In the FCC documentation we can see a photo of the Swift coil (
link)... it appears to be ~ 17 turns around a ferrite. I estimate the ferrite to be approximately 6.35mm diameter by 35mm length and the 'magnet wire' to be approximately 24 AWG (0.5mm dia), coil length around 8.5mm.
Been a very long time since I did any RF stuff so be warned I may be completely wrong but...
To get a high ‘Q’ factor in the 1 KHz to 1 MHz frequency range it is likely the ferrite is MnZn (Mix 31, 73, 75) with high permeability above 800 µ (plus fairly low volume resistivity and moderate saturation flux density).
Looking at available ferrites something like this might do
digikey (Fair-Rite 4077266011) or
Mouser (Fair-Rite 4077276011) . Putting these into an online calculator [
link] (Rod length 38mm; 6.35mm dia; Initial magnetic permeability 2000; Wire dia AWG-24; turns 17) gives an inductance of around 15 μH. Another calculator [
link] states at 38 kHz an inductive reactance XL= 3.58 kΩ.
For a resonant LC circuit, yet more calculators [
link] [5] suggests a 1.1 µF parallel capacitor to achieve resonance at around 39 kHz. with a characteristic impedance of 4 Ohms.
Because of the low duty cycle it might be possible to drive it push-pull from two output port pins (4 Ohm load!), or maybe use an op-amp arrangement per attached image. For sure some playing around will be required to optimise power coupling into the coil.