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Problem 10.21PP
Run-to-Run Control: Consider the RTP system shown in Fig. We wish to heat up a 
semiconductor wafer, and control the wafer surface temperature accurately using rings of 
tungsten halogen lamps. The output of the system is temperature Tas a function of time: y = T(t). 
The system reference input R is a desired step in temperature {700° C). and the control input is 
lamp power. A pyrometer is used to measure the wafer center temperature. The model of the 
system is first order, and an integral controller is used as shown in Fig. Normally, there is no 
sensor bias {b = 0).
Figure RTP system
(a) Suppose the system suddenly develops a sensor bias b^O . where b is known. What can be 
done to ensure zero steady-state tracking of temperature command R. despite the presence of 
the sensor bias?
(b) Now assume b = 0. In reality, we are trying to control the thickness of the oxide film grown 
(Ox) on the wafer and not the temperature. At present, no sensor can measure Ox in real time. 
The semiconductor process engineer must use off-line equipment (called metrology) to measure 
the thickness of the oxide film grown on the wafer. The relationship between the system output 
temperature and Ox is nonlinear and given by the integral of the Arrhenius equation;
4r
Oxide diickiiess = J pe~'^dt.
where ff is the process duration, and p and c are known constants. Suggest a scheme in which 
the center wafer oxide thickness Ox can be controlled to a desired value (say. Ox = 5000 A) by 
employing the temperature controller and the output of the metrology.
Step-by-step solution
step 1 of 2
a) We just increase R by + b L e. replaces R by (i? +b) to cancel the sensor bias.
Step 2 of 2
■ y - T
T =
5 R
s a