Metamaterial-Inspired Complementary Split Ring Resonator Sensor and Second-Order Approximation for Dielectric Characterization of Fluid

Abstract

A metamaterial-based complementary split ring resonator (CSRR) structure is used to develop a sensor for dielectric characterization of fluids. The fluid present in the vertical column interacts with the fields around the CSRR causing a shift in the transmission coefficient curve ($S_{21}$). An empirical relationship can be established between the dielectric properties and the resonance frequency and Q-factor. This relationship is used for the dielectric characterization of the fluid. A second-order polynomial function is employed for a better curve fitting of the data to achieve higher accuracy in the prediction of complex permittivity ($\varepsilon '$ and $\varepsilon ''$). Multi-variate polynomial regression is used to determine the coefficients of the polynomial function. The proposed sensor predicts the permittivity of the sample with high accuracy. The design is very simple and the sample can be easily changed by replacing the glass tube. The sensor has very high sensitivity and requires a very little volume of the sample.

Appendix A: Pseudo Code for Multi-Variate Polynomial Regression

• 1. Import Libraries [For python Pandas Numpy sklearn]

• 2. Read data from csv (comma separated values) file and convert it into a dataframe (say df)

  df = pandas . read_csv ("filename_with_extension")

• 3. There are 2 training data columns of 2 target columns to be predicted. Extract only required columns from dataframe.

  $$ \begin{gathered} {\text{train \_features = ['column1', 'column2']}} \hfill \\ {\text{target\_column\_1 = df ['target1']}} \hfill \\ {\text{target\_column\_2 = df ['target2']}} \hfill \\ {\text{data = df ['train\_features']}} \hfill \\ \end{gathered} $$

• 4. Take the input of two train_features values (f and Q) to predict target values ($\varepsilon '$ and $\varepsilon ''$)

  $$ \begin{gathered} {\text{f = float (input (Enter the value of first input feature))}} \hfill \\ {\text{Q = float (input (Enter the value of second input feature))}} \hfill \\ \end{gathered} $$

• 5. Generate polynomial and interaction features. The degree-2 polynomial features are [1, a, b, $a^{2}$, ab, $b^{2}$]. [Use PolynomialFeatures class from sklearn.preprocessing library for python]

  $$ \begin{gathered} {\text{poly = sklearn}}{\text{.preprocessing}}{\text{.Polynomial Features (degree = DEGREE OF POLYNOMIAL}} \hfill \\ \quad \quad \quad {\text{ YOU WANT TO CREATE)}} \hfill \\ {\text{poly}}\_{\text{variables = poly}}.{\text{fit}}\_{\text{transform}}({\text{arrayofvaluesofinputfeatures}}) \hfill \\ \end{gathered} $$

• 6. Repeat the above process for test values for which we are predicting output values:

  $$ {\text{test\_variables = poly}}{\text{.fit\_transform (an array}}\;{\text{of input values f and Q)}} $$

• 7. To create model, use simple linear regression. [Use LinearRegression from library sklearn.linear_model for python]

  $$ \begin{gathered} {\text{regression = sklearn}}{\text{.linear\_model}}{\text{.LinearRegression ( )}} \hfill \\ {\text{model = regression}}{\text{.fit(poly\_variables,target\_column\_1 )}} \hfill \\ \end{gathered} $$

• 8. Obtain the coefficients and intercept of fitted equation.

  $$ \begin{gathered} {\text{print}}({\text{coefficients}}\;{\text{of}}\;{\text{fitted}}/{\text{learned}}\;{\text{equation}}) \hfill \\ {\text{print}}({\text{intercept}}\;{\text{of}}\;{\text{fitted}}/{\text{learned equation}}) \hfill \\ \end{gathered} $$

• 9. Predict the model. [Use model.predict() for python]

  $$ \begin{gathered} {\text{prediction = model}}{\text{.predict(test\_variables)}} \hfill \\ {\text{print(prediction)}} \hfill \\ \end{gathered} $$

• 10. Repeat 7, 8 and 9 for second target column to predict another target value. We have to use 2 separate models for training, as it is multi-target regression problem.