Question

A heated process is used to heat a semiconductor wafer operates with first-order dynamics, that is, the T'(s)/P'(s)=K/ts+1.

where K has units [°C/Kw] and t has units [minutes]. The process is at steady state when an engineer changes the power input stepwise from 1 to 1.5 Kw. She notes the following:
(i) The process temperature initially is 80°C.
(ii) Four minutes after changing the power input, the temperature is 230 °C.
(iii) Thirty minutes later the temperature is 280 °C.
(a) What are K and tin the process transfer function?
(b) If at another time the engineer changes the power input linearly at a rate of 0.5kW/min, what can you say about the maximum rate of change of process temperature: When will it occur? How large will it be?

Answer 1

To find the values of K and t in the semiconductor wafer heating process transfer function, two equations are set up using the data points provided from the first-order response. Once the system constants are determined, it is noted that the maximum rate of temperature change for a linear power input change occurs immediately after the change begins.

Step-by-step explanation:

The question involves determining the constants in a first-order dynamic system representing a semiconductor wafer heating process. The transfer function is given as T'(s)/P'(s)=K/(ts+1), where K is the process gain with units °C/kW, and t is the time constant with units minutes.

Part (a): To find K and t, we analyze the given data points. At steady state, a step change in power input from 1 kW to 1.5 kW occurs, and the temperature rises from 80°C to 230°C in 4 minutes, reaching 280°C after 30 minutes.

Assuming a first-order response, the temperature change (ΔT) due to a change in power (ΔP) after a certain time (4 minutes) can be expressed as ΔT(t) = KΔP(1 - e^(-t/τ)), where τ is the time constant. Using the data points, we can set up two equations and solve for K and t.

Part (b): When power input changes linearly at a rate of 0.5 kW/min, the process temperature will change at a rate determined by the derived K and t. The maximum rate of temperature change will occur immediately after the power change starts, as first-order systems respond most quickly to changes at the beginning.