A plasma etching batch process has a process gain of E Ȧ /min (but no dynamics). The manipulated variable is the etch time, so the controlled variable is the film thickness. There are no time constants that need to be included. Assume Gv = Gm = 1. Derive the closed-loop transfer function for a

Question

A plasma etching batch process has a process gain of $E \dot{A} / \min$ (but no dynamics). The manipulated variable is the etch time, so the controlled variable is the film thickness. There are no time constants that need to be included. Assume $G_{v}=G_{m}=1$ . Derive the closed-loop transfer function for a set-point change for two different controllers:

(a) $G_{c}=K_{c}$

(b) $G_{c}=\frac{1}{\tau_{I} s}$

In both cases, analyze the effect of a unit step set-point change. Sketch the response and show whether there is offset or not.

(a) [latex]G_{c}=K_{c}[/latex]

(b) [latex]G_{c}=\frac{1}{ au_{I} s}[/latex]

In both cases, analyze the effect of a unit step set-point change. Sketch the response and show whether there is offset or not.

Accepted Answer

The closed loop transfer function for a set point change (Eq. 11-26), is given by

[latex]\frac{Y}{Y_{s p}}=\frac{K_{m} G_{c} G_{v} G_{p}}{1+G_{c} G_{v} G_{p} G_{m}}[/latex] (11-26)

[latex]\frac{Y}{Y_{s p}}=\frac{K_{m} G_{c} G_{v} G_{p}}{1+G_{c} G_{v} G_{p} G_{m}}[/latex]

Substituting the values from above and in part (a), we get,

[latex]\frac{Y}{Y_{s p}}=\frac{K_{c} E}{1+K_{c} E}[/latex]

Multiplying by a unit step change in set point gives

[latex]Y(s)=\frac{K_{c} E}{1+K_{c} E} \frac{1}{s} \Rightarrow y(t)=\frac{K_{c} E}{1+K_{c} E}[/latex]

A sketch might look like this (the step change at [latex]t=5[/latex] )

As evidenced by the sketch, there is offset for this controller.

For part (b), we substitute the values into Eq. 11-26 to get

[latex]\frac{Y}{Y_{s p}}=\frac{E}{ au_{I} s+E}=\frac{1}{\frac{ au_{I}}{E} s+1}[/latex]

Multiplying by a unit step change in set point gives

[latex]Y(s)=\frac{1}{\frac{ au_{I}}{E} s+1} \frac{1}{s} \Rightarrow y(t)=1-\exp \left(-\frac{E}{ au_{I}} t ight)[/latex]

A sketch would look like this

As evident in the sketch, there is no offset for this controller.

Question 11.26: A plasma etching batch process has a process gain of E Ȧ /m...

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