Empirical Modeling of Heating Element Power for the Czochralski Crystallization Process
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Original versionModeling,Identification and Control 31(2010) Nr. 1, S.19-34
The Czochralski (CZ) crystallization process is used to produce monocrystalline silicon. Monocrystalline silicon is used in solar cell wafers and in computers and electronics. The CZ process is a batch process, where multicrystalline silicon is melted in a crucible and later solidi es on a monocrystalline seed crystal. The crucible is heated using a heating element where the power is manipulated using a triode for alternating current (TRIAC). As the electric resistance of the heating element increases by increased temperature, there are signi cant dynamics from the TRIAC input signal (control system output) to the actual (measured) heating element power. The present paper focuses on empirical modeling of these dynamics. The modeling is based on a dataset logged from a real-life CZ process. Initially the dataset is preprocessed by detrending and handling outliers. Next, linear ARX, ARMAX, and output error (OE) models are identi ed. As the linear models do not fully explain the process' behavior, nonlinear system identi cation is applied. The Hammerstein-Wiener (HW) model structure is chosen. The nal model identi ed is a Hammerstein model, i.e. a HW model with nonlinearity at the input, but not at the output. This model has only one more identi ed parameter than the linear OE model, but still improves the optimization criterion (mean squared ballistic simulation errors) by a factor of six. As there is no nonlinearity at the output, the dynamics from the prediction error to the model output are linear, which allows a noise model to be added. Comparison of a Hammerstein model with noise model and the linear ARMAX model, both optimized for mean squared one-step-ahead prediction errors, shows that this optimization criterion is 42% lower for the Hammerstein model. Minimizing the number of parameters to be identi ed has been an important consideration throughout the modeling work.