Figure 3: Vectorial (cone) visualization of the current distribution on a patch antenna.
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=== Computing The Near Fields ===
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Once all the current distributions are known in a planar structure, the electric and magnetic fields can be calculated everywhere in that structure using the dyadic Greens's functions of the background structure:
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:<math> \begin{align} \mathbf{E(r) = E_{inc}(r)} + & \sum_{n=1}^N I_n^{(J)} \iiint_V \overline{\overline{G}}_{EJ}(r|r') \cdot f_n^{(J)}(r') \, d\nu' + \\ & \sum_{k=1}^K V_n^{(M)} \iiint_V \overline{\overline{G}}_{EM}(r|r') \cdot f_k^{(M)}(r') \, d\nu' \end{align} </math>
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:<math> \begin{align} \mathbf{H(r) = H_{inc}(r)} + & \sum_{n=1}^N I_n^{(J)} \iiint_V \overline{\overline{G}}_{HJ}(r|r') \cdot f_n^{(J)}(r') \, d\nu' + \\ & \sum_{k=1}^K V_n^{(M)} \iiint_V \overline{\overline{G}} {HM}(r|r') \cdot f_k^{(M)}(r') \, d\nu' \end{align} </math>
<!--[[File:PMOM92(2).png]]-->
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The above equations can be cast into the spectral domain as follows:
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:<math> \begin{align} \mathbf{E(r) = E_{inc}(r)} + \frac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \bigg[ & \sum_{n=1}^N I_n^{(J)} \tilde{\overline{\overline{G}}}_{EJ}(k_{\rho}, z|z') \cdot \tilde{f}_n^{(J)}(k_x, k_y) + \\ & \sum_{k=1}^K V_n^{(M)} \tilde{\overline{\overline{G}}}_{EM}(k_{\rho}, z|z') \cdot \tilde{f}_k^{(M)}(k_x, k_y) \bigg] \, dk_x \, dk_y \end{align} </math>
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:<math> \begin{align} \mathbf{H(r) = H_{inc}(r)} + \frac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \bigg[ & \sum_{n=1}^N I_n^{(J)} \tilde{\overline{\overline{G}}}_{HJ}(k_{\rho}, z|z') \cdot \tilde{f}_n^{(J)}(k_x, k_y) + \\ & \sum_{k=1}^K V_n^{(M)} \tilde{\overline{\overline{G}}}_{HM}(k_{\rho}, z|z') \cdot \tilde{f}_k^{(M)}(k_x, k_y) \bigg] \, dk_x \, dk_y \end{align} </math>
<!--[[File:PMOM93(1).png]]-->
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Calculation of the near-zone fields (fields at the vicinity of the unknown currents) is done at the post-processing stage and in a Cartesian coordinate systems. These calculations involve doubly infinite spectral-domain integrals, which are computed numerically. As was mentioned earlier, [[EM.Cube]]'s planar MoM engine rather uses a polar integration scheme, where the radial spectral variable k<sub>ρ</sub> is integrated over the interval [0, Mk<sub>0</sub>], M being a large enough number to represent infinity, and the angular spectral variable t is integrated over the interval [0, 2π]. You also saw some of the numerical [[parameters]] related to this spectral-domain integration scheme.
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{{Note|When the observation plane is placed very close to the radiating J and M currents, the Green's functions exhibit singularities, which translate to very slow convergence or divergence of the integrals. You need to be careful to place field sensors at adequate distances from these radiating sources.}}
=== Visualizing The Near Fields ===
Near-zone magnetic field map above a microstrip-fed patch antenna.
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=== Computing The Far Fields ===
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Unlike differential-based methods, MoM simulators do not need a radiation box to calculate the far field data. The far-zone fields are calculated directly by integrating the currents on the traces and across the embedded objects using the asymptotic form of the background structureâs dyadic Green's functions:
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:<math> \mathbf{E^{ff}(r)} = \iiint_V \mathbf{ \overline{\overline{G}}_{EJ,ff}(r|r') \cdot J(r') } \, d\nu ' + \iiint_V \mathbf{ \overline{\overline{G}}_{EM,ff}(r|r') \cdot M(r') } \, d\nu '</math>
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:<math> \mathbf{H^{ff}(r)} = \dfrac{1}{\eta_0} \mathbf{ \hat{r} \times E^{ff}(r) }</math>
<!--[[File:PMOM112.png]]-->
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where η<sub>0</sub> = 120π is the characteristic impedance of the free space. As can be seen from the above equations, the far fields have the form of a TEM wave propagating in the radial direction away from the origin of coordinates. This means that the far-field magnetic field is always perpendicular to the electric field and the propagation vector, which in this case happens to be the radial unit vector in the spherical coordinate system. In other words, one only needs to know the far-zone electric field and can easily calculate the far-zone magnetic field from it. In [[EM.Cube]]'s mixed potential integral equation formulation, the far-zone electric field can be expressed in terms of the asymptotic form of the vector electric and magnetic potentials '''A''' and '''F''':
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:<math>\mathbf{E^{ff}}(x,y,z) = j k_0 \eta_0 \hat{r} \times [\hat{r} \times \mathbf{A}(r \to \infty)] + j k_0 \hat{r} \times \mathbf{F}(r \to \infty)</math>
<!--[[File:PMOM113.png]]-->
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The asymptotic form of these vector potentials are calculated using the "'''Method of Stationary Phase'''" when k<sub>0</sub>r → ∞. In that case, one can use the approximation:
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:<math> k_0 |\mathbf{r-r'}| \approx k_0 (r - \mathbf{\hat{r} \cdot r'}) </math>
<!--[[File:PMOM115.png]]-->
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After applying the stationary phase method, one can extract the spherical wave factor exp(-jk<sub>0</sub>r)/r from the far-zone electric field, leaving the rest as functions of the spherical angles θ and φ. In other words, the far field is normalized to r, the distance from the field observation point to the origin. It is customary to express the far fields in spherical components E<sub>θ</sub> and E<sub>φ</sub>. Note that the outward propagating, TEM-type, far fields do not have radial components, i.e. E<sub>r</sub> = 0.
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:<math> \mathbf{E_{\theta}}(\theta, \phi) = \cos\theta \cos\phi E_x + \cos\theta \sin\phi E_y - \sin\theta E_z </math>
:<math> \mathbf{E_{\phi}}(\theta, \phi) = -\sin\phi E_x + \cos\phi E_y </math>
<!--[[File:PMOM114.png]]-->
=== Visualizing The Far Fields ===