# Wheeler Method for Evaluation of Antennas Submerged in Lossy Media
## 2 Antenna Radiation Efficiency for Lossy Media  
Measurement using the gain comparison method in lossy media is presented in Figure 1. The propagation region between Tx and Rx is divided into two physical layers: lossy medium and air. Inevitably, the radiated field is scattered at the boundary. Attenuation region R1 is different depending on the conditions of the lossy medium such as shape, size, and antenna location. In this case, the radiation efficiency based on combining radiated power in all directions cannot be specified [21].  
A modified radiation efficiency \(\eta_{MR}\) of the antenna for lossy media is defined as follows:  
\[\eta_{MR}(d_{r})=P_{rad}(d_{r})/P_{acc}=P_{rad}(d_{r})/\Big{(}P_{in}\Big{(}1-|S _{11}|^{2}\Big{)}\Big{)}, \tag{1}\]  
where \(P_{rad}\) is the total radiated power related to the propagation distance due to the path loss in the lossy media (Figure 2), \(P_{in}\) is the incident power, and \(P_{rad}\) is the power incident to the antenna terminals. It is calculated by integrating the power flow normal to the virtual sphere surface according to the radial distance from the antenna center [4]. However, all previously reported works related to the modified radiation efficiency have been demonstrated and verified without experimental results [16; 17; 26; 27].  
Figure 1: Illustration of conventional gain comparison method which evaluates radiation characteristics for antenna in lossy media.  
The Wheeler method is used to assess the efficiency \(\eta_{MR}\) of the insulated antenna in lossy media (Figure 3a). When \(d_{r}\) is equal to the radius of the Wheeler cap, the modified radiation efficiency can be briefly obtained using the Wheeler method. When the metallic cap with a radius of radial length is covered, the radiation resistance can be removed with an equivalent circuit, as shown in Figure 3b. The radiation efficiency \(\eta_{WR}\) is obtained as follows:  
\[\eta_{WR}=\frac{Re\{Z_{in}\}-Re\{Z_{WC}\}}{Re\{Z_{in}\}}, \tag{2}\]  
where \(Z_{in}\) is the antenna input impedance, \(Z_{WC}\) is the input impedance by enclosing the antenna with a radiation shield [8]. The radius of the radiation shield is equal to the radian length, where the far-field and near-field have the same intensity. The radian length \(r_{l}\) in free space is as follows:  
\[r_{l}=\frac{\lambda_{g}}{2\pi}=\frac{\lambda_{0}}{2\pi\sqrt{\varepsilon_{r}}}, \tag{3}\]  
where \(\lambda_{0}\) is the wavelength in a vacuum, \(\lambda_{g}\) is the wavelength in the media. In lossy media, the radian length \(r_{l}\)\({}^{\prime}\) can be calculated as follows:  
\[r_{l}\)\({}^{\prime}=\frac{1}{\left\{\left(\varepsilon\mu\omega^{2}\right)^{2}+\left( \omega\mu\sigma\right)^{2}\right\}^{0.25}}, \tag{4}\]  
where \(\varepsilon\) is the permittivity, \(\mu\) is the permeability, \(\sigma\) is the conductivity, \(\omega\) is the angular frequency.  
Figure 3: (**a**) Structure of the antenna isolated from the lossy medium by the insulating layer and the Wheeler cap. (**b**) Equivalent circuit model of antenna within the Wheeler cap.  
Figure 2: Simulation setup for calculation of radiated power.