Response to step function

Inversion of equation 7.79 gives the response of the thermocouple to a step change in temperature of the surrounding fluid, i.e.  

Plotting JM vs /t illustrates the distinctive characteristics of the step response of a first-order system (Fig. 7.27). The response is immediate and the slope is a maximum at t = 0 and decreases with increasing time. If the response is measured experimentally, the time constant may be evaluated thus  

Hence the response will have reached 63.2 per cent of its final value when t - r. 

Therefore at t = 0, the slope of the graph is 1 /r. Hence, if the initial rate of change were maintained, the response would be completed in a time equivalent to one time constant. 

If the value of the time constant obtained by both methods is not the same within the limits of experimental error, the response is not truly first order. 

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Hyuga, H., Kinosita, K., and Wakabayashi, N. (1991) Transient and steady-state deformations of a vesicle with an insulating membrane in response to step-function or alternating electric fields. Japanese Journal of Applied Physics Part 1, 30 (10), 2649-2656. 

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