It is not surprising, that the large cap in parallel with the signal source reflects so much energy back and causes a high SWR. Simulate it in PSpice and see... The goal is to drive a high current through the pancake coils. In cases when the frequency is low or the coil has a low inductance (or both), it is straightforward to drive high AC current through this coil using a low-impedance high-current amplifier, such as the EL2009. In such cases, the coil’s impedance is low enough whereby it can be driven by an amplifier directly. However, the impedance of a coil increases with frequency as |Z|=√(R 2 + (2πfL) 2), so at high-frequency, the coil's impedance is assumed to be very high. See the impedance of my coil here as it reaches over 7kΩ. Thus, a high-voltage driver is needed to drive high current into this coil. These capacitors try to do that by swinging the front-end of your RF amplifier to high voltages. That's why you are measuring almost 200V at these coils' terminals (the current would be more interesting though, because it is the current which generates the magnetic field - the goal). Anyway, what SWR do you get when the secondary of that 1:1 balun is loaded only with a 50Ω dummy load ?
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