1. A thermocouple is a device commonly used to measure the temperature in, for example, boilers and engines. It has the advantage of directly giving out a voltage that is approximately proportional to the measured temperature. The thermo-voltage is, however, only a few millivolts.

In the block diagram of FIGURE 4, the output of the thermocouple, vT(t), is fed into an amplifier to give a sufficiently large signal to drive a signal processing stage. The signal processing stage includes an analogue-to-digital converter (ADC) to give an output to drive a digital display. A low-pass filter (LPF) is placed between the amplifier and the signal processing stage to remove noise that may be picked up from adjacent 'noisy' equipment and from 'mains' interference.

The thermocouple should ideally have a step change output to a step change in temperature. In practice this is not the case because the mass of the device introduces a thermal delay as it warms up or cools down to a new temperature. It can be shown that the temperature of the thermocouple is related to a step change in temperature by the equation:

The time constant depends upon the mass (in) and surface area (A) of the thermocouple, the thermal specific heat (c) of the material it is made from and also the heat transfer coefficient (h) between the thermocouple and its surroundings. Thus:

ζ= mc/Ah

2. (a) In FIGURE 5(a) the thermocouple is shown with an initial temperature the same as that for the ambient air, TA. It is then immersed into a tub of hot water at a constant temperature of Tw. Thus the applied step change in temperature is ΔT = Tw - TA.

(i) Write down the Laplace equation for equation (1).

(ii) Hence obtain the equation of the transient response of the thermocouple.

(iii) Plot the transient response using the data in TABLE B over a time period of about 4ζ.