Analog interface based on capacitance multiplier for capacitive sensors and application to evaluate the quality of oils

1.   Introduction

Nowadays, capacitive sensors are widely used in many areas of science and industry. Simple sensors for humidity, pressure, gases and other physical and chemical quantities ^[1], as well as more complex biosensors, such as for the detection of DNA ^[2], proteins ^[3], bacteria or cells ^[4,5,6], blood analysis ^[7], water analysis ^[8] and other biological and medical agents used in biological, medical and other fields of science.

The operating principle is simple, any change in the material under test (MUT), which is used as the dielectric material of a capacitor structure, such as the interdigital capacitor (IDC) structure, will change the relative electrical permittivity (${\varepsilon }_{r}$), and consequently, the capacitance of the structure.

Several methods have been proposed to measure the capacitance of similar structures ^[9,10,11,12], such as the double differential principle ^[13], differential measurement with current sense amplifiers ^[14], RC phase delay ^[15], charging and discharging method ^[16,17], oscillators ^[18,19], capacitance-to-phase converters ^[20], capacitance-to-frequency (C/F) converters ^[6,21], capacitance-to-time (C/T) converters ^{[22,23,24,25,26,27,28,29,30,31]}, switched-capacitors (SC) and the charge-transfer method ^[18,32,33]. Some of these methods use frequencies from kHz to MHz, higher frequencies for small capacitance measurements ^{[9,10,11,12,26,27,31,35]}, and some of them implemented in CMOS technology ^{[6,9,10,11,12,26,27]} are more complex. Direct sensor-to-microcontroller interfaces have the advantage of simplicity but are used to measure capacitances of tens and hundreds of pF ^[17], or with a specific algorithm to reduce the range to 1 pF on high frequency ^[35]. Simple ring and relaxation oscillators to measure the capacitance lower than 2 pF need a large value of resistor, even in MΩ range ^{[27,31,32,36]}, which is more vulnerable to noise. Also, operational amplifiers (OpAmp) with high SR (slew-rate) and GB (gain bandwidth) are needed, as well as high-speed comparators ^[34,37]. RC relaxation oscillators exhibit non-linear behavior owing to the inverse relationship between capacitance and frequency ^[34,37]. In particular, for capacitances of several pF, non-linearity was observed, as reported in ^[30]. Other techniques employ operational transconductance amplifiers (OTAs) or transimpedance amplifiers (TIAs) to measure capacitance below 1 pF ^{[9,10,11,12,24,29]}. High-performance methods, such as the capacitance-to-digital (C/D) converter ^[9,10,11,12], employ complex circuits integrated into CMOS technology ^{[9,10,11,12,29,30,31]}. In reference [38], an application for real-time monitoring of transformer oil condition was presented, using the AD7150. As an IC, it is a more complex circuit and typical works in the excitation frequency of 32 KHz as specified in its datasheet. The ICs in the AD7745/46/47 and AD7150/51/52/53 families exhibit a limited capacitance range, typically up to 20 pF ^[32].

In this study, the use of a capacitance multipliers are studied and used as the main stage for the implementation of a novel capacitance measuring technique with high sensitivity classified within the C/T converter category. Wide ranges of capacitances, from fF to μF can be measured. It can detect small changes in capacitance without the need for high frequencies, switching devices or more complex circuits.

Following that, the implementation of this method will be presented through an application for measuring the quality/degradation of edible oils or other liquid MUT, such as lubricant oils. According to ^[39,40,41], the increase of Total Polar Compounds (TPC) in edible oils due to their repeated thermal process is an indicator of their degradation and quality. Kumar et al. ^[42] showed that the electrical properties of oils can be used as indicators of the condition and quality of edible oils because they are inherently dependent on TPC and are well correlated with physical properties such as the viscosity of the oils.

Pérez and Hadfield ^[43] demonstrated that the same applies to lubricating oils, and can be used to estimate their quality. The authors implemented a sensor and high-frequency analog interface using CFA (current feedback amplifier) to measure the capacitance of the sensor.

These physical and chemical changes indicate the degradation of edible or lubricating oils, and are expressed as changes in the relative complex electrical permittivity ${\varepsilon }_{r}^{*}$ of the oils.

Thus, oils are the MUT to be measured, which are placed on the surface of an IDC sensor. According to ^[43], changes in the relative complex electrical permittivity ${\varepsilon }_{r}^{*}$ of the oils will also change the total capacitance of the sensor, in our case the IDC.

Pérez and Hadfield ^[43], shows that the electrical permittivity is equal to (1):

where ${\varepsilon }_{r}^{{\mathrm{\text{'}}}}$ is the real part of the relative complex electrical permittivity and represents the energy storage, while respectively ${\varepsilon }_{r}^{{\mathrm{\text{'}}}{\mathrm{\text{'}}}}$ is the imaginary part and expresses the losses of the dielectric material. In lubricating oils ${\varepsilon }_{r}^{{\mathrm{\text{'}}}}$≫${\varepsilon }_{r}^{{\mathrm{\text{'}}}{\mathrm{\text{'}}}}$, and in low frequencies the electrical permittivity has higher variations. Thus, the sensitivity of the sensor is higher when the measurement frequency remains constant. In the proposed circuit, the operation principle of measuring the capacitance is based on applying the desired adjustable constant DC current to the capacitive sensor to achieve the maximum sensitivity of the measurement circuit and measure the maximum capacitance, because of the effect of the relative complex electrical permittivity, which is maximum at 0 Hz. For certain materials under test on the IDC, such as water ^[44,45], the real and imaginary parts of (1) strongly affect the measurement capacitance and decrease as the frequency is increased from a few Hz to a few KHz. The authors of ^[44,45] showed that the measured capacitance of water can be dramatically changed in the range of 100 Hz to 2 KHz. Consequently, some materials under testing require extremely low frequencies (or 0 Hz) to measure the maximum capacitance with the highest attainable sensitivity.

The proposed circuit comprises a widely used 555 timer configured as a monostable multivibrator specifically designed to operate with DC current. A capacitance multiplier circuit and an adjustable constant current source are provided on a 555 timer as a novel modified monostable multivibrator to extend the capability of capacitance measurement circuits, enabling them to effectively measure capacitances as low as a few fF or even aF while maintaining high sensitivity. A constant current source is chosen because is more immune to noise. Additionally, the capability to adjust the desired current allows for control over the chosen measurement range within which the sensor operates, enabling a wide measurement range spanning from fF to μF. These changes adapt the circuit for measuring the capacitance of any tested material. Remarkably, this level of performance is comparable, if not superior, to more complex circuits. Additionally, this circuit is simple and cost-effective, necessitating only one OpAmp, three resistors and a single transistor in its basic configuration. It can be easily implemented using discrete components.

In the subsequent sections of this paper, we elaborate on a simple modification of the capacitance multiplier, which enhances the accuracy of the capacitance measurement circuit, and also elaborate on the error sources in measurement and linearity. This novel circuit and modification also offers the flexibility to easily upgrade existing monostable or astable multivibrator circuits, enabling them to operate effectively in scenarios involving low capacitances, whether using a constant current source or a voltage source.

2.   Materials and methods

The basic concept of the proposed capacitance-to-time converter (CTC) circuit illustrated in Figure 1 consists of a constant current source to provide more immunity to noise that linearly charges the capacitive sensor ${C}_{x}$ and, with the help of a timer, the charging time is converted into a pulse of the same duration as that of the charging time.

Sensor ${C}_{x}$ is linearly charged up to the timer threshold. Depending on the timer used, the threshold can be adjusted by using one of its terminals.

The 555 timer is a highly popular, exceptionally versatile and widely used analog IC. Its utility spans a diverse range of applications in electronic circuits, remaining a subject of ongoing research interest to this day, frequently incorporated as a component part in novel circuit designs ^[46].

Common circuits with the 555 timer as shown in Figure 2 are simple and work well for high values of capacitance ${C}_{x}$. However, when the capacitance ${C}_{x}$ becomes sufficiently small, as is the actual capacitance of an IDC sensor, the input parasitic capacitances of the circuit components become comparable or even higher than the capacitance we want to measure, which creates problems in the operation of the circuit and errors introduced in the charging time measurements.

The parasitic capacitances are connected in parallel to the capacitance of the sensor and form a current divider with respect to the current source. This means that the charging current of ${C}_{x}$ becomes significantly different (smaller) from the desired value when the parasitic capacitances have a value comparable to that of ${C}_{x}$. The total capacitance that will eventually be charged is ${C}_{x}+{C}_{stray}$, with ${C}_{stray}$ being the parasitic capacitances.

The simulation results using OrCAD PSpice for capacitance ${C}_{x}$ at 10 nF and 5 pF are shown in Figure 3. For the case of 5 pF, the charging current is no longer the desired one. Hence, the charging time of the capacitance is different than expected. The constant current source was set to 100 μA; however, as depicted in Figure 3(b), the current through capacitor ${C}_{x}$ is non-constant and does not exceed 6.6 μA.

The aforementioned reasons indicate that common and simple circuits utilizing the 555 timer are not suitable for accurately measuring small capacitances. Various alternative techniques have been developed to address this issue. In this work, the proposed circuit retains the simplicity of a typical 555-based capacitance measurement circuit with excellent linearity. Avoiding the use of oscillators, switched capacitor methods, complex switch topology, large MΩ resistors susceptible to noise, TIA or CFA configurations or other more complex techniques.

2.1.   Capacitance multiplier

In order to measure very low capacitance values (pF or fF) at low frequencies, a capacitance-to-voltage (CTV) converter, which consists of a capacitance multiplier and a constant current source, is used. The capacitance multiplier is a circuit that behaves as a multiplier of the capacitance ${C}_{x}$ and is shown in Figure 4(a).

The resulting equivalent circuit is illustrated in Figure 4(b), where the equivalent capacitor ${C}_{x}^{{\mathrm{\text{'}}}}$ appears to charge from ${V}_{in}$ through resistor ${R}_{S}$.

The resulting equivalent capacitance ${C}_{x}^{{\mathrm{\text{'}}}}$ and resistor ${R}_{S}$ are equal to:

The total current from ${V}_{in}$ is the sum of the currents ${I}_{1}$ and ${I}_{2}$ through resistors ${R}_{1}$ and ${R}_{2}$, respectively. The current ${I}_{2}$ is ${R}_{2}⁄{R}_{1}$ times smaller than ${I}_{1}$. The current of the equivalent capacitor is ${I}_{1}$. Current ${I}_{2}$ creates a voltage drop across resistor ${R}_{2}$ and is an error factor in the circuit. Ignoring this error, the equivalent capacitor charges at input voltage ${V}_{in}$ start at zero, which is desirable.

If the ratio ${R}_{2}⁄{R}_{1}$ is very high (greater than 100), ${R}_{S}$ can be equal to ${R}_{1}$.

In the case of applying a constant current source with a current ${I}_{s}$ through ${R}_{s}$ and ${C}_{x}^{{\mathrm{\text{'}}}}$, ${I}_{s}$ is equal to the current of the equivalent capacitance ${C}_{x}^{{\mathrm{\text{'}}}}$ and is given by:

The sum of the voltage drops across ${R}_{s}$ and ${C}_{x}^{{\mathrm{\text{'}}}}$ is denoted as ${V}_{in}$. This was determined using Kirchhoff's law and solving (3) for the voltage across ${C}_{x}^{{\mathrm{\text{'}}}}$. It is time dependent and is given by:

where $\varDelta t$ is the time elapsed since ${C}_{x}^{{\mathrm{\text{'}}}}$ is charging linearly.

Similarly, the ${V}_{in}$ on Figure 2(a) is given by:

2.2.   The overall circuit

Figure 5 shows the overall circuit with the implementation of the capacitance multiplier and use as timer a typical CMOS 555. The circuit's supply voltage is denoted as ${V}_{CC}$. The microcontroller (MCU) provides the trigger pulse to begin the charging of ${C}_{x}$ (transition of the voltage level at the TRIGGER input below $1/3{V}_{CC}$). A constant current source causes the charging; optionally, its current can be regulated by the MCU's digital-to-analog converter (DAC). The output of the 555 is HIGH until the voltage drop across ${C}_{x}$ exceeds the threshold of $2/3{V}_{CC}$. Then, capacitor ${C}_{x}$ discharges through the internal transistor of 555 and its output switches to LOW. The duration of the output pulse, that is, the charging time ${t}_{c}$, is timed by the MCU.

The estimated value of $\widehat{{C}_{x}}$ is calculated from the following, when the current ${I}_{s}$ of the linear charge and the charging time ${t}_{c}$ are known,

where ${V}_{c} = 2/3{V}_{CC}$ is the threshold voltage in the CONTROL pin of 555 and $N = {R}_{2}/{R}_{1}$ is the capacitance multiplication factor.

The voltage waveforms across ${C}_{x}$ at charging, across the equivalent ${C}_{x}^{{\mathrm{\text{'}}}}$ (which is the input voltage of the capacitance multiplier), the trigger signal and the output pulse of the monostable multivibrator, are shown in Figure 6.

3.   Measurement errors

The main goal is the measurement of the charging time of the equivalent capacitance ${C}_{x}^{{\mathrm{\text{'}}}}$ formed by the capacitance multiplier circuit. This time is translated by the monostable multivibrator as the time duration of the output pulse. The measurement errors that can occur are mainly due to the parasitic capacitances of the components connected to the same node as the capacitor ${C}_{x}$, the non-ideal characteristics of the operational amplifier (OpAmp), the transient effect at the beginning of charging and noise.

Charging starts from the voltage ${I}_{s}{R}_{s}$ to the threshold voltage ${V}_{c}$ of the monostable multivibrator. This leads to a shorter charging time than is actually needed. The error introduced by ${R}_{s}$ is an initial condition in charging ${C}_{x}$, and consequently, the equivalent ${C}_{x}^{{\mathrm{\text{'}}}}$.

3.1.   Effect of parasitic capacitances

Parasitic capacitances are often connected in parallel to the same nodes as capacitor ${C}_{x}$ and equivalent capacitance ${C}_{x}^{{\mathrm{\text{'}}}}$, causing significant errors.

These are the parasitic capacitances of the threshold and discharge inputs of 555, which are parallel to the equivalent capacitance ${C}_{x}^{{\mathrm{\text{'}}}}$. In addition, the parasitic capacitance of the non-inverting input of OpAmp is parallel to the capacitor ${C}_{x}$. The circuit that includes parasitic capacitances is shown in Figure 7.

${C}_{stray1}$ is the parasitic capacitance connected in parallel with the equivalent capacitance ${C}_{x}^{{\mathrm{\text{'}}}}$, and ${C}_{stray2}$ is the parasitic capacitance connected in parallel with the capacitor ${C}_{x}$. The parasitic capacitances of the traces on the PCB connections are added to both ${C}_{stray1}$ and ${C}_{stray2}$ and should be considered as well.

The equivalent unknown capacitance ${C}_{x}^{{\mathrm{\text{'}}}}$ can be calculated using the following:

Considering that the values of the parasitic capacitances are in the same order, the impact of ${C}_{stray1}$ is negligible compared to (${C}_{x}+{C}_{stray2}$) as it is multiplied by the large factor $N$. To face this problem, an OpAmp with very low input parasitic capacitance is chosen. Also, careful design on the PCB level should take place in order to minimize the impact of the conductors that connect ${C}_{x}$ with the OpAmp (e.g., be as short as possible to achieve a lower parasitic capacitance).

3.2.   Transient phenomena and error caused by non-ideal OpAmp

The transient effect that occurs at the beginning of charging in each measurement cycle, as well as the constant error generated by ${R}_{2}$, creates an initial non-zero voltage condition on the equivalent capacitance ${C}_{x}^{{\mathrm{\text{'}}}}$, and thus charging starts from this voltage up to $2/3{V}_{CC}$. This causes the charging time to decrease, as shown in Figure 8, and the result is an underestimation of the capacitance when it is calculated by the microcontroller.

The minimum output voltage of a non-ideal OpAmp may not be zero when both inputs are shorted together. This is defined by the parameter ${V}_{OL}$. When the non-inverting input ${V}_{+}$ is equal to zero, because the capacitor ${C}_{x}$ is initially uncharged and the circuit is a voltage follower, the inverting input ${V}_{-}$ and the output are equal to ${V}_{OL}$. ${V}_{OL}$ changes when the ${V}_{+}$ input is greater than the ${V}_{-}$ input, which is equivalent to ${V}_{OL}$.

The equivalent circuit is shown in Figure 9. The superposition theorem is used to calculate the voltage ${V}_{in}$, which is equal to ${I}_{2}{R}_{2}+{V}_{OL}$, where ${R}_{2}\left|\right|{R}_{1}$ with respect to the current source.

When ${V}_{+} < {V}_{OL}$, where ${V}_{-} = {V}_{OL}$, ${C}_{x}$ is charged exponentially by the voltage source ${V}_{in}$ after $5{R}_{2}{C}_{x}$ time constants. When ${V}_{+} > {V}_{OL}$, the output voltage of OpAmp follows ${V}_{+}$. Consequently, ${V}_{+}$ is equal to ${V}_{-}$ and the current source charges ${C}_{x}$ linearly.

Because the impact of ${I}_{2}{R}_{2}$ is negligible owing to the very high $N = 1000$, the minimum output voltage ${V}_{OL}$ can be reduced by adding a filter to the feedback of the capacitance multiplier. RC networks are formed in the circuit, which can cause a transient oscillation effect at the beginning of the charging.

The time constant for the RC network of resistor ${R}_{2}$ in series with capacitor ${C}_{x}$ is equal to:

and the time constant for the RC network of resistor ${R}_{1}$ in series with the equivalent capacitance ${C}_{x}^{{\mathrm{\text{'}}}}$ is:

If ${R}_{2} = N{R}_{1}$ and ${C}_{x}^{{\mathrm{\text{'}}}} = N{C}_{x}$, then (8) and (9) will apply:

Therefore, $\tau = \tau {\mathrm{\text{'}}}$ and these RC circuits are equal.

An additional RC network is formed. It consists of an equivalent capacitance ${C}_{x}^{{\mathrm{\text{'}}}}$ and a resistor ${R}_{2}$. The time constant $\tau {\mathrm{\text{'}}}{\mathrm{\text{'}}}$ for this network is equal to:

If an RC circuit is placed in the feedback of the capacitance multiplier, which is equal to or larger than $\tau {\mathrm{\text{'}}}{\mathrm{\text{'}}}$, the transient effect will be eliminated. That is, when:

where ${\tau }_{f}$ is the time constant of the feedback network, while ${R}_{3}$ and ${C}_{1}$ are the feedback components.

The feedback network affects the value of the voltage ${V}_{OL}$. Resistors ${R}_{3}$ and ${R}_{1}$ form a voltage divider with respect to ${V}_{OL}$. Hence, the inverting input voltage and the error introduced by ${V}_{OL}$ are decreased. Therefore, the non-inverting input should only exceed a fraction of the value of ${V}_{OL}$. The capacitance multiplier based on the ${R}_{3}{C}_{1}$ network is shown in Figure 10.

An improvement in linearity at the start of charging was observed using the ${R}_{3}{C}_{1}$ feedback network, as shown in Figure 11(b). The only remaining error is that due to ${I}_{2}{R}_{2}$. The error from ${I}_{2}{R}_{2}$ can be reduced by decreasing the ${R}_{2}$ value. The $N$ factor remains constant when ${R}_{1}$ is adjusted properly.

The capacitance ${C}_{x}$ is variable, as is ${C}_{x}^{{\mathrm{\text{'}}}}$ (the variability of ${C}_{x}$ leads to the variability of ${C}_{x}^{{\mathrm{\text{'}}}}$). Therefore, the constant feedback network induces an error in the circuit because the capacitor ${C}_{1}$ of the feedback network has a constant value.

To avoid this error, condition (13) should be applied for the maximum variation of ${C}_{x}^{{\mathrm{\text{'}}}}$. That is, ${C}_{1}\ge {C}_{x\left(max\right)}^{{\mathrm{\text{'}}}}$, since ${R}_{3} = {R}_{2}$. In this case, the pulse width at an output of 555 is closer to the expected value.

3.3.   Input offset error and ${V}_{OL}$

For a single supply operation, the values of the supply rails are 0 V and ${V}_{CC}$ in OpAmp. However, the minimum possible output ${V}_{OL}$ of OpAmp may not be 0 V and is given in the manufacturer's datasheet. For the TLC271 used in the capacitance multiplier circuit, the minimum voltage may be non-zero, and in the worst case, ${V}_{OL\left(max\right)} = 50\;{\mathrm{m}}{\mathrm{V}}$.

The input offset voltage ${V}_{IO}$, owing to the asymmetry of the input bias currents of the OpAmp inputs multiplied by the gain, appears as an output offset voltage.

Combined with the error caused by ${I}_{2}{R}_{2}$, this produces a total error in the OpAmp output. The error caused by resistor ${R}_{2}$ is the highest; therefore, it is necessary to eliminate all errors for high precision applications.

By choosing an OpAmp with lower ${V}_{IO}$ and ${V}_{OL\left(max\right)}$, it is possible to limit or eliminate the error due to asymmetry and the minimum output voltage of the OpAmp.

3.4.   Effect of noise

To measure small values of ${C}_{x}$, a high value of the factor $N$ is required so that the duration of the pulse can be measured by the microcontroller. This means that the ${R}_{2}$ resistor needs to be increased; simultaneously, the current through it will decrease, divided by $N$ times. As a result, ${R}_{2}$ becomes very vulnerable to thermal noise and any type of noise. A small current is more sensitive to charge quantization as well as to the noise figure (NF) caused by components in the overall circuit.

In the circuit shown in Figure 12, a noise source at the frequency of the power supply network (50 or 60 Hz) has been added to the circuit and connected in series with the resistor ${R}_{2}$. This noise source represents the susceptibility of the resistor ${R}_{2}$ to noise. The effect of this source on the duration of the output pulses is shown in Figure 13. The period of the TRIGGER signal was set as 12 ms.

When the period of the TRIGGER signal is changed to 20 ms, the noise effect on the circuit remains constant during each charging cycle and, consequently, during the charging time, i.e., the duration of the output pulse. The results are presented in Figure 14.

When the pulse of the TRIGGER signal has the same period as the noise signal (${T}_{noise} = 1⁄\left(50\;{\mathrm{H}}{\mathrm{z}}\right) = 20\;{\mathrm{m}}{\mathrm{s}}$), the noise is sampled and practically captured at the same value as it had or will have in the previous or next period $T$. A stable effect of noise in the measurement has been achieved and, consequently, the noise can be eliminated easily, provided that the amplitude of the noise in $T$ does not vary from period to period, and there is no phase difference between the TRIGGER and noise signals.

It should be noted that for short charging times (small values of ${C}_{x}$), a small portion of the noise period will affect the charging signal. The smaller this part is, the more linear the effect tends to become, as shown in Figure 15. For a longer charging time, the effect of noise occurs for a longer period, as shown in Figure 16.

In both Figure 15 and Figure 16, it is observed that increasing the noise voltage reduces the potential difference across the resistor ${R}_{2}$ and results in a reduction of the charging current, thus also increasing the charging time, and vice versa.

The effect of noise can be stabilized by synchronizing the measurement in the noise frequency and phase. In addition, to limit noise, digital filters can be implemented in the MCU to filter the time values of the output pulse width, i.e., the charging time of capacitor ${C}_{x}$.

3.5.   Error caused by current source

The current source of the proposed circuit was chosen among four different current source configurations: modified Howland, Wyatt, simple BJT and simple JFET, as shown in Figure 17.

Comparing these configurations of current sources on the maximum load that can be driven, it was found that Howland and simple BJT current sources yield the best results. In the constant current region, current sources are ordered from the highest current stability with respect to the load changes to the lowest, as follows: simple JFET, Howland, and simple BJT current sources. Wyatt current source, on the other hand, exhibits a tendency to significantly diverge as the load increases, rendering it unsuitable for this application.

Another aspect of comparison pertains to the stability of current sources in the constant current region over a temperature range of 0 to 80℃. The desired current was set at 100 μA. The results indicate that Howland is more stable than the others with a variation of only 1 nA. However, both the BJT and JFET current sources exhibited relatively small variations in the current within this temperature range. The variations in the desired current were 3.31 μA for the BJT current source, and 5 μA for the JFET current source.

Furthermore, a comparison of the current source stability under varying power supply voltages reveals that the BJT current source is more stable than all other configurations. In contrast, Howland is more sensitive to changes in the power supply voltage. The JFET remains stable, but dramatically reduces the load range in which it can operate as a constant current.

Simple BJT and Howland current sources emerged as optimal choices. However, Howland, in the range of zero to low loads, presented a significant current over the desired current owing to the ${V}_{OL}$ parameter. Therefore, to ensure simplicity and cost-effectiveness, a simple BJT current source was selected. Any diversion of the desired constant current can be compensated through calibration using a binary search algorithm.

The current source was set to 100 μA using a binary search algorithm based on the output voltage of the ESP32 DAC before initiating the measurement process. Additionally, the power supply voltage ${V}_{CC}$, emitter voltage ${V}_{E}$ of the current source and 555 threshold voltage ${V}_{c}$ were continuously monitored by the ESP32 through its ADC, which was software-linearized for enhanced linearity and accuracy.

The ESP32 internal 8-bit DAC offered a resolution of 12.941 mV. With a maximum error of 1 LSB, the resulting error at the current source was ±0.863%, which is considered acceptable. However, this error can be further reduced by employing an external, high-resolution DAC.

Nevertheless, precise monitoring of ${V}_{CC}$ and voltage ${V}_{E}$ across the emitter resistor ${R}_{E}$ (±0.1% tolerance) can eliminate the DAC error, enabling the calculation of current ${I}_{s}$ and determination of unknown capacitance ${C}_{x}$ with greater accuracy. Therefore, the only errors at the current source introduced by the tolerance of ${R}_{E}$ are ±0.99% and from 1 LSB error of the 12-bit ADC to monitor ${V}_{CC}$ and ${V}_{E}$, whereas in the worst-case scenario it is ±0.0806% for ${V}_{CC}$ and ${V}_{E}$. The monitored ${V}_{c}$ introduces error directly in the capacitance calculations, and is ±0.0244%. The sum of these errors, the total error in current, is ±1.0707%, and all errors affect the capacitance with a total error ±1.0948%. The fixed error of tolerance of ${R}_{E}$ is easy to eliminate with a correction factor given by calibration with low tolerance capacitors to pF or by accurately measuring the ${R}_{E}$ value and using this value in calculations for ${I}_{S}$. Thus, the remaining random error that affects the capacitance is only ±0.105% and is caused by ADC. The averaging process described in the next section can reduce this error further.

3.6.   Error caused by ${R}_{1}$ and ${R}_{2}$ tolerance

The multiplication factor $N$ is the ratio of ${R}_{2}/{R}_{1}$. The change of $N$ evidently affects ${C}_{x}^{{\mathrm{\text{'}}}}$. According to (5), the tolerance of these resistors affects the current ${I}_{2}$ and thus the charging time ${t}_{c}$. According to (6), this impacts ${C}_{x}^{{\mathrm{\text{'}}}}$ and finally the calculation of $\widehat{{C}_{x}}$.

The tolerance of ${R}_{1}$ and ${R}_{2}$ influence the accuracy of the measurement circuit. Specifically, for $N = 1000$ and 1% tolerance resistors (${R}_{1} = 100\;{\mathrm{\Omega }}$, ${R}_{2} = 100\;{\mathrm{K}}{\mathrm{\Omega }}$), in the worst case, the multiplication factor error deviates ±2.02% from the desired value. According to simulations for ${C}_{x} = 1\;{\mathrm{p}}{\mathrm{F}}$, this introduces ±1.494% error in ${I}_{2}$ and ±1.256% in ${t}_{c}$ from the values without tolerance. The total error of $\widehat{{C}_{x}}$ becomes ±0.393%. The error caused by resistor tolerance can be reduced with low tolerance resistors, such as 0.1%, which is the same as that used in ${R}_{E}$, or can be eliminated with a correction factor given by calibration with low tolerance capacitors as ${C}_{x}$ in pF.

4.   Simulation and experimental results

4.1.   Simulation results

The capacitor ${C}_{c}$ in Figure 5 is replaced by an appropriate resistor. This change causes a decrease in the threshold voltage in the CONTROL pin of 555 from $2/3{V}_{CC}$ to 2.626 V in order to remain within the dynamic input and/or output range of OpAmp.

The simulation results for capacitance variations of ${C}_{x}$ from 10 pF to 11 pF, with a step of 100 fF, and for $N = 1000$ are shown in Figure 18. The corresponding changes of the equivalent capacitance ${C}_{x}^{{\mathrm{\text{'}}}}$ for this range are from 10 nF to 11 nF, with a step of 100 pF.

According to Figure 18, the total pulse width will change incrementally by about 23.49 µs from 10 pF to 11 pF. A change of 1 pF corresponds to approximately 2.35 µs/100 fF. If a wider dynamic range of input and output of an OpAmp is chosen, then changes of 3.3 μs/100 fF can be achieved when the voltage supply remains at ${V}_{CC} = 5\;{\mathrm{V}}$ and the threshold voltage in the CONTROL pin of 555 is set to $2/3{V}_{CC}$.

These time variations can be measured easily by a microcontroller. The capacitance ${C}_{x}$ of the sensor can then be estimated.

4.1.1.   Linearity of interface

According to equations (5) and (6), it is expected that the measuring circuit will have a linear response (pulse width) to the changes of the capacitance ${C}_{x}$. Chart the pulse width line ${PW}_{line}\left({C}_{x}\right)$ extending from the minimum pulse width value $P{W}_{min}$ to the maximum pulse width value $P{W}_{max}$ of the measured pulse width. Following this, determine the discrepancies between ${PW}_{line}\left({C}_{x}\right)$ and the measured pulse width data ${PW}_{measur.}\left({C}_{x}\right)$. The non-linearity error can be calculated and expressed as:

In the range of 1 pF to 100 pF, a maximum non-linearity of 0.063% is observed, with a notable spike occurring at 30 pF, where the non-linearity reaches 0.085% (the value of ${R}^{2} = 1$), as illustrated by the simulation results in Figure 19.

4.2.   Experimental results

The experimental procedure used edible sunflower oil that was purchased from a local market. The oil has been divided into eight vials with a 5 ml capacity. Seven of these vials were placed in an oven at a temperature of 200℃, below the smoking point of the oil, and one vial was used as a reference for fresh sunflower oil.

Every 2 hours a vial was removed from the oven, and the total time the vial remained in the oven was recorded on the vial label. The last vial was removed after 14 h. The available samples are shown in Figure 20.

A 200 µl amount of oil from each sample was placed on the surface of the sensor using a precision micropipette. After each measurement, the sensor was cleaned before applying a new sample. Initially, the cleaning process was carried out with isopropyl alcohol; however, in practice, this procedure shifted the initial capacitance value of the sensor for measurement in air, possibly because of the various sorption phenomena occurring on the PCB substrate. Therefore, the cleaning process of the sensor surface was performed by thoroughly wiping it without the use of any solvent. This brings the capacitance of the sensor closer to its initial value.

Measurements of the sensor capacitance charge ramp waveforms and the corresponding 555 output pulse were performed using a SIGLENT SDS 1202X-E oscilloscope. The waveforms are shown in Figure 21.

The experimental setup was designed on a PCB and implemented as an integrated portable device with wireless communication via Bluetooth to transfer the measurements and display them on a smartphone. The device uses an ESP32 microcontroller with built-in wireless communication capabilities. The input time resolution of ESP32 is 12.5 ns at a clock frequency of 240 MHz. According to Eq. (6), the time resolution translates into detectable changes in ${C}_{x}$ of 476 aF @ ${V}_{c}$ = 2.626 V.

The IDC sensor had an initial capacitance of approximately 9.014 pF when measured in air, i.e., without oil on its surface. The geometrical specifications of the IDC are shown in Table 1.

The following parameters were used: $N = 1000$ (chosen for the resistor values of the capacitance multiplier: ${R}_{1} = 100\;{\mathrm{\Omega }}$ and ${R}_{2} = 100\;{\mathrm{K}}{\mathrm{\Omega }}$), ${I}_{S} = 100\;{\mathtt{μ}}{\mathrm{A}}$ and the threshold voltage in the CONTROL pin of 555 was set to ${V}_{C} = 2.626\;{\mathrm{V}}$ using an appropriate resistor value. The IDC sensor and the experimental setup as portable device are shown in Figure 22.

The noise standard deviations ${\sigma }_{{n}_{0}}$ for all the parameters measured by the device (raw data) were calculated and are listed in Table 2. The standard deviation after the averaging procedure was also observed.

The appropriate number of samples for the average was determined using the Allan deviation curve, as shown in Figure 23, for the charging time ${t}_{c}$.

All sample measurements were performed at room temperature (24℃) using the portable device and are shown in Table 3. 10,000 measurements were taken using ESP32. Outlier values exceeding two standard deviations were replaced, and an average of 10,000 valid measurements were calculated. The data is sent via Bluetooth to a smartphone.

An application using Android Studio was developed to communicate with the portable device and measure the duration of the TIMER output pulse, as well as other system parameters. In addition, the smartphone application was used to start the calibration process, and the current source was set to 100 μA from the output voltage of the ESP32 DAC by using a binary search algorithm, before starting the measurement process.

The experimental protocol can be summarized in the following steps: (a) Cleaning the IDC surface, (b) calibrate the constant current source to the desired current, (c) perform 10,000 measurements using an ESP32, outliers are rejected and replaced with new values, then averaging to calculate the capacitance of the IDC, (d) capacitance and other parameters of the circuit sent to a smartphone via Bluetooth, (e) 200 µl of oil sample was placed on IDC surface and (e) measure the new capacitance affected by the oil sample as described in (c). These steps are repeated for each new oil sample. The capacitance of oil samples was compared to that of fresh oil.

Figure 24 shows the corresponding graph of the measurements (pulse duration/width time), which appears as a light orange line. The light blue line represents the measurements where the difference between every initial value of the sensor before each measurement takes place and the initial reference value before the first sample (i.e., fresh sunflower oil) is taken into account. This difference is added algebraically to each pulse width time measurement to remove any offset error of the initial value of the sensor from the measurement. This error was caused by the non-exact repeated cleaning process, in which the initial capacitance value of the sensor shifted slightly each time.

The estimated capacitance of the IDC sensor for the respective measurements has been calculated using (6) and is also presented in Figure 24. The light orange and light blue lines for the time measurements and the orange and blue lines for the corresponding calculated capacitances of the sensor are shown in the graph.

The blue line in Figure 24 shows that the capacitance decreases and does not follow the expected increase, owing to the formation of polar compounds (increase in TPC), which occurs from time 0 to 4 hours. This probably occurred because of the decrease in moisture and the amount of water contained in the oil, simultaneously with the increase in TPC. The presence of water and moisture in the sample, owing to the high relative electrical permittivity of water (${\varepsilon }_{r}\cong 80$ @ 20℃), significantly varies the total capacitance of the sensor, and as it evaporates from the sample, the capacitance decreases. Once the water evaporated, which seems to occur after 4 hours, an increase in capacitance is observed, as expected, to follow the increase in TPCs.

Scatter plots and box plots depict the distribution of the measurements for the IDC sensor's minimum and maximum values, as depicted in Figure 25. Sensor repeatability was also calculated, with a value of 0.01436 observed within the 9 - 10 pF range.

Table 4 presents a comparison of the sensitivity of the proposed method with that of previous studies on capacitance-measurement circuits based on capacitance-to-period (time) converters.

In this study, the sensitivity was adjustable and defined by the factor $N$ of the capacitance multiplier. The comparison in Table 4 refers to the single-sensor measurements. The resolution of the proposed interface depends directly on the microcontroller clock and can be improved further.

The measurement range can be adjusted with two degrees-of-freedom (DoF) by varying the current supplied by the current source and the value of factor $N$. Specifically, for the system parameters set at ${I}_{s} = 100\;{\mathtt{μ}}{\mathrm{A}}$ and $N = 1000$, the theoretical lower value that can be measured is 476 aF, dependent upon the time resolution (clock) of the ESP32 or any other microcontroller. Zero sensor dead-zone was observed.

The influence of noise negatively affects the limit-of-detection (LoD) and the limit-of-quantification (LoQ). Noise stands as the primary limiting factor, particularly when ${I}_{2}$ is extremely low, owing to a desired reduction in ${I}_{s}$ and/or a high value of the $N$ factor, such as 10,000. Consequently, calculation of the LoD and LoQ becomes necessary. The LoD can be expressed as follows:

where $\sigma$ is the standard deviation of the sensor response in 0.25 pF capacitor (four 1 pF in series), and $s$ is the slope of the calibration process. The concertation of TPCs in oil samples can not be defined without expensive instruments. Common SMD capacitors within the range of 0.25 pF to 10 pF were employed for calibration. Subsequently, 20 final measurements were taken from the interface for each capacitor, and these measurements were averaged to determine the final value. Thanks to the linearity of the sensor, $s\cong 1$ for the output $\widehat{{C}_{x}}$ in pF. The LoD is calculated as 0.002090, in pF. The LoQ given by:

Similarly, the LoQ value was determined as 0.006334, in pF.

5.   Conclusions

An analog interface based on Capacitance Multiplier and an IDC sensor was implemented to measure the quality of edible oils. For $N = 1000$, the noise in the device was at an acceptable level and small differences in the capacitance were measured. Therefore, the oil quality was estimated. The total change in the sensor capacitance for the sunflower oil samples from the initial value of approximately 9.014 pF showed a maximum range of change from 1.338 pF to 1.382 pF for the 14-hour thermal processed pure sample without frying any edible material. The system measured total changes of 44 fF. The output pulse time variation was 1.15 µs/44 fF. The normal usage of frying edible materials with oil is expected to result in higher TPC production as the sensor's capacitance increases, making it more easily measurable. An improvement for a smaller effect of noise and denser structure of the IDC sensor may also allow for higher values of $N$, such as 10,000, which could further increase the sensitivity of the interface and improve the LoD and LoQ. In future work, the presence of moisture and water in the sample should be eliminated, for example, by thermal treatment or by covering or coating the surface of the sensor with hydrophobic films or polymers that selectively bind the chemical compounds that need to be measured.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.