Proprietà dinamiche degli strumenti di misura Marco Tarabini Politecnico di Milano Dipartimento di...

Proprietà dinamiche degli strumenti di misura

Marco Tarabini

Politecnico di Milano

Dipartimento di Meccanica, Sezione di Misure e Tecniche sperimentali

© Misure e Tecniche Sperimentali – laboratorio

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.

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Gi Go Instrument

go (t) = Gstrum gi (t)


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

3

t

Gradino:

0 tg

0 t0(t)g

ii

gi

t

Impulso:

0 t

0t 0(t)g i

gi

t

Rampa

0t tm

0 0(t)g i

t

gi

dt dt


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.

4

t gradino

gi

t impulso

gi

t rampa

gi

t t


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.

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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.

6

)()( tgtgc iu

1 2

3

V

x


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.

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21

21

31

31

R

V

R

Vi

xcV 31

cg

gS

i

u 1


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.

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gicgdt

dgb u

u

)( fluidotermterm Ahtd

dcm


Thermal Sensors

With simple computations

Said

One can write

is said time constant of the thermometer

9

fluidotermterm

td

d

Ah

mc

][sAh

Vc

Ah

mc

fluidotermterm

td

d


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=:

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at attermterm term term

da e e

d t

1

a 1

,)(

et iniztermterm

t

fluidoterm et

)(

iniztermfluido ,

368.0)( 1fluidofluidoterm et

fluidotermterm

td

d


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

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• 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)

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00

)(

t

t

t

term edt

td

t

fluidoterm et

)(


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

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t

iniztermfluido

fluido et

,

)(

iniztermfluido

fluido tt

,

)(ln

Misure meccaniche e termicheMisure di temperatura e determinazione sperimentale della prontezza di uno strumento del primo ordine

Marco Tarabini [email protected]


15Scopo 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.


16Esperienza 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!);


17Esperienza 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 / 3


18Modello 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

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Proprietà dinamiche degli strumenti di misura Marco Tarabini Politecnico di Milano Dipartimento di...

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