Enhancing sensor linearity through the translinear circuit implementation of piecewise and neural network models

1.   Introduction

Sensors play a crucial role in various industries including automotive, electronics, mining, and waste management. The properties of a sensor directly impact important parameters such as process productivity, resilience and accuracy, particularly within the automation industry. However, the nonlinearity of sensors, influenced by factors like the properties of the metallic components, aging effects and hysteresis errors, imposes limitations on the performance of processes in automation. In the literature, numerous methods for linearization have been proposed, categorized as either hardware-based or software-based approaches.

Thermocouples hold significant importance within the manufacturing industry, primarily employed for temperature measurement in furnaces and molten metal applications. The nonlinearity errors observed in thermocouples stem from two major factors: the cold junction temperature and the properties of the metal, specifically the Seebeck coefficients. The hardware circuit, as described in reference [1], processes the output signal from the thermocouple using a voltage-controlled oscillator (VCO). The resulting signal is then transmitted to a PC for further processing, where a suitable algorithm is employed to extract the cold junction temperature. Utilizing a two-dimensional continued-fraction-based algorithm, the hot junction temperature is accurately measured while minimizing non-linearity within a range of $\pm$1.4$^{\circ}$C. The proposed hardware plus software-based approach in the study ^[2] employs the inverse reference function for linearizing the K-type thermocouple in the range of -100$^{\circ}$C to 1372$^{\circ}$C. A decent improvement in the linearity of $\pm$1$^{\circ}$C is obtained by referring to the thermo-emf vs temperature reference table. A dual input logarithmic amplifier is proposed in the work ^[3] to linearize the T-type thermocouple in the range of 0-100$^{\circ}$C. Appropriately choosing the value of $\lambda$ which is the ratio of feedback resistance to the bridge resistance can improve the linearity. The normalized linearity error of 0.0018 is achieved in the work ^[3]. The logarithmic amplifier built with the ratiometric function is proposed in the work by the investigators ^[4] to linearize the various types of thermocouples. The linearity improvement in terms of less root mean square error of 0.051 is achieved for T-type thermocouple over the temperature range of 0-400$^{\circ}$C. The optimal local linear models are used via a data-driven approach to linearize various thermocouples, as discussed in the paper ^[5]. Excellent improvement in the linearity of $\pm$0.12% and $\pm$0.41% is achieved for J-type, and K-type thermocouple is achieved in the proposed multi-linear model approach ^[5]. To linearize the thermocouple, the authors ^[6] use a radial basis function-based neural network technique with a gradient descent algorithm for tuning its parameters. Despite implementation problems, the work ^[6] addresses a significant amount of linearity improvement. The MOSFET-based translinear circuit for realizing the non-linear function to linearize the T-type thermocouple is proposed in the work ^[7]. A decent improvement in the linearity of 75% is reported with less power consumption.

The Pt100 sensor, which has a resistance of 100$\Omega$ at 0$^{\circ}$C (32$^{\circ}$F), is the most prevalent sensor in the process industry. The error between the temperature and the resistance curve is substantially high in the low-temperature region (from 50$^{\circ}$K to 200$^{\circ}$K) and small in the high-temperature region. One of the methods proposed in the literature ^[8] employs the analog circuit with a positive feedback path with calibration performed as transfer function fragmentation yields better results. The bridge circuit-based linearization of RTD is proposed in the literature ^[9]. The circuit is built with the transfer function of RTD with an absolute error term introduced and the error is reduced with the aid of a bridge circuit, differential amplifier, and supply voltage former. A decent amount of linearity of 0.1$^{\circ}$C is reported in the work ^[9]. The logarithmic ratiometric function with its parameters is optimized using the evolutionary algorithm employed for linearizing the RTD proposed in the work ^[10]. The LABVIEW-based implementation ^[10] of non-linear function completely eliminates the problem of the non-linearity of any sensor. The linearization of RTD with a voltage divider and feedback compensation is proposed ^[11] with the estimated error after linearization being 0.5$^{\circ}$C.

The implementation of two-stage piecewise linearization proposed in the work ^[12] for the linearization of thermocouples results in a better reduction of error. The study ^[13] reports a non-linear regression approach for K-type thermocouples with good linearity of 0.02%. The polynomial estimation method of linearizing T-type thermocouple, discussed in the work ^[14], results in an accuracy of 0.3$^{\circ}$C. In the study referenced as ^[15], Artificial Neural Networks (ANN) are suggested as a linearization technique, demonstrating effective linearity through inverse modeling. However, challenges related to implementation have been encountered.

In this study, the application of piecewise and neural network regression models through the analog mode circuit (Translinear) is introduced to achieve linearization of thermocouples and RTDs. The effectiveness of the linearization approach is assessed by quantifying the full-scale error (FSE). The research ^[16] introduces an approach that utilizes translinear circuitry to implement an evolutionary optimized non-linear function, specifically for the purpose of linearizing different thermocouples. This work serves as a notable reference point for comparing the results of the proposed translinear implementation.

2.   Materials and method

2.1.   Piecewise regression

Piecewise regression, also referred to as segmented regression or broken-stick regression, is a regression analysis method that divides the independent variable into intervals and fits each interval with a separate line segment. It is particularly useful for modeling data that exhibits a change point or breakpoint. The precise location within the variable where the functional form changes plays a crucial role in segmented regression, as it signifies a shift in the data pattern. This shift is typically represented by a node, a shared point, or a change point. In this study, the parameterization of segmented linearization is defined as follows.

where $\beta_{0}$ is the y intercept, $\beta_{1}$ and $\beta_{2}$ are the regression coefficients and $\gamma$ is the break point.

2.2.   Neural network

Artificial neural networks (ANN) are a form of machine learning that leverages artificial intelligence to identify patterns within extensive datasets. They are capable of predicting outputs based on given inputs using either neural networks or linear regression models. The effectiveness of regression models relies heavily on the suitability of the regression equation for the data at hand. Despite their computational complexity, neural networks possess versatility as they can dynamically adapt and select the most suitable regression approach. The process of linearizing a sensor can be perceived as a regression problem, aiming to forecast sensor output by utilizing target or straight-line data.

2.2.1.   Levenberg-Marquardt algorithm

A widely used technique for training a linear regression problem is the back-propagation algorithm, which employs the gradient descent method (GD) in a supervised learning setting. Its objective is to minimize the network's error by propagating the gradient of the error curve, as described in reference [17]. To achieve error function minimization, the back-propagation algorithm utilizes the Levenberg-Marquardt (LM) algorithm, where a conservative step size is maintained to ensure the accuracy of the linear approximation.

The LM algorithm combines elements of both the steepest descent and Gauss-Newton methods. During training, the error at the output is computed by comparing the desired output vector (linear output) with the actual output vector (nonlinear output) resulting from the sensor's input (nonlinear output).

where $e_{m}$ - Training Error in voltage, $d_{v}$ - Desired output vector in volts and $a_{v}$ - Actual output vector from sensor in volts. The sum square error($E_{v}$) is calculated from the training error as follows.

The steepest descent algorithm uses the first-order derivative of the error function with respect to weight and can be called a gradient. Thus, the change in weight of the steepest descent algorithm is the function of the previous weight and the gradient (g). Unlike the steepest descent algorithm, Newton's method implies that all gradient components are functions of the weights, as follows.

Consequently, the second-order derivatives of the total error function must be determined for each component of the gradient vector. The resulting matrix is called the Hessian matrix. However, the computation of the Hessian matrix is complex and can be simplified by introducing the Jacobian matrix, which represents the first-order partial derivatives of an error function. Thus, the algorithm LM is constructed using the Jacobian matrix and a combination coefficient $\mu$. The weight change in each iteration using the LM algorithm is as follows.

Thus, the convergence speed of the LM algorithm is stable and fast due to the introduction of the combination coefficient $\mu$. It is challenging to determine the ideal number of hidden neurons, but a workable solution has been developed and is described in the paper ^[18].

2.3.   Translinear circuits

The term "translinear" denotes the linear correlation between the drain current and transconductance in MOSFETs. Figure 1 showcases a circuit that possesses the ability to embody fundamental mathematical expressions by adhering to the translinear principle in both stacked and folded configurations ^[19]. According to the translinear principle of MOSFETs, assuming a zero-body effect (connecting bulk of the NMOS to its source in a process with deep N well), the product of clockwise and counterclockwise currents remains constant. The MOSFET is dependent on the exponential current-voltage relationship of a circuit element. The ideal element follows the current relationship as follows.

where $I_{S}$ is the saturation current, $V_{gs}$ is the gate-source voltage, $V_{Th}$ is the threshold voltage, $V_{T}$ is the thermal voltage, $\mu$ is the mobility of the electrons, $C_{ox}$ is the oxide capacitance per unit area and n is the weak inversion slope factor. According to Kirchhoff's law applied to the loop, the sum of the voltages around the device, connected clockwise and counterclockwise respectively, is equal to zero.

Because of the exponential characteristics as expressed in the above Eq 2.6, the translinear principle can be expressed as follows.

Eqs (2.7) and (2.8) enables us to realize simple mathematical operations such as multiplication and division using translinear or current mode circuits.

2.4.   Field programmable array

A Field Programmable Array (FPGA) is a type of programmable logic device (PLD) that offers a high degree of flexibility and customization for digital circuit design. It consists of an array of configurable logic blocks interconnected by programmable interconnects. The logic blocks typically include lookup tables, registers, multiplexers and other digital components. The key feature of an FPA is its ability to be reconfigured or programmed by the user to implement specific digital functions or circuits. This programming is done using hardware description languages (HDLs) or graphical tools that allow designers to define the desired functionality and interconnections.

3.   Results

In this study, translinear circuits are utilized to implement piecewise and neural regression models for sensor linearization. The chosen sensors for evaluation are T, K, N and J-type thermocouples and an RTD (Pt-100). Table 1 presents the measurement range, full-scale error (FSE) and mean square error (MSE) of these sensors prior to linearization. To facilitate the translinear implementations, the sensor outputs were scaled within the range of 0-5$\mu$A.

3.1.   Implementation of piecewise regression model

The process of piecewise or segmented regression involves partitioning sensor data into distinct segments with predetermined breakpoints or nodes distributed uniformly. A function is then fitted to each segment to capture the underlying relationship. This study employs the 'R' programming language to identify the breakpoints and establish the piecewise regression model for aligning the sensor output with a straight line. 'R' serves as a statistical computation and analysis tool, enabling the creation of an optimized regression model for the sensor. Specifically, the "segmented function" in 'R' is utilized to identify the breakpoint within the sensor data and provide the corresponding fitted piecewise regression model for values above or below the breakpoint. This research uses translinear circuits to implement the fitted piecewise model. As depicted in Figure 2, the block diagram illustrates the integration of a multiplexer (MUX) to switch circuit operations based on the identified breakpoints.

3.2.   Description of the circuit

3.2.1.   Circuit for piecewise linearization for input less and greater than the Breakpoint

Figure 3 displays the comprehensive circuit diagram illustrating the translinear implementation of the piecewise model. The mathematical representation of the piecewise regression model for a sensor output ('x') below the breakpoint is as follows.

In this case, $\beta_{0}$ and $\beta_{1}$ represent the regression coefficients or slopes of the line segments between 'x' = 0 and the breakpoint. Within the circuit, these regression coefficients, denoted as '$I_{1}$' and '$I_2$', are combined with the sensor input 'x', as depicted in Figure 3. The mathematical operations necessary for inputs below the breakpoint are executed using the translinear principle. Translinear circuits, as current-mode circuits, adhere to the translinear principle effectively within the sub-threshold region, as outlined in reference [20]. The transistors in the circuit operate within the sub-threshold region, exhibiting a relationship between $I_D$ and $V_{GS}$ described by the following equation.

The saturation current $I_S$ can be calculated as follows.

where $V_{gs}$ is the gate-source voltage, $V_{Th}$ is the threshold voltage and $V_{T}$ is the thermo-electric voltage, $\mu$ is the electron mobility, $C_{ox}$ is the oxide capacitance per unit area and n is the weak inversion slope factor. The output of the sensor 'x' is supplied to the multiplication circuit through a low voltage cascade current mirror ($M_{32}$ to $M_{44}$). The supply voltage for the circuit is 2.5 V. Transistors $M_{46}$, $M_{48}$, $M_{51}$, and $M_{53}$ perform multiplication with the regression coefficients of say '$I_1$', while $M_{45}$, $M_{49}$, $M_{50}$, and $M_{54}$ are native NMOS (NMOS with negative $V_{th}$) devices, which act as cascode devices and are necessary to reduce the mirror error caused by channel length modulation in translinear multiplication devices. Transistors $M_{47}$ and $M_{52}$ are employed to realize the current sources of '$I_1$A' and 1$\mu$A. The implemented equation for linearizing the sensor input greater than breakpoint is as follows.

Through the substitution of the determined coefficient values, the aforementioned equation can be simplified to reduce circuit complexity. As a result, the required calculation for implementation involves solely the product of the breakpoint and 'x', combined with a constant term. The multiplication operation is carried out by transistors $M_{21}$, $M_{23}$, $M_{26}$, and $M_{28}$. The error correction is achieved through the utilization of native NMOS devices, denoted by a bubble at the gates, represented by transistors $M_{20}$, $M_{24}$, $M_{27}$, and $M_{29}$. Transistors from $M_7$ to $M_{19}$ are employed as low-voltage cascode current mirror.

3.3.   Switching of circuits operation as per break points

This section describes the process of switching circuit operations based on whether the sensor input is below or above the breakpoints. The identified breakpoint, obtained through the use of the 'R' language, is represented in the circuit as 'k', which falls within the microampere range and is controlled by transistors $M_3$ and $M_4$. The sensor input, denoted as 'x', is compared to the breakpoint 'k' using transistors $M_5$ and $M_6$. The input of the inverter, formed by the combination of transistors $M_1$ and $M_2$, becomes either high or low depending on whether the current 'x' exceeds or is below the current 'k'. Consequently, when the input is high, the output of the inverter becomes low, and vice versa.

Furthermore, the output of the inverter serves the purpose of circuit selection through a multiplexer (MUX). The MUX is constructed using transmission gates spanning from $M_{58}$ to $M_{61}$. A transmission gate, similar to a relay, operates as an electrical device based on CMOS technology. It consists of a parallel connection between nMOS and pMOS transistors, wherein the gate input of one transistor (C) is the complement of the gate input of the other transistor ($\bar{C}$). The inverter configuration involving transistors $M_{56}$ and $M_{57}$ facilitates the generation of states such as C and $\bar{C}$. Consequently, the output of the piecewise fitted polynomial circuits, representing the current flowing through the port named as $I_{OUT}$, is selected based on the states of C and $\bar{C}$.

The CADENCE/VIRUTOSO software tools are employed for the design and simulation of the circuit. The breakpoint in the output of a T-type thermocouple, which is 1.909, is detected by the 'R' programming language. The fitted regression model for a value less than 1.909 is shown below.

Equations (3.6) and (3.7) give the regression model for more than the breakpoint at 1.909.

where $x_1$ is equal to

The W/L ratio of MOSFET's employed in the Figure 3 is as follows.

$M_1$, $M_2$ and $M_{58}$ to $M_{61}$are 150n/280n

$M_3$, $M_4$ are 2$\mu$/4$\mu$

$M_5$, $M_6$ are 5$\mu$/900n

The W/L ratio MOSFET's employed for circuit less than and greater than breakpoints are employed as similar to Neural Network implementation as mentioned in the forthcoming section.

Table 2 illustrates the FSE of before and after linearization of the T-type thermocouple during the translinear implementation of the piecewise regression model. The linearization process yields a remarkable enhancement in linearity, with an FSE improvement of approximately $\pm$1.38% compared to the calculated FSE of $\pm$0.91%. Consequently, the piecewise translinear model achieves a significant boost in linearity, surpassing 75% improvement for the T-type thermocouple.

3.4.   Implementation of neural regression model

The proposed scheme, depicted in Figure 4, illustrates the LM-based Artificial Neural Network (ANN) algorithm utilized for the linearization of the chosen sensors. To reduce the circuit's complexity, the network is constructed with only two hidden neurons, accompanied by an input and output neuron. The weights from the input neuron to the hidden neurons are denoted as $W_1$ and $W_2$ respectively, while the weights from the hidden neurons to the output neuron are represented as $W_3$ and $W_4$. In this study, a linear activation function is employed, ensuring that the neurons' output is directly proportional to the input. The subsequent steps are executed within the 'R' programming language to determine the appropriate weights and biases.

● The network is created and configured.

● The scaled output of the sensor is fed into the input neuron.

● The weights and biases are initialized.

● The network is trained with the target vector.

● The results are predicted for the given data set.

Table 3 and Figure 5 demonstrate the optimal weights and biases determined for linearizing the specified sensor with the least amount of trained error.

3.4.1.   Description of the circuit

The detailed circuit diagram of the ANN model is presented in Figure 6. The output signal from the sensor, f(x), is first scaled down to a range of 0-5 $\mu$A and then fed into the circuit through the utilization of a cascode current mirror consisting of transistors $M_2$ to $M_5$. However, the standard cascode current mirror suffers from the drawback of larger input and output compliance voltage. To overcome this limitation, a low-voltage cascode current mirror, as depicted in Figure 6, is employed in this study to achieve the desired current-matching characteristics without a significant compliance voltage or voltage margin. In order to reduce the minimum allowable output voltage, the gate potential of transistor $M_5$ is decreased by an amount equal to the threshold voltage $V_{Th}$. To accomplish this, transistor $M_1$ is employed, with a current of 5 $\mu$A, to generate the necessary bias voltage for the cascode transistor. The width of $M_1$ is typically chosen to be 1/4th of the width of $M_3$. Equation 3.8 provides a description of the bias voltage generated by the cascode transistors.

Here, $V_{OD}$ represents the overdrive voltage, which is calculated as $V_B - V_{Th}$. The same approach is applied to determine the current through the PMOS transistors ($M_9$ to $M_{12}$). The mirrored current flowing through transistors $M_{3-4}$ is utilized to bias the current-sourcing PMOS transistors $M_7$ and $M_8$. The translinear principle, implemented using transistors $M_{14}$, $M_{15}$, $M_{17}$ and $M_{19}$, facilitates the multiplication of the current from $M_{11}$ by $W_1$. Error correction is performed through the use of native MOS transistors ($M_{13}$, $M_{16}$, $M_{18}$, and $M_{20}$). These transistors have a negative or close-to-zero threshold voltage, which helps reduce the difference in $V_{ds}$ and thus ensures that the current replicates the input source. The output of the hidden neuron ($y_1$) is observed at the drain of transistor $M_{20}$, and it is combined with the bias current ($b_1$).

Furthermore, the calculation of the output neuron is accomplished by copying the drain current from $M_20$ via $M_{21}$ to $M_{23}$A. Since the activation function used in this study is a simple linear activation function, the current of the PMOS mirror is directly multiplied by the weight $W_3$ using translinear multipliers ($M_{25}$ to $M_{30}$). To reduce the error, the native NMOS devices marked with a bubble at the gates ($M_{24}$, $M_{27}$, $M_{29}$, $M_{31}$) serve as cascode devices. The circuit employed in this work for multiplying the currents(i.e f(x) x W1) can also able carry out the division of two currents by replacing the 1$\mu$ with a divisible value. The circuit utilizes transistors, specifically $M_{15A}$, $M-{17A}$, $M_{19A}$, $M_{26A}$ and $M_{28A}$, to realize the current sources responsible for representing weights and biases with respect to the ratio of width to length of the copying transistor($M_1$).

Similarly, the calculation of the second hidden neuron and its output neuron computation is performed with the corresponding weight (not shown in the figure) and added to the previous result ($y_2$). The transistors performing the translinear multiplication/division operate in the sub-threshold region as expressed in the equation.

The ratio of W/L of MOSFET's employed in Figure 6 are as follows.

$M_1$, $M_7$, $M_8$ are 2$\mu$/4$\mu$

$M_3$, $M_4$ are 2$\mu$/1.25$\mu$

$M_2$, $M_5$ are 2.5$\mu$/300n

$M_9$, $M_11$ are 9$\mu$/1$\mu$

$M_9$, $M_11$ are 1$\mu$/1$\mu$

$M_14$, $M_19$, $M_25$, $M_38$, are 30$\mu$/600n

$M_15$, $M_17$ are 30$\mu$/600n

The circuit design and simulation of the ANN Model for the T-type thermocouple were performed using the CADENCE/VIRTUOSO tools. The results of the translinear implementation are presented in Table 4, covering a temperature range of 0$^{\circ}$C to 440$^{\circ}$C. The maximum Full-Scale Error (FSE) achieved during the implementation is 1.16%, significantly reduced from the initial value of 5.5% before linearization. Consequently, the linearity improvement for the T-type thermocouple amounts to nearly 80% after linearization.

3.5.   FPGA implementation

In this study, to enable a comparison between analog and digital domain results, the implementation of ANN and Piecewise models was also carried out on an FPGA. In FPGA devices, arithmetic operations can be executed using floating-point or fixed-point arithmetic. In this study, fixed-point arithmetic is employed, utilizing varying bit lengths for the input and output. The input thermo-EMF and linearizing parameters, such as weights, biases, and polynomial coefficients, are assigned bit lengths of seven and eight bits for the integer and fractional parts respectively. For enhanced accuracy and precision, the output bit length is set between 8 to 16 bits. To implement the function of ANN and piecewise models, an algorithm is devised that encompasses various mathematical operations. The binary output generated by Xilinx-Verilog is loaded into the FPGA device for execution. Table 6 summarises the FPGA-based ANN and piecewise linearization of all the chosen sensors. theoretically computed FSE is 0.171

4.   Discussion

The performance evaluation of the translinear implementation for both the piecewise fitted regression and ANN models applied to thermocouples and RTD are illustrated in Figure 7–11, accompanied by the corresponding data presented in Table 5. The outcomes of our proposed implementation are compared with the results obtained by other researchers for comprehensive analysis and comparison.

The piecewise regression model divides the sensor data into segments and fits a linear regression within each segment. In analyzing the linearization results of the selected sensors, we can assess the proximity of each Thermo EMF data point to the corresponding temperature value. For the T-type thermocouple and the Pt-100, the data points (representing Thermo EMF values over the temperature range) are widely spread apart, allowing the piecewise method to generate an excellent regression model. Consequently, the translinear implementation achieves a percentage linearity of approximately 75% for the T-type thermocouple and 85% for the Pt-100. These sensors exhibit consistent exponential characteristics across their entire temperature range.

In contrast, other sensors such as K, N and J-type thermocouples have data points that are closely clustered together and exhibit different trends within their measurement range. As a result, the percent linearity improvement achieved for these sensors is lower: 45% for the K-type thermocouple, 15% for the N-type thermocouple and 55% for the J-type thermocouple. It becomes evident that the piecewise regression model is less effective in capturing the nonlinearity of these sensors. To enhance the linearity of such sensors, employing more breakpoints becomes necessary. However, increasing the number of breakpoints or equations also leads to a rise in circuit complexity.

On the contrary, the neural network proves to be effective in addressing nonlinearity issues, as it can implicitly capture the nonlinear relationships between variables with minimal training. Implementing the translinear approach for the T-type thermocouple results in an impressive 80% improvement in linearity compared to the desired predicted output of the neural network model, which stands at 90%. Given the exponential nature of the T-type thermocouple's ideal properties, the network is able to train efficiently across its entire range. In contrast, the characteristics of temperature and thermoelectric voltage of the K-type thermocouple demonstrate good linearity at lower temperatures but deviate from a straight line at higher temperatures. The problem of overfitting becomes more prevalent in the low-temperature range compared to the high-temperature range. Thus during the simulation, the linearity of the K-type thermocouple is enhanced only by approximately 61%, while the theoretically predicted FSE improvement stands at 79%.

The linearity improvement resulting from the translinear implementation of the N and J-type thermocouples is consistent, as both thermocouples exhibit persistent deviations from a straight line and lack exponential characteristics. During the simulation, the linearity of the N and J-type thermocouples improves by approximately 60% compared to the theoretically expected FSE of 70% and 75%, respectively.

In the case of the Pt-100, despite its ideal features often deviating from a straight line, the LM algorithm successfully identifies the optimal model to accurately fit these characteristics. As a result, the translinear implementation achieves a remarkable linearity improvement of about 82%, surpassing the theoretically predicted outcome of 90%.

According to Table 6, the FPGA-based linearization demonstrates superior suppression of non-linearity compared to the translinear circuit. However, the main sources of error in the FPGA implementation are associated with bit errors, which can be mitigated by increasing the bit length. It is important to note that the FPGA-based approach has certain drawbacks, such as limited response speed, quantization errors, and higher power consumption.

The FSE accuracy is affected in the MOSFET-based translinear implementation because the MOSFET translinear principle is not well-suited for mixed transistor loops. As a result, the matching of N and P devices is impacted by scaling through the device's Width and Length. The weak inversion region in which the MOSFET operates poses a challenge to achieving linearity and contributes to the high error rate observed in the translinear implementation. Additionally, the MOSFET-based current mirrors utilized in the circuit, which serve to replicate current from one section to another, introduce their own sources of error. However, the inherent correction mechanisms and the implementation of a low-voltage cascode arrangement effectively mitigate the inaccuracies arising from the current mirrors, resulting in reduced error. The performance of the proposed implementation is compared with the existing techniques in Table 7.

5.   Conclusions

The proposed implementation of translinear circuits for both the piecewise regression and artificial neural network (ANN) regression models provides a practical approach for enhancing the linearity of transducers in analog mode. A comparison is made between the results obtained from this implementation and those of previous studies. In terms of reducing non-linearity errors in the selected sensors, the ANN-based translinear circuit outperforms the piecewise regression-based circuit. The performance of current mode circuits based on the piecewise regression model is hindered by challenges in accurately detecting and identifying appropriate breakpoints. Increasing the number of breakpoints can improve linearity for sensors with diverse trends throughout their operating range, but it also leads to increased circuit complexity.

On the other hand, the neural network technique, with a limited number of neurons, excels in minimizing non-linearity errors in the sensors. Optimized weights are obtained through offline training using the Levenberg-Marquardt (LM) technique to achieve the lowest relative error. While increasing the number of neurons can enhance linearity, it also increases circuit complexity. The issue of overfitting negatively impacts the results for K, N and J type thermocouples.

At the circuit level, errors arise from variations in threshold voltage and aspect ratio of transistors, as well as assumptions of exponential relationships between drain current and gate-to-source voltage. Despite these challenges, the MOSFET-based translinear circuit implementation offers advantages over previous works in terms of response time and power consumption. As compared to FPGA devices(30$\mu$), the proposed translinear implementation consumes significantly lower power, approximately 1$\mu$W. To further improve linearity, the MOSFET-based translinear circuit can be replaced with a BJT, which follows the exponential law across a wide current range and exhibits excellent matching characteristics. The advantages in power consumption and response speed position analog computation as superior to digital computation for edge computing applications.

Use of AI tools declaration

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

Acknowledgments

The authors would like to thank the Federal Institute of Science and Technology and Cochin University for providing the necessary resources and the conducive environment to carry out this research work.

Conflict of interest

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