Hybrid MoM-PO Method for Metal Antennas with Large Scatterers

Hybrid method of moments (MoM) and physical optics (PO) computational technique in Antenna Toolbox™ allows you to model antennas near large scatterers such as parabolic reflectors. The antenna element is modeled using MoM while the effect of electrically large structures is considered using PO.

Subdomain RWG Basis Functions and Extra Dimensions

The familiar Rao-Wilton-Glisson (RWG) basis functions on triangles are based on [2].

In the image, for two arbitrary triangular patches tr_n⁺ and tr_n^- having areas A_n⁺ and A_n^- and sharing a common edge l_n the basis functions has the form

$${\overrightarrow{f}}_n(\overrightarrow{r})=\left\{\begin{array}{l}\frac{l_n}{2A_n^{+}}{\overrightarrow{\rho }}_n^{+}\overrightarrow{r}\text{ in }t{r}_n^{+}\\ \frac{l_n}{2A_n^{-}}{\overrightarrow{\rho }}_n^{-}\overrightarrow{r}\text{ in }t{r}_n^{-}\end{array}\right\}\text{ (1)}$$

where $${\overrightarrow{\rho }}_n^{+}=\overrightarrow{r}-{\overrightarrow{r}}_n^{-}$$ is the vector drawn from the free vertex of the triangle tr_n⁺ to the observation point $$\overrightarrow{r}$$; $${\overrightarrow{\rho }}_n^{-}={\overrightarrow{r}}_n^{-}-\overrightarrow{r}$$ is the vector drawn from the observation point to the free vertex of the triangle tr_n^-. The basis function is zero outside the two adjacent triangles. The RWG vector basis function is linear and has no flux (that is, has no normal component) through its boundary.

From [1], along with the standard definition, this method requires two unit normal vectors $${\overrightarrow{n}}_n^{\pm }$$ and two-unit vectors $${\overrightarrow{t}}_n^{\pm }$$ also shown in the figure. Vector $${\overrightarrow{t}}_n^{+}$$ is the plane of triangle tr_n⁺; both vectors are perpendicular to the edge l_n. They are defined at the center of edge , which isl_n denoted by $${\overrightarrow{r}}_n$$. Directions of $${\overrightarrow{t}}_n^{\pm }$$

are also shown in the figure. This technique assumes that the normal vectors are properly (angle between adjacent $${\overrightarrow{n}}_n^{\pm }$$ must be less than 180 degrees) and uniquely defined. Specific vector orientation (e.g. outer or inner normal vectors) does not matter. We then form two cross product vectors $${\overrightarrow{l}}_n^{\pm }$$,

$${\overrightarrow{l}}_n^{\pm }={\overrightarrow{t}}_n^{\pm }\times {\overrightarrow{n}}_n^{\pm }\text{ (2)}$$

and establish that both such unit vectors directed along the edge are identical,

$${\overrightarrow{l}}_n^{\pm }={\overrightarrow{l}}_n^{-}={\overrightarrow{l}}_n\text{ (3)}$$

Only vector $${\overrightarrow{l}}_n$$ is eventually needed.

MoM Region and PO Region

The surface current density, $$\overrightarrow{J}(\overrightarrow{r})$$, on the entire metal surface is expanded into N RWG basis functions. However, a part of such basis functions belongs to the MoM region (or "exact region") while another part will belong to the PO region (or "approximate region"). These basis functions (or regions) can overlap and be arbitrarily distributed in space (not necessarily be contiguous). The method assumes that N_MoM basis functions from the MoM region up front in the list and N_PO basis functions for the PO region afterward. Therefore, you have $$(N_{PO}+N_{MoM}=N)$$

$$\overrightarrow{J}(\overrightarrow{r})={\displaystyle \sum _{n=1}^{N_{MoM}}{I}_n^{MoM}}{\overrightarrow{f}}_n(\overrightarrow{r}),\overrightarrow{J}(\overrightarrow{r})={\displaystyle \sum _{n=1}^{N_{PO}}{I}_n^{PO}}{\overrightarrow{f}}_{n+N_{MoM}}(\overrightarrow{r})\text{ (4)}$$

MoM Solution and PO Solution

If there is no PO region, you can solve the entire problem using the MoM with single square MoM system matrix $$\widehat{Z}$$, which may be subdivided into 4 matrices as shown.

$$\widehat{Z}=\left(\begin{array}{cc}{\widehat{Z}}_{11}& {\widehat{Z}}_{12}\\ {\widehat{Z}}_{21}& {\widehat{Z}}_{22}\end{array}\right),\dim ({\widehat{Z}}_{11})=N_{MoM}\times N_{MoM},\dim ({\widehat{Z}}_{12})=N_{MoM}\times N_{PO}\text{ (5)}$$

The figure shows the matrix interpretation of the hybrid MoM-PO solution and its comparison with plain MoM solution. The method assumes the antenna feeds gives the vector, $$\overrightarrow{V}$$ that describes the excitation, which belongs to the MoM region only.

The hybrid solution keeps submatrices $${\widehat{Z}}_{11}$$ and $${\widehat{Z}}_{12}$$. In other words, the method solves the standard system of the linear equations for the MoM region where radiation from the PO region via $${\widehat{Z}}_{12}$$ is considered.

The hybrid solution ignores the submatrices, $${\widehat{Z}}_{22}$$ entirely. Here, the currents in the PO region do not interact with each other. They are found via the radiated magnetic field, $$\overrightarrow{H}(\overrightarrow{r})$$, from the MoM region, using PO approximation [1]. A new matrix describes this operation, $${\widehat{Z}}_{PO}$$, and negative identity matrix, E, which replaces $${\widehat{Z}}_{22}$$.

Finding Z_PO

The suitable PO approximation has the form [1]

$$\overrightarrow{J}(\overrightarrow{r})=2\delta (\overrightarrow{r})[\overrightarrow{n}(\overrightarrow{r})\times \overrightarrow{H}(\overrightarrow{r})]\text{ (6)}$$

where δ accounts for the shadowing effects. If the observation point lies in the shadow region, δ must be zero. Otherwise it equals ±1 depending on the direction of incidence with respect to the orientation normal vector $$\overrightarrow{n}(\overrightarrow{r})$$. Using second Eq.(4) yields:

$${\displaystyle \sum _{n=1}^{N_{PO}}{I}_n^{PO}{\overrightarrow{f}}_{n+N_{MoM}}(\overrightarrow{r})}=2\delta (\overrightarrow{r})[\overrightarrow{n}(\overrightarrow{r})\times \overrightarrow{H}(\overrightarrow{r})]\text{ (7)}$$

Reference [1] outlines an elegant way to express unknowns I_n^PO explicitly, using an interesting variation of the collocation method. First, we consider a collocation point that tends to the edge center $${\overrightarrow{r}}_{n+N_{MoM}}$$ of a certain basis function $${\overrightarrow{f}}_{n+N_{MoM}}(\overrightarrow{r})$$ and is in its plus triangle. We then multiply Eq. (7) by vector $${\overrightarrow{t}}_{n+N_{MoM}}^{+}$$. Since the normal component of the basis function under interest at the edge is one and all other basis functions sharing the same triangle have no normal component at this edge, the result becomes

$$I_n^{PO}=2\delta ({\overrightarrow{r}}_{n+N_{MoM}}){\overrightarrow{t}}_{n+N_{MoM}}^{+}\cdot \lfloor {\overrightarrow{n}}_{n+N_{MoM}}^{+}\times \overrightarrow{H}({\overrightarrow{r}}_{n+N_{MoM}})\rfloor \text{ (8a)}$$

Repeat the same operation with the minus triangle and obtain

$$I_n^{PO}=2\delta ({\overrightarrow{r}}_{n+N_{MoM}}){\overrightarrow{t}}_{n+N_{MoM}}^{-}\cdot \lfloor {\overrightarrow{n}}_{n+N_{MoM}}^{-}\times \overrightarrow{H}({\overrightarrow{r}}_{n+N_{MoM}})\rfloor \text{ (8b) }$$

Add both Eqs. (8a) and (8b) together, divide the result by two, and transform the triple vector product to obtain

$$I_n^{PO}=2\delta ({\overrightarrow{r}}_{n+N_{MoM}})\overrightarrow{H}({\overrightarrow{r}}_{n+N_{MoM}})\cdot (\lfloor {\overrightarrow{t}}_{n+N_{MoM}}^{+}\times {\overrightarrow{n}}_{n+N_{MoM}}^{+}\rfloor +\lfloor {\overrightarrow{t}}_{n+N_{MoM}}^{-}\times {\overrightarrow{n}}_{n+N_{MoM}}^{-}\rfloor )/2\text{ (9)}$$

Therefore, according to Eqs. (2) and (3),

$$I_n^{PO}=2\delta ({\overrightarrow{r}}_{n+N_{MoM}})\overrightarrow{H}({\overrightarrow{r}}_{n+N_{MoM}})\cdot {\overrightarrow{l}}_{n+N_{MoM}}(10)$$

To complete the derivation, the H-field radiated by the MoM region is always written in the form

$$\overrightarrow{H}(\overrightarrow{r})={\displaystyle \sum _{n=1}^{N_{MoM}}{\overrightarrow{C}}_n(\overrightarrow{r})I_n^{MoM}}\text{ (11)}$$

where $${\overrightarrow{C}}_n(r)$$are given by individual basis function contributions. In the simplest case, every such contribution is the dipole radiation [3]. Substitution of Eq. (11) into Eq. (10) yields

$$\begin{array}{l}{I}_n^{PO}={\displaystyle \sum _{n=1}^{N_{MoM}}{\widehat{Z}}_{POmn}}{I}_n^{MoM}\text{ (12)}\\ {\widehat{Z}}_{POmn}=2\delta ({\overrightarrow{r}}_{n+N_{MoM}})\overrightarrow{C}({\overrightarrow{r}}_{n+N_{MoM}})\cdot {\overrightarrow{l}}_{m+N_{MoM}}m=1,\mathrm{....},N_{PO},n=1,\mathrm{...},N_{MoM}\end{array}$$

Direct Solution Method

According to the second figure, the coupled system of equations has the form

$$\begin{array}{l}{\widehat{Z}}_{11}{\overrightarrow{I}}^{MoM}+{\widehat{Z}}_{12}{\overrightarrow{I}}^{PO}=\overrightarrow{V}\text{ (13)}\\ {\overrightarrow{I}}^{PO}={\widehat{Z}}_{PO}{\overrightarrow{I}}^{MoM}\end{array}$$

The direct solution method results in the substitution of the expression for the PO current into the first equation,

$$({\widehat{Z}}_{11}+{\widehat{Z}}_{12}{\widehat{Z}}_{PO}){\overrightarrow{I}}^{MoM}=\overrightarrow{V}\text{ (14)}$$

Note

The classic physical optics (PO) formulation does not support multiple reflections from a physical structure illuminated by a plane wave. The PO current density is valid only in the illuminated region of the structure. This formulation does not handle any reflections from the illuminated region that result in secondary illumination of a different region of the structure.

• Case 1: When the direction of the incident plane wave results in a reflection back in the direction of the incoming source.

  

• Case 2: When the angle of the incident plane wave causes a second reflection from a different part of the structure, this reflection contributes significantly to the scattered field and is not be considered by the PO solver.

  

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References

[1] U. Jakobus and F. M. Landstorfer, “Improved PO-MM Formulation for Scattering from Three-Dimensional Perfectly Conducting Bodies of Arbitrary Shape,” IEEE Trans. Antennas and Propagation, vol. AP-43, no. 2, pp. 162-169, Feb. 1995.

[2] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas and Propagation, vol. AP-30, no. 3, pp. 409-418, May 1982.

[3] S. Makarov, Antenna and EM Modeling in MATLAB, Wiley, New York, 2002.