Third-order Polynomial Equation Research Articles

Microwave Sensor for Sodium Chloride Density Measurement in Aqueous Solutions

Accurate determination of sodium chloride (NaCl) density in water is vital for assessing environmental impact, preventing soil salinization in agriculture, ensuring quality and consistency in industrial processes, facilitating medical treatments, and maintaining taste and preservation standards in the food and beverage industry. This paper introduces a novel microwave sensor design specifically tailored to accurately assess NaCl density in aqueous solutions. Starting with a standard solution of 10 g of salt dissolved in 100 ml of water, resulting in a molarity of approximately 1.71 M, five distinct samples are meticulously prepared. These samples cover a range of NaCl concentrations, with different ratios of salt solution and drinking water, including pure water, 10 ml of salt solution with 90 ml of water, 20 ml of salt solution with 80 ml of water, 30 ml of salt solution with 70 ml of water, and 40 ml of salt solution with 60 ml of water. Each sample undergoes analysis using the developed microwave sensor to determine its transmission coefficient. The magnitude of the transmission coefficient is closely tied to the density of the salt solution based on molarity. Through a detailed regression analysis, a strong quantitative relationship between the transmission coefficient and salt solution density is revealed. This correlation can be accurately represented by a third-order polynomial equation. This research is significant as it advances microwave sensor technology, allowing for accurate and efficient measurement of NaCl density in water.

Outflow risks of antibiotic-resistant bacteria in stormwater bioretention cells: understanding roles of adsorption and transmission.

In this study, lab-scale bioretention cells were designed for the investigation of antibiotic-resistant bacteria (ARB) outflow profiles at different depths, effects of adsorption and transmission, as well as modelling evaluation of ARB outflow risks using the common decay models (e.g., first-order decay models). ARB outflow was first found in the upper layers (after 100 days of the operation) with the lowest transmission frequencies of antibiotic resistance. Although the adsorption of ARB onto the substrate and its surface biofilms was effective with the maximum amount of ARB adsorbed (Qmax) reaching 108 CFU/g of the substrate and 107 CFU/g of biofilms, ARB outflow was detected in the bottom outlets after over 4 months of operation, reflecting that there was still a risk of antibiotic resistance through the treatment of bioretention cells. ARB outflow for both upper and middle outlets could be well described by third-order polynomial equations with correlation coefficients 0.9067 (p = 0.0002) and 0.9780 (p < 0.0001), respectively, where there were both positive and negative relationships between outflow ARB and inflow ARB, confirming the combined action of mechanisms blocking ARB outflow (e.g., substrate adsorption) and promoting ARB outflow (like transmission). These suggested two potential controlling approaches for ARB outflow from stormwater bioretention cells.

Effects of Leaf Wetness Duration, Temperature, and Host Phenological Stage on Infection of Walnut by Xanthomonas arboricola pv. juglandis.

Bacterial blight, caused by Xanthomonas arboricola pv. juglandis, is a significant disease affecting walnut production worldwide. Outbreaks are most severe in spring, and closely tied to host phenology and weather conditions. Pathogen infections are mainly observed in catkins, female flowers, leaves, and fruits. In this study, the effect of wetness duration and temperature on walnut infections by X. arboricola pv. juglandis was determined through two independent experiments conducted under controlled environmental conditions. The combined effect of both climatic parameters on disease severity was quantified using a third-order polynomial equation. The model obtained by linear regression and backward elimination technique fitted well to the data (R2 = 0.94 and R2adj = 0.93). The predictive capacity of the forecasting model was evaluated on pathogen-inoculated walnut plants exposed to different wetness duration-temperature combinations under Mediterranean field conditions. Observed disease severity in all events aligned with predicted infection risk. Additionally, the relationship between leaf and fruit age and the disease severity was quantified and modelled. A prediction model, which has been referred to as the WalBlight-risk model, is proposed for evaluation as an advisory system for timing bactericide sprays to manage bacterial blight in Mediterranean walnut orchards.

Smart Framework for Quality Check and Determination of Adulterants in Saffron Using Sensors and AquaCrop

Saffron is a rare and valuable crop that is only cultivated in specific regions with suitable topographical conditions. To improve saffron cultivation, it is crucial to monitor and precisely control the crop’s agronomic variables over at least one growth cycle to create a fully automated environment. To this end, agronomic variables in the Punjab region of India were analyzed and set points were calculated using third-order polynomial equations through the application of image processing techniques. The relationship between canopy cover, growth percentage, and agronomic variables was also investigated for optimal yield and quality. The addition of adulterants, such as turmeric and artificial colorants, to saffron is a major concern due to the potential for quality compromise and fraud by supply chain vendors. Hence, there is a need for devising an easy, reliable, and user-friendly mechanism to help in the detection of adulterants added to the saffron stigmas. This paper proposes an automated IoT-based saffron cultivation environment using sensors for determining set points of agronomical variables. In addition, a sensor-based chamber has been proposed to provide quality and adulteration checks of saffron and to eliminate product counterfeiting. The AquaCrop simulator was employed to evaluate the proposed framework’s performance. The results of the simulation show improved biomass, yield, and harvest index compared with the existing solutions in precision agriculture. Given the high value and demand for saffron, ensuring its purity and quality is essential to sustain its cultivation and the economic viability of the market.

Comparison of single catheter versus dual catheter-based EBT3 film calibration for the Ir-192 beam energy

Purpose. Films and TLDs have been the common choices for passive in-vivo dose measurement in radiotherapy. In the brachytherapy applications, it is very difficult to report and verify the dose at multiple localized high dose gradient regions and also the dose to organ at risk. This study was carried out to introduce a new and accurate calibration method for GafChromic EBT3 films irradiated using Ir-192 photon energy from miniature High Dose Rate (HDR) Brachytherapy source. Materials and methods. Film holder made of Styrofoam was used to hold the EBT3 film at its center. It was placed inside the mini water phantom and the films were irradiated by Ir-192 source of microSelectron HDR afterloading brachytherapy system. Two different setups: Single catheter-based film exposure and dual catheter-based film exposure were compared. The films scanned on a flatbed scanner were analysed in three different color channels: red, green, and blue using Image J software. The dose calibration graphs were generated using the third-order polynomial equations fitted on the data points from two different methods of calibration procedure. Maximum and mean dose difference between TPS calculated and measured was analyzed. Results. The measured dose difference from the TPS calculated doses were evaluated for the three groups of dose ranges (low, medium and high). In the high dose range, standard uncertainty of dose difference are ±2.3%, ±2.9%, and ±2.4% respectively for the red, green, and blue color channel when the TPS calculated dose was compared with single catheter based film calibration equation. Whereas it is observed as 1.3%, 1.4% and 3.1% for the red, green, and blue color channels respectively when compared with the dual catheter based film calibration equation. A test film was exposed to a TPS calculated dose of 666 cGy to validate the calibration equations, single catheter based film calibration equation estimated the dose difference as −9.2%, −7.8% and −3.6% respectively in the red, green, and blue color channels whereas the same were observed as 0.1%, 0.2% and 6.1% respectively when dual catheter based film calibration equation was applied. Conclusion. Source miniature size, reproducible positioning of the film and catheter system inside water medium are the major challenges in the film calibration with Ir-192 beam. To overcome these situations dual catheter-based film calibration was found more accurate and reproducible as compare to the single catheter based film calibration.

ОПЫТНО-АНАЛИТИЧЕСКОЕ ОПРЕДЕЛЕНИЕ КРИТИЧЕСКИХ ТЕМПЕРАТУРНЫХ ТОЧЕК ТЕПЛИЦЫ

In greenhouse, soil is the main heat accumulator, and its rational use is a very urgent task, which can be successfully solved only with full automation of the transfer of the missing thermal energy. To do this, it is enough to determine the following parameters in the greenhouse: the maximum daily temperatures of air and soil inside and outside, the distribution of temperature maxima during the day, the relative position and time of balancing the temperatures of the air and soil half-space in the greenhouse in the morning and in the evening, or the point of reversal of the heat flow. The temperature parameters were determined experimentally over a continuous 11 days, then approximated to third-order polynomial equations with a reliability of 0.92 ... 0.96, and analytical studies were carried out on them. The daily temperature maxima were calculated by equating the first derivatives to zero and solving the quadratic equation. The daily points of heat reversal to the soil and from the soil in time were determined by solving the system of equations for changing the temperature of the soil and the air half-space inside the greenhouse (points of intersection of the graphs). Similarly, from the system of equations, critical points were determined in the time of transition of the temperature of the air half-space inside the greenhouse to the negative zone (t=0ºC). The same method was used to calculate the optimal temperatures in the greenhouse or the moments in time of switching on and off the heat source, for which topt was introduced into the system of equations for each cultivated plant. A differentiated heating system for the greenhouse is proposed in terms of soil heat consumption, which consists in ensuring the minimum supply of additional heat during the daytime optimum and the maximum at nighttime minimum points.

Landau levels and snake states of pseudo-spin-1 Dirac-like electrons in gapped Lieb lattices

This work reports the three-band structure associated with a Lieb lattice with arbitrary nearest and next-nearest neighbors hopping interactions. For specific configurations, the system admits a flat band located between two dispersion bands, where three inequivalent Dirac valleys are identified. Furthermore, quasi-particles are effectively described by a spin-1 Dirac-type equation. Under external homogeneous magnetic fields, the Landau levels are exactly determined as the third-order polynomial equation for the energy can be solved using Cardano’s formula. It is also shown that an external anti-symmetric field promotes the existence of current-carrying states, so-called snake states, confined at the interface where the external field changes its sign.

COVID-19 peak estimation and effect of nationwide lockdown in India

There was a fury of the pandemic because of novel coronavirus (2019-nCoV/SARS-CoV-2) that happened in Wuhan, Hubei province, in China in December 2019. Since then, many model predictions on the COVID-19 pandemic in Wuhan and other parts of China have been reported. The first incident of coronavirus disease 2019 (COVID-19) in India was reported on 30 January 2020, which was a student from Wuhan. The number of reported cases started to increase day by day after 30 February 2020. The purpose of this investigation is to provide a prediction of the epidemic peak for COVID-19 in India by utilizing real-time data from 30 February to 14 April 2020. We apply the well-known epidemic compartmental model “SEIR” to predict the epidemic peak of COVID-19, India. Since we do not have the complete detail of the infective population, using the available infected population data, we identify the R0 by using polynomial regression. By using the third-order polynomial equation, we estimate that the basic reproduction number for the epidemic in India is R0 = 3.3 (95%CI, 3.1–3.5), and the epidemic peak could be reached by September 2020.

Kinetics, energy efficiency and mathematical modeling of thin layer solar drying of figs (Ficus carica L.)

First convectional thin layer drying of two fig (Ficus carica L.) varieties growing in Morocco, using partially indirect convective dryer, was performed. The experimental design combined three air temperature levels (60, 70 and 80 °C) and two air-flow rates (150 and 300 m3/h). Fig drying curve was defined as a third-order polynomial equation linking the sample moisture content to the effective moisture diffusivity. The average activation energy ranged between 4699.41 and 7502.37 kJ/kg. It raised proportionally with the air flow velocity, and the same patterns were observed for effective moisture diffusivity regarding drying time and velocity. High levels of temperature (80 °C) and velocity (300 m3/h) lead to shorten drying time (200 min) and improve the slices physical quality. Among the nine tested models, Modified Handerson and Pabis exhibited the highest correlation coefficient value with the lowest chi-square for both varieties, and then give the best prediction performance. Energetic investigation of the dryer prototype showed that the total use of energy alongside with the specific energy utilization (13.12 and 44.55 MWh/kg) were inversely proportional to the velocity and drying temperature. Likewise, the energy efficiency was greater (3.98%) in drying conditions.

Empirical path loss models at 433 MHz in Himalayan snow for health monitoring

PurposeThe purpose of this paper is to investigate the effect of snow on the radio link performance of wireless sensor nodes in Indian Himalayan conditions and to propose empirical path loss models for radio wave propagation.Design/methodology/approachAt the remote test site, one source and three listening wireless sensor nodes were deployed at frequency of 433 MHz. The path loss models are derived from experimental data collected during the period of snowfall and clear weather conditions. Linear, exponential, second and third-order polynomials path loss models have been investigated along with experimental data.FindingsWith the help of curve fitting and goodness-of-fit tests, it is found that path loss can be modelled through third-order polynomial equation during the snowfall period. However, if sensor is buried, the acceptable path loss model is exponential. Similarly, for unified modelling requirement, exponential path loss model over linear can be a preferred choice.Originality/valueResults show that path loss can be estimated priori for deciding optimum transmission energy in wireless sensor network. Presented work is usable in extending the lifetime of health monitoring devices buried in snowy environment.

Design and Analysis of Minimally Invasive Surgical Robot

Minimally Invasive Surgical robots are commonly seen in medical centres for surgical purposes. However, the major challenges seen with such robotic systems are its heavy weight, costly and lack of capability to reconfigure according to the patient-specific needs. In this paper, a robot is proposed to overcome the difficulties with existing robotic systems. Six different concepts are proposed in this paper which are modelled in SolidWorks software. The best design has been selected using the concept scoring technique. Besides, the forward and inverse kinematics for the best concept is achieved using the Denavit-Hartenberg (D-H) modelling technique. Finally, results obtained from the analytical model has been verified using Peter Corke Toolbox in MATLAB. The trajectory planning of the selected concept has be done using the third-order polynomial equation and it is plotted using MATLAB.

Slowness vector versus ray direction in polar anisotropic media

SUMMARY The inverse problem of finding the slowness vector from a known ray direction in general anisotropic elastic media is a challenging task, needed in many wave/ray-based methods, in particular, solving two-point ray bending problems. The conventional resolving equation set for general (triclinic) anisotropy consists of two fifth-degree polynomials and a sixth-degree polynomial, resulting in a single physical solution for quasi-compressional (qP) waves and up to 18 physical solutions for quasi-shear waves (qS). For polar anisotropy (transverse isotropy with a tilted symmetry axis), the resolving equations are formulated for the slowness vectors of the coupled qP and qSV waves (quasi-shear waves polarized in the axial symmetry plane), and independently for the decoupled pure shear waves polarized in the normal (to the axis) isotropic plane (SH). The novelty of our approach is the introduction of the geometric constraint that holds for any wave mode in polar anisotropic media: The three vectors—the slowness, ray velocity and medium symmetry axis—are coplanar. Thus, the slowness vector (to be found) can be presented as a linear combination of two unit-length vectors: the polar axis and the ray velocity directions, with two unknown scalar coefficients. The axial energy propagation is considered as a limit case. The problem is formulated as a set of two polynomial equations describing: (i) the collinearity of the slowness-related Hamiltonian gradient and the ray velocity direction (third-order polynomial equation) and (ii) the vanishing Hamiltonian (fourth-order polynomial equation). Such a system has up to twelve real and complex-conjugate solutions, which appear in pairs of the opposite slowness directions. The common additional constraint, that the angle between the slowness and ray directions does not exceed ${90^{\rm{o}}}$, cuts off one half of the solutions. We rearrange the two bivariate polynomial equations and the above-mentioned constraint as a single univariate polynomial equation of degree six for qP and qSV waves, where the unknown parameter is the phase angle between the slowness vector and the medium symmetry axis. The slowness magnitude is then computed from the quadratic Christoffel equation, with a clear separation of compressional and shear roots. The final set of slowness solutions consists of a unique real solution for qP wave and one or three real solutions for qSV (due to possible triplications). The indication for a qSV triplication is a negative discriminant of the sixth-order polynomial equation, and this discriminant is computed and analysed directly in the ray-angle domain. The roots of the governing univariate sixth-order polynomial are computed as eigenvalues of its companion matrix. The slowness of the SH wave is obtained from a separate equation with a unique analytic solution. We first present the resolving equation using the stiffness components, and then show its equivalent forms with the well-known parametrizations: Thomsen, Alkhalifah and ‘weak-anisotropy’. For the Thomsen and Alkhalifah forms, we also consider the (essentially simplified) acoustic approximation for qP waves governed by the quartic polynomials. The proposed method is coordinate-free and can be applied directly in the global Cartesian frame. Numerical examples demonstrate the advantages of the method.

Thermal stability and thermal expansion behavior of FeCoCrNi2Al high entropy alloy

Present work reports the thermal stability and thermal expansion behavior of dual-phase FeCoCrNi2Al HEA prepared by Mechanical Activated Synthesis and consolidated by hot pressing. The thermal stability of the phases present in FeCoCrNi2Al HEA has been extensively studied using in-situ high-temperature X-ray diffraction (HT-XRD) in conjunction with dilatometry and differential scanning calorimetry (DSC). The DSC thermogram shows a single endothermic peak at 1430 °C (1703 K) which belongs to the melting point of the alloy. HT-XRD and dilatometry experiments were carried out from room temperature to 1000 °C (1273 K). HT-XRD study has shown that the room temperature FCC + BCC (face-centred cubic + body-centred cubic) phases remains stable up to 1000 °C (1273 K). Although the amount of BCC phase has increased above 800 °C (1073 K), no additional phase formation was observed in HT-XRD. The coefficient of thermal expansion (CTE) curve shows linear increment up to 1000 °C (1273 K) with a slight change in slope beyond 800 °C (1073 K). Theoretical CTE was computed using the lattice parameter of the FCC phase, obtained from HT-XRD, as a function of temperature and compared with experimental CTE. Third-order polynomial equation was fitted to the experimental CTE data and the constants were evaluated which can be used to predict the coefficient of thermal expansion of the alloy.

Modeling of the Temperature Dependence of the Dielectric Function of Biaxial α-SnS

The dielectric functions of biaxial α-SnS at temperatures from 27 to 350 K are modeled with an analytic expression for Tauc-Lorentz oscillators. The temperature dependence of all the obtained parameters are fitted to third-order polynomial equations. The results provide a comprehensive database of the coefficients, which allows the construction of the dielectric function of biaxial α-SnS at arbitrary temperatures from 27 to 350 K and in the spectra range from 0.73 to 6.0 eV, which can be converted to optical constant values such as refractive indices and the absorption coefficients along the armchair-, zigzag-, and c-axes directions for designing optoelectronic devices based on the SnS material.

Estimation of body surface area in neonates, infants, and children using body weight alone.

BackgroundThe aim of this study was to use Body Surface Area (BSA) data calculated with the Mosteller equation to test potential new equations that estimate BSA using Body Weight (BW) alone in children aged 0–18 years.Mosteller’s equation, the golden standard at our hospital, was used to calculate the BSA in infants and children aged 0–18 years using BW and height data from 27,440 hospital visits by 20,635 patients over one year.MethodsThe best fit of three nonlinear regression equations (third-order polynomial, Meeh-type, and modified Boyd self-adjusting-type) to a plot of the calculated Mosteller BSA values versus BW was then investigated. The correlation between the BSA values estimated by these equations and the Mosteller BSA values was established by the Spearman rank correlation test. Bias and precision were evaluated as outlined by Sheiner and Beal. Measured and estimated BSA values were compared using the Eksborg plot.ResultsThe estimated BSA values from all three equations and the BSA values from the Mosteller equation were closely correlated (P < .0001). The third-order polynomial and Meeh-type equations overestimated BSA by 0.13% and 0.40%, respectively, while the Boyd self-adjusted-type equation underestimated BSA by 0.060%. For the entire pediatric population, the best fit was obtained with the Meeh-type equation: 99.2% of the Meeh/Mosteller BSA ratios were within the range of 0.9–1.1 when compared with 98.3% and 97.2% for the polynomial and Boyd-type equations, respectively.ConclusionA single Meeh-type equation can be used to predict the results of Mosteller equation when H is not available with high precision and accuracy in children aged 0–18 years, including term neonates. We now plan to include the results of this study in CPOE systems in Sweden to improve drug dosage in all children.

Quantification of maximal power output in well-trained cyclists

ABSTRACT This study aimed to compare mechanical variables derived from torque-cadence and power-cadence profiles established from different cycle ergometer modes (isoinertial and isokinetic) and modelling procedures (second- and third-order polynomials), whilst employing a novel method to validate the theoretical maximal power output (Pmax). Nineteen well-trained cyclists (n = 12 males) completed two experimental sessions comprising six, 6-s maximal isoinertial or isokinetic cycling sprints. Maximal pedal strokes were extracted to construct power–cadence relationships using second- and third-order polynomials. A 6-s sprint at the optimal cadence (Fopt) or optimal resistance (Topt) was performed to assess construct validity of Pmax. No differences were found in the mechanical parameters when derived from isokinetic (Pmax = 1311 ± 415, Fopt = 118 ± 12) or isoinertial modes (Pmax = 1320 ± 421, Fopt = 116 ± 19). However, R2 improved (P < 0.02) when derived from isoinertial sprints. Third-order polynomial modelling improved goodness of fit values (Standard Error, adjusted R2), but derived similar mechanical parameters. Finally, peak power output during the optimised sprint did not significantly differ from the theoretical Pmax in both cycling modes, thus providing construct validity. The most accurate P-C profile can be derived from isoinertial cycling sprints, modelled using third-order polynomial equations.

An Attempt to Modify the Crop Coefficient Value of Wheat and Chickpea for Humid Subtropical Climate of Varanasi District, Uttar Pradesh

Crop water requirement is one of the essential components that should always be taken into account wherever water management strategies are adopted for effective utilization of water. In India agriculture is considered as the backbone of economy but nothing technical or advance technology has being adopted for its advancement and still major portion of agriculture are dependent on the verge of monsoon. Therefore crop coefficient (Kc) value for wheat and Chickpea as a function of relative humidity, wind speed and crop height has been formalized in this study for efficient use of available water. The empirical formula was applied for estimating Kc(mid) and Kc(end) values for humid subtropical climate of Varanasi. The corrected crop coefficient value of wheat for Kc(mid) was found to be 1.12 and for end season Kc(end) corrected was found to be 0.42. Mean maximum height of wheat crop in mid-season was obtained as 0.80 m and for the end season mean maximum height was found to be 0.99 m. The corrected crop coefficient value of Chickpea for midseason was found to be 0.98 whereas it was found out to be 0.38 at Kc(end). Mean maximum height of chickpea crop in mid-season was obtained as 0.39 m and for the end season of the crop mean maximum height was found to be 0.52 m. Crop coefficient curve for wheat crop was prepared in which X-axis represents time in days and Y- axis represents crop coefficient value for wheat. A third-order polynomial equation (y = -3E - 06x3 + 0.0005x2 - 0.0039x + 0.3702) has been obtained for wheat crop with correlation coefficient of 0.97. Similarly, graphical plot was constructed for chickpea crop and a third-order polynomial equation (y = -2E - 06x3 + 0.0002x2 - 0.0004x + 0.3732) has been obtained with correlation coefficient of 0.98. Furthermore, third-order polynomials were fitted well to predict the crop coefficient values as function of growing degree-days (GDD).

Velocity Kinematics Analysis and Trajectory Planning of 5 DOF Robotic Arm

Trajectory planning is important in robots to achieve smooth path planning. This paper presents the velocity kinematics analysis and trajectory planning of a 5 DOF robotic arm. The Jacobian matrix is utilized to analyze the velocity kinematics and the third-order polynomial equation is used to determine the path of angle, velocity, and acceleration of the robotic arm. A 5 DOF robotic arm used with revolute joints and the motion of it is performed using the Arduino Mega2560 (microcontroller) which controlling on servo motors. The results of the velocity kinematics indicated the maximum linear velocity occurs with the z-direction and the cubic polynomial equation satisfied a smooth path for angle and velocity but a discontinuous path for acceleration.

Effect of Structural Uncertainty in Passive Microwave Soil Moisture Retrieval Algorithm.

Passive microwave sensors use a radiative transfer model (RTM) to retrieve soil moisture (SM) using brightness temperatures (TB) at low microwave frequencies. Vegetation optical depth (VOD) is a key input to the RTM. Retrieval algorithms can analytically invert the RTM using dual-polarized TB measurements to retrieve the VOD and SM concurrently. Algorithms in this regard typically use the τ-ω types of models, which consist of two third-order polynomial equations and, thus, can have multiple solutions. Through this work, we find that uncertainty occurs due to the structural indeterminacy that is inherent in all τ-ω types of models in passive microwave SM retrieval algorithms. In the process, a new analytical solution for concurrent VOD and SM retrieval is presented, along with two widely used existing analytical solutions. All three solutions are applied to a fixed framework of RTM to retrieve VOD and SM on a global scale, using X-band Advanced Microwave Scanning Radiometer-Earth Observing System (AMSR-E) TB data. Results indicate that, with structural uncertainty, there ensues a noticeable impact on the VOD and SM retrievals. In an era where the sensitivity of retrieval algorithms is still being researched, we believe the structural indeterminacy of RTM identified here would contribute to uncertainty in the soil moisture retrievals.

Wave Digital Modeling of Nonlinear 3-terminal Devices for Virtual Analog Applications

We propose a novel modeling method for circuits containing arbitrary nonlinear 3-terminal devices, which operates in the wave digital (WD) domain. This approach leads to the definition of a general and flexible WD model for 3-terminal devices, whose number of ports varies from 1 to 6. The generality of the method is confirmed by the fact that the WD models of 3-terminal devices already discussed in the literature can be seen as particular cases of the model that we present here. As examples of applications of our method, we develop WD models of the three most widespread types of transistors in audio circuitry, i.e., the MOSFET, the JFET and the BJT. These models are here designed to be used in Virtual Analog audio applications; therefore, their derivation is aimed at minimizing computational complexity while avoiding implicit relations between port variables, as far as possible. Proposed MOSFET and JFET models are characterized by third-order polynomial equations; hence, explicit closed-form wave scattering relations are obtained. On the other hand, the Ebers–Moll model describing the BJT results in transcendental equations in the WD domain that cannot be solved analytically. In order to cope with this problem, we propose a modified Newton–Raphson (NR) method for solving the implicit Ebers–Moll equations in the WD domain. Such iterative method exhibits a significantly higher robustness and convergence rate with respect to the traditional NR method, without compromising its efficiency. Finally, WD implementations of some audio circuits containing transistors are discussed.