Why systems containing positive feedback loops will always be more prone to bouncing, wobbling, oscillating, jumping, vibrating, overshooting, and crashing than a passively stable design, no matter what you do to control them.
> The actual operation of control rods in a nuclear reactor is a bit more complicated than a standard linear feedback controller, because rather than just directly subtracting neutrons, the control rod position modulates the (average) gain of the reaction feedback loop itself, so they can shift the reaction from being intrinsically stable to intrinsically unstable.
Nuclear reactors are also frighteningly complicated control systems because of delayed neutrons. The classic image of a bomb-like chain reaction is one of "prompt criticality," where nuclear fission produces new neutrons near-instantly that (can) cause more fission reactions, and so on. The timescales of this reaction are so quick that physical control would be impossible — nuclear core detonations are faster than the speed of sound in the core, after all.
However, nuclear reactors are saved through delayed neutrons; a small fraction of neutrons flying around the core come not from the fission itself, but from the decay of short-lived fission products. This timescale is still relatively fast (a few fractions of a second), but it's slow enough for control systems (passive or active) to maintain the reactor in a stable state. The reactor is "prompt subcritical" with respect to the direct fission neutrons, but held at criticality after the contribution of delayed neutrons.
So, just like distance along the real axis corresponds to growth rate, distance along the imaginary axis corresponds to sinusoidal oscillation frequency. A pole on the imaginary axis close to the origin resonates at a very low frequency; the higher up the imaginary axis you go, the higher the oscillation frequency gets.
As you increase the proportional controller gain, the resonant frequency goes up like tensioning a guitar string.
> The actual operation of control rods in a nuclear reactor is a bit more complicated than a standard linear feedback controller, because rather than just directly subtracting neutrons, the control rod position modulates the (average) gain of the reaction feedback loop itself, so they can shift the reaction from being intrinsically stable to intrinsically unstable.
Nuclear reactors are also frighteningly complicated control systems because of delayed neutrons. The classic image of a bomb-like chain reaction is one of "prompt criticality," where nuclear fission produces new neutrons near-instantly that (can) cause more fission reactions, and so on. The timescales of this reaction are so quick that physical control would be impossible — nuclear core detonations are faster than the speed of sound in the core, after all.
However, nuclear reactors are saved through delayed neutrons; a small fraction of neutrons flying around the core come not from the fission itself, but from the decay of short-lived fission products. This timescale is still relatively fast (a few fractions of a second), but it's slow enough for control systems (passive or active) to maintain the reactor in a stable state. The reactor is "prompt subcritical" with respect to the direct fission neutrons, but held at criticality after the contribution of delayed neutrons.
Definitely! And the reaction rate is also not homogeneous throughout the reactor. It's a complex system in every sense.
You explained what the distance along the real axis implies, but what about the distance along the imaginary axis?
Great question!
So, just like distance along the real axis corresponds to growth rate, distance along the imaginary axis corresponds to sinusoidal oscillation frequency. A pole on the imaginary axis close to the origin resonates at a very low frequency; the higher up the imaginary axis you go, the higher the oscillation frequency gets.
As you increase the proportional controller gain, the resonant frequency goes up like tensioning a guitar string.