Proprietà dinamiche degli strumenti di misura Marco Tarabini Politecnico di Milano Dipartimento di Meccanica, Sezione di Misure e Tecniche sperimentali

Introduction When choosing a transducer, one must take care not only of the static properties of instruments (sensitivity, bias, floor noise and so on) but also of how this instrument describes time varying signals. The dynamic properties of instruments are not expressed by single numbers, while the static properties are. For instance, the sensitivity of a displacement transducer is 100 mV/mm, its full scale value is ±20 mm, its accuracy can be  0.5 % of the full scale and so on. With the term “dynamic properties” we commonly indicate the response of the instrument to input signals gi. The output of the instrument gu will depend on the inputs and on the transfer function of the instrument Gstrum.

What about G? Gstrum can be determined both analytically (with a deep knowledge ofthe physical phenomena governing the instrument behaviour) or experimentally, with the dynamic characterization of the transducer. The dynamic properties are commonly studied with Impulse Step Ramp Sine

In practice? It is also necessary to distinguish between the ideal signal (with which one can study the mathematical model) and the actual signal (that one can apply to a real instrument in order to observe his behaviour). In particular, it is not possible to create the impulse and the step signals, since sudden variation of the input quantity are nearly impossible. In view of this difference, the dynamic experimental characterization of the instrument may differ from the analytical one.

Study of the dynamic response The study of the responses to the impulse, ramp and step inputs is usually performed in time domain, while the study to the sinusoidal response is carried out in frequency domain. One can observe that the ideal step is the derivative of the ideal ramp, and the ideal impulse is the derivative of the step. This means that it is theoretically possible to switch between different inputs with proper differentiations or integrations. Depending on their dynamic behaviour, instruments can be classified on the base of their order, i.e. on the order of the differential equation describing their performances. Instruments are classified in: Zero order instruments; First order instruments; and Second order instruments.

Zero Order Instruments Are also said ideal instruments, and are all those instruments for which the following equation is valid These instruments ideally reproduce each kind of input. These instrument does not exist, but in some cases, for specific input configurations (for instance, in a specific frequency range), their behaviour can be approximated with the zero order one. A typical example is a resistive displacement sensor.

How does it work? The measurement is obtained with the reading of the 1-3 voltage. V1-3 is proportional to the resistance R1-3, and consequently: The voltage 1-3 depends only on the position of the transducer and consequently the displacement is proportional to V1-3. The static sensitivity of the instrument is therefore defined as The resistive behaviour of the transducer is an approximation of the actual one. At high frequency also the cable capacitance and the resistance inductance cannot be neglected and, consequently, the zero order approximation is no more valid.

First order Instruments First order instruments are governed by a differential equation similar to the following one: An example of these instruments is the thermometer. Let us consider a RTD (resistance thermal device), that is initially at a temperature term,iniz and afterwards placed in a fluid whose temperature is fluido. The RTD will change its temperature and reach the fluid one. The transitory temperature can be explained using the thermal exchange equation. Neglecting the radiating and conductive terms, one can write: Where c is the specific heat of the thermometer, m its mass, A its surface and h is the thermal exchange coefficient.

Thermal Sensors With simple computations Said  One can write  is said time constant of the thermometer

Step Response If the temperature input is a step, it is possible to impose the exponential solution By introducing the initial and final conditions, we obtain: Where The instrument behaviour depends on : if  increases, a larger time is needed to reach the final temperature value. The time constant meaning can be studied by setting t=:

Dynamic Calibration How can we experimentally identify the time constant tau?  is the time for which the measurand is has still to cover the 36.8% of In other words, the 63,2% of imposed by the step is achieved after a time equal to  It seems quite easy, we take the thermometer from a water + ice bath and we put it into boiling water. The time requested to reach 63.2° is tau Method based on one single point

Second way... Another physical meaning of the time constants given by the derivative at time t=0 The curve tangent at t=0 s is inversely proportional to the time constant and proportional to the temperature step. Consequently we can derive tau starting from the derivative at time equal to zero by knowing the temperature difference. Method based on several points (tangent at zero)

Third way Logarithmic regression: we know that the response is exponential We know that the solution is And consequently, we can compute the logarithm of the above expression obtaining And now? After changing the variable, go to excel and use the regr.lin command

Misure meccaniche e termiche Misure di temperatura e determinazione sperimentale della prontezza di uno strumento del primo ordine Marco Tarabini marco.tarabini@polimi.it

Scopo del laboratorio Scopo dell’esercitazione è la determinazione sperimentale della prontezza di uno strumento del primo ordine: Perché? Perché nella misura di quantità variabili nel tempo insorgono una serie di problemi legati alle caratteristiche degli strumenti. E’ necessario indagare se un determinato strumento è valido per misure dinamiche  prontezza. Prima di determinare la prontezza dello strumento, impariamo ad eseguire una misura di temperatura con una termocoppia.

Esperienza di laboratorio – prima parte Misurare con la termoresistenza la temperatura ambiente, ed assumerla pari a quella di riferimento, ovvero dei morsetti del multimetro (si ricorda che misura = numero+udm+incertezza!); riscaldare una resistenza riscaldante alimentandola con una tensione costante, e lasciare andare a regime (5-10 min); misurare con la termocoppia la temperatura della resistenza riscaldante (si ricorda che misura = numero+udm+incertezza!);

Esperienza di laboratorio – prima parte Calcolo dell’incertezza: Termocoppia: consultare la tabella dei potenziali termoelettrici e considerare l'indicazione "limits of error" come accuratezza; dei due numeri indicati, considerare il maggiore. La termocoppia del laboratorio è di classe "standard". Termoresistenza: la termoresistenza del laboratorio è di classe B (nel range ‑50 °C/+500 °C). Si riportano di seguito le tabelle per il calcolo delle accuratezze. Calcolare l'incertezza a partire dall'accuratezza, con i = a /

Modello di uno strumento del primo ordine Nota importante sulle unità di misura: Tin per misure di temperatura è espressa in [°C] Vout, grandezza in uscita dallo strumento può essere in [V] per una termocoppia o in [Ω] per una termoresistenza. Nel laboratorio in oggetto, la centralina di acquisizione dei dati NON provvede anche a fare la conversione in [°C] della misura in uscita dello strumento. Pertanto la sensibilità statica k andrà ricavata dalla tabella della termocoppia utilizzata.

Seconda Parte Identificazione della costante di tempo mediante i tre metodi proposti Intercetta con il livello 63.2% Tangente all’origine Regr Ressione Logaritmica Verifica dei dati per transitorio in salita e in discesa