Willy-nilly, you'll get maybe two decades of range.

To maximize bandwidth, you need to use a proper transmission line transformer.

Consider the ideal transmission line: energy goes in one end, and comes out the other (after a delay). There is only the differential mode, no common mode current between the two ports.

We can stack one port in series with the other, and thus get a 1:2 transformer. Problem: half the signal is immediate, half delayed. So it will have nulls at high frequency, at frequencies that are multiples of the delay period.

So let's simply delay the original signal too, and then stack them. We get a pair of TLs, one pair of ports in parallel at one end, the other ports in series at the other end. Done and done.

Now we have a transmission line transformer, that is wholly independent of frequency! The delays are matched, so the EM wave in the transmission line (from either side) doesn't know anything different is going on.

Which gives us the continuity relationship: for a 1:2 TLT, we must have Zout = 4*Zin, and Z(TL) = 2*Zin. For a 50 ohm input, we use 100 ohm transmission line, and get 200 ohms out. Two 100 ohm TLs in series makes 200 ohms, and two in parallel makes 50 ohms.

In reality, there is a common mode current, so we will not have infinite bandwidth. The LF cutoff can be extended arbitrarily far down, by putting more and more cores around the TLs.

The basic design uses either a TL wrapped around a toroid, or P, or U type core; or a stack of toroid cores on the line. The first has the downside that the TL is close to itself, between turns, which changes the impedance and velocity slightly (preferably, you should keep the turns well-spaced apart). The second has the downside that, aside from simply taking *a lot of cores* -- all those cores also introduce a lot of capacitance (ferrite has a fair dielectric constant), which either loads down the TL (reducing Zo), or introduces more common mode impedance than you might otherwise be expecting (particularly important to high impedance transformers).

But given those limitations, you can build a matched delay type (Guanella) TLT, with real materials, for 3 or 4 decades, pretty easily, without extreme consideration of materials, no algebra whatsoever*, no simulation.

*I'd say no math whatsoever, to the extent that what you're doing is more basic, like geometry. But geometry and math are pretty close buds so that would be a stretch.

To push further, you need very good cores. A few high frequency cores (usually from the #61 or #43 range, or similar) at each end will keep things happy near the top. A few low frequency cores (#77, W, etc. -- higher permeability, the better) will get you below a MHz. To go further, you need still better materials: amorphous or nanocrystalline material, or perhaps supermalloy.

"VLF to 2ghz" sounds, well it sounds doubly like you don't know what you want/need, honestly, but this is the principle required to approach that.

Tim