15 Dynamic Temperature Measurement 15 .1 General Information The term dynamic temperature measurement covers all measurements during which thermal transients occur in a sensor irrespective of whether the transient is caused by temperature variations in the medium, whose temperature is to be measured, or in the temperature sensor, itself . Thus, unavoidable dynamic errors occur during the measurement of any temperatures, changing with time . Errors also arise during the temperature measurement of a medium at a constant temperature using a temperature sensor immersed in the medium . Determination of the dynamic errors of a thennometer, requires knowledge of its dynamic properties . In many non-electric thermometers where the sensor and indicator form one inseparable unit, the dynamic properties to be described must refer to the whole device . Electric thermometers are mostly used when it is essential to know the dynamic error so that it can be taken into consideration . Consequently, the dynamic parameters of electric sensors will be the main topic for discussion in this chapter . It must be stressed that dynamic errors in temperature measurement are principally caused by the sensor . For this reason, any influence of the dynamic properties of indicating instruments may be neglected in most cases . Knowledge of the dynamic properties of a temperature sensor is necessary for the following main cases : " to determine the necessary immersion time, while measuring a constant medium temperature, " to determine the dynamic errors while measuring temperatures changing with time, " to compare the dynamic properties of different temperature sensors, so that the one best suited for a specific application, may be chosen, " to determine the true temperature variations of temperatures changing in time by correcting known indicated values, " to describe the dynamics of a sensor when it is part of a closed loop temperature control system as described by Michalski and Eckersdorf (1987), " to choose the type and optimum settings of a corrector of dynamic errors . Temperature Measurement Second Edition L. Michalski, K. Eckersdorf, J. Kucharski, J. McGhee Copyright © 2001 John Wiley & Sons Ltd ISBNs: 0-471-86779-9 (Hardback); 0-470-84613-5 (Electronic) 280 DYNAMIC TEMPERATURE MEASUREMENT As the problem of the dynamics of temperature sensors is approached in different ways, the number of existing references concerned with these dynamics is very large . Consequently only concise principles are presented in this chapter . 15 .1 .1 Dynamic errors Dynamic error can be defined as the difference between the sensor temperature, t9 T (t), and the temperature, 0(t) , measured by another inertia free sensor, exhibiting the same static errors . Consequently the dynamic error is that part of the systematic error which varies with time . To simplify the problem, assume that the static error equals zero . Using Figure 15 .1, the dynamic error is then defined as : AO dy . (t) = OT (t) - t9(t) (15 .1) Measuring the stepwise changing temperature, the relative dynamic error is defined as : Sz9dyn(t)= A6dy ( t ) - t9T(t)-6(t) (15 .2) A6 A6 where A9 is the value of the temperature step . The response time, t r , after which the relative dynamic error does not exceed a certain value, is closely connected with the relative dynamic error . For instance at t >_ tr,5 % the absolute value of the relative error is 60d,(t) <<-5% . The dynamic error can also be defined for other, non-periodictemperature variations . Its value is then mostly related to the maximum change of the measured temperature . In dynamic temperature measurement Hofmann (1976) asserts that it is necessary to determine the dynamic errors in two cases . The first occurs when the measured, indicated temperature and the sensor's dynamic properties are known . Another occurs when the medium temperature, the input, is known as a function of time as well as the dynamic properties of the sensor . In both cases it is convenient to represent the dynamic error using the Laplace transform to obtain : A6 dyn (s) = 6 r (s) - 6(s) (15 .3) TEMPERATURE "Tit) G~ d ,it)=4 T ( tok tl-~(t) SENSOR ~ - Figure 15 .1 Definition of dynamic error GENERAL INFORMATION 281 When measuring a sinusoidally changing temperature 6(CV) = A6sin 0ot +ti p the dynamic errors consist of amplitude and phase errors . Amplitude error, AA(o)), is given by the difference of the amplitudes of the sensor temperature 019T (c)) and its true value A6, so that : AA(o)) = Otfr((o) - A6 (15 .4) or the relative error, 8A(Co), related to the amplitude of the measured temperature may be written as : SA(o))= AA ( co) = AOT(o))-019 (15 .5) AO A6 It is also given by the amplitude ratio : 11(co) = A QU O)) (15 .6) Phase error is defined as the phase shift between the sensor temperature, O T (t), and the measured temperature 6(t) . The cut-off frequency, f, of a temperature sensor defines a frequency belowwhich the amplitude error does not exceed a given value (e.g . if f <fc,5 then 1,5AJ < 5 %)when a state of stationary oscillations exists . 15 .1 .2 Dynamic properties of temperature sensors Modelling temperature sensors consists of four main steps decribed by Jackowska-Strumillo et al . (1997) . Applying this process eventually allows the dynamic properties of a temperature sensor to be described by the following equation : F[Yn(t),Yn-1(t) . . . . . . Y(t) ;6 m (t),6 m-1 (t) . . . . . . t9(t)] = 0 (15 .7) where y(t) . . . . . . y" (t) are the sensor output signal and its time derivatives, and 13(t) . . . . . . 0m (t) are the measured temperature and its time derivatives . In the case when the dynamic behaviour is linear, equation (15 .7) becomes : d l Y(t) ja i _ Yb j d ~ tg(t) (15 .8) t-o dt` j-o dti where a i and bj are constant coefficients and m < n . The transfer function, G 7 (s), can be used to describe and present the dynamic properties of a temperature sensor . It is the ratio of the Laplace transform of the sensor output signal, 282 DYNAMIC TEMPERATURE MEASUREMENT y(s), to the Laplace transform of the measured temperature signal, 9(s), when the initial conditions are zero so that : GT(s)= As) (15 .9) Taking equation (15 .8) into account the transfer function can then be given by : m Eb j sj GT (s) = J n 0 = L(s) E als i M(s) (15 .10) i=0 The operational transfer function, GT(s), is thus presented as the ratio of the polynomials L(s) and M(s) . Poles of the transfer function are the roots of the equation M(s) = 0 and the zeros of the transfer function are the roots of the equation L(s) = 0 . If the temperature input signal, 9(s), is known, a knowledge of the transfer function, GT(s), of a temperature sensor, enables its output signal to be determined as a function of time (Doetsch, 1961) . Each electric temperature sensor may be regarded as composed of a thermal conversion stage and an electrical conversion stage as shown in Figure 15 .2 . In the thermal conversion stage, the temperature, 9(t), of the medium whose temperature is being measured is converted into the sensor's temperature, t9 T (t) . The sensor's temperature, O T (t), is converted into the electrical output signal y(t) (e .g . thermal emf) in the electrical conversion stage . This second conversion stage has a purely static character . Thus, the sensor transfer function GT(s) of equation (15 .9) can be expressed as a product of the transfer function of the thermal conversion stage, F T (s), and of the coefficient K T , representing the properties of the electrical conversion stage, and called the sensor gain so that equation (15 .11) is obtained : (a) 4(t) THERMAL (t) ELECTRICAL y (t) CONVERSION CONVERSION STAGE STAGE t-DOMAIN PRESENTATION (b) ~(s) 4 T (s) y(s) FT(s) K T s-DOMAIN } PRESENTATION Figure 15 .2 Block diagram of a temperature sensor GENERAL INFORMATION 283 G T (s) = KTFT(s) (15 .11) where : K T = d~ (15 .11 a) T F T = (s)) (15 .11 b) This approach to the presentation of sensor dynamics, makes it possible to limit further discussions of the dynamics to those of the thermal conversion stage . In the case of steady-state periodic variations of measured temperature, the frequency response, G T (jco) , of the sensor may be considered instead of the sensor transfer function, GT(s) . The sensor frequency response is the ratio of the phasor values of the output signal y(jro) to the phasor value of the variable component of the sinusoidally changing measured temperature 6(jo)) GT(jCO) = y(jw) (15 .12) Nja)) where co is the angular frequency and j = . The frequency response of a sensor can be obtained by substitutingjco in equation (15 .9) in place of the operator, s . Consequently from equation (15 .10) the frequency response becomes : m Y,bi(jo)) j GT(jco)= j=0 = L(jo)) (15 .13) ~) nYai (jco)i M(j ~ ) i=0 In a similar fashion, the transfer function of equation (15 .11) can be rewritten as the frequency response function : GT(jO)) = KTFT(jo)) (15 .14) where K T is the sensor gain, and FT(I w) is the frequency response of the thermal stage . Another way of expressing the frequency response of the sensor is : GT (jo)) = KT [P(o)) + jQ(o))) (15 .15) 284 DYNAMIC TEMPERATURE MEASUREMENT where P(o)) = J2e [F T (jco)] (15 .15 a) Q(co) = Jm [F T (j co)] (15 .15b) where JZe means "take the real part of and Dm means "take the imaginary part of' . Using the polar form for the complex variable GT(j(O) = AY((D) Ad exp[j(VO)] (15 .16) where Ay((o) is the amplitude of the output signal y(jw), AO is the amplitude of the first harmonic of measured temperature O(jco) and (p(o)) is the phase shift between y(jO)) and 6(j)) . Equations (15 .15) and (15 .16) are related as : Ay A~ ) = JGT(jp)j = KT P 2 (p)+Q 2 (0 ) (15 .17) (p (co) = argGT Go)) _ -arctan Q ( ) (15 .18) Ay((o) l AO of equation (15 .17) and (p(o-)) of equation (15 .18) are respectively called the amplitude and phase characteristics of a temperature sensor . Sometimes it is more convenient to use the amplitude of the sensor temperature AOT((O) instead of the amplitude of the output signal Ay((o) . The ratio : A 0~CO ) = PZ((0)+QZ((0) =IFT(j(o)I (15 .19) is called the amplitude characteristic of the thermal conversion stage of the temperature sensor . 15 .2 Idealised Sensor The existence of an idealised temperature sensor, designed as a homogeneous cylinder made of a material having infinitely great thermal conductivity ~ , will be assumed . Let this sensor have the mass m, specific heat c and a surface area, A, for heat exchange with its surroundings . During the temperature measurement of a liquid or gaseous medium, the sensor is completely immersed, so that it does not exchange heat with any other medium with a different temperature . As an example, any electric temperature sensor can be taken, provided that it is connected with the indicating instrument by extremely thin wires . It is IDEALISED SENSOR 285 further assumed, that the thermal capacity, mc, of the sensor is negligibly small compared with the total thermal capacity of the medium and that the heat transfer coefficient a , between the sensor and the medium is constant . 15 .2 .1 Transfer function To set up the differential equation, which describes the sensor's dynamics and thus its transfer function, the method of heat balances will be used . Assume, at the time t = 0 - , an infinitesimally small time before zero, that the sensor is in a steady state, with its temperature equal to the ambient temperature OT = 0, At t = 0 + immerse the sensor in the medium at temperature 0 , higher than the ambient temperature so that 0 > Oa . For the temperature excess over the given reference value the notation O is introduced . In this book O is also simply referred to as temperature . The initial conditions at t = 0 - are given by : O T =Z T -O a =0 and O=6-6 a >0 According to Newton's law, when the sensor is immersed in the medium, the heat transferred to the sensor in the time interval dt will be : dQ = a4(E) -O T )dt (15 .20) where a is the heat transfer coefficient between sensor and medium and A is the heat exchange area . The heat stored in the sensor is : dQ=mcdOT (15 .21) where m is the mass of the sensor, and c is the specific heat of the sensor material . From equations (15 .20) and (15 .21) it follows that a4(0- O T )dt = mcdO T (15 .22) or nic deT +OT =O (15 .23) a4 dt Introducing the notation : me = NT (15 .24) aA Equation (15 .23) can be expressed as : 286 DYNAMIC TEMPERATURE MEASUREMENT NT d d T + O T = O (15 .25) where N T , which is called the sensor time constant at the given heat transfer conditions (a = constant), is expressed in time units . Taking the Laplace transform, equation (15 .25) becomes : N T SO T (s)+O T (s)=0(s) (15 .26) where s is the Laplace operator . Defining the transfer function of the thermal stage of the sensor as : F T (s) = O(sj) (15 .27) Equation (15 .26) becomes : FT (s) 1+sNT (15 .28) The sensor transfer function is : GT(s)= O(s) (15 .29) or G T (s) = K T F T (s) = KT 1 + sNT (15 .30) Equation (15 .30) shows that the transfer function of an idealised temperature sensor is that of a first order inertia . The frequency response of the thermal stage of an idealised sensor may be written as : FT(j(o) = OT(jCO) = 1 (15 .31) O(jco) 1 + j(oNT and of the sensor as a whole : GT(j(O) _ O(jco) - KT 1 + jcoNT (15 .32) IDEALISED SENSOR 287 15 .2 .2 Measurement of time varying temperature If the changes of measured time varying temperature, 0(t), can be described by elementary functions, the temperature indicated by the sensor 6r (t) , and the dynamic measurement error can be determined in a simple way by using the Laplace transform . As an example, consider a step temperature input from an initial temperature, 6b, to a final temperature oe . In practice this case corresponds to the immersion of a temperature sensor with a temperature, Ob, into a medium with a temperature, 6e , which is written mathematically as : ~9(t) O b fort <_ 0 - {6 e ort>0 or z9(t) = (Oe - O b )1 (t) + Ob where 1(t) is a unit step at t = 0 (15 .33) As it is necessary to obtain zero initial conditions for the Laplace transform, the excess temperature O = O e - 6b , will be used, to obtain : O(t) = O a 1(t) (15 .34) for which the Laplace transform is : O(s) = Oe (15 .35) s From (15 .27) it follows that : OT(s) = F T (s)O(s) (15 .36) Inserting F T (s) from equation (15 .28) and O(s) from equation (15 .35) into equation (15 .36) the Laplace transform of the sensor temperature will be : OT(s) s(1_ O NT) (15 .37) After the inverse-transformation in accordance with Doetsch (1961) the temperature time dependence will be : O T (t) =-C-1 [aT(s)l (15 .38) 288 DYNAMIC TEMPERATURE MEASUREMENT OT(t) = OX - e -t/N T) (15 .39) The final result is then : OT (t) = (Ve - Ob)(I - e _tIAI T) + Ob (15 .40) From equations (15 .38) and (15 .40) it follows that the step input response of an idealised temperature sensor is an exponential curve, having the time constant N T as shown in Figure 15 .3 . From this curve, the time constant N T can be found in a graphical way from the tangent to the curve OT(t)=f(t) at any point, or as the time after which O T (t = N T ) = 0 .6320, . Also the half-value time, or 50 % rise time to .5, which is the time when O T = 0 .50, can be used to determine the time constant . From equation (15 .39) at t = to .5, it is clear that : 0 .50, = 0,(I - e to 5 I NT ) After some transformations the time constant is obtained as : NT = 1 2 tO .5 = I 0 .693 tO .5 (15 .41) The nine-tenth value time, t o .9 , is also a characteristic value of the temperature step response . From the exponential function it can be shown that : to-5 = 3 .32 (15 .42) to .9 la) (b N T TIME CONSTANT 'e A e4d (t) e ,"' * T = f (t) m 0 .99, e j U, T =f(t) a 0,6328, 0,59, W n w E H W N H - - - _- - - - W F x 0 4 ' to s TIME t N T 0 TIME t to .s Figure 15 .3 Step response of an idealised temperature sensor