A PI controller is to be used in a temperature control system for a bioreactor. For nominal conditions, it has been determined that the closed$-$loop system is stable when

$\backslash$tau I $=10$ min and $10<$ Kc $<0.$ Would you expect these stability limits to change for any of the following instrumentation

changes? Justify your answers using qualitative arguments.

$($a$)$ The span on the temperature transmitter is increased from

$20$C to $40$C$.$

$($b$)$ The zero on the temperature transmitter is decreased from

$30$C to $10$C$.$

$($c$)$ The control valve $$trim$$ is changed from linear to equal

percentage.