# Broadband Flexible Microstrip Antenna Array with Conformal Load-Bearing Structure
## 3 Theoretical Analysis  
### The Equivalent Medium Model of the Honeycomb Core Layer  
The typical structure of the honeycomb core layer with a periodic cell and three-dimensional (3D) geometry has been shown in Figure 4. The periodic cell is a regular hexagon, where \(C_{1}\) and \(h_{2}\) are applied to describe the side length and thickness of the ortho-hexagonal cell, and \(t\) is the thickness of the cell wall, respectively. The honeycomb layer consists of the host material (Nomex, \(\varepsilon_{r}=4\)) and filling material (air, \(\varepsilon_{r}=1\)) in terms of material. The honeycomb core layer is an EM transmission device. Consequently, we mainly focus on its radial-effective permittivity and permeability in this work. The solution physical model and the process of the solution method are shown in Figure 5. In the solution physical model, it should be noted that port 1 and port 2 are immediately below and above the honeycomb core layer, respectively. Then, in process of the presented solution method, the equivalent EM parameters of the homogeneous medium plane can be obtained from the scattering parameters of the two-port. Illuminated by an incident plane wave, the layer of a homogeneous material with the characteristic impedance \(Z\), and the refractive index \(n\) can be expressed as:  
\[Z=\sqrt{\frac{\left(1+S_{11}\right)^{2}-S_{21}^{2}}{\left(1-S_{11}\right)^{2} -S_{21}^{2}}} \tag{7}\]  
\[n=\frac{1}{kh_{2}}\cos^{-1}\biggl{[}\frac{1}{2S_{21}}\Bigl{(}1-S_{11}^{2}+S_{2 1}^{2}\Bigr{)}\biggr{]} \tag{8}\]  
Figure 4: The typical structure of a honeycomb core layer: (**a**) cellular structure and (**b**) 3D conformed view. Figure 3: The coordinate geometry and dimensions of the proposed antenna array: (**a**) explosion view of the structure, (**b**) the feeding network with Wilkinson power divider and ground with four coupling apertures on the middle layer. where \(h_{2}\) is the thickness of the honeycomb layer, and \(k\) denotes the wave number in free space. The effective permittivity \(\varepsilon_{e}\) and permeability \(\mu_{e}\) of the planar homogeneous material is given by:  
\[\varepsilon_{e}=\frac{n}{Z} \tag{9}\]  
\[\mu_{e}=n\cdot Z \tag{10}\]  
According to a two-port scattering parameter retrieval method (Nicolson-Ross-Weir) [23], the honeycomb periodic cell structure and layer are established as physical models for the simulation first. In these models, the honeycomb layer is composed of 5 \(\times\) 10 periodic cells, and the dimension of the cell structure is as follows: \(C_{1}\) = 1.83 mm, \(h_{2}\) = 2 mm, and \(t\) = 0.035 mm. Following this, these models are both simulated by CST MWS at a frequency band from 10 GHz to 15 GHz. The results of the honeycomb periodic cell and layer, including effective permittivity in the real part (\(\varepsilon_{e}\)), effective permittivity in the imaginary part (\(\varepsilon_{e}\)), effective permeability in the real part (\(\mu_{e}\)), and effective permeability in the imaginary part (\(\mu_{e}\)), are depicted in Figure 6. It can be seen that there are some slight differences between the periodic cell and layer in the calculated effective permittivity and permeability. To compare the validity and reasonableness of the EM parameters in the equivalent homogeneous medium plane, the reflection and transmission coefficients of these equivalent plane and honeycomb core layer structures are simulated in flat state and under different bending states. The results of the comparison have been shown in Figure 7. The maximum  
Figure 5.0 The solution process of the equivalent model, (**a**) the solution physical model, and (**b**) the process of the presented solution method. Figure 5.1 The EM parameters of periodic cell and layer in real and imaginary parts, (**a**) effective permittivity (**b**) effective permeability. errors in the reflection coefficient of the cell equation and layer equation are within 3 dB and 1 dB, respectively.