Measurement Techniques, Vol. 45, No. 4, 2002

METHOD OF CALCULATING THE ADDITIONAL ERROR OF MEASUREMENT TRANSDUCERS FOR STOCHASTIC SIGNALS

R. L. Pinkhusovich and B. F. Kuznetsov

UDC 681.2.008

We consider measurement transducers for which the input signal can be represented as a Gaussian steady-state random process. A method is proposed for calculating a multiplicative additional error caused by influencing quantities. The models of the signal at the input of the transducer and also of the signal forming the additional error consists of harmonic components and steady-state random processes.

Among the error components of measurement transducers are additional errors [1] caused by a change in the conditions of measurement, i.e., of the parameters of the environment (temperature, pressure, humidity, etc.) or unmeasured signal parameters (interfering components in gas analyzers, the presence of higher harmonic components for electrical measurements). Under technical conditions, it is usual to assign so-called influence functions to measurement transducers, i.e., dependences linking the additional error with the deviation of some particular quantity or value taken during calibration: δi = Fi(εi – ki),

(1)

where δi is the additional error from the ith influencing quantity; εi is the actual value of the influencing quantity; ki is the value of the influencing quantity taken from the calibration; Fi(·) is the influence function. The total error, for certain assumptions (statistical independence of the components, possibility of describing them by a normal distribution law) can be calculated in accordance with the following expression: n

δΣ = δ + 2

∑ Fi2 (εi − ki ) ,

(2)

i =1

where δΣ is the total measurement error; δ is the basic error; n is the number of influencing quantities taken into account. Such an approach to determining the error is legitimate for statistical measurements but requires the determination of all the influencing quantities εi forming the vector of the additional errors. For dynamic measurements, when the quantity changes during the measurement process, such a determination loses all meaning. In addition, the environmental parameters related to the influencing quantities are also functions of time. From these considerations, it can be stated that δi(t) = Fi[εi(t) – ki].

(3)

In a number of cases, for a known influence function, the error δi(t) can be eliminated from the results of the measurements by using automatic correction methods. However, the implementation of this possibility requires the introduction Translated from Izmeritel’naya Tekhnika, No. 4, pp. 14–16, April, 2002. Original article submitted October 29, 2001.

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of additional elements into the information-measurement system such as computing facilities, sensors, interfacing and other devices, and this makes the system more expensive. It is possible to adopt a correct solution concerning the need to introduce compensation of an additional error only when a priori information is available on the actual value of the additional error. It follows from Eq. (3) that under unfavorable conditions the additional error δi can considerably exceed the basic error of the transducer. An illustration of this position was given by considering the metrological characteristics of certain mass produced instruments of the Angarsk Experimental Design Bureau of Automation. Thus the Baikal 5Ts hygrometer has basic errors of 4 and 2.5% on different scales. An additional error due to the temperature is 2% for each 10°C in the range 5–50°C. It is not difficult to calculate that a deviation of the ambient temperature of 20°C from that assumed during calibration will produce an additional error equal to the basic error (for the 4% error) and larger by more than a factor one and a half than the basic error (for the 2.5% error). This example demonstrates that the additional error can reach unacceptably large values. Let us consider a method which makes it possible to perform calculations of the additional error caused by an influencing quantity for dynamic measurements. We assume that both the input actions and the influencing quantities are steady-state or quasisteady-state random processes. Let us adopt the following model of the input action of the measurement transducer [3]: x(t) = µx + x0(t) + xh(t), (4) where µx = M{x(t)} is the mathematical expectation of the input random process; x0(t) is a centered steady-state random process having a normal distribution law and a variance σx2; xh(t) = Cx sin(ωxt) is a harmonic component. The authors consider the adoption of a model of the input action in the form of a random process to be the most satisfactory description of practical cases, at least for the measurement of the parameters of technochemical processes. The legitimacy of this approach is also confirmed by a number of publications, for example [1, 3–5]. The proposed property of a steady-state nature of the input random process results from the presence of automatic regulators which maintain the value of the technology parameter at the set level of the regulator on account of the presence of the integrating component in the regulation law. This can be explained moreover by the presence of the harmonic component in the model of Eq. (4) since the characteristic operating regime of the regulator, so-called readjustment, provides for the most accurate attainment of the given quantity. Let us represent the mathematical model of the influencing quantity in the form ε(t) = µε + ε0(t) + εh(t),

(5)

where µε is the displacement of the mathematical expectation of the influencing quantity relative to the normal value adopted during calibration; ε0(t) is a centered steady-state random process having a normal distribution law and a variance σε2; εh(t) = Cε sin(ωεt) is the harmonic component resulting from cyclic trends of the mathematical expectation of the environmental parameters (for example, the daily temperature cycle). We assume that the input action x0(t) and the influencing quantity ε0(t) are statistically independent. Let us consider the case of estimating the additional error taking one influencing factor into account. We shall consider that the influence function F(·) is linear and that the error is purely multiplicative. In addition, we shall neglect the finite response time along the channels of the influencing quantity and the measurement signal, considering it to be insignificant. In this case, the output signal of the measurement transducer is determined by the relationship y(t) = x(t)[1 + aε(t)], where a is an influence coefficient. Figure 1 shows a structural scheme illustrating the process of forming an additional error. In this case, the square of the error is defined as δ2(t) = [y(t) – x(t)]2 = a2x2(t)ε2(t). 355

On the basis of the expression obtained and of the mathematical models of Eqs. (4) and (5) given above, we write δ2(t) = a2[(µx + x0(t) + xh(t))(µε + ε0(t) + εh(t))]2. After performing some straightforward transformations, let us apply the operation of the mathematical expectation, eliminating the centered component of the first power (whose mathematical expectation is equal to zero). At the same time we eliminate terms with harmonic components of the first power whose average value tends to zero over a sufficiently long time interval. Consequently the approximations made lead to the following relationship: µ 2x µ 2ε + x 02 (t )ε 20 (t ) + x h2 (t )ε 2h (t ) + x 02 (t )µ 2ε + µ 2x ε 20 (t ) +  M{δ 2 (t )} = a 2 M  . 2 2 2 2 2 2 2 2 + x h (t )µ ε + µ x ε h (t ) + x 0 (t )ε h (t ) + x h (t )ε 0 (t ) 

(6)

Let us introduce the notation z1, z2(t), z3(t), ..., z9(t) to the components of the additional error in their consecutive order in Eq. (6) and using the well-known property of a mathematical expectation (that the mathematical expectation of random quantities is equal to the sum of the mathematical expectations of each quantity [6]), we write 9

M{δ 2 (t )} = a 2

∑ M{zi2 (t )}.

(7)

i =1

Let us consider all the components of Eq. (7). It is trivial to find the mathematical expectation z1 = µxµε since µx = const and µε = const: z12 = µ 2x µ 2ε. In order to find the mathematical expectation of the square of the components, let us utilize the well-known relationships of probability theory [6]: M{X 2 Y 2} = M{X 2}M{Y 2}, M{X 2} = D{X} + µ 2X,

M{Y 2} = D{Y} + µ 2Y,

where X, Y are independent random quantities; µX and µY are respectively their mathematical expectations; D{·} is the variance. Then the mathematical expectation z2(t) = x02(t)ε02(t) in accordance with the given relationships is M{z2(t)} = σx2 σε2. In order to find the mathematical expectation of the component z3(t) = xh2(t)εh2(t), we utilize the fact that both factors are determinate functions of time, and therefore Cx2 Cε2 T →∞ T

M{z3 (t )} = lim

T

∫

sin 2 (ω x t )sin 2 (ω ε t ) dt =

0

Cx2 Cε2 . 4

(8)

It should be noted that the relationship obtained is valid when ωx ≠ ωε. If ωx = ωε and the phase difference is zero, calculations using Eq. (8) give the following result: M{z3 (t )} =

356

3 2 2 C C , 8 x ε

(9)

ε(t) δ(t) = ax(t)ε(t)

y(t) = x(t) + ax(t)ε(t)

x(t) MT

Fig. 1. Structural schematic diagram of the additional error formation process: MT is the measurement transducer.

and when the phase difference ϕ is not equal to zero: M{z3 (t )} =

1 2 2 Cx Cε [1 + 2 cos 2 (ϕ)]. 8

(10)

The mathematical expectations of the components z4(t) and z5(t) are calculated in a similar way to z2(t) and are defined by the expressions M{z4 (t )} = σ 2x µ 2ε , M{z5 (t )} = σ 2ε µ 2x . In order to calculate the mathematical expectations of the components z6(t), z7(t), z8(t) and z9(t), we assume that the harmonic signal with an amplitude C is a particular case of a steady-state random process with zero mathematical expectation and a variance: 1 D = lim T →∞ T

T

∫C

2

sin 2 (ωt ) dt =

0

1 2 C . 2

Then in a similar way to the calculation performed for z2(t), we arrive at the following relationships: M{z6 (t )} =

µ 2ε Cx2 , 2

M{z7 (t )} =

µ 2x Cε2 , 2

M{z8 (t )} =

σ 2x Cε2 , 2

M{z9 (t )} =

σ 2ε Cx2 . 2

Let us write, on the basis of the results obtained, the final expression for the mathematical expectation of the square of the additional error:   µ 2x µ 2ε + σ 2x σ 2ε + µ 2x σ 2ε + µ 2ε σ 2x +   2 2 M{δ (t )} = a  3C 2 C 2 µ 2 C 2 + µ 2 C 2 σ 2 C 2 + σ 2 C 2  . x ε + x ε ε x + x ε ε x +  8 2 2  

(11)

The maximum value corresponding to the in-phase action of the periodic components of the useful signal and the influencing quantity are chosen for the fifth term. In the case when this condition is not observed, this term can be described by expressions (8) or (10). The obtained expression (11) makes it possible the calculate the additional error caused by the influencing quantity for dynamic measurements. 357

We recall that it was assumed above that the measurement signal and the influencing quantity are independent. This limitation is realized most frequently in practice although cases exist (especially when one is concerned with such influencing quantities as the temperature, the ambient humidity, atmospheric pressure) when the condition of statistical independence is not observed and the use of the above formula gives an underestimate of the error. Taking account of a correlation between the measured parameters and the influencing quantities considerably complicates the calculations.

REFERENCES 1. 2. 3. 4. 5. 6.

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A. G. Sergeev and V. V. Krokhin, Metrology [in Russian], Logos, Moscow (2000). Investigation Procedure IM 2247-93, Metrology. Fundamental Terms and Definitions [in Russian], State System for the Unity of Measurements. V. V. Volgin and R. N. Karimov, Estimate of Correlation Functions in Industrial Control Systems [in Russian], Énergiya, Moscow (1979). M. P. Samesenko, Random Processes and Control Systems [in Russian], Vishcha Shkola, Kiev, Donetsk (1986). A. M. Onishchenko, Izmerit. Tekh., No. 9, 7 (1996). E. S. Ventsel’, Probability Theory [in Russian], Nauka, Moscow (1969).