Fig. P4.3-9 illustrates a thermal stress chamber. This system is described in Problem 1.6-1. The system output c(t) is the chamber temperature in degrees Celsius, and the control-voltage input m(t) operates a valve on a steam line. The sensor is based on a thermistor, which is a temperature sensitive resistor. The disturbance input d(t) models the opening of the door into the chamber. With the door closed,

d(t) = 0 ; if the door is opened at t = t₀,

d(t) = u (t − t₀), a unit-step function.

(a) Suppose that the sampling-and-data-reconstruction process is implemented with an analog-todigital converter (A/D) and a digital-to-analog converter (D/A). Redraw Fig. P4.3-9 showing the A/D and the D/A.

(b) Derive the transfer function C(z)/E(z).

(c) A constant voltage of e(t) = 10 V is applied for a long period. Find the steady-state temperature of the chamber, with the door closed. Note that this problem can be solved without knowing the sample period T .

(d) Find the steady-state effect on the chamber temperature of leaving the door open.

(e) Find the expression for C(s) as a function of the Laplace-transform variable s, the control input, and the disturbance input. The z-transform variable z cannot appear in this expression.

Fig. P4.3-9

Problem 1.6-1

A thermal test chamber is illustrated in Fig. P1.6-1(a). This chamber, which is a large room, is used to test large devices under various thermal stresses. The chamber is heated with steam, which is controlled by an electrically activated valve. The temperature of the chamber is measured by a sensor based on a thermistor, which is a semiconductor resistor whose resistance varies with temperature. Opening the door into the chamber affects the chamber temperature and thus must be considered as a disturbance.

Transcribed Image Text:

E(s) en T 1-e-3 D(s) (1) M(s) m(t) Voltage Chamber 2.5 s+0.5 2.5 $+0.5 Sensor 0.04 Temperature (C)