# Quasi Single Point Calibration Method for High-Speed Measurements of Resistive Sensors

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{DD}, and the threshold voltage of the pins of the PDDs, V

_{f}. All these possible sources of error can be compensated thanks to calibration resistors. Another possible source of error would be the self-heating of the resistors. However, this phenomenon can be neglected in DICs, since the current does not pass through the resistors continuously and its value is low due to the high resistance normally shown by resistive sensors.

_{E}(R), can result in a high value. If the resistance value of a sensor is only to be measured sporadically, this problem may not be particularly important. However, if repetitive measurements are to be made for a single sensor or information can be obtained for different sensors (or even if these two situations occur simultaneously), then T

_{E}(R) becomes a fundamental parameter that can considerably slow down an application, or, in certain cases, even prevent the use of DICs. Finding a method to reduce T

_{E}(R) can be crucial in some practical applications, such as tactile sensors, where the number of resistive sensors to be read can be very high and, moreover, a high reading frequency of the sensors is required in order to calculate characteristics such as grips or slippages [29,30].

_{E}(R) to be reduced without the need for additional hardware while maintaining adequate accuracy in the measurement, making it possible to use DICs in applications such as those indicated above.

## 2. Evaluating R Estimation Time in Classical Calibration Methods

_{DD}through the Pp pin of the PDD (resistor Rp is optional and only necessary if the current through the pin needs to be limited, although in the literature we can also find that this resistor can reduce the influence of the power-supply noise [31]). After charging, a discharge is made either through R or through a calibration resistor, R

_{c0}. This is done by configuring the pin connected to the resistor that discharges as a logic 0 output, while the pin of the other resistor is configured as a high-impedance output. This way of proceeding in the discharge of a resistor will be called the normal discharge procedure. The discharge ends when the Pp pin (configured as the high-impedance input during discharge) detects a change to the logic 0 input. The capacitor is then re-charged and finally discharged again through the other resistor until a new logic 0 input is detected.

_{R}and T

_{Rc0}are the discharge times from V

_{DD}to V

_{f}through R and R

_{c0}, respectively (the sub-index of the time measurement will always indicate which resistor it is discharged through). These times will be expressed as the number of PDD clock cycles occurring during the discharge process. Since R

_{o}is unknown, we cannot obtain the value of R based on Equation (1). However, if, as usually occurs, R

_{o}is a small value compared to those of the different resistors to be measured, it is possible to approximate:

_{o}requires methods and circuits that use two calibration resistors: two-point calibration method (TPCM) and three-signal calibration method (TSCM) [5]. The TPCM, as shown in Figure 1b, carries out the same processes as the SPCM, but with two calibration resistors, R

_{c1}and R

_{c2}. In this method, the calculation of R is given [5] by:

_{c1}, R+R

_{c1}, and R

_{c2}+R

_{c1}. Proceeding in this way makes the equation for determining R simpler [32], which can be written as:

_{E}(R) is different for each of the aforementioned methods. If we define T

_{charge}as the time necessary for the capacitor charge, the values of T

_{E}(R) for the different methods are given by:

_{E}indicates which calibration method it is calculated with.

_{charge}is much lower than the other times of these equations, since, as mentioned, R

_{p}is either very small or not necessary, meaning we can eliminate this term from these equations. Secondly, calibration resistors are not the same in all methods. A common choice for the SPCM is to place R

_{c0}in the middle of the range of resistors to be measured [R

_{min}, R

_{max}], in order to minimize the maximum error when using Equation (2). Given this, Equation (5) becomes:

_{o}is very small compared to those of the resistors to be measured.

_{c1}and R

_{c2}to be, respectively, in 15% and 85% of the resistance value range to be measured in order to minimize errors in estimation [26], meaning Equation (6) can be written as:

_{c1}and R

_{c2}in the TSCM, and, although it is at least possible to maintain the 85% criterion for R

_{c2}, it is obvious that R

_{c1}cannot be in 15% of the range as its minimum value is R

_{min}+R

_{c1}. The systematic error made by this method will therefore always be greater than that obtained by the TPCM [33]. In any case, if we maintain the same criteria of 15% and 85% to situate the calibration resistors, we will have:

_{E}(R,TSCM) > T

_{E}(R,TPCM), meaning that, in terms of temporal performances for the same accuracy, the TPCM outperforms the TSCM [33]. It is also obvious that T

_{E}(R,SPCM) < T

_{E}(R,TPCM), although the TPCM is more accurate and thus, the application for which the DIC is used will determine which one to use. Given the foregoing, the ideal situation would be to find a calibration method with a T

_{E}(R) similar to or lower than T

_{E}(R,SPCM) and an accuracy equivalent to that obtained by the TPCM. The following section presents two new calibration methods that meet these two requirements when measuring the resistance value of a large number of sensors using the same DIC or carrying out repetitive measurements of the same sensor. We call these new methods quasi single-point calibration methods (QSPCMs).

## 3. Quasi Single-Point Calibration Methods

#### 3.1. Quasi Single-Point Calibration Method

^{th}estimation of R, R

^{n}(the superscript will be used to indicate the estimation number in all variables). Since the calibration resistors are fixed, we have the following relationship if Equation (3) is used for an initial estimation of R:

_{DD}, V

_{f}, and R are considered to be the same for the three measurements, since the temporal moments of the discharges of the series of measurements for an estimation are very close to each other (this approximation is one of the sources of error in any type of DIC). ${T}_{R}^{n}$, ${T}_{Rc1}^{n}$, and ${T}_{Rc2}^{n}$ are measured again in any new estimation of R and R

^{n}, thus updating the values of the voltages and R

_{o}.

^{n}, we can also use the data from the initial estimation of R as follows:

^{n}:

_{o}<< R

_{c1}, and, moreover, the variations of R

_{o}over time (due to the circuit conditions) are even smaller, meaning ${R}_{o}^{0}\approx {R}_{o}^{n}$. Hence, if we define:

^{0}we will need to evaluate the discharges through R

_{c1}and R

_{c2}; for any other estimate of R, we only need to evaluate the discharge through R

_{c1}. Equation (15) can be applied not only for a succession of measurements of the same sensor, but also when a series of resistive sensors is being measured (in this situation, R

^{n}would be the n

^{th}resistive sensor of the series).

_{c2}is close to the highest value to be measured and its discharge time is only necessary in evaluating R

^{0}, it is significantly time-saving in the estimations of R

^{n}. Moreover, it is obvious that the hardware necessary to use Equation (15) in a DIC is the same as for the TPCM, meaning there is no additional hardware cost. There may only be a computational cost in estimating R

^{0}derived from a quotient and an added multiplication in Equation (15) compared to with Equation (3). However, if the product $A\left({R}_{c2}-{R}_{c1}\right)$ is stored in the PDD memory, starting from R

^{1}, the number of arithmetic operations is lower in the QSPCM than in the TPCM, since subtraction of the denominator of the quotient is removed.

^{j}, similar to A that appears in Equation (14), taking these means into account:

#### 3.2. Fast Quasi Single-Point Calibration Method

_{x}(T

_{x}> T

_{Rc1}), followed immediately by the discharge of the capacitor through R

_{c1}until completion. This discharge procedure will be referred to as the R accelerates the discharge procedure. There is therefore a reduction in the measurement times of all resistors with T

_{R}> T

_{x}, and this reduction increases as the value of R increases. Using the FCM I, the value that T

_{R}would have with the normal discharge procedure can be found, according to the expression:

_{c1}after discharging through R for time T

_{x}. If we call the time used in the accelerated discharge procedure ${T}_{R}^{*}$, this can be calculated using the following expression:

_{R}= ΔT

_{R}(R,R

_{c1},T

_{x}) be the difference in measurement times between the normal discharge procedure and the accelerated discharge procedure for R. Its value is therefore given by:

_{x}decreases. However, the choice of T

_{x}also has implications for the maximum error in estimating R. Thus, the smaller T

_{x}(and also, in consequence, the time needed to find T

_{R}), the greater the error in estimating R may be, although Reference [33] shows that there may be an optimal T

_{x}values zone where this phenomenon does not occur.

_{c2}, with T

_{E}(R) decreasing even more. This method will be used to reduce T

_{E}(R), even though it also comes at a small cost in terms of accuracy of results. Obviously, T

_{Rc1}< T

_{x}< T

_{Rc2}must occur in order to apply the method.

_{c2}, the estimation of R is given by:

_{c1}after having done so through R in the n

^{th}estimation. For its part, A

^{*}is described as:

^{*j}is the same as A defined in Equation (17) but is with T

_{Rc2}of each estimation calculated from the measurements of the accelerated discharge procedure, similarly to that in Equation (24).

_{E}(R) in the new methods to those obtained in Equations (8) and (9), we are going to use the mean values of T

_{E}(R) (R being constant) in estimating n resistive sensors (or n estimations of the same sensor), $\mu \left({T}_{E}(R)\right)$. In the case of the SPCM and the TPCM: $\mu \left({T}_{E}(R),SPCM\right)$ and $\mu \left({T}_{E}(R),TPCM\right)$ match the values of T

_{E}(R) in Equations (8) and (9), since T

_{E}(R) is the same in any estimation of R

^{n}. However, in the QSPCM and in the Fast-QSPCM, the values of T

_{E}(R) differ in the first estimation. For its part, in QSPCM-j and Fast-QSPCM-j, the first j estimations have a different T

_{E}(R) value from those of the others. Table 1 shows the $\mu \left({T}_{E}(R)\right)$ for each method when n estimations of R are made.

## 4. Experimental Results and Discussion

_{c0}= 3486.8 Ω was used as a calibration resistor for the SPCM, and R

_{c1}= 1098.1 Ω and R

_{c2}= 6165.3 Ω were used for the other methods. All the resistors were measured using an Agilent 34401A digital multimeter. A number of 500 measurement cycles were performed for each of the 23 resistors used in order to measure the maximum errors and uncertainty. These 500 cycles were repeated again each time the resistors are estimated using a different method. In each cycle, the discharge time was measured through the resistor to be estimated and through one or both calibration resistors, depending on the method used and the measurement cycle in question. Thus, for example, for the QSPCM, discharge is via R

_{c2}only in the first measurement cycle, while in the remaining 499 cycles discharge is only through R

_{c1}and R.

_{x}= 163.84 µs was chosen for the comparison, i.e., half the time that can be measured with the 14-bit counter. This value was chosen as it is suitable for monitoring the value of the most significant bit of the counter, in order to know if time T

_{x}was reached during the discharge, thus facilitating the hardware to be designed in the FPGA. This choice implies that all resistors with values under approximately 4000 Ω discharge the capacitor by themselves, while larger resistors use the accelerated discharge procedure to do so. However, as shown in Figure 3a, resistors with values below 4000 Ω do not present the same errors using TPCM and FCM II since, in this second method, R

_{c2}is also evaluated using R

_{c1}. Although this reduces T

_{E}(R), it also increases the error. For its part, the Fast-QSPCM in Figure 3a presents very similar errors to FCM II, and only shows to some degree greater errors than FCM II for some of the higher resistance values. It is again important to note that the TPCM can be used to measure a maximum resistance value of slightly more than 7500 Ω; however, thanks to the decrease in discharge times for large resistors, the FCM II and the Fast-QSPCM can be used to measure resistors of up to 10 kΩ. For this reason, the graphs in Figure 3 only show the results obtained with FCM II and Fast-QSPCM for resistors with values greater than 7500 Ω. Figure 3b shows the relative errors made by these methods, again in a log2 scale, where it is observed that the relative errors remain practically constant for large resistance values, with very close values in all three methods.

_{max}= 7464.5 Ω and R

_{min}= 267.56 Ω. The results are calculated based on data used for Figure 2, Figure 3, Figure 4 and Figure 5 and equations listed in Table 1. The T

_{x}= 163.84 µs was retained for the Fast-QSPCM and Fast-QSPCM-4 methods.

_{max}according to the different methods when estimating a single sensor once. For its part, the last column shows the case in which 20 estimations are made (for a single sensor or 20 different sensors). Particularly striking is the fact that the Fast-QSPCM reduces the mean time for 20 estimations by 61% compared to the TPCM, or by 47% compared to the SPCM. However, the maximum relative error only increases by 0.1% compared to that with the TPCM and is 3.65 times lower than that with the SPCM. For their part, the QSPCM and the QSPCM-2 present the lowest relative errors of all the methods, and yet show important reductions in the mean estimation times compared to the SPCM and the TPCM.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Different types of direct interface circuits (DICs): (

**a**) single-point calibration method (SPCM); (

**b**) two-point calibration method (TPCM); (

**c**) three-signal calibration method (TSCM).

**Figure 2.**Errors in estimating resistance values using SPCM, TPCM, and QSPCM: (

**a**) absolute maximum errors (linear scale); (

**b**) relative maximum errors (log2 scale).

**Figure 3.**Errors in estimating resistance values using the TPCM, FCM II, and Fast-QSPCM: (

**a**) absolute maximum errors (linear scale); (

**b**) relative maximum errors (log2 scale).

**Figure 4.**Comparison of errors made in estimating resistance values using the QSPCM and different values of j in QSPCM-j: (

**a**) absolute maximum errors (linear scale); (

**b**) relative maximum errors (log2 scale).

**Figure 5.**Comparison of errors made in estimating resistance values using Fast-QSPCM and different values of j in Fast-QSPCM-j: (

**a**) absolute maximum errors (linear scale); (

**b**) relative maximum errors (log2 scale).

**Table 1.**Mean time for an estimation of R, if n estimations are made, both in traditional methods and in the methods presented in this paper.

Method | $\mathbf{Mean}\text{}\mathbf{of}\text{}{\mathit{T}}_{\mathit{E}}\mathit{R}\text{}\mathbf{for}\text{}\mathit{n}\text{}\mathbf{Estimations},\text{}\mathit{\mu}\left({\mathit{T}}_{\mathit{E}}(\mathit{R})\right)$ |
---|---|

SPCM | ${T}_{R}+{T}_{Rc0}$ |

TPCM | ${T}_{R}+{T}_{R\mathrm{max}}+{T}_{R\mathrm{min}}$ |

Fast calibration method II (FCM II) | $\{\begin{array}{c}\mu \left({T}_{E}(R,TPCM)\right)-\Delta {T}_{R}\left(R,{R}_{c1},{T}_{x}\right),\hspace{1em}{T}_{R}>{T}_{x}\\ \mu \left({T}_{E}(R,TPCM)\right),\hspace{1em}{T}_{R}\le {T}_{x}\end{array}$ |

Quasi single-point calibration method (QSPCM) | ${T}_{R}+{T}_{Rc1}+\frac{{T}_{Rc2}}{n}$ |

Fast single-point calibration method (Fast-QSPCM) | $\{\begin{array}{c}\mu \left({T}_{E}(R,QSPCM)\right)-\Delta {T}_{R}\left(R,{R}_{c1},{T}_{x}\right)-\frac{\Delta {T}_{R}\left({R}_{c2},{R}_{c1},{T}_{x}\right)}{n},\hspace{1em}{T}_{R}>{T}_{x}\\ \mu \left({T}_{E}(R,QSPCM)\right),\hspace{1em}{T}_{R}\le {T}_{x}\end{array}$ |

QSPCM-j | ${T}_{R}+{T}_{Rc1}+j\frac{{T}_{Rc2}}{n}$ |

Fast-QSPCM-j | $\{\begin{array}{c}\mu \left({T}_{E}(R,QSPCM-j)\right)-\Delta {T}_{R}\left(R,{R}_{c1},{T}_{x}\right)-j\frac{\Delta {T}_{R}\left({R}_{c2},{R}_{c1},{T}_{x}\right)}{n},\hspace{1em}{T}_{R}>{T}_{x}\\ \mu \left({T}_{E}(R,QSPCM-j)\right),\hspace{1em}{T}_{R}\le {T}_{x}\end{array}$ |

**Table 2.**Comparison of the performance of the different methods for a range of resistors with values between 267.56 Ω and 7464.5 Ω.

Method | Max. Absolute Error (Ω) | Max. Relative Error (%) | $\mathit{\mu}\left({\mathit{T}}_{\mathit{E}}({\mathit{R}}_{\mathbf{max}})\right)$ µs | |
---|---|---|---|---|

1 estimation | 20 estimations | |||

SPCM | 10.07 | 2.63 | 449.83 | 449.83 |

TPCM | 5.11 | 0.62 | 605.00 | 605.00 |

FCM II | 6.00 | 0.92 | 409.11 | 409.11 |

QSPCM | 5.56 | 0.52 | 605.00 | 364.54 |

Fast-QSPCM (T_{x} = 163.84 µs) | 7.42 | 0.72 | 409.11 | 238.36 |

QSPCM-2 | 5.10 | 0.54 | 605.00 | 377.20 |

Fast-QSPCM-4 (T_{x} = 163.84 µs) | 6.79 | 0.71 | 409.11 | 265.32 |

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**MDPI and ACS Style**

Botín-Córdoba, J.A.; Oballe-Peinado, Ó.; Sánchez-Durán, J.A.; Hidalgo-López, J.A.
Quasi Single Point Calibration Method for High-Speed Measurements of Resistive Sensors. *Micromachines* **2019**, *10*, 664.
https://doi.org/10.3390/mi10100664

**AMA Style**

Botín-Córdoba JA, Oballe-Peinado Ó, Sánchez-Durán JA, Hidalgo-López JA.
Quasi Single Point Calibration Method for High-Speed Measurements of Resistive Sensors. *Micromachines*. 2019; 10(10):664.
https://doi.org/10.3390/mi10100664

**Chicago/Turabian Style**

Botín-Córdoba, Jesús A., Óscar Oballe-Peinado, José A. Sánchez-Durán, and José A. Hidalgo-López.
2019. "Quasi Single Point Calibration Method for High-Speed Measurements of Resistive Sensors" *Micromachines* 10, no. 10: 664.
https://doi.org/10.3390/mi10100664