A heater for a semiconductor wafer has first-order dynamics, that is, the transfer function relating changes in temperature T to changes in the heater input power level P is

where K has units [^ˆ˜C/Kw] andÏ„has units [min].

The process is at steady state when an engineer changes the power input step wise from 1 to 2 kW. She notes the following:
(i) The process temperature initially is 100 ^ˆ˜C.
(ii) Four minutes after changing the power input, the temperature is 400^ˆ˜C.
(iii) Thirty minutes later the temperature is 500^ˆ˜C.
(a) What are K and Ï„ in the process transfer function?
(b) If at another time the engineer changes the power input linearly at a rate of 0.5 kW/min, what can you say about the maximum rate of change of process temperature: When will it occur? How large will it be?

K T'(s) P'(s) ts +1