Procedure for Determining the Upper Bound on Absolute Error for LVDT Sensors

This paper presents the procedure for determining the upper bound on absolute error (UBAE) for linear variable differential transformer (LVDT) sensors without inter-winding capacitance. This procedure is based on the results of a parametric identification of the sensor, from which the coefficient values of the corresponding transfer function are obtained. The UBAE is then determined by a simulation experiment using dedicated computational software. The error value and the shape of the input signal, which is constrained in magnitude, are determined. The calculations and additional validation of the procedure presented in this article are carried out using the MathCad 5.0 program. As the basis for the proposed procedure, we review issues related to the modelling of LVDT sensors without inter-winding capacitance. The solutions presented here can be used to determine the UBAE, which can then be used to assess the accuracy of these types of sensor and their mutual comparability in terms of dynamic accuracy.


Introduction
Linear variable differential transformer (LVDT) sensors are the type of devices that are widely used in industry to measure displacement or linear position, due to their precision, reliability, and ability to operate in harsh environmental conditions [1,2].They are typically deployed in measuring instruments where the accuracy of displacement or distance measurement is crucial, for example in thickness gauges, micrometers and other measuring tools [3], and can be used to monitor and control the positions of moving parts in industrial machines, such as pistons, shafts, and guides, or to control the positions of valves, trolleys, robot arms and other components.This allows for precise adjustment of production processes and increased machine efficiency [4].In medicine, LVDT sensors are used in a variety of applications, for example in medical apparatus for the precise positioning of surgical instruments, measuring displacements during diagnostic tests, or motion control for prosthetics [5].In research laboratories, these types of sensors are used in various scientific experiments where precise measurements of displacement or deformation of materials are required [6].A high level of accuracy is required from LVDT sensors due to their important roles in industrial applications.In engineering practice, this accuracy is ensured by calibration, and is most often determined based on the sensitivity of the tested sensor and the dependence of the output voltage on the position of the core [7,8].In many cases, the phase shift between the voltages output from the sensor's secondary circuits is also determined [9].The accuracy of LVDT sensors can be increased further by determining the upper bound on absolute error (UBAE) [10][11][12], using a mathematical model (transfer function) obtained from a parametric identification of the sensor [13].This identification is performed in the time domain by stimulating the sensor with a harmonic signal of constant amplitude but variable frequency [14].As a result of this practical identification experiment, the parameters of the mathematical model of the sensor are obtained.An example of the modelling of an LVDT sensor without inter-winding capacitance is presented in Section 2 of this paper.The next step, which is based on the results of parametric identification, is a simulation experiment aimed at determining the UBAE [10,11].This experiment involves determining the magnitudeconstrained input signal [15] for which the maximum value of the dynamic error is obtained [10].It should be emphasised that this is the highest error value that can be obtained for the tested LVDT sensor.The procedure for determining the UBAE is discussed in detail in Section 3 of this paper.

Mathematical model of the LVDT sensor without inter-winding capacitance
We consider the LVDT sensor for which an equivalent circuit diagram is shown in Fig. 1.This circuit includes one primary winding and two secondary windings, wound opposite to each other [1,2].

Fig. 1. Equivalent circuit diagram for an LVDT sensor without inter-winding capacitance 𝑅 , 𝑅
resistances of the primary and secondary circuits,  output resistance,  ,  self-inductances of the primary and secondary circuits,  ,  mutual inductances of the primary and secondary circuits.
The voltages  and  are the input and output of the sensor, respectively, while the  and  are the voltages generated by the secondary coils.The currents denoted by  and  are related to the primary and secondary circuits of the sensor.A core connecting rod allows to move the magnetic core to be moved.The core displacement is denoted as  in Fig. 1.
The output voltage  in domain  is where  j, and j √ 1 and  2π are the imaginary number and angular frequency in rad/s, respectively.From an analysis of the primary circuit of the sensor in the  domain, we have The transformation in Eq. ( 2) yields Analysing the secondary circuit of the sensor gives where    .The transformation in Eq. ( 4) gives Comparing the right-hand sides of Eqs. ( 3) and ( 5), we have where     ,  is the proportionality constant, and    .By substituting Eq. ( 6) into Eq.( 1) and applying a simple transformation, we obtain the transfer function   for the LVDT sensor without inter-winding capacitance as follows   (7)

Procedure for determining the absolute error
The procedure for determining the UBAE is based on the following formula where  and  denote the magnitude constraint and the interval of the input signal, respectively, and       .The functions   and   are the impulse responses of the sensor and the reference, respectively [10,11].The response   can be obtained as follows where ℒ denotes the inverse Laplace transformation,  ∈ ℝ ,  ∈ ℝ and  ∈ ℝ are the state, input and output matrices associated with the sensor transfer function, and , , ,  are the denominator order, numerator order, number of inputs, number of outputs, respectively.The canonical form of the state-space representation for the sensor transfer function given in Eq. ( 7) is as follows: The impulse response    is chosen in advance as a mathematical description of the analogue filter, and is expressed as where  ∈ ℝ ,  ∈ ℝ and  ∈ ℝ .
In this paper, we use a sixth-order Bessel filter, which in the canonical form of the state-space representation is and The signal with amplitude constraint  which produces the UBAE is given by the formula where sgn denotes a signum function.
The error as a function of time is given by the formula The error   corresponds to the value of the UBAE for  .

Implementation of the proposed procedure
We consider the mathematical model of the LVDT sensor shown in Fig. 1 with parameters obtained based on the parametric identification and included in [14].These parameters were determined using the measurement points of both frequency responses (i.e., the amplitude and phase), via a practical measurement experiment.The parameters of the LVDT sensor are given in Table 1.The calculations were carried out for a time , magnitude constraint  and cut-off frequency  with values of 50 μs, 2 mV and 100 Hz, respectively.The parameter  corresponds to the sensitivity of the sensor.The sixth-order Bessel filter given in Eqs. ( 12) ( 14) is applied as a reference.Fig. 2 shows the impulse response for the parameters given in Table 1, which were obtained for time  = 50 μs.

Fig. 2. Impulse response 𝑘 𝑡
The impulse response   for the reference is shown in Fig. 3.This response was obtained based on the formula given in Eq. ( 11) and with values for the magnitude constraint  and cut-off frequency  of 50 μs and 2 mV, respectively.

Fig. 3. Impulse response 𝑘 𝑡
Fig. 4 shows the shape of the signal   with magnitude constraint .This signal undergoes one time switching which is equal to 38.23 μs and produces the UBAE.The signal   was obtained using a simulation method based on Eq. ( 15).For any other dynamic signal included in this constraint, a dynamic error value that is lower than the UBAE will be obtained.

Fig. 4. Shape of the signal 𝑥 𝑡
Fig. 5 shows the relationship between the error  and time .The value of this error at   corresponds to the UBAE, and is equal to 9.00 10 Vs.The UBAE is calculated using Eq. ( 8).

Fig. 5. Error 𝑒 𝑡
The UBAE has the property that it reaches its maximum value at time  , as can be seen from Fig. 5.The figure shows that after switching of the signal   , the error   starts to increase exponentially, and reaches a maximum value at time .This is the highest possible error value that can be obtained in response to any input signal with a given magnitude constraint.The value of this error can be taken into account as an additional comparative criterion when assessing the accuracy of LVDT sensors with analogous catalogue parameters.Fig. 6 shows the relationship between the UBAE and the time .This error was determined for  ∈ 0, 300 μs .We can see here that the UBAE increases exponentially, and reaches a steady state at a value of 1.28 10 Vs.The solid line approximating the points determined for the UBAE was obtained using a tenth-order polynomial with the coefficients given in Table 2.The polynomial equation determined here for the UBAE allows us to obtain the value of the error which is the intermediate between the particular times .

Validation of the proposed procedure
In order to validate the results obtained here for the case of one constraint on the signal which maximises the error, we consider the harmonic signal described by the following function: where  4,095,024 Hz, which is included in the constraints  of the signal   .The maximum value of the error   is equal to 9.23 10 Vs.This error is therefore 10. 26 10 times lower than the maximum value of the UBAE, and was obtained for the LVDT sensor with the parameters given in Table 1 in response to a simulation signal with one constraint, denoted as   .

Conclusions
This paper presents the procedure for determining the UBAE for an LVDT sensor without interwinding capacitance when one constraint is imposed on the simulation signal exciting the input of this sensor.The results obtained in this work and our validation process confirm that the UBAE has the highest value for a signal with one constraint on its magnitude.The value of this error is 9.00 10 Vs.It should be emphasised that the UBAE and corresponding signal with one constraint were obtained through simulation; we note that for real signals, it is only possible to obtain errors with values which are lower than the UBDE.From the relationship between the UBAE and the time, it can be seen that this error first increases exponentially and then reaches a steady state.This relationship can form the basis for a comprehensive comparison of the accuracy of different types of LVDT sensors.

Fig. 6 .
Fig. 6.Relationship between UBAE and time Figure 7 shows the signals   and   .

Table 2 .
Coefficients of the polynomial used to approximate the UBAE