Analysis of the Upper Bound of Dynamic Error Obtained during Temperature Measurements

Abstract

This paper presents an analysis of the upper bound of the dynamic error obtained during temperature measurements. This analysis was carried out for the case of the absolute error criterion and for the numerically determined excitation signals, with one and two constraints. The negative temperature coefficient (NTC) and K-type thermocouple sensors were tested, and the upper bound of the dynamic error was determined for the case of one and two constraints imposed on the input signal. The influence of the sensor modelling uncertainty on the values of the upper bound of the dynamic error has also been taken into account in this paper. Numerical calculations and the corresponding analysis were carried out using the MathCad 14 program. The solutions presented in this paper make it possible to obtain precise solutions in the field of classic calibration of temperature sensors—but, above all, they allow for a mutual comparison of the accuracy of widely used sensors in the energy industry.

1. Introduction

Temperature measurements carried out by means of relevant sensors are widely used in the energy industry to monitor outside temperature [1]; to control generators [2], transformers [3], bearings [4], brakes [5] and hydraulics temperature [6]; and as diagnostics of electrical machines [7,8]. The accuracy of the sensors [9,10] is extremely important for these measurements. Manufacturers determine the accuracy and provide this information in the corresponding data sheets; these data are then verified by using the standard calibration [11,12]. With regard to the instruments intended for static measurements (measurement of signals that are unchangeable or repeated over time) [13,14], this method of determining the accuracy seems to be insufficient. The accuracy class [15] is defined for the instruments that perform static measurements, and it is possible to compare these devices in a meaningful way. However, in the case of sensors intended for temperature measurements, the above approach is not possible because temperature is a physical quantity that is subject to dynamic changes [16,17]. Hence, this quantity is indeterminate in time.

Previous papers [18,19] proposed a procedure for the extended calibration of temperature sensors, carried out using the upper bound of dynamic error (UBDE) and for the case of the integer-square criterion [20,21,22]. It should be emphasised that this criterion is one of the main quality functional used in widely understood measurements. The UBDE depends on both the dynamic properties of the temperature sensors and the shape of the dynamically variable input signal [23,24]. Therefore, it can be compared with the accuracy class of instruments intended for static measurements (analogue or digital instruments). Researchers have shown that, by using the UBDE, it is possible to compare sensors that have analogous catalogue parameters but that are produced by competing companies. This approach also applies to sensors used in the energy industry. It should be noted that in this industry, sensors are required to be both highly accurate and reliable [25].

This paper presents the analysis of the UBDE for the absolute error criterion [26,27] and for the case of two types of signals with constraints [28], namely with constrained magnitude and rate of change. The first constraint concerns the measuring range of the sensor, while the second one results from its dynamic properties and is determined on the basis of the corresponding impulse response [29,30]. Such signals are determined on the basis of dedicated algorithms [26], which also allow one to determine the UBDE. The impulse response is obtained on the basis of the corresponding transfer function determined during the parametric identification [31,32]. This paper presents a detailed analysis that was carried out for two exemplary temperature sensors, namely the negative temperature coefficient (NTC) [33] and K-type thermocouple sensors [34]. The analysis presented in Section 4 allows one to perform similar tests for other types of sensors intended for temperature measurement and used in various fields of energy. In this way, one can obtain a summary of the UBDE values for sensors used in the energy industry that have similar parameters but are produced by different manufacturers. Such a summary may constitute the basis for a mutual comparison of temperature sensors and may increase their price competitiveness [35,36], which can significantly reduce the costs of measurements used in the power industry.

2. General Assumptions

Analysis of the UBDE obtained during the temperature measurement for the absolute error includes the seven main stages shown in Figure 1.

Stage 1 concerns determining the measurement points of the step response associated with the considered temperature sensor. This procedure is carried out by performing a dedicated measurement experiment supported by a DAQ card [37] and control and measurement software (e.g., LabVIEW [38]) or by using a specialised measurement systems [39]. This stage is not considered in this paper. The parameters underlying the numerical analysis are the step-response measurement points determined for the NTC and K-type thermocouple sensors.

Stages 2–7 are realised numerically by applying the relevant calculation methods.

Stage 2 concerns the approximation of the measurement points of the step responses obtained for the considered temperature sensors by using the polynomial method [40,41]. The implemented procedure is aimed at minimising the corresponding approximation error.

Stage 3 is directly related to Stage 2, which concerns determining the equation that represents the step response of the considered sensor. This equation is determined iteratively by a step-by-step increase in the order of the potential step-response equation. The uncertainty value of the polynomial approximation is also obtained. After obtaining the optimal order of the approximating equation, the values of the corresponding parameters and the associated uncertainties are determined.

Stages 4 and 5 include determining the UBDE alongside the sensor’s input signals, with one constraint (Stage 4) or two constraints (Stage 5). The procedures for determining both types of signals are based on the transformation of the step responses, except that the Stage 5 procedure, for obvious reasons, is more computationally complex. In the case of signals with one constraint, only the limitation related to the signal magnitude, which results from the measuring range of the sensor, is considered. Thus, such signals have a rectangular shape and the number of switchings and the duration of these switchings are determined [15,20]. For signals with two constraints, both the magnitude and the rate of change constraints are considered [15,20]. The second constraint results from the dynamic properties of the considered sensor and is determined on the basis of its impulse response (see Section 3). Such signals are, therefore, triangular or trapezoidal in shape. The UBDE is determined for the ranges of variability of the values of the step-response parameters, determined by the values of the associated uncertainties.

Stages 6 and 7 are devoted to summarising the UBDE values obtained during Stages 4 and 5.

The general guidelines for the methods and materials included in Figure 1 are presented in Section 3.

3. Methods and Materials

Let $\Lambda =\left[t_0 t_1 \dots t_{N-1}\right]$ denote the vector of the times for which the measurement points of the sensor’s step response were determined by the practical experiment. These points are included in the following vector $H=\left[h_0 h_1 \dots h_{N-1}\right]$, where $N$ denotes the number of measurement points. Then, the polynomial equation has the following form [32,33]:

$$h_{\mathrm{s}}\left(t\right)=a_0+a_{1}t+a_2{t}^2+\cdots +a_{\alpha }{t}^{\alpha },$$

(1)

where $h_{\mathrm{s}}\left(t\right)$ is the sensor step response; $a_0,$ $a_1,\dots , a_{\alpha }$ are the polynomial coefficients; $\alpha$ is the polynomial order; and $\alpha +1$ is the number of parameters of the polynomial. The estimates $\tilde{a}$ of the polynomial coefficients are obtained using the following matrix equation:

$$\tilde{A}={\left({\Phi }^{\mathrm{T}}\Phi \right)}^{-1}{\Phi }^{\mathrm{T}}H,$$

(2)

where

$$\Phi =\left[\begin{array}{cccc}1& t_{0 }& \cdots & {t_0}^{\alpha }\\ 1& t_{1 }& \cdots & {t_1}^{\alpha }\\ ⋮& ⋮ & ⋮& ⋮\\ 1& t_{N-1}& \cdots & {t_{N-1}}^{\alpha }\end{array}\right]$$

(3)

The standard uncertainty associated with the polynomial approximation is given by

$$u\left[h\left(t\right)\right]=\sqrt{\frac{{\left(\Phi \tilde{A}-H\right)}^T\left(\Phi \tilde{A}-H\right)}{N-\mathsf{\alpha }-1}},$$

(4)

while the standard uncertainties associated with the particular polynomial coefficients $a_{0,}$ $a_{1,}\dots , a_{\alpha }$ are calculated using the following equation:

$$u\left[a_i\right]=u\left[h\left(t\right)\right]\sqrt{{\mathsf{\Theta }}_{i,i}},$$

(5)

where

$$\mathsf{\Theta }={\left({\Phi }^T\Phi \right)}^{-1},$$

(6)

and $i=0,1,\dots ,\alpha .$

The relative uncertainties associated with the coefficients $a_{0,}$ $a_{1,}\dots , a_{\alpha }$ can be calculated using the following simple formula:

$$\delta (a_i)=\frac{u\left(a_i\right)}{a_i}$$

(7)

The optimal polynomial order $\alpha$ is determined by checking the chi-squared test given by the following formula:

$$\chi ={\sum }_{n=0}^{N-1}{\left[{\Delta }_n\right]}^2/{\sigma }_{\Delta }{}^2,$$

(8)

where

$${\Delta }_n=h_n\left(t, a_{0, } a_1,\dots ,a_{\alpha }\right)-H_n$$

(9)

and $n=0,1,\dots , N-1$, while $\sigma$ denotes the standard deviation calculated for the vector $\Delta$, which represents the approximation error. Equations (1)–(9) apply to Stages 2 and 3, which are included in the block diagram shown in Figure 1.

Let us denote ${\mathrm{UBDE}}_1$ as the upper bound of dynamic error, which is the product of the signal with one constraint defined by $x_{01}\left(t\right)$. The procedure for determining ${\mathrm{UBDE}}_1$ and the corresponding signal $x_{01}\left(t\right)$ (Stage 4 in Figure 1) is based on the following formulae [15,20]:

$${\mathrm{UBDE}}_1=A{\int }_0^T\left|k_{\mathrm{s}}\left(t\right)-k_{\mathrm{r}}\left(t\right)\right|\mathrm{d}t$$

(10)

and

$$x_{01}\left(t\right)=A \mathrm{sgn}\left[k_{\mathrm{s}}\left(T-t\right)-k_{\mathrm{r}}\left(T-t\right)\right],$$

(11)

where

$$k_{\mathrm{s}}\left(t\right)={\int }_0^t{h}_{\mathrm{s}}\left(t\right)\mathrm{d}t$$

(12)

and

$$k_{\mathrm{r}}\left(t\right)={\int }_0^t{h}_{\mathrm{r}}\left(t\right)\mathrm{d}t$$

(13)

are the impulse responses associated with the sensor and corresponding reference, while $A,$ $T$, $\mathrm{sgn}$ and $h_{\mathrm{r}}\left(t\right)$ denote the magnitude constraint, time of the sensor testing, signum function and the reference step response, respectively. The time $T$ corresponds to the time when the sensor’s step response $h_{\mathrm{s}}\left(t\right)$ reaches the steady-state value.

The reference denoted above by the subscript r is the mathematical model of a system with higher accuracy than the temperature sensor under consideration. This mathematical model can be represented by the impulse response $k_{\mathrm{r}}\left(t\right),$ the step response $h_{\mathrm{r}}\left(t\right)$ or the transfer function $K_{\mathrm{r}}\left(s\right),$ where $s=\mathrm{j}\omega$ is the Laplace operator, while $\omega =2\mathsf{\pi }f,$ $\mathrm{j}$ is an imaginary number and $f$ denotes the corresponding frequency. When the mathematical model of reference is given by the transfer function, then the impulse response $k_{\mathrm{r}}\left(t\right)$ can be obtained by applying the inverse Laplace transformation as follows:

$$k_{\mathrm{r}}\left(t\right)={\mathcal{L}}^{-1 }\left[K_{\mathrm{r}}\left(s\right)\right],$$

(14)

where ${\mathcal{L}}^{-1 }$ is the symbol of the inverse Laplace transformation. It should be emphasised here that the frequency response of the reference must be the same as the frequency response of the temperature sensor under consideration.

Let us denote ${\mathrm{UBDE}}_2$ as the upper bound of dynamic error, which is obtained as a result of the signal with two constraints denoted below by $x_{02}\left(t\right)$ (Stage 5 in Figure 1). ${\mathrm{UBDE}}_2$ is determined based on the following formula:

$${\mathrm{UBDE}}_2={\int }_0^T\left[k_{\mathrm{s}}\left(t-\tau \right)-k_{\mathrm{r}}\left(t-\tau \right)\right]x_{02}\left(\tau \right)\mathrm{d}\tau .$$

(15)

We can easily see here that in order to obtain the value of the ${\mathrm{UBDE}}_2$, it is necessary to determine the signal $x_{02}\left(t\right).$ This signal can only be obtained by transforming the additional function $f\left(t\right)$ as given by the following formula:

$$x_{02}\left(t\right)={\int }_0^{t}f\left(\tau \right)\mathrm{d}\tau .$$

(16)

The function $f\left(t\right)$ can be obtained based on the transformation of the impulse response

$$k\left(t\right)=k_{\mathrm{s}}\left(t\right)-k_{\mathrm{r}}\left(t\right)$$

(17)

with the following six main stages [15,20]:

1.

Determine the pulse function $f_1\left(t\right)$, the duration of which is equal to $2A/\vartheta ,$ where $\vartheta$ is the rate of change constraint. This pulse function can be obtained using the following formula:

$$\begin{gathered} f_1\left(t\right)=\vartheta /2A\mathrm{if}0<t\le 2A/\vartheta \\ {f}_1\left(t\right)=0\mathrm{if}2A/\vartheta <t\le T. \end{gathered}$$

(18)

The rate of change constraint is determined as the maximum deviations of the sensor’s impulse response from the steady state.

2.

Calculate the function $f_2\left(t\right)$ based on the function $f_1\left(t\right)$ and the impulse response given by Equation (17). The function $f_2\left(t\right)$ has the form of the following convolution integral:

$$f_2\left(t\right)={\int }_0^t{f}_1\left(t-\tau \right)k\left(\tau \right)\mathrm{d}\tau .$$

(19)

3.

Determine the rectangular function $f_3\left(t\right)$ as follows:

$$f_3\left(t\right)=A \mathrm{sgn}\left[f_2\left(T-t\right)\right].$$

(20)

4.

Determine the function $f_4\left(t\right)$ as follows:

$$\begin{gathered} f_4\left(t\right)=f_3\left(t\right)\mathrm{if}0<t\le 2A/\vartheta \\ {f}_4\left(t\right)=f_3\left(t\right)-f\left(t-2A/\vartheta \right)\mathrm{if}2A/\vartheta <t\le T. \end{gathered}$$

(21)

5.

Determine the function $f_5\left(t\right)$ as follows:

$$\begin{gathered} f_5\left(t\right)=f_4\left(t\right)\vartheta /A\mathrm{if}0<t\le A/\vartheta \\ {f}_5\left(t\right)=0\mathrm{if}A/\vartheta <t\le 2A/\vartheta \\ {f}_5\left(t\right)=f\left(t\right)\vartheta /2A\mathrm{if}2A/\vartheta <t\le T. \end{gathered}$$

(22)

6.

Determine the signal $x_{02}\left(t\right)$ by substituting the function $f_5\left(t\right)$ into the function $f\left(t\right)$ in Equation (16). The signal $x_{02}\left(t\right)$ is the one that produces ${\mathrm{UBDE}}_2$.

4. Example of the Analysis

The measurement points of the NTC and K-type thermocouple temperature sensors (Stage 1 in Figure 1) were determined by using a measurement system to study temperature transducers (Figure 2) [31].

Table 1. Specifications of the measuring equipment.

Name of the Device	Type	Operating Conditions	Manufacturer
Scientech 2302 TechBook, Study of Temperature Transducers	Scientech 2302	0–40 °C 85% Rh	Scientech Technologies

includes the specifications of the measuring equipment used in the tests. The NTC and K-type sensors are included in the Scientech 2302 TechBook, Study of Temperature Transducers, manufactured by the Scientech Technologies company.

The measurement results of the step responses for the NTC and K-type thermocouple temperature sensors are shown in

Table 2. Measured points of the step response.

Negative Temperature Coefficient (NTC) Sensor
t (min)	0	1	2	3	4	5	6	7	8	9	10
VNTC (V)	2.38	2.59	2.83	3.00	3.19	3.31	3.41	3.48	3.54	3.60	3.61
K-type Thermocouple Sensor
t (min)	0	1	2	3	4	5	6	7	8	9	10
VK (mV)	2.90	3.62	3.94	4.12	4.27	4.39	4.43	4.54	4.54	4.55	4.56

.

The times for both temperature sensors and the corresponding voltage responses (V_NTC and V_K, respectively, for the NTC and K-type thermocouple sensors) are included in Table 2. Figure 3 shows the step responses: $h_{\mathrm{NTC}}\left(t\right)$ and $h_{\mathrm{K}}\left(t\right)$ obtained for the NTC (a) and K-type thermocouple (b) sensors, respectively, based on the measurement points presented in Table 2. These step responses were obtained by using the polynomial method (Stage 2 in Figure 1). Equations (2)–(9) were used for this purpose.

The polynomial method was applied by successively increasing the corresponding order, with simultaneous checking using the chi-squared test given by Equation (8). The obtained polynomial equations, which are the step response of both sensors, have the following forms:

$$\begin{gathered} h_{\mathrm{NTC}}\left(t\right)=a_0+a_{1}t+a_2{t}^2+a_3{t}^3+a_4{t}^4+a_5{t}^5= \\ 2.38+0.192t+0.0321t^2-0.0117t^3+1.23 {10}^{-3}{t}^4-4.49 {10}^{-5}{t}^5 \end{gathered}$$

(23)

and

$$\begin{gathered} h_{\mathrm{K}}\left(t\right)=a_0+a_{1}t+a_2{t}^2+a_3{t}^3+a_4{t}^4= \\ 2.93+0.794t-0.178t^2+0.0192t^3-7.75\times {10}^{-4}{t}^4. \end{gathered}$$

(24)

We can easily notice that the optimal polynomial equations are fifth and fourth order, respectively, for the NTC and K-type thermocouple sensors. The values of the parameters are represented by the same number of significant figures, which is equal to three for each temperature sensor. The standard uncertainties that Equation (4) associated with the polynomial approximation are 0.0128 and 0.0461, respectively, for the NTC and K-type thermocouple sensors. The standard and relative uncertainties that Equations (4) and (5) associated with the step-response parameters of the NTC sensor are: $u\left(a_0\right)=0.0127,$ $u\left(a_1\right)=0.0307,$ $u\left(a_2\right)=0.0216,$ $u\left(a_3\right)=5.76\times {10}^{-3},$ $u\left(a_4\right)=6.46\times {10}^{-4},$ $u\left(a_5\right)=2.57\times {10}^{-5},$ $\delta \left(a_0\right)=0.532\%,$ $\delta \left(a_1\right)=16.0\%,$ $\delta \left(a_2\right)=67.1\%,$ $\delta \left(a_3\right)=49.3\%,$ $\delta \left(a_4\right)=52.6\%$ and $\delta \left(a_5\right)=57.3\%.$ The same uncertainties associated with the parameters of the K-type thermocouple sensor are: $u\left(a_0\right)=0.0442,$ $u\left(a_1\right)=0.0680,$ $u\left(a_2\right)=0.0298,$ $u\left(a_3\right)=4.59\times {10}^{-3},$ $u\left(a_4\right)=2.27\times {10}^{-4},$ $\delta \left(a_0\right)=1.51\%,$ $\delta \left(a_1\right)=8.56\%,$ $\delta \left(a_2\right)=16.8\%,$ $\delta \left(a_3\right)=23.9\%$ and $\delta \left(a_4\right)=29.3\%.$ The values of the chi-squared test are 24.4 and 38.5 for the NTC and K-type thermocouple sensors, respectively. It can be seen that the relative uncertainties obtained for the NTC sensor have higher numerical values than those uncertainties obtained for the K-type sensor. The maximum values of the relative uncertainties are equal to 67.1% (parameter $a_2$) and 29.3% (parameter $a_4$) for the NTC and K-type sensors, respectively.

Figure 4 shows the distribution of the approximation error $∆$, which is given by Equation (9), for the NTC and K-type thermocouple temperature sensors.

The maximum values of the error $∆$ are 0.0185 (3 min) and 0.0586 (1 min), while the minimum values of this error are 0.0017 (5 min) and 0.0023 (6 min), for the NTC and K-type thermocouple sensors, respectively.

For the purpose of determining ${\mathrm{UBDE}}_1$ and ${\mathrm{UBDE}}_2$, an eighth-order low-pass Bessel filter, with the transfer function defined as below, was assumed as the model of standard

$$\begin{gathered} K_{\mathrm{r}}\left(s\right)\text{:} \\ =\frac{198.8}{\left[{\left(\frac{s}{2\mathsf{\pi }{f}_{\mathrm{c}}}\right)}^2+3.53\frac{s}{2\mathsf{\pi }{f}_{\mathrm{c}}}+3.18\right]\left[{\left(\frac{s}{2\mathsf{\pi }{f}_{\mathrm{c}}}\right)}^2+1.79\frac{s}{2\mathsf{\pi }{f}_{\mathrm{c}}}+4.19\right]} \\ \frac{1}{\left[{\left(\frac{s}{2\mathsf{\pi }{f}_{\mathrm{c}}}\right)}^2+2.76\frac{s}{2\mathsf{\pi }{f}_{\mathrm{c}}}+3.84\right]\left[{\left(\frac{s}{2\mathsf{\pi }{f}_{\mathrm{c}}}\right)}^2+3.28\frac{s}{2\mathsf{\pi }{f}_{\mathrm{c}}}+3.38\right]}, \end{gathered}$$

(25)

where $f_{\mathrm{c}}$ denotes the filter cut-off frequency. The value of this frequency is determined on the basis of the Fourier transform module based on the following relation:

$$K_{\mathrm{s}}\left(f\right)=\left|\mathcal{F}\left[k_{\mathrm{s}}\left(t\right)\right]\right|=\left|{\int }_0^T{k}_{\mathrm{s}}\left(t\right) {\mathrm{e}}^{-\mathrm{j}2\mathsf{\pi }ft}\mathrm{d}t\right|,$$

(26)

where $\mathcal{F}$ denotes the symbol of the Fourier transform.

Figure 5 shows the result of the conversion of the step responses $h_{\mathrm{NTC}}\left(t\right)$ and $h_{\mathrm{K}}\left(t\right)$ to the impulse responses $k_{\mathrm{NTC}}\left(t\right)$ and $k_{\mathrm{K}}\left(t\right)$, for the NTC and K-type thermocouple sensors, respectively, using Equations (12) and (26).

Figure 5 shows that the impulse responses of both sensors increase. In the initial phase, they are non-linear, and, after 5 min, they are linear. It can also be seen that the impulse responses of both sensors reach final values that are many times higher than in the case of the corresponding step responses.

Figure 6 shows the result of the transformation of the impulse responses $k_{\mathrm{NTC}}\left(t\right)$ and $k_{\mathrm{K}}\left(t\right)$ of the NTC and K-type thermocouple sensors, respectively, executed using Equation (26).

From Figure 6, it follows that the frequency $f_{\mathrm{c}}$ to be substituted into Equation (25) is 0.0316 and 0.0317 Hz for the NTC and K-type thermocouple sensors, respectively. This frequency corresponds to the decrease in the characteristic $K_{\mathrm{s}}\left(f\right)$ (vertical dashed lines in Figure 6), with respect to the value of $K_{\mathrm{s}}\left(0\right)$.

Based on Equation (14), the impulse responses $k_{\mathrm{r}}\left(t\right)$ are determined for both sensors. These responses then form the basis for determining the impulse response $k\left(t\right)$, which is given by Equation (17).

For the purpose of determining ${\mathrm{UBDE}}_1$ and ${\mathrm{UBDE}}_2$ in Equations (23) and (24), the ranges of change of the parameters $a_0,a_1,\dots , a_{\alpha }$, which are determined by the associated uncertainties $u\left(a_0\right),$ $u(a_1),\dots , a_{\alpha }$, were taken into account. The Monte Carlo method, based on a pseudo-random number generator with a uniform distribution, was used for this purpose [14].

Figure 7 shows the signals $x_{01}\left(t\right)$—determined by Equation (11)—and $x_{02}\left(t\right)$—determined by Equation (16)—obtained for both considered sensors by applying the methods described in detail in Section 3. These signals produce ${\mathrm{UBDE}}_1$ and ${\mathrm{UBDE}}_2$. The signals $x_{01}\left(t\right)$ and $x_{02}\left(t\right)$ are defined by $x_{01\mathrm{NTC}}\left(t\right),$ $x_{01\mathrm{NTC}}\left(t\right)$ and $x_{01\mathrm{K}}\left(t\right),$ $x_{02\mathrm{K}}\left(t\right)$ for the NTC and K-type thermocouple sensors, respectively. The magnitude constraint $A$ was assumed to be the highest value of the step responses $h_{\mathrm{NTC}}\left(t\right)$ and $h_{\mathrm{K}}\left(t\right).$ Hence, the values of constraints $A$ are 3.61 and 4.56 V for the NTC and K-type thermocouple sensors, respectively.

The rate of change constraint $\vartheta$ was determined as the maximum deviation of the impulse responses $k_{\mathrm{NTC}}\left(t\right)$ and $k_{\mathrm{K}}\left(t\right)$ from the steady state. The values of these constraints are 31.97 and 42.17 m/V for the NTC and K-type thermocouple sensors, respectively. It should be emphasised that the values of the rate of change constraint affect the steepness of the slopes of the trapezoidal signal with two constraints.

It can be seen from Figure 7 that the signals $x_{01}\left(t\right)$ and $x_{02}\left(t\right)$ obtained for both considered sensors are signals with a maximum of two switches. This is due to the fact that the time responses (both step and impulse) for the temperature sensors are exponential characteristics. Signals with a much greater number of switching times are obtained, for example, for acceleration sensors [14], with time responses that are oscillating. It should be emphasised here that any other signal $x\left(t\right)$ included in the constraints of signals $x_{01}\left(t\right)$ and $x_{02}\left(t\right)$ could generate an error that, at most, is equal to the value of the UBDE [15].

The values of the ${\mathrm{UBDE}}_1$ and ${\mathrm{UBDE}}_2$ are: ${\mathrm{UBDE}}_{1\mathrm{NTC}}=681.9 \mathrm{V} \min$, ${\mathrm{UBDE}}_{1\mathrm{K}}=915.0 \mathrm{V} \min$ and ${\mathrm{UBDE}}_{2\mathrm{NTC}}=518.7 \mathrm{V} \min$, ${\mathrm{UBDE}}_{2\mathrm{K}}=835.6 \mathrm{V} \min$ for the NTC and K-type thermocouple sensors, respectively. The values of these errors correspond to the error defined as

$$\epsilon \left(t\right)={\int }_0^{t}k\left(t-\tau \right) x\left(t\right)\mathrm{d}t,$$

(27)

and determined for $t=T$.

Figure 8 shows the errors ${\epsilon }_{\mathrm{NTC}1}\left(t\right)$ and ${\epsilon }_{\mathrm{K}1}\left(t\right)$, determined for the NTC and K-type thermocouple sensors as a result of the signal $x_{01}\left(t\right)$, and the errors ${\epsilon }_{\mathrm{NTC}2}\left(t\right)$ and ${\epsilon }_{\mathrm{K}2}\left(t\right)$, determined for these sensors as a result of the signal $x_{02}\left(t\right)$.

It should be emphasised here that only for $t=T$ is it possible to obtain the UBDE for the absolute error criterion. Higher values of the UBDE are obtained for signals with one constraint $x_{01}\left(t\right)$, due to the vertical edges of this signal.

5. Conclusions

The results of the analysis described in Section 4 show that for the absolute error criterion, the signals producing the UBDE show a small number of switching times. This is due to the fact that the time responses (impulse and step) of sensors intended for temperature measurements have an exponential shape. However, this significantly reduces the implementation time of the numerical procedures aimed at determining the UBDE and corresponding input signals. In the case of sensors with an oscillating shape of time responses (e.g., sensors intended for acceleration or vibration measurement), the values of the implementation times of numerical procedures may be several times higher. This is due to the need to implement procedures based on evolutionary techniques, e.g., using the genetic algorithm [18]. Additionally, it can be seen that for signals with one constraint, higher values of the UBDE are obtained than for signals with two constraints. The ${\mathrm{UBDE}}_1$ values obtained for both sensors are 1.3 and 1.1 times higher than the ${\mathrm{UBDE}}_2$ values obtained for the NTC and K-type sensors, respectively. This is because signals with one constraint (rectangular shape) have steeper slopes than signals with two constraints (triangular or trapezoidal shape). It should also be emphasised that the analysis presented in this article could be applied successfully to different types of temperature sensors than the two considered in this paper. Thus, it is possible to compare the accuracy of sensors with analogous catalogue parameters but produced by different manufacturers. Such a comparison, as well as the selection of sensors with higher accuracy, is the basis for increasing the safety and reliability of the components and integrated systems used in the energy industry.

It should be emphasised that the solutions presented in this paper can be used to test all other sensors, for which it is possible to determine a mathematical description of the corresponding impulse response. The procedure for determining the UBDE for the absolute error criterion can be considered as the final product. For other types of sensors, it is an open issue to develop the appropriate methods for determining the mathematical description of the impulse response of the sensor under consideration.