# An Ultra-Wideband Frequency Selective Rasorber with Low Infrared Emissivity
## 2 Analysis, Design and Experimental Verification
### Transmission/Absorption Conditions Based on EC Analysis Method  
The FSR is a composite structure comprising a resistive sheet (RS), a dielectric spacer (DS), and a band-pass FSS [2]. It ensures the proper transmission of signals within the working frequency band and absorbs, rather than reflects, out-of-band radar waves. This effectively reduces the omnidirectional RCS. In addition, the FSR can be considered a combination of a circuit analog absorber (CAA) and a bandpass FSS [17]. According to the equivalent transmission line theory [3], the microwave system consisting of FSR and its surrounding free space can be represented as a two-port microwave network, with the dielectric layer and free space being equivalent to a two-wire transmission line. Port 1 and Port 2 serve as plane wave incident ports in free space. The lossy RS and lossless FSS are equivalent to circuit loads composed of resistors, capacitors, and inductors in parallel on the transmission line. The series resonance represented by \(RL_{1}\)\(C_{1}\) in RS functions as low-frequency wave absorption, while \(L_{2}\)\(C_{2}\) in RS and \(L_{3}\)\(C_{3}\) in FSS represent parallel resonance, playing a role in high-frequency wave transmission. Therefore, the structure and equivalent circuit can be illustrated as Figure 2, where \(Z_{0}=120\)\(\pi\Omega\), represents the free-space impedance of Port 1 and Port 2. \(Z_{RS}\), \(Z_{C}\), and \(Z_{FSS}\) denote the equivalent impedances of RS, DS, and FSS respectively. Actually, the intermediate DS typically employs a high transmission layer such as air or PMI foam, so it can be assumed that \(Z_{c}=Z_{0}\). According to transmission line theory, the ABCD matrix of the two-port network can be expressed as:  
\[ \begin{bmatrix}A&B\\ C&D\end{bmatrix} = \begin{bmatrix}1&0\\ \frac{1}{Z_{RS}}&1\end{bmatrix}\begin{bmatrix}cos\delta&jZ_{0}sin\delta\\ j\frac{1}{Z_{0}}sin\delta&cos\delta\end{bmatrix}\begin{bmatrix}1&0\\ \frac{1}{Z_{FSS}}&1\end{bmatrix}  \tag{1}\]  
In Equation (1), \(\delta=\frac{2\pi}{\lambda}tcos\theta\), while the incident wavelength is denoted as \(\lambda\), the thickness of the DC layer is represented by \(t\), and the refraction angle of the wave in DS is indicated as \(\theta\). Figure 2: Sectional structure (**a**) and equivalent circuit of FSR (**b**). Therefore, the elements A, B, C, D in the matrix can be represented as follows:  
\[A=cos\delta+j\frac{Z_{0}}{Z_{FSS}}sin\delta \tag{2}\]  
\[B=jZ_{0}sin\delta \tag{3}\]  
\[C=\frac{Z_{RS}+Z_{FSS}}{Z_{RS}Z_{FSS}}cos\delta+j\frac{sin\delta cos\delta}{Z_{0 }(1+1/Z_{RS}Z_{FSS})} \tag{4}\]  
\[D=cos\delta+j\frac{Z_{0}}{Z_{RS}}sin\delta \tag{5}\]  
The transmission coefficient \(S_{21}\) and reflection coefficient \(S_{11}\) can be expressed as:  
\[S_{11}=\frac{AZ_{0}+B-CZ_{0}^{2}-DZ_{0}}{AZ_{0}+B+CZ_{0}^{2}+DZ_{0}} \tag{6}\]  
\[S_{21}=\frac{2Z_{0}}{AZ_{0}+B+CZ_{0}^{2}+DZ_{0}} \tag{7}\]
* _Absorption conditions_ At the optimal absorption frequency \(f_{a}\), the transmission coefficient \(S_{21}\) and reflection coefficient \(S_{11}\) of the two-port network are interrelated as follows: \(|S_{11}|=0\), \(|S_{21}|=0\). In the above conditions, substitute Equations (2)-(5) into (6) and (7), respectively, and then we can obtain:  
\[Z_{RS}=sin^{2}\delta(Z_{0}+Z_{FSS})-Z_{FSS}cos^{2}\delta-j(Z_{0}+2Z_{FSS})sin \delta cos\delta \tag{8}\]  
\[\frac{Z_{0}Z_{FSS}^{2}}{tan\delta}+jZ_{0}Z_{FSS}(Z_{0}+Z_{FSS})=0 \tag{9}\]  
According to Equation (9), \(Z_{FSS}\) has two solutions:  
\[Z_{FSS}= \begin{cases}0\\ -\frac{Z_{0}(tan\delta+j)}{tan\delta+1}\end{cases}  \tag{10}\]  
It is obvious that Re(\(Z_{FSS}\)) can not be negative, so according to \(Z_{FSS}=0\), FSS behaves similarly to a metallic ground plane at the frequency of absorption. Then, substitute Equation (10) into Equation (8):  
\[Z_{RS}=Z_{0}sin^{2}\delta-jZ_{0}sin\delta cos\delta \tag{11}\]  
While \(Z_{RS}=Z_{0}\), the incident wave enters DS through RS with no insertion loss, which is the condition of the premise of optimal absorption.