## 2 Materials and Methods
### Design of the Proposed Sensor  
A SMA connector is one of connectors used in radio frequency and microwave devices [17; 18; 19]. The connector has the impedance of 50 \(\Omega\), and is typically designed for uses in the frequency range of 0-18 GHz. Figure 2 illustrates a photograph of the SMA connectors compared to a US quarter dollar. The sensor used with the honey samples is a modified SMA connector. The low-cost modification involves trimming the pins to be an open-ended coaxial sensor, which can measure the complex relative permittivity of materials. Also, the results can be used to calculate the electrical conductivity, and the phase constant. The structure of the proposed sensor is shown in Figure 3.  
In Figure 3a, A is the diameter of the inner coaxial core of 1.30 mm; B is the diameter of the outer coaxial core of 4.35 mm. The dielectric material in the connector is made of Teflon with \(\varepsilon_{r}\) of 2.08. The impedance of this connector is 50 \(\Omega\) that can be calculated by Equation (5) [20] as follows:  
\[z=\frac{60}{\sqrt{\varepsilon_{r}}}\ln\left(\frac{B}{A}\right) \tag{5}\]  
The measurement of the complex relative permittivity (\(\varepsilon_{r}\)) of honey samples is based on the measurement of reflection coefficient (\(S_{11}\)) when the connector is immersed in the honey samples. The change of the complex relative permittivity of the honey sample results in the change of the equivalent circuit of capacitor, reflection coefficient (\(S_{11}\)) and admittance (\(Y\)) as the model in Figure 4 [20]. Figure 3: (**a**) The design of the modified SMA connector. (**b**) A photograph of the proposed sensor. Figure 2: A photograph of SMA connectors. From the equivalent circuit of the proposed sensor, \(C_{f}\) is the capacitance independent of the material and \(C_{0}\) is the capacitance of the capacitor filled with the air. The admittance (\(Y\)) of this sensor depends on the complex relative permittivity (\(\varepsilon_{r}\)) of material as in Equation (6) [20; 21; 22]:  
\[Y=j\omega\varepsilon_{r}C_{0}+j\omega C_{f}, \tag{6}\]  
when \(Y\) is related to the reflection coefficient (\(S_{11}\)) as in Equation (7) [20; 21]:  
\[Y=\frac{1}{Z_{0}}(\frac{1-S_{11}}{1+S_{11}}). \tag{7}\]  
From Equations (6) and (7), the relationship of the complex relative permittivity and reflection coefficient can be derived by measuring the three reference materials with known complex relative permittivity and a material with unknown complex relative permittivity as in Equation (8) [5]:  
\[\frac{(\varepsilon_{UN}-\varepsilon_{A})(\varepsilon_{B}-\varepsilon_{C})}{( \varepsilon_{UN}-\varepsilon_{B})(\varepsilon_{C}-\varepsilon_{A})}=\frac{(S _{11,UN}-S_{11,A})(S_{11,B}-S_{11,C})}{(S_{11,UN}-S_{11,B})(S_{11,C}-S_{11,A})} \tag{8}\]  
Equation (8) is used to determine the complex relative permittivity (\(\varepsilon_{UN}\)) of the unknown material when the complex relative permittivity of the three materials (\(\varepsilon_{A},\varepsilon_{B},\varepsilon_{C}\)) and the reflection coefficient of all four materials (A, B, C, UN) are determined. \(S_{11,A},S_{11,B},S_{11,C}\) and \(S_{11,UN}\) values can be measured with a vector network analyzer and \(\varepsilon_{A},\varepsilon_{B},\varepsilon_{C}\) can be found in [5; 6]. Therefore, the equation only needs to calculate \(\varepsilon_{UN}\). The sensor we designed and built must be tested for validation before using it with honey samples. The testing procedures are shown in Figure 5.  
The validation of the sensor is a test to provide the reflection coefficient of four materials: air (A), distilled water (B), ethanol (C) and methanol (UN) which can be determined in the experiments as shown in Figure 5 by measuring it when the temperature of material is constant at the temperature of 25 \({}^{\circ}\)C. The complex relative permittivity of air (A), distilled water (B) and ethanol (C), which is used in Equation (8) can be found in [5; 6]. Figure 4: Capacitance model and equivalent circuit of the open-ended sensor [20]. Figure 5: A sketch of the experimental setup.