Relation between Source Excitation and Far-Field Pattern¶

Release:	0.0.0
Date:	July 03, 2011

Many students either dislike or afraid of Electromagnetic theory! Obviously, the reason is the mathematical complexity involved in the EM theory. At the same time, nobody has that much trouble with subjects such as signal processing, communication theory, etc. But, if observed closely, the fundamental theory behind all these subjects is nothing but the Fourier analysis. In fact, Fourier analysis plays crucial role in understanding many areas of science and engineering.

So, it is the authors’ opinion that if the books on antenna theory starts with a slightly different approach using Fourier analysis as the starting point, then the learning process becomes easier and more fun. In this brief introductory article, authors use this approach to explain the antenna theory (and of course, array antenna theory too).

Symmetric Maxwell’s Equations¶

Symmetric Maxwell’s equations are given as

$\nabla \cdot \mathbf{D} &= \rho_e \\ \nabla \cdot \mathbf{B} &= \rho_m \\ \nabla \times \mathbf{E} &= -\mathbf{J}_m-\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \times \mathbf{H} &= +\mathbf{J}_e+\frac{\partial \mathbf{D}}{\partial t}$

where $\mathbf{D}=\epsilon\mathbf{E}$ and $\mathbf{B}=\mu\mathbf{H}$ .

Vector Potentials and Helmholtz Wave Equations¶

Assuming the entire system is linear, any source can be separated into electric and magnetic currents (sources). The vector potentials $\mathbf{A}$ and $\mathbf{F}$ corresponding to these given electric and magnetic currents $\mathbf{J}_e$ and $\mathbf{J}_m$ are given as

$\nabla^{2}\mathbf{A}+k^{2}\mathbf{A} &= -\mu\mathbf{J}_{e} \\ \nabla^{2}\mathbf{F}+k^{2}\mathbf{F} &= -\epsilon\mathbf{J}_{m}.$

The above equations are derived from the Maxwell’s equations using simple vector identities. Using the Sommerfeld radiation condition, solutions to the above inhomogeneous Hemlholtz equations are given as

$\mathbf{A} &= \frac{\mu}{4\pi}\int_\mathrm{volume}\mathbf{J}_e\frac{e^{-jk_0|\mathbf{r-r^\prime}|}}{|\mathbf{r-r^\prime}|}dv^\prime \\ \mathbf{F} &= \frac{\epsilon}{4\pi}\int_\mathrm{volume}\mathbf{J}_m\frac{e^{-jk_0|\mathbf{r-r^\prime}|}}{|\mathbf{r-r^\prime}|}dv^\prime.$

And of course, from the definitions, electric and magnetic fields can be written in terms of the vector potentials as

$\mathbf{H}_A &= \frac{1}{\mu}\left(\nabla\times\mathbf{A}\right) \\ \mathbf{E}_A &= \frac{1}{j\omega\epsilon\mu}\nabla\left(\nabla.\mathbf{A}\right)-j\omega\mathbf{A} \\ \mathbf{E}_F &= -\frac{1}{\epsilon}\left(\nabla\times\mathbf{F}\right) \\ \mathbf{H}_F &= \frac{1}{j\omega\epsilon\mu}\nabla\left(\nabla.\mathbf{F}\right)-j\omega\mathbf{F}$

where $\mathbf{E}_{\mathrm{tot}}=\mathbf{E}_A+\mathbf{E}_F$ and $\mathbf{H}_{\mathrm{tot}}=\mathbf{H}_A+\mathbf{H}_F$ . For further details, please refer to (page.135, [Balanis]).

Far-field Approximations¶

In rectangular co-ordinate system, when $r \gg r'$ , the term $|\mathbf{r-r^\prime}|$ can be approximated as

$|\mathbf{r-r^{\prime}}| &\approx \left(r-\hat{\mathbf{r}}\cdot\mathbf{r^{\prime}}\right) \\ &\approx r-\left(\sin\theta\cos\phi x'+\sin\theta\sin\phi y'+\cos\theta z'\right) \\ &\approx r-\left(ux'+vy'+wz'\right).$

So, far-field vector potentials are given as

$\mathbf{A} &\approx \frac{\mu}{4\pi}\frac{e^{-jk_0r}}{r}\int_\mathrm{volume}\mathbf{J}_e{e^{+jk_0(ux'+vy'+wz')}}dv^\prime \\ \mathbf{F} &\approx \frac{\epsilon}{4\pi}\frac{e^{-jk_0r}}{r}\int_\mathrm{volume}\mathbf{J}_m{e^{+jk_0(ux'+vy'+wz')}}dv^\prime.$

From the above equations, it is evident that $\mathbf{A} \& \mathbf{J}_e$ and $\mathbf{F} \& \mathbf{J}_m$ form Fourier transform pairs. In comparison to the signal processing terminology, $(x,y,z)$ and $(u,v,w)$ are analogous to time $t$ and frequency $f$ , respectively.

Also, far-field electric and magnetic field components can be approximated as (for the $\theta$ and $\phi$ components only since $E_r\simeq0$ and $H_r\simeq0$ )

$\mathbf{E}_A &\simeq -j\omega\mathbf{A} \\ \mathbf{H}_F &\simeq -j\omega\mathbf{F}.$

The far-field components $\mathbf{H}_A$ and $\mathbf{E}_F$ are related to the above components as $\mathbf{H}_A\simeq\frac{1}{\eta}\left(\hat{\mathbf{a}}_r\times\mathbf{E}_A\right)$ and $\mathbf{E}_F\simeq-\eta\left(\hat{\mathbf{a}}_r\times\mathbf{H}_F\right)$ , where $\eta=\sqrt{\mu / \epsilon}$ is the free space wave impedance.

Far-field Green’s Functions of Infinitesimal Dipoles¶

Now, a simple example will be considered. This example deals with evaluation of the far-field components corresponding to a infinitesimal electric dipole placed at the origin and oriented along the $z$ -axis. The corresponding vector potential is given as

$\mathbf{A} = \left(\frac{\mu}{4\pi}\frac{e^{-jk_0r}}{r}\right)\hat{\mathbf{z}}$

Converting the above equation into spherical co-ordinate system gives

$\mathbf{A} = \left(-\frac{\mu}{4\pi}\frac{e^{-jk_0r}}{r}\sin\theta\right)\hat{\mathbf{\theta}}.$

In deriving the above equation, radial component of the vector potential is neglected. Finally, far-field electric field is given as

$\mathbf{E} = -j\omega\mathbf{A} = \left(j\omega\frac{\mu}{4\pi}\frac{e^{-jk_0r}}{r}\sin\theta\right)\hat{\mathbf{\theta}}.$

Similar far-field Green’s functions corresponding to infinitesimal dipoles oriented along various directions are given below.

Relation between Source Excitation and Far-Field Pattern¶

Symmetric Maxwell’s Equations¶

Vector Potentials and Helmholtz Wave Equations¶

Far-field Approximations¶

Far-field Green’s Functions of Infinitesimal Dipoles¶

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Relation between Source Excitation and Far-Field Pattern¶

Symmetric Maxwell’s Equations¶

Vector Potentials and Helmholtz Wave Equations¶

Far-field Approximations¶

Far-field Green’s Functions of Infinitesimal Dipoles¶

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