Generally speaking, you are applying HV AC to the kanthal rod and looking for the rectification property of the flame. This is thermionic.
Excitation is around 50-100VAC and very low current under 10uA. Older equipment used mains sine-wave and modern gear uses a square-wave driven by MCU as safety codes require a lot of self-test for flame detection circuit.
The rod must not have soot etc. buildup which is easily cleaned off with steel wool. Also, any leakage current to ground due to rust or soot, or shoddy/melted/wet wiring or cracked porcelain can be a problem. Note this leakage does not count as a false-positive for flame present because it's just an ohmic resistance seen and not rectification.
If you're having problems, check the rod is clean with no leakage current path, and the pilot flame is centered on it. Not to the side or lifting up off the burner, which happens if the pilot burner is dirty or backdraft or chimney/negative pressure in the building issue. The flame detector can be working fine but if the flame is poor, flickering, off to the side of the rod etc. due to ash/soot buildup, it will constantly trip out.
If you are very careful and need to spoof flame-present, then use a 1N4007+10MEG series resistor switched in at the moment when the controller needs to see flame, and of course manually light the pilot. I do this testing burner firmware. It's a dangerous procedure.
There's two types, as far as a single rod - flame sensor, or igniter+flame sense function. If it's the combo function, the igniter will put many kV at the flame rod and the flame-sense circuit portion has clamp diodes so it does not get damaged during ignition.