rautio
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Hi Loucy, Ok to just call me Jim. Quite a discussion we have going!
As for the gap excitation, due to the overlapping roof-top basis functions (I hope that has some meaning to you, if not, I will explain), the voltage over the subsection that gets excited is restricted to the very center of the subsection, the peak of the roof-top. This roof-top peak is the gap excitation. This is true for both unshielded and shielded EM tools.
This gap excitation does indeed excite many many transmission line modes. If more than one mode is propagating on the transmission line, this is a big problem. You will get odd results in the analysis. If you build it, your connectors will also excite all the propagating modes and generate odd results there too.
So, let's restrict the transmission line to one single propagating mode. All other modes are cut-off (wave number and Zo are both imaginary). All of these evanescent modes add together to create a net capacitance across the gap. (If there is loss, you also have resistance.) Since you are not going to use this same gap port when you actually build the circuit, the gap capacitance must be removed. This is easily done by the double delay de-embedding described in a paper I referenced in a previous post in this thread. The value of the capacitance is printed out by Sonnet. Feel free to try it for different subsectioning. You will see some change when you change the subsection size, but not much. At any rate, all of that capacitance is removed from your results. Only the effect of the single propagating mode is left.
The important thing to realize is that for a given analysis result, the voltage across the gap is unique. This is because the integration path is zero length. It does not matter what path you take across the gap, you always get the same answer.
The current is unique because that is what MoM solves for. That is the answer that comes out of the matrix inversion. There is only one number, and that is the number we use. This makes sense. There should be a single number for the current. All you have to do is sit there and count the number of electrons flowing past in one second. There is a single integer number that is the exact and unique answer.
You may be familar with finite elements. My understanding is that they determine the current from the fields. OK, now you can have different answers depending on the path of integration taken. Not so for MoM, both shielded and unshielded.
As I said above, there is no problem for calculating the voltage across a gap. You get a single answer. No one can pick a different path across the gap for the integral and come up with a different answer. In sharp contrast, when we have a transmission line and want to calculate the line voltage for a given mode, now we have to pick a path for integration from the line to the ground return. If the line is lossless and homogenous (i.e., only one dielectric), then the answer you get is unique, no problem. Pick any path for integration that you want. Always the same answer. (This is for a given analysis result.)
If the line is lossy or there is more than one dielectric, now the answer you get depends on the path you pick. This is for the same single analysis result, no changing subsectioning or anything else. This is where the ambiguity comes in. This is also where approximation comes in if you try get calculate S-parameters directly by setting up a transmission line and looking for incident and reflected waves.
Turns out that port voltages and currents are equivalent data sets to forward and reflected waves. If you have data for one set of variables, you can do arithmetic and get the data for the other set. I simply noted (about 20 years ago) that it is realtively easy to get exact data (to within EM accuracy) for the current and voltage data set (i.e., Y-parameters), and that is what we use internally. When the user wants S-parameters, it is just some quick arithmetic to get the results converted. No transmission lines needed. No uncertainty in seperating incident and reflected, and no ambiguity in calculating voltage between the line and whatever we arbitrarily say is "ground". Pretty neat, huh?
Added after 49 minutes:
OK Jian, yes the conj(Zc) in the numerator only does make mag(Gamma) <= 1 for passive one-port circuits. That's nice. However, in this case, Gamma = 0 now corresponds to a case where the reflected wave is not zero, it can even be quite large. That is because this is a "reflection coefficient" used for optimizing to a conjugate match. It is not used to calculate the actual reflected wave.
My understanding of your postion is that waveguide theory is invalid for complex Zo because you can not seperate the incident and reflected waves, and you used the unconjugated gamma definition to illustrate your point (it gave Gamma > 1), at least for the case when the complex Zo comes from "transverse loss".
I gave a counter example on how it would be easy to identify the incident and reflected waves on any transmission line and you did not respond.
I further illustrated how we can easily calculate S-parameters without having to identify incident and reflected waves on a transmission line, and we can get these strange values of Gamma without loss and without transmission lines. And further, these strange ("ridiculous" as you describe them) values of gamma are easily understood as making sense and being genuinely useful. Your position (as I understand it) is that complex Zo (at least certain kinds of complex Zo) causes these problems because of a certain kind of loss in transmission lines can not then be true.
As for the gap excitation, due to the overlapping roof-top basis functions (I hope that has some meaning to you, if not, I will explain), the voltage over the subsection that gets excited is restricted to the very center of the subsection, the peak of the roof-top. This roof-top peak is the gap excitation. This is true for both unshielded and shielded EM tools.
This gap excitation does indeed excite many many transmission line modes. If more than one mode is propagating on the transmission line, this is a big problem. You will get odd results in the analysis. If you build it, your connectors will also excite all the propagating modes and generate odd results there too.
So, let's restrict the transmission line to one single propagating mode. All other modes are cut-off (wave number and Zo are both imaginary). All of these evanescent modes add together to create a net capacitance across the gap. (If there is loss, you also have resistance.) Since you are not going to use this same gap port when you actually build the circuit, the gap capacitance must be removed. This is easily done by the double delay de-embedding described in a paper I referenced in a previous post in this thread. The value of the capacitance is printed out by Sonnet. Feel free to try it for different subsectioning. You will see some change when you change the subsection size, but not much. At any rate, all of that capacitance is removed from your results. Only the effect of the single propagating mode is left.
The important thing to realize is that for a given analysis result, the voltage across the gap is unique. This is because the integration path is zero length. It does not matter what path you take across the gap, you always get the same answer.
The current is unique because that is what MoM solves for. That is the answer that comes out of the matrix inversion. There is only one number, and that is the number we use. This makes sense. There should be a single number for the current. All you have to do is sit there and count the number of electrons flowing past in one second. There is a single integer number that is the exact and unique answer.
You may be familar with finite elements. My understanding is that they determine the current from the fields. OK, now you can have different answers depending on the path of integration taken. Not so for MoM, both shielded and unshielded.
As I said above, there is no problem for calculating the voltage across a gap. You get a single answer. No one can pick a different path across the gap for the integral and come up with a different answer. In sharp contrast, when we have a transmission line and want to calculate the line voltage for a given mode, now we have to pick a path for integration from the line to the ground return. If the line is lossless and homogenous (i.e., only one dielectric), then the answer you get is unique, no problem. Pick any path for integration that you want. Always the same answer. (This is for a given analysis result.)
If the line is lossy or there is more than one dielectric, now the answer you get depends on the path you pick. This is for the same single analysis result, no changing subsectioning or anything else. This is where the ambiguity comes in. This is also where approximation comes in if you try get calculate S-parameters directly by setting up a transmission line and looking for incident and reflected waves.
Turns out that port voltages and currents are equivalent data sets to forward and reflected waves. If you have data for one set of variables, you can do arithmetic and get the data for the other set. I simply noted (about 20 years ago) that it is realtively easy to get exact data (to within EM accuracy) for the current and voltage data set (i.e., Y-parameters), and that is what we use internally. When the user wants S-parameters, it is just some quick arithmetic to get the results converted. No transmission lines needed. No uncertainty in seperating incident and reflected, and no ambiguity in calculating voltage between the line and whatever we arbitrarily say is "ground". Pretty neat, huh?
Added after 49 minutes:
OK Jian, yes the conj(Zc) in the numerator only does make mag(Gamma) <= 1 for passive one-port circuits. That's nice. However, in this case, Gamma = 0 now corresponds to a case where the reflected wave is not zero, it can even be quite large. That is because this is a "reflection coefficient" used for optimizing to a conjugate match. It is not used to calculate the actual reflected wave.
My understanding of your postion is that waveguide theory is invalid for complex Zo because you can not seperate the incident and reflected waves, and you used the unconjugated gamma definition to illustrate your point (it gave Gamma > 1), at least for the case when the complex Zo comes from "transverse loss".
I gave a counter example on how it would be easy to identify the incident and reflected waves on any transmission line and you did not respond.
I further illustrated how we can easily calculate S-parameters without having to identify incident and reflected waves on a transmission line, and we can get these strange values of Gamma without loss and without transmission lines. And further, these strange ("ridiculous" as you describe them) values of gamma are easily understood as making sense and being genuinely useful. Your position (as I understand it) is that complex Zo (at least certain kinds of complex Zo) causes these problems because of a certain kind of loss in transmission lines can not then be true.
