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I am going to build a small low power AM (broadcast band) transmitter, I was thinking if I vary the number of turns on the coil can I get it to transmit at different frequencies, or will their have to be more involved modifications?
You can vary the "C" with a variable capacitor. The "C" can be a single variable - or a fixed capacitor and smaller variable in parallel. Sometimes there will be a larger variable as sort of a coarse tune and a smaller variable in parallel as a fine tune.
The "L" can be varied any number of ways. Physically deforming the coil will change it somewhat. Coils with multiple taps work. You might also use a coil with a slug or core that can be moved in or out - slug tuned is one way of describing it.
Keep in mind that you are responsible for whatever signals you emit. In many countries low power broadcasting (within the rules) is allowed but only within the broadcast band. If you tune and operate your transmitter out of band you've violated the rules - even if you aren't aware of it.
My only AM BCB transmitter I've made was the "Wireless Rebroadcaster" from Popular Electronics back around 1967 or so, using a single 12AT7 dual triode. Interesting that I've never considered building a solid-state model.
Dean...why does it say RIP in your profile?...mkes you sound like you are dead....but if yoou are posting you must be undead