What determines spool up time?

[barit1]

The short answer:

At a given speed, the compressor can only pump a certain exit pressure. More precisely a certain pressure ratio Pout/Pin.

And since the turbine nozzle is a choke point, varying the temperature there will affect the pressure,which reflects back to the compressor Pout.

So, more fuel (more temperature) causes the compressor to work harder - and the compressor at any given point can only stand so much of this before it will burp, back-flow, surge, stall... Let alone the incremental hardware distress from thermal shock.

So - advancing the go-handle quickly DOES NOT give you unlimited fuel. You get enough to start acceleration, and the acceleration lets the MEC/PMC/FADEC know that now a bit more fuel is permissible, and the cycle continues until you reach TO or whatever desired thrust.