# Effect of Buildings on the Radiation Characteristics of MF Broadcast Antennas
## 2 Materials and Methods
### Radiation of the Antennas in the Presence of Buildings  
Let us consider an antenna that radiates in free space. It is well known [13] that the space surrounding the antenna is subdivided into three regions: (a) the reactive near-field, where \(0<r<0.62\sqrt{\left(2h\right)^{3}/\lambda}\); (b) the radiating near-field, where \(0.62\sqrt{\left(2h\right)^{3}/\lambda}\leq r<2\left(2h\right)^{2}/\lambda\); and (c) the far-field, where \(r\geq 2\left(2h\right)^{2}/\lambda\). In terms of the monopole that radiates at 1494 kHz, \(2h=0.836\lambda\) with \(\lambda=200.803\) m. Thus, the boundaries for the reactive and radiating near-field are \(0<r<95.16\) m and \(95.16\leq r<280.68\) m, respectively, while the far-field is at a distance \(r\geq 280.68\) m. For the array operating at 1008 kHz, \(2h\approx\lambda=297.619\) m and the boundaries for the reactive and radiating near-field are \(0<r<184.88\) m and \(184.88\leq r<596.76\) m, respectively. For \(r\geq 596.76\) m, the space may be considered as far-field. Explicit expressions for the electric field intensity at the aforementioned distinct regions, for both antennas, may be found in [25] and are given below. The non-zero components of the far field of the monopole are:  
\[E_{\theta,mon}\cong j\zeta\frac{I_{m}e^{-jkr}}{2\pi r}\left[\frac{\cos(kh \cos\theta)-\cos(kh)}{\sin\theta}\right],\ \ H_{\varphi,mon}=\frac{E_{\theta,mon}}{\zeta} \tag{5}\]  
where \(\zeta=120\pi\) (\(\Omega\)) stands for the free-space impedance. The non-zero components of the radiating near-field of the monopole are given by:  
\[E_{\rho,mon}=j\zeta\frac{I_{m}}{2\pi\rho}\left[(z-h)\frac{e^{-jkR_{1}}}{R_{1} }+(z+h)\frac{e^{-jkR_{2}}}{R_{2}}-2z\cos(kh)\frac{e^{-jkr}}{r}\right] \tag{6a}\] \[E_{z,mon}=-j\zeta\frac{I_{m}}{4\pi}\left[\frac{e^{-jkR_{1}}}{R_{1}}+\frac{e^{- jkR_{2}}}{R_{2}}-2\cos(kh)\frac{e^{-jkr}}{r}\right]\] (6b) \[H_{\varphi,mon}=j\frac{I_{m}}{4\pi\rho}\left[e^{-jkR_{1}}+e^{-jkR_{2}}-2\cos (kh)e^{-jkr}\right] \tag{6c}\]  
with \(\rho^{2}=x^{2}+y^{2}\); \(R_{1}=\sqrt{\rho^{2}+\left(z-h\right)^{2}}\), \(R_{2}=\sqrt{\rho^{2}+\left(z+h\right)^{2}}\), and \(r=\sqrt{\rho^{2}+z^{2}}\) are depicted in Figure 1a. For the array, the corresponding non-zero components of the far field are:  
\[E_{\theta,ar}=E_{\theta,mon}S(\theta,\varphi),\ \ H_{\varphi,ar}=\frac{E_{ \theta,ar}}{\zeta} \tag{7}\]whereas the non-zero components of the radiating near-field are written as follows:  
\[E_{\rho,\sigma}=E_{\rho,mon}S(\theta,\varphi),\ \ E_{z,\sigma}=E_{z,mon}S(\theta, \varphi),\ \ H_{\varphi,\sigma}=H_{\varphi,mon}S(\theta,\varphi) \tag{8}\]  
\(S(\theta,\varphi)\) in Equation (8) stands for the array factor and is given by:  
\[S(\theta,\varphi)=1-j\frac{I_{m1}}{I_{m2}}e^{ikd\sin\theta\cos\varphi} \tag{9}\]  
The appropriate Equations (5)-(8), should substitute \(E_{i}^{inc}\) in Equation (2) in order to formulate the linear set of equations for the unknown currents \(I_{j}\). Obviously, it is the distance \(R\) between the antenna and the building that determines the appropriate equations, depending on the region of space. \(R\) is defined as the length of the perpendicular bisector from the antenna to the facade of the building, and it is shown in Figure 3. All results presented in Section 3 were produced by assuming that the buildings are in the radiating near-field region of the antenna. Thus, the unknown currents \(I_{j}\) for the monopole are calculated from the linear set of equations obtained by substituting \(E_{i}^{inc}\) in Equation (2) by Equation (6), whereas, for the array, \(E_{i}^{inc}\) in Equation (2) is substituted by the electric-field intensity given by Equation (8). The set of linear equations resulting from Equation (2) is solved numerically by truncation and matrix inversion after checking the convergence of the solution. For this purpose, a custom computer code was developed in Fortran 90.