[[Image:Info_icon.png|30px]] Click here to access '''[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types]]'''.
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== Computing Port Characteristics From Time-Domain Data ==
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For the computation of the S-parameters in [[EM.Tempo]], the source associated with each port is excited separately with all the other ports turned off. Due to the existence of internal source resistances, tuning the other sources off is indeed equivalent to terminating the other ports in their characteristics impedances (matched ports). When the jth port is excited, all the S<sub>ij</sub> parameters are calculated together based on the following definition:
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:<math> S_{ij} = \sqrt{\frac{Re(Z_i)}{Re(Z_j)}} \cdot \frac{V_j - Z_j^*I_j}{V_i+Z_i I_i} </math>
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where V<sub>i</sub> is the voltage across Port i, I<sub>i</sub> is the current flowing into Port i and Z<sub>i</sub> is the characteristic impedance of Port i. The sweep loop then moves to the next port until all ports have been excited. Besides the scattering parameters, the admittance (Y) and impedance (Z) parameters are also calculated based on the following relationships:
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:<math>\mathbf{ [Z] = [\sqrt{Z_0}] \cdot ([U]+[S]) \cdot ([U]-[S])^{-1} \cdot [\sqrt{Z_0}] }</math>
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:<math> \mathbf{ [Y] = [Z]^{-1} } </math>
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where <math>\mathbf{[U]}</math> is the identity matrix of order N, and <math>\mathbf{\sqrt{Z_0}}</math> is a diagonal matrix whose diagonal elements are the square roots of port characteristic impedances. The voltage standing wave ratio (VSWR) of the structure at the first port is also computed based on the following definition:
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:<math> \text{VSWR} = \frac{|V_{max}|}{|V_{min}|} = \frac{1+|S_{11}|}{1-|S_{11}|} </math>
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== Admittance/Impedance Characteristics of Gap Sources ==
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[[EM.Cube]]'s MoM-based computational modules, [[EM.Picasso]] and [[EM.Libera]], provide a number of gap source types for which you can compute the port characteristics, <i>i.e.</i> S/Z/Y parameters. These source types are:
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{| class="wikitable"
|-
! scope="col"| Soure Type
! scope="col"| Module
! scope="col"| Host Types
! scope="col"| Computed Parameters
|-
| Strip Gap Circuit
| [[EM.Picasso]]
| Metal traces
| Y-parameters
|-
| Strip Gap Circuit
| [[EM.Picasso]]
| Slot traces
| Z-parameters
|-
| Probe Gap Circuit
| [[EM.Picasso]]
| Embedded PEC vias
| Y-parameters
|-
| Strip Gap Circuit
| [[EM.Libera]] - Surface MoM
| PEC strips
| Y-parameters
|-
| Wire Gap Circuit
| [[EM.Libera]] - Wire MoM
| PEC lines and thin wires
| Y-parameters
|}
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Once one set of Y or Z parameters are computed, the other parameters are calculated based on the matrix relationships given in the previous section.
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A gap source on a metal strip or a metal wire behaves 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 (<b>J</b><sub>s</sub>). At the end of the MoM simulation, the electric current passing through the voltage source (V<sub>s</sub>) is computed and integrated to find the total input current (I<sub>in</sub>). From this one can calculate the input admittance as
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:<math> Y_{in} = \frac{I_{in}}{V_s} = \frac {\int_W \hat{y} \cdot \mathbf{J_s} \, dy} {V_s} </math>
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for gap sources on metal traces, where the line integration is performed across the width of the metal strip. In the case of a wire gap, the current solution is indeed the total current.
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It is important to note that the input admittance 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 that one may normally define for a microstrip port, which is referenced to the substrate's ground. To resolve this problem, you can place a gap source on a metal strip line by a distance of a quarter guide wavelength (λ<sub>g</sub>/4) away from its open end. Note that λ<sub>g</sub> = 2π/β, where β 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 in this case is identical to the input admittance of the planar structure.
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<table>
<tr>
<td> [[Image:PMOM59(1).png|thumb|left|480px|Definition of different input impedances at the gap location.]] </td>
</tr>
<tr>
<td> [[Image:PMOM60(1).png|thumb|left|480px|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.]] </td>
</tr>
</table>
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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 (<b>M</b><sub>s</sub>). At the end of a planar MoM simulation, the magnetic current passing through the current source (I<sub>s</sub>) is computed and integrated to find the total input voltage across the current filament (V<sub>in</sub>). From this one can calculate the input impedance as
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:<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>
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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.
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<table>
<tr>
<td> [[Image:PMOM61(1).png|thumb|left|480px|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.]] </td>
</tr>
</table>
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In [[EM.Picasso]], a probe source on a PEC via behaves 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 the vertical electric current flow (<b>J</b><sub>p</sub>). At the end of the 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
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:<math> Y_{in} = \frac{I_{in}}{V_s} = \frac {\int_S \hat{z} \cdot \mathbf{J_p} \, ds} {V_s} </math>
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where the surface integration is performed over the cross section of the via. 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 is equal to that of the structure at a reference plane that passed through the host via.
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<table>
<tr>
<td> [[Image:PMOM62(2).png|thumb|left|480px|Input impedance of a probe source on a PEC via connected to a ground plane.]] </td>
</tr>
</table>
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Once the admittance or impedance parameters are determined, one can find their complementary parameter set:
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:<math> \mathbf{ [Z] = [Y]^{-1} } </math>
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:<math> \mathbf{ [Y] = [Z]^{-1} } </math>
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The scattering parameters can then be computed from the impedance parameters using the following relationship:
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:<math>\mathbf{ [S] = [Y_0] \cdot ([Z]-[Z_0]) \cdot ([Z]+[Z_0])^{-1} \cdot [Z_0] }</math>
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where ['''Z<sub>0</sub>'''] is a diagonal matrix whose diagonal elements are the port characteristic impedances, and ['''Y<sub>0</sub>'''] is a diagonal matrix whose diagonal elements are the port characteristic admittances.
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== Calculating Scattering Parameters Using Prony's Method ==
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[[EM.Picasso]] provides a special source type called '''[[Glossary of EM.Cube's Excitation Sources#Scattering Wave Port | Scattering Wave Port]]''' that is specifically intended for computing the S-parameters of planar structures. This is done by analyzing the current distribution patterns on the port transmission lines. The discontinuity at the end of a port line (junction region) gives rise to a standing wave pattern 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>i.e.</i> the S<sub>11</sub> parameter. A more rigorous technique is Pronyâs 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>
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where c<sub>i</sub> are complex coefficients and γ<sub>i</sub> are, in general, complex exponents. From the physics of transmission lines, we know that lossless lines may support one or more propagating modes with pure real propagation constants (real γ<sub>i</sub> exponents). Moreover, line discontinuities generate evanescent modes with pure imaginary propagation constants (imaginary γ<sub>i</sub> exponents) that decay along the line as you move away from the location of such discontinuities.
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In practical planar structures, each port line normally supports one, and only one, dominant propagating mode. Multi-mode 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 is used by [[EM.Picasso]] to extract the incident and reflected propagating and evanescent exponential waves from the standing wave data. From the 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.
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<table>
<tr>
<td>
[[Image:OBS MAN10.png|thumb|left|640px|Minimum and maximum current locations of the standing wave pattern on a microstrip line feeding a patch antenna.]]
</td>
</tr>
</table>
== Exciting Multiport Structures Using Linear Superposition ==