1. A process input is given by the following expression
Y(s) = K/s(τs+1)
Because we know that the Laplace transform of the output can express the product of the system transfer function and the Laplace transform of the input, give all possible combinations of G(s) and U(s) that yield the above Y (s) and interpret them.
2. A heated process is used to heat a semiconductor wafer operates with first-order dynamics, that is, the transfer function relating changes in temperature T to changes in the heater input power level P is
T'(s)/P'(s) = K/τs+1
where K has units [0C/Kw] and τ 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:
1. The process temperature initially is 80 0C.
2. Four minutes after changing the power input, the temperature is 2300C.
3. Thirty minutes later the temperature is 280 0C.
(a) What are K and 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 process temperature. When will it occur? How large will it be?
3. The dynamic behavior of the liquid level in each leg of a manometer tube, responding to a change in pressure, is given by
d2h'/dt2 + (6μ/R2ρ)dh'/dt + (3g/2L)h' = (3/4ρL)p'(t)
where h'(t) is the level of fluid measured with respect to the initial steady-state value, p'(t) is the pressure change, and R, L, q, ρ and μ are constants.
(a) Rearrange this equation into standard gain-time constant form and find expressions for K, τ, ε in terms of the physical constants.
(b) For what values of the physical constants does the manometer response oscillate?
(c) Would changing the manometer fluid so that ρ (density) is larger make its response more or less oscillatory? Repeat the analysis for an increase in μ (viscosity).
4. A second-order critically damped process has the transfer function
Y(s)/U(s) = K/(τs+1)2
(a) For a step change in input of magnitude M, what is the time (ts) required for such a process to settle to within 5% of the total change in the output?
(b) For K = 1 and a ramp change in input, u(t) = at, by what time period does y(t) lag behind u(t) once the output is changing linearly with time?
5. The following transfer function is not written in a standard form
G(s) = 2(s+0.5)e-5s/(s+2)(2s+1)
(a) Put it in standard gain/time constant form.
(b) Determine the gain, poles and zeros.
(c) If the time-delay term is replaced by a 1/1 Pad´e approximation, repeat part (b).