A Linear, Millimetre Displacement-to-Frequency Transducer

The paper presents a novel linear, high-fidelity millimetre displacement-to-frequency transducer, based on the resistive conversion of displacement into a proportional voltage, and then frequency. The derivation of the nonlinearity, fidelity and sensitivity of the transducer is presented. Experimental results confirm that a displacement of 0–100 mm is converted into a frequency range of 0–100 kHz, with a normalised fidelity factor of 99.91%, and a worst-case nonlinearity of less than 0.08%. Tests using laboratory standards show that a displacement of 10 mm is transduced with an accuracy of ±0.6%, and a standard deviation of 530 Hz. Estimates included in the paper show that the transducer could cost less than 1% of existing systems for millimeter displacement measurement.


Introduction
Several ultrasound, optical or laser-based devices exist for the measurement of displacements larger than one metre [1][2][3][4]. The cost of the modifications required for the use of these systems for measuring displacements in the range of a few micrometres to millimetres (submetre) is only justifiable in a few circumstances. For affordable submetre displacement measurements, capacitive and inductive position sensors are often used. However, the frequency dependence of capacitive and inductive sensors limits their domains of application [5][6][7]. In fact, a comparative discourse relating the range of displacement measurable versus the sensor recommended, could be found in [8].

OPEN ACCESS
In process and industrial instrumentation systems, several variables are detected using elastic sensors as primary sensing elements. Elastic sensors often generate displacements in the range of several micrometres to millimeters, which have to be conditioned further. Table 1, derived from information available in Chapter 8 of [9], shows example applications of elastic sensors resulting in an intermediate displacement variable. Moreover, physiological changes in biological tissues resulting from dehydration, accumulation of fluid due to disease, etc., can be studied using submetre displacement measurements [10]. Millimeter displacement is also encountered in the analysis of the integrity of civil structures [11,12], where such measurement systems as the GPS-RTS are currently used. A key challenge in the current systems for millimeter-displacement measurement is the high cost of acquisition of such measurement systems. Hence, there is significant motivation for the exploration of cheaper systems for use in small displacement measurement. Moreover, the transducers most suitable for the conditioning of such small displacement signals must have high sensitivity, high fidelity and minimum nonlinearity for acceptable accuracy of transduction. This paper presents the design, analysis and experimental validation of a submetre displacement-to-frequency transducer. The system is based on the sensitivity of some resistive elements to displacement. Resistive sensors are relatively cheap; and their zero-order dynamics make them suitable for both static and dynamic measurements. Unlike time-of-flight devices or phase-based measurement systems, resistive millimeter displacement transducers need be coupled physically to the displacement being measured.
In the rest of the paper, the circuit design, analysis of the basic displacement-to-voltage converter, and the implementation of the primary conditioning amplifier circuit are presented in Section 2. The voltage-to-frequency conversion design is presented in Section 3 of the paper. Circuit realisation, experimental results, and discussions of these results form Section 4 of the paper. Section 5 presents conclusions and the limitation of the transducer circuit. A list of references concludes the paper.

The Basic Displacement Sensor, and the Design and Analysis of the Primary Conditioning Circuit
The basic displacement-to-voltage sensor is shown in Figure 1. The sensor consists of a three-terminal potentiometer of total resistance R P , supplied by a DC voltage V s . The resistance between terminals A and B of the potentiometer is directly related to the displacement d(t) (alternatively, the normalised displacement x), where 0 ≤ x = ≤ 1; and d(t) = xd T . Then: (1) Note that, the maximum value of E TH is V s , when x = 1. The Thevenin's resistance of the equivalent sensor circuit, R TH , is evaluated to yield [9]: The sensitivity of E TH to the normalised displacement x is given by: To avoid excessive power dissipation in the resistance of the potentiometer, V s is usually kept small. Consequently, the sensitivity of this basic sensor is small. Additional conditioning is required to improve the sensitivity of the sensor. Now, the normalised sensitivity is given by:

Primary Conditioning of Sensor Output
The equivalent circuit resulting from the connection of a primary amplifier of input resistance R L , across terminals AB of the sensor circuit is shown in Figure 2. Loading effects tend to degrade the performance of amplifiers. The loading effect of the conditioning circuit modifies the Thevenin's voltage to: The normalised value of this voltage is also derived to be: In the next sub-section of the paper, Equations (5) and (6) are used to analyse the quality of the displacement-to-voltage conversion amplifier, and to show any additional condition(s) that could be imposed on the conditioning circuit to further improve the performance of the transducer.

Quality Analysis of the Primary Conditioning Circuit
In this subsection, the analysis of the quality of the primary signal amplification, based on Equation (5), is presented.

Fidelity of Primary Amplifier
Fidelity is a measure of how faithfully a circuit has processed a given signal to minimize distortions. The concept of fidelity is usually used in the analysis of high frequency amplifiers. In the current paper, the concept of fidelity is used to quantify the loading effect of the primary conditioning amplifier on the signal produced by the sensor. Now, the voltage drop due to the loading effect of R L is obtained to be: (7) or, in normalised form: (8) The normalised fidelity factor is then given as: For perfect fidelity, K = 0. Practically, this requires that:

Sensitivity Analysis
The sensitivity of the conditioned output is derived to yield: (11) The normalised sensitivity is found to be: (12) It is required to select the value of K in such a manner, as to minimize variations in within the range of measurements.

Nonlinearity Effects
From Equation (5), V L is nonlinear in x. The nonlinearity N(x) can be quantified by using: (13) where the linear part of V L is defined by the following parameters: (14) and the nonlinearity N(x) is given as: Nonlinearity is not desirable, and is eliminated as in Equation (10). In fact, it is evident from Equations (9), (10) and (15) that K 0 improves linearity and fidelity. This contradicts the requirement for enhanced sensitivity as in Equation (12), for which K ∞.The approach in this paper is to select K 0 for fidelity and linearity enhancement; and to effect sensitivity improvement using voltage to frequency conversion.

Practical Realisation of the Signal Conditioning Amplifier
The practical implementation of the primary conditioning amplifier uses the summing amplifier shown in Figure 3 [13,14], with the amplified voltage given by: An amplifier gain of 10 was used for the current work. This yields the normalised sensitivity parameter given in Equation (17).
(17) Table 2 summarises the parameters of the sensor and the amplifying circuit.   Table 2 with Equations (5) and (10) Similarly: (20) and the sensitivity of the amplified voltage is given by: In Section 3, we present a technique to further improve the sensitivity of the transducer, using voltage-to-frequency conversion.

Sensitivity Enhancement
As observed above, a small value of K (0.0056) was required to both minimize nonlinearity effects, and to enhance fidelity of the primary conditioning circuit. This value of K however, lowers the sensitivity of the transduction process. Since submetre displacements can be very small, a very high sensitivity transducer is required (as shown in Table 2, ideal sensitivity required is ∞). In the sequel, we present a voltage-to-frequency converter circuit that is used to further enhance the sensitivity of the developed transducer.

Voltage-to-Frequency (VFC) Conversion
Apart from sensitivity enhancement, the conversion of v 0 into a frequency signal has several other advantages, including: high noise immunity, high output power, wide dynamic range, and ease of interfacing with digital data acquisition systems. Table 3 shows key values of v 0 and their corresponding frequency representations. Table 3. Voltage-to-frequency table.

d (mm)
x v 0 (V) Frequency (kHz) 0 0 0 0 100 1 10 100 The linear relationship between v 0 and frequency in Table 3 is expressed mathematically as: Applying Equation (20) in Equation (22) we obtain: The AD 650 voltage-to-frequency converter (VFC) was used for the implementation of the displacement-to-frequency conversion circuit satisfying Equation (23). The pin layout of the AD 650 VFC is obtained from the manufacturer's manual for the device [15]. The selection of components for the VFC circuit is presented in the sequel.

Component's Selection for the AD 650 VFC
For the AD 650, only four component values must be selected by the user [15]. Using the manufacturer's notation, these are the input resistance R IN , the timing capacitor C OS , the logic resistor R 2 and the integration capacitor C INT . The first two are determined by the input voltage range and full-scale frequency. Additional relationship between R IN and C OS is provided through graphs obtainable in [15]. Sample design for a maximum frequency of 100 KHZ in the data sheet of the AD 650 VFC used R IN = 40 kΩ and this has been adopted for the realization in this study. Table 4 summarizes the components used for the design of the VFC circuit, with C INT calculated using the equation: and has also improved the resolution to = 10 −3 mm/bit.

Simulations, Experimental Validation, Results Presentation and Discussions
The indices assessing the quality of the transducer were evaluated by simulation. The rest of the results were obtained through experimental measurements.

Experimental Validation
The experimental setup is shown in Figure 7. For the experiments, a slide wire potentiometer was used as the submeter displacement sensor. It had a maximum displacement d T = 100 mm = 10 −1 m, and x Percentage Nonlinearity a total resistance of 11.2 kΩ (instead of the design maximum resistance of 10 kΩ). The potentiometer was supplied by a 1volt DC supply. The Thevenin voltage of the sensor, as a function of displacement, is shown in Figure 8. A plot of the amplified sensor voltage as function of detected displacement is shown in Figure 9. The overall displacement-to-frequency transduction is shown in Figure 10. For the analysis of the accuracy and precision of transducing displacement inputs into frequency, repeated measurements of 10 mm displacement were undertaken. The results are shown in Figures 11 and 12.    To put the above costs in perspective, Table 6 compares the cost of the reported transducer with those of existing displacement sensors. It is evident from Table 6, that the reported transducer has a very significant financial advantage over several existing systems for displacement measurements.

Conclusions
It is concluded that a cheap, linear, millimetre displacement-to-frequency transducer with both high sensitivity and high fidelity has been successfully realised.

Limitations
The design sensor resistance of 10 kΩ was not available. A sensor of total resistance of 11.2 kΩ was used instead. Whereas this larger resistance value did not directly affect the accurate performance of the transducer, it was observed that, the maximum output frequency was 120 kHz (instead of the design maximum frequency of 100 kHz). Temperature variations constitute a significant random impact on sensor performance. Temperature effects have not yet being characterized. The effect of supply voltage variation is also still under investigation. Test measurements were undertaken using laboratory standards. Traceability of accuracy shall be undertaken in subsequent development, using facilities at a national metrology centre.