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EM.Picasso

10,321 bytes removed, 21:21, 3 June 2015
/* Excitation Sources */
Figure 1: Coupling gap sources in the Port Definition dialog by associating more than one source with a single port.
 
=== Calculating Port Characteristics At Gap Discontinuities ===
 
A gap source on a metal trace and a probe source on a PEC via behave like a series voltage source with a prescribed strength (of 1V and zero phase by default) that creates a localized discontinuity on the path of electric current flow. At the end of a planar MoM simulation, the electric current passing through the voltage source is computed and integrated to find the total input current. From this one can calculate the input admittance as
 
:<math> Y_{in} = \frac{I_{in}}{V_s} = \frac {\int_W \hat{y} \cdot \mathbf{J_s} \, dy} {V_s} </math>
<!--[[File:PMOM54(1).png]]-->
 
for gap sources on metal traces, where the line integration is performed across the width of the metal strip, and
 
:<math> Y_{in} = \frac{I_{in}}{V_s} = \frac {\int_S \hat{z} \cdot \mathbf{J_p} \, ds} {V_s} </math>
<!--[[File:PMOM55.png]]-->
 
for probe sources on PEC vias, where the surface integration is performed over the cross section of the via. On the other hand, a gap source on a slot trace behaves like a shunt current source with a prescribed strength (of 1A and zero phase by default) that creates a localized discontinuity on the path of magnetic current flow. At the end of a planar MoM simulation, the magnetic current passing through the current source is computed and integrated to find the total input voltage across the current filament. From this one can calculate the input impedance as
 
:<math> Z_{in} = \frac{V_{in}}{I_s} = \frac{\int_W \hat{y} \cdot \mathbf{M_s} \,dy} {V_s} = \frac{\int_W E_y \, dy}{V_s} </math>
<!--[[File:PMOM56.png]]-->
 
Note that the input admittance or impedance defined at a gap source port is referenced to the two terminals of the voltage source connected across the gap as shown in the figure below. This is different than the input admittance or impedance that one may normally define for a microstrip port, which is referenced to the substrate's ground.
 
[[File:PMOM59(1).png|800px]]
 
Definition of different input impedances at the gap location.
 
To resolve this problem, you can place a gap source on a metal strip line by a distance of a quarter guide wavelength (&lambda;<sub>g</sub>/4) away from its open end. Note that (&lambda;<sub>g</sub> = 2p/ß), where ß is the propagation constant of the metallic transmission line. As show in the figure below, the impedance looking into an open quarter-wave line segment is zero, which effectively shorts the gap source to the planar structure's ground. The gap admittance or impedance in this case is identical to the input admittance or impedance of the planar structure.
 
[[File:PMOM60(1).png|800px]]
 
Placing a gap source a quarter guide wavelength away from the open end of a feed line to effectively short it to the ground at the gap location.
 
The same principle applies to the gap sources on slot traces. The figure below shows how to place two gap sources with opposite polarities a quarter guide wavelength away from their shorted ends to calculate the correct input impedance of the CPW line looking to the left of the gap sources. Note that in this case, you deal with shunt filament current sources across the two slot lines and that the slot line carry magnetic currents. The end of the slot lines look open to the magnetic currents, but in reality they short the electric field. The quarter-wave CPW line acts as an open circuit to the current sources.
 
[[File:PMOM61(1).png|800px]]
 
Placing two oppositely polarized gap sources a quarter guide wavelength away from the short end of a CPW line to effectively create an open circuit beyond the gap location.
 
The case of a probe source placed on a PEC via that is connected to a ground plane is more straightforward. In this case, the probe source's gap discontinuity is placed at the middle plane of the PEC via. If the via is short, it is meshed using a single prismatic element, which is connected to the ground from one side and to the metal strip line from the other. Therefore, the probe admittance or impedance is equal to that of the structure at a reference plane that passed through the host via.
 
[[File:PMOM62(2).png|800px]]
 
Input impedance of a probe source on a PEC via connected to a ground plane.
 
=== Exciting Multiport Structures Using Linear Superposition ===
 
If your planar structure has two or more sources, but you have not defined any ports, all the lumped sources excite the structure locally and contribute to the excitation vector needed for the MoM solution of the problem. However, when you assign N ports to the sources, then you have a multiport structure that is characterized by an N×N admittance matrix (instead of a single Y<sub>in</sub> parameter), or an N×N impedance matrix, or an N×N scattering matrix. To calculate these matrices, [[EM.Cube]] uses a binary excitation scheme in conjunction with the principle of linear superposition. In this binary scheme, the structure is analyzed N times. Each time one of the N port-assigned sources is excited, and all the other port-assigned sources are turned off.
 
In the case of gap sources on metal traces and probe sources on PEC vias, turning a source off means shorting a series voltage source. The electric currents passing through these sources are then found at each port location, and the admittance [[parameters]] are found as follows:
 
:<math> I_m = \sum_{n=1}^N Y_{mn} V_n, \quad \quad Y_{mn} = \frac{I_m}{V_n} \bigg|_{V_k=0, k \ne n}</math>
<!--[[File:PMOM57.png]]-->
 
In the case of gap sources on slot traces, turning a source off means opening a shunt filament current source. The magnetic currents passing through the source locations, and thus the voltages across them, are then found at all ports, and the impedance [[parameters]] are found as follows:
 
:<math> V_m = \sum_{n=1}^N Z_{mn} I_n, \quad \quad Z_{mn} = \frac{V_m}{I_n} \bigg|_{I_k=0, k \ne n}</math>
<!--[[File:PMOM58.png]]-->
 
The N solution vectors that are generated through the N binary excitation analyses are finally superposed to produce the actual solution to the problem. However, in this process, [[EM.Cube]] also calculates all the port characteristics. Keep in mind that the impedance (Z) and admittance (Y) matrices are inverse of each other. From the impedance matrix, the scattering matrix is calculated using the following relation:
 
:<math> \mathbf{[S] = [Y_0] \cdot ([Z]-[Z_0]) \cdot ([Z]+[Z_0])^{-1} \cdot [Z_0]} </math>
<!--[[File:PMOM63.png]]-->
 
where <math>\mathbf{[Z_0]}</math> and <math>\mathbf{[Y_0]}</math> are diagonal matrices whose diagonal elements are the port characteristic impedances and admittances, respectively.
=== Modeling Lumped Elements In Planar MoM ===
{{Note|The impedance of the lumped circuit is calculated at the operating frequency of the project using the specified R, L and C values. As you change the frequency, the value of the impedance that is passed to the Planar MoM engine will change.}}
 
=== Calculating Scattering Parameters Using Prony's Method ===
 
The calculation of the scattering (S) [[parameters]] is usually an important objective of modeling planar structures especially for planar circuits like filters, couplers, etc. As you saw earlier, you can use lumped sources like gaps and probes and even active lumped elements to calculate the circuit characteristics of planar structures. The admittance / impedance calculations based on the gap voltages and currents are accurate at RF and lower microwave frequencies or when the port [[Transmission Lines|transmission lines]] are narrow. In such cases, the electric or magnetic current distributions across the width of the port line are usually smooth, and quite uniform current or voltage profiles can easily be realized. At higher frequencies, however, a more robust method is needed for calculating the port [[parameters]].
 
One can calculate the scattering [[parameters]] of a planar structure directly by analyzing the current distribution patterns on the port [[Transmission Lines|transmission lines]]. The discontinuity at the end of a port line typically gives rise to a standing wave pattern that can clearly be discerned in the line's current distribution. From the location of the current minima and maxima and their relative levels, one can determine the reflection coefficient at the discontinuity, i.e. the S<sub>11</sub> parameter. A more robust technique is Prony’s method, which is used for exponential approximation of functions. A complex function f(x) can be expanded as a sum of complex exponentials in the following form:
 
:<math> f(x) \approx \sum_{n=1}^N c_i e^{-j\gamma_i x} </math>
<!--[[File:PMOM73.png]]-->
 
where c<sub>i</sub> are complex coefficients and &gamma;<sub>i</sub> are, in general, complex exponents. From the physics of [[Transmission Lines|transmission lines]], we know that lossless lines may support one or more propagating modes with pure real propagation constants (real &gamma;<sub>i</sub> exponents). Moreover, line discontinuities generate evanescent modes with pure imaginary propagation constants (imaginary &gamma;<sub>i</sub> exponents) that decay along the line as you move away from the location of such discontinuities.
 
In practical planar structures for which you want to calculate the scattering [[parameters]], each port line normally supports one, and only one, dominant propagating mode. Multi-mode [[Transmission Lines|transmission lines]] are seldom used for practical RF and microwave applications. Nonetheless, each port line carries a superposition of incident and reflected dominant-mode propagating signals. An incident signal, by convention, is one that propagates along the line towards the discontinuity, where the phase reference plane is usually established. A reflected signal is one that propagates away from the port plane. Prony's method can be used to extract the incident and reflected propagating and evanescent exponential waves from the standing wave data. From a knowledge of the amplitudes (expansion coefficients) of the incident and reflected dominant propagating modes at all ports, the scattering matrix of the multi-port structure is then calculated. In Prony's method, the quality of the S parameter extraction results depends on the quality of the current samples and whether the port lines exhibit a dominant single-mode behavior. Clean current samples can be drawn in a region far from sources or discontinuities, typically a quarter wavelength away from the two ends of a feed line.
 
[[File:PMOM71.png|800px]]
 
Figure 1: Minimum and maximum current locations of the standing wave pattern on a microstrip line feeding a patch antenna.
== Running Planar MoM Simulations ==
Kazem Sabet
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