# Measurement of Dielectric Properties of Thin Materials for Radomes Using Waveguide Cavities
## 2 Theory and Materials
\[e^{\prime}=\bigg{(}\frac{(f_{c}-f_{s})}{2f_{s}}\bigg{)}\bigg{(}\frac{V_{\mathrm{ C}}}{V_{s}}\bigg{)}+1, \tag{3}\]  
\[e^{\prime\prime}=\bigg{(}\frac{V_{\mathrm{C}}}{4V_{\mathrm{S}}}\bigg{)}\bigg{(} \bigg{(}\frac{1}{Q_{\mathrm{S}}}\bigg{)}-\bigg{(}\frac{1}{Q_{c}}\bigg{)}\bigg{)}, \tag{4}\]  
\[\tan\delta=\frac{e^{\prime\prime}}{e^{\prime}}, \tag{5}\]  
where the subscripts \(c\) and \(s\) represent the empty cavity and the cavity with the sample, respectively. The resonant frequency is represented by \(f\), while \(Q\) is the Q-factor, and \(V_{\mathrm{C}}\) and \(V_{\mathrm{S}}\) represent the volume of the cavity and volume of the sample, respectively. We also required certain equations to account for the loss of the foam that was used to hold the sample in place within the cavity. Therefore, Equations (5)-(10) were used for the actual measurement process. First, the real and imaginary parts of permittivity and loss tangent for the empty cavity and the filled cavity, which was filled with foam only, were calculated [20]. From there, the loss tangent and real and imaginary parts of the permittivity could be determined for the sample material (Equations (5)-(7)). Then, with Equations (9)-(11), we first found a filling factor based on the orientation of the sample in the cavity and then the real part of permittivity and the loss tangent were calculated. The real part of permittivity was found by assuming that the real part of permittivity was much larger than the imaginary part (\(e^{\prime}\gg\epsilon^{\prime\prime}\)). The permittivity and loss tangent were determined by treating the cavity as partially filled, as the sample material only partially filled the cavity [20]. These simplified equations rely on approximations dependent on the loss tangent being very low and the resonant mode amplitude being much higher than any other modes. They were sufficiently accurate for this application. Figure 5: Material under test orientations: (**a**) sample in y-z (used in this work); (**b**) sample in x-y plane (traditionally used) [24]. These equations are known as simplified equations, as they assume that the loss tangent is very low and the resonant mode is much greater than the other modes. However, they were sufficiently accurate for this research application. \[\epsilon_{1}^{\prime}=\left(\frac{f_{0}}{f_{1}}\right)^{2}, \tag{6}\]  
\[\epsilon_{1}^{\prime\prime}=\left(\frac{1}{Q_{1}}-\frac{1}{Q_{0}}\sqrt{\frac{f_ {0}}{f_{1}}}\right)\left(\frac{f_{0}}{f_{1}}\right)^{2}, \tag{7}\]  
\[\tan\delta_{1}=\frac{\epsilon_{1}^{\prime\prime}}{\epsilon_{1}^{\prime}}, \tag{8}\]  
Figure 4:0  
Figure 4:1  
Figure 4:2  
where subscript 0 represents an empty cavity, subscript 1 represents the cavity filled with only foam, and subscript 2 represents the cavity with both foam and the sample. As previously stated, \(C_{yz}\) is the filling factor based on the orientation of the sample within the cavity, and \(a\) is the width of the cavity. The thickness of the sample is represented by \(t\), and the various resonant frequencies are represented by \(f\). The units of the parameters need only be consistent as all dimensions are ratios of the same physical quantities. The Q-factors are shown by \(Q\), and \(C\) denotes whichever filling factor is being used. Some cost-effective materials are available in the form of weatherproof canvas or similar fabrics. However, as these are not created specifically for radome applications, the dielectric properties of these materials are unknown. We initially ordered seven fabrics of various thicknesses, rigidity, and types from ExtremTextil in Germany [25]. These fabrics and their thicknesses are noted in Table 2, each with a shortened name for ease of reading throughout this report. The thicknesses of these materials were manually measured with digital calipers. Some features of these fabrics are summarized in Figure 4:3 These features are either important or useful for a radome fabric to ensure that the covered systems are protected from weather and/or debris. For example, a material being flame-retard