids, but I have plotted a few resonances on a 100% coverage, single layer toroid winding. Here, the boundary condition forces the first mode to be a full wave, and it is series resonant. Again, the resonant impedance is quite high, which makes it difficult to dampen at the terminals -- if we're using this toroid for a current transformer, we can't afford much burden resistance. Maybe the resistor needs to be a few ohms for the application, but the correct value for damping would be 300 ohms, or a few kohm even -- highly impractical to dampen.
Knowing that it is a resonant mode, however, it can be shunted by simply distributing the applied current around the core. Instead of using one loop in one place, use two loops, wired in parallel but positioned in quadrature (at 90 degrees to each other on the toroid). Or hexature, etc. (I was using a high-mu toroid, which is rather lossy at the 10s of MHz these modes were showing up, so the 4th mode was already very weak; I didn't try looking any higher.)
Anyway, what we're after, is knocking out series resonances (impedance dips at the EUT/RF port) and replacing them with resistances, hopefully resistances that are high enough not to worry about (i.e., R >> Zo).
If we have discrete inductors that manifest as lumped single RLC networks (i.e., effectively some simple equivalent parallel capacitance, without having ugly modes at higher frequencies), we can employ the same methods, applied to a lumped-constant circuit, without having to worry about waves and modes necessarily. Here's a bias tee by Picotronics, flat to, as their name suggests, some picoseconds (many GHz):
https://www.seventransistorlabs.com/Images/Picotronics/MVC-349X.JPGhttps://www.seventransistorlabs.com/Images/Picotronics/Schematic.JPGIt seems they opted for the parallel damping resistance instead. I think that can save a little on impedance -- you don't have the lossy cap-to-ground loading it down. Which one is best probably depends on the inductors used, and desired flatness.
Don't forget, too -- inductors have capacitance between their terminals, and to ground or free space; a 2-terminal model of an inductor or capacitor can be erroneous at these frequencies. This may also be a factor.
As for some other questions -- air core is preferred over ferrite because ferrite saturates. Rod cores aren't too bad here, as they can be made with quite high saturation currents, and the advantage is small anyway (mu_eff maybe only 2-5 say), enough to be helpful without getting in the way.
Saturation current should be quite high indeed for a mains LISN or CDN -- loads with poor power factor can draw quite high current peaks, and those current peaks are also likely where most of your EMI is transmitted (the FWB acts as a PIN diode), so you don't want your network pooping out on the peaks!
Tim