# Dynamics of Measurements

If the measured quantity is constantly changing with time, the measure­ment result always follows the measured quantity with a time lag. This fact re­lates to all types of measurements, and can be called the “Second Law of Measurements.” The reason for this Second Law is that any event requires a finite time in order for it to be executed. The event can be, for example, heat transfer from air to a temperature probe, the movement of a tracer sample to FIGURE 12.12 Quantities related to the target value fulfillment

The analyzer, or light traveling from the measurement volume to a laser ane­mometer detector. Obviously these events all require very different times, from several seconds to approximately 10’9 seconds, but regardless of the time span no event can be executed in zero time. The important fact is the inertia of the instrument compared with the speed (frequency) of variations in the quantity value. If the instrument is slow, information regarding the behavior of the measured quantity is lost. If the mean value is of primary interest, this may not be critical. However, if there is a need to closely follow the changes of the measured quantity, it is essential that the instrument is selected in an approved manner.

The time difference (delay) between the measured quantity and the mea­surement result is called the Inertial error. A definition2 is the error due to iner tia (mechanical, thermal, etc.) of the parts of a measuring instrument. In ventilation equipment the critical component in the measuring chain, from the dynamic point of view, is often the sensor or the measuring transducer (probe).

The sensor is the element of an instrument directly influenced by the mea­sured quantity.2 In temperature measurement the thermal mass (capacity) of the sensor usually determines the meter’s dynamics. The same applies to ther­mal anemometers. In IR analyzers used for concentration measurement, the volume of the flow cell and the sample flow rate are the critical factors. Some instruments, like sound-level meters, respond very fast, and follow the pres­sure changes up to several kHz.

Depending on the sensor/probe construction, there may be from one to several capacities, each increasing the inertial error. The inertial error depends not only on the features of the instrument, but also on the character of the

Change in the measured quantity. There are of course many types of variation the measured quantity can follow. To estimate the inertial error using a simple computational approach, the changes can roughly be classified as either a sin­gle step-type change; an exponential-type change; or a continuous up-and — down, oscillating, sine-type change.

The step change is close to the situation where the sensor is suddenly moved from one place to another having a different stare of the measured quantity. The exponential change could, for example, be the temperature change of a heating coil or some other first-order system. Finally, the velocity fluctuations of room air can be approximated with a sine or cosine function.

12.3.4.2 Systems of First Order

Step Response

In an ideal first-order system, only one capacity causes a time lag between the measured quantity and the measurement result. Typically, an unshielded thermometer sensor behaves as a first-order system. If this sensor is rapidly moved from one place having temperature T1 to another place of temperature T2, the change in the measured quantity is close to an ideal step. In such cases, the sensor temperature indicated by the instrument has a time history as shown in Fig. 12.13. The sensor temperature T = T(t) as a function of time can easily be shown to be

T(n = TJ+(T2-T,)(l-^/r) , (‘12.15)

Where T is time and xis the sensor time constant. From Eq. (12.15) the sensor temperature can be computed at any time instant, provided the initial temper ­ature, the final temperature, and the time constant are known. In the above case the temperature measurement is used as an example, but the same princi ­ple and equation apply to any quantity, provided the system is first order.

Continuing the above example, the inertial error obviously is the differ­ence between the final temperature and the sensor temperature. From Eq. (12.15), the inertial error of a first-order system is

E,(t) = (T2 — T, )(1 — E~t/T) . (12.16i

The error has its maximum value E, = T2 — Tj at T = 0 and decreases toward E, =0 when the time approaches infinity. From Eq. (12.16), the de­sired time for the inertial error to reach a certain value can be solved for.

The time constant in the previous equations is, in principle, the product of the sensor capacity and the resistance of the flow into the sensor,

T=RC. (12.17)

In the case of a temperature probe, the capacity is a heat capacity C — Me, Where M is the mass and C the material heat capacity, and the resistance is a thermal resistance R = /(hA), where H is the heat transfer coefficient and A Is the sensor surface area. Thus the time constant of a temperature probe is r = Mc/(hA). Note that the time constant depends not only on the probe, but also on the environment in which the probe is located. According to the same principle, the time constant, for example, of the flow cell of a gas analyzer is r = V/qt„ where V is the volume of the cell and Qv the sample flow rate.

Response Time

The time constant is one way of determining the dynamic features of a measurement system. Not all instrument manufacturers use the time constant; some use the response time instead. The response time is the time between a step change of the measured quantity and the instant when the instrument’s response does not differ from its final value by more than a specified amount.1’2 The response time is defined according to a deviation from the fi­nal value. Often response times for the relative deviation of 1.%, 5%, 10%, or 37% are used. The corresponding response times are denoted by 99%, 95%, 90%, or 63% response time, respectively. The response time for a first-order system can be solved from Eq. (12.15). Note that the 63% response time of a first-order system is the same as the time constant R of the system.

Not all instruments behave as a first-order system. If several capacities exist in a series connection and none dominates, i. e., is much larger than the others, then the response differs from a first-order system. The step response of a higher-order sys —

Lem is shown in Fig. 12.14. The curve is typically S-shaped and deviates more from the first order exponential curve, the higher the number of the active capacities.

For example, a temperature-measuring device, having its sensor placed in a protecting tube, is a system of second order. For such a system no single time constant exists in the same way as a first-order system. The behavior of such a system is often given by a response time. Another concept is to give the appar­ent time constant rfl, which can be constructed by placing a line in the inflec­tion point of the step response curve; see Fig. 12.14.