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You are correct. The AMT5000 does not have a filter other than the series loading coil. Spectral analysis shows a very small spike at the second harmonic, but it is 43 dB below the carrier. All other points in the spectrum are well below that. Since the FCC rules allow 20 dB spurious emissions, I chose not to add any additional filter components that would add some losses.
The circuit does use two capacitors in the RF output circuit: a 470 pF cap across the MOSFET for optimizing the Class E waveforms at the MOSFET drain and a .047 uF series coupling capacitor. Both are C0G (NP0) ceramic capacitors. C0G ceramics are the lowest loss capacitors. They have a dissipation factor of 0.1% compared to 0.3% for Mica, but they are not typically stocked by distributors in voltage ratings above 100 V. The ESRs at 1600 kHz for the two caps I mentioned are 0.212 ohm for the 470 pF and 0.002 ohm for the .047 uF cap.
The voltage before any series loading coil in a transmitter are low and 50 V C0G ceramics would be your best choice for lowest loss, but you also have to deal with inductor loss in the filter circuit, which likely will be the higher contributor to loss.
Another factor to consider for low-pass filters is that they are designed for certain input and output impedances. The resistive impedance at the input of the series loading coil can range from maybe 10 ohms up to maybe 100 ohms, depending mostly on the antenna ground resistance but also on the loss resistance of the loading coil. Designing the filter for a “standard” 50 ohm load value may cause problems at the high and low resistance extremes of the tuned antenna. I don’t have much experience running the equations for low pass filters, so I don’t really know how sensitive they are to the output impedance. I’m just pointing it out as a possible issue.
Spurious filtering performance of the series-tuned coil/antenna gets better as the coil and ground loss resistances drop (Q is higher) and gets a little worse at 100 ohms, but is still better than 40 dB. The resistance would have to go up to somewhere in the 500 to 1000 ohm range before the spurious filtering would hit 20 dB. That wouldn’t even be a usable antenna.