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In the application note https://www.analog.com/media/en/technical-documentation/application-notes/an118fb.pdf there is a MOSFET used for linear control of an oscillator. The MOSFET has Rg = 10k. Why is the value so high? How does it affect stability?
The \$10k\Omega\$ resistor in series with the gate of the FET is likely there to limit the maximum current drawn from the op-amp and to isolate the FET input capacitance from the op-amp.
Although designated a rail-to-rail op-amp (both input and output), the load current must be small enough to allow approaching the rails.
The op-amp has two input branches, one for dc feedback and one for ac feedback.
The op-amp is configured as an integrator to eliminate dc steady state error of the dc HV output.
The gate capacitance (Ciss) is approximately \$1nF\$. This ground connected load capacitor can destabilize the op-amp, so the series resistor provides isolation.
The ac feedback is designed to syncronously pulse the oscillator twice per cycle to maintain oscillation. The ripple from the bridge rectifier is twice the oscillation frequency and has several phase shifting elements on its path to the inductor. The input RC network provides a phase lead, while the others are phase lags.
The gate resistor together with Ciss creates a low pass filter with a corner of about 16KHz. The oscilloscope image from the application note indicates an oscillation frequency of about 38kHz. This puts the 76kHz ripple frequency well above the 16kHz corner introducing asignificant phase lag.
To summarize, the \$10k\Omega\$ resistor:
