Lumped Analysis

A lumped parameter analysis of probes is given below. The effect of spatial variability is discussed through specific examples later in this chapter. Let be the flow input and the probe output. The order of a probe, a transducer or a measurement system is determined by the order of the differential equation relating and with time as the independent variable. Hence we have:

In the above equations is the static sensitivity of the probe that can be determined once-and-for-all from a steady state experiment. Consider the response of these systems to a step input , a constant and a periodic input . Here is frequency and the imaginary unit . For a step input, we assume the initial conditions to be quiescent, i.e., . For a periodic input we assume that the system has reached a dynamic steady state and the output oscillates with the same frequency as the forcing frequency . The second part of this assumption is strictly true only for linear systems, i.e. coefficients , , and are independent of , and . In both laboratory and field experiments the fluctuations in the input will displace the measurement system only marginally with respect to the operating point and so its performance can be locally linearized. Hence the linear analysis presented here is not severely restrictive.