# Revisiting the Analysis of Radiative Mid-Range Wireless Link for Pacemakers
## 2 Materials and Methods
### Asymptotic Form of Gain Reduction Factor  
The radiation characteristics can be varied from the near-field region to the far-field region of the antenna of interest. There are two categories in the near-field region: reactive near-field region and radiative near-field region. The reactive near-field region is known as non-radiative near-field region or inductive near-field region. The boundary to distinguish reactive near-field from radiative near-field is known as \(\lambda/2\pi\) for an electrically small antenna [38]. This paper revisits the formulas that are effective in the mid-range region, including the radiative near-field and far-field region. The distinctive radiation characteristics in near-field and far-field regions are investigated through an example of a large aperture antenna, represented by a reflector antenna with a radius of 10 \(\lambda\).  
Figure 1 describes the variation of the H-plane radiation pattern obtained from full-wave simulation, FEKO. The H-plane radiation pattern shows that there is a rare radiating part close to the antenna; as the distance increases, the radiation towards a boresight direction appears, and, finally, a uniform far-field pattern is created. For the on-axis direction, as the distance from the antenna increases, the magnitude of the power densities oscillates and then monotonically decreases, as presented in the previous work [29; 33; 36]. The effective wavelength \(\lambda_{e}\) in the biological tissue presented in [7] can be defined as  
\[\lambda_{e}=\frac{\lambda_{0}}{\text{Re}\bigg{[}\sqrt{\varepsilon_{r}-\frac{ j^{\prime}}{\text{i}\omega_{0}}}\bigg{]}} \tag{1}\]  
where \(\lambda_{0}\) and \(\varepsilon_{0}\) are wavelength and permittivity in a free space, respectively, and \(\varepsilon_{r}\) and \(\sigma\) are relative permittivity and conductivity of the biological tissue. The surrounding environment for the implanted antenna is selected as the human skin, where its electrical properties are summarized in Table 1. The effective wavelength \(\lambda_{e}\) in the human skin is 93.6 mm, from which the thickness of the human skin of 4 mm and 8 mm can be estimated as 0.043 \(\lambda_{e}\) and 0.085 \(\lambda_{e}\), respectively. The electrical distance from the implanted antenna can be reduced based on the \(\lambda_{e}\), compared to the one in a free space. The effective wavelength is negligible in the far-field region, while it affects the electrical distance in the close part of the radiative near-field region and the reactive near-field region. The boundary between the radiative near-field and far-field region is suggested as \(2D^{2}/\lambda\) or \(2\alpha G/\pi^{2}\)[29]. The different radiation characteristics in each region can be applied to the derivation of the gain reduction factor \(\gamma\), which is defined as  
\[\gamma(\Delta)=\frac{G_{F}(\Delta)}{G}=1-\alpha\Delta^{-2} \tag{2}\]  
where the \(G\) and \(G_{F}\) are the far-field gain and Fresnel gain of the antenna, respectively, and the coefficient \(\alpha\) is related to the rate of the reduction in the far-field gain \(G\), which is found to be \(\alpha=0.066\). The normalized distance \(\Delta\) can be defined as  
\[\Delta=\left\{\begin{array}{l}\frac{R}{2D^{2}/\lambda}\left(\text{Traditional definition}\right)\\ \frac{R}{2\lambda G/\pi^{2}}\left(\text{Proposed definition}\right)\end{array}\right. \tag{3}\]In this paper, the gain reduction factor with the proposed separation distance integrated with the Friis formula is used to enhance the accuracy in the radiative near-field region.