7.2.1 Calculation of the far field from the slot field distribution

The electromagnetic field from electric and magnetic current sources in an unbounded homogenous region can be expressed generally from

$$\vec{E}=-\vec{\nabla}\times\vec{F}+ \frac{1}{j\omega\varepsilon}(\vec{\nabla}\times\vec{\nabla}\times\vec{A}-\vec{J})$$ (7.20)

and

$$\vec{H}=\vec{\nabla}\times\vec{A}+ \frac{1}{j\omega\mu}(\vec{\nabla}\times\vec{\nabla}\times\vec{F}-\vec{M})$$ (7.21)

with

$$\vec{A}(\vec{r})=\frac{1}{4\pi}\iiint\frac{\vec{J}(\vec{r}\,')e^{-jk\vert\vec{r}-\vec{r}\,'\vert}}{\vert\vec{r}-\vec{r}\,'\vert} \textrm{d}\tau '$$ (7.22)

and

$$\vec{F}(\vec{r})=\frac{1}{4\pi}\iiint\frac{\vec{M}(\vec{r}\,')e^{-jk\vert\vec{r}-\vec{r}\,'\vert}}{\vert\vec{r}-\vec{r}\,'\vert} \textrm{d}\tau ',$$ (7.23)

where $\vec{E}$ is the electric field density, $\vec{H}$ is the magnetic field density, $\vec{A}$ is the magnetic vector potential, $\vec{F}$ is the electric vector potential, $\vec{r}\,'$ is the vector to the magnetic and the electric current sources in (7.22) and (7.23) respectively, $\mu$ is the permeability, and $\varepsilon$ is the permittivity of the homogenous region $\tau '$. This is described in more detail in [102].
With the angle $\xi$ between the vectors $\vec{r}$ and $\vec{r}\,'$, depicted in Figure 7.2, the distance $\vert\vec{r}_{1}\vert$ can be approximated with

$$\vert\vec{r}\vert\geq\frac{2D_{a}^2}{\lambda_{0}}$$ (7.24)

in the far field, where $\vert\vec{r}\vert$ becomes large compared to $\vert\vec{r}\,'\vert$. The direction of $\vert\vec{r}\vert$ is $\vec{e}_{r}$. For antennas with an active dimension $D_{a}$, such as, for example, the length of a dipole, or the length of an aperture, the far field region condition is

$$\vert\vec{r}\vert\geq\frac{2D_{a}^2}{\lambda_{0}}$$ (7.25)

according to [104]. The wavelength in air is $\lambda_{0}$.

Figure 7.2: Angle $\xi$ between the vectors $\vec{r}$ and $\vec{r}\,'$.

With (7.24) the electric vector potential (7.23) in the far field region becomes

$$\vec{F}(\vec{r})\approx\frac{1}{4\pi}\frac{e^{-jk\vert\vec{r}\ver... ...r}\vert} \iiint\vec{M}(\vec{r}\,')e^{jk\vec{r}\,'\vec{e}_{r}}\textrm{d}\tau '.$$ (7.26)

The magnetic vector potential becomes

$$\vec{A}(\vec{r})\approx\frac{1}{4\pi}\frac{e^{-jk\vert\vec{r}\ver... ...r}\vert} \iiint\vec{J}(\vec{r}\,')e^{jk\vec{r}\,'\vec{e}_{r}}\textrm{d}\tau '.$$ (7.27)

The radiation from the enclosure is mainly determined by the electric voltage distribution at the slot [59]. From this voltage distribution an equivalent magnetic source current on the slot is obtained as depicted in Figure 7.3 for the calculation of the radiated electric far field. With (7.20) the electric far field from magnetic current sources

$$\vec{E}=-\vec{\nabla}\times\vec{F}$$ (7.28)

is applied on (7.26) to obtain the far field approximation for the electric field density

$$\vec{E}(\vec{r})\approx\frac{jk}{4\pi}\frac{e^{-jk\vert\vec{r}\ve... ...r}\,')e^{jk\vec{r}\,'\vec{e}_{r}}(\vec{e}_{r}\times\vec{e}_{m})\textrm{d}\tau',$$ (7.29)

according to [59], [69], where $\vec{e}_{m}$ is the direction of the magnetic current density $\vec{M}(\vec{r}\,')$.

Figure 7.3: Equivalent magnetic current sources at the enclosure slot for the derivation of the radiated far field from the slot. This spherical angle definition was used, because it enables simpler radiation field expressions.

With the coordinate system definition and the equivalent magnetic current sources at the slot depicted in Figure 7.3, (7.29) for the electric far field becomes

$$\vec{E}_{far}=-\frac{jk}{4\pi}\frac{e^{-jkr}}{r}\sin(\vartheta)\v... ..._{\varphi} \int_{x=0}^{L}\left\{U(x)e^{jkx\cos(\vartheta)}\textrm{d}x\right\}.$$ (7.30)

The magnetic far field is described with

$$\vec{H}_{far}=-\frac{jk^{2}}{4\pi}\frac{e^{-jkr}}{r}\frac{1}{\ome... ...}_{\vartheta}\int_{x=0}^{L}\left\{U(x)e^{jkx\cos(\vartheta)}\textrm{d}x\right\}$$ (7.31)

accordingly. With a declaration of $p$ interface ports at the slot of the enclosure, (7.30) is discretized to

$$\vec{E}_{far}=-\frac{jk}{4\pi}\frac{e^{-jkr}}{r}\sin(\vartheta)\v... ...arphi}\frac{L}{p} \sum_{i=1}^{p}\left\{U_{i}e^{jkx_{i}\cos(\vartheta)}\right\}$$ (7.32)

and (7.31) is discretized to

$$\vec{H}_{far}=-\frac{jk^{2}}{4\pi}\frac{e^{-jkr}}{r}\frac{1}{\ome... ...heta}\frac{L}{p} \sum_{i=1}^{p}\left\{U_{i}e^{jkx_{i}\cos(\vartheta)}\right\}.$$ (7.33)

$U_{i}$ denote the voltages at the $p$ slot ports with the integer index $i\in[1,p]$. The far field condition (7.25) for the enclosure depicted in Figure 7.3 becomes

$$\vert\vec{r}\vert\geq\frac{2L^2}{\lambda_{0}}$$ (7.34)

C. Poschalko: The Simulation of Emission from Printed Circuit Boards under a Metallic Cover