6.2.1   S-Parameters

S-parameters are calculated in the post-processing called S-Parameters. S-Parameters are calculated from Direct Fourier Transforms (DFT) of signals at ports using Differential method, which is an original method for S-parameters calculation developed by QWED team (see ref. [9] [44]). It has proven to be accurate, effective, and versatile. It is called "differential" because it extracts the S-parameters from the derivatives of field quantities. Because of the necessity to extract spatial derivatives it cannot be applied directly at ports, but only at reference planes situated at some distance from the ports. To assure accurate S-parameters extraction it is recommended to keep 4-5 cells distance between the reference plane and the port plane.

We calculate the S-matrix of an N-port where N is the number of virtual ports defined in the structure. The system can include both Transmission Line Ports and Point Sources/Probe ports. For each transmission line port, the total fields at its respective reference plane are filtered through the modal template and decomposed into incident and reflected waves via the differential method. Reference impedances for S-parameters extraction are the actual frequency-dependent wave impedances of respective transmission lines. At a Point Source/Probe port, the S-Parameters system operates similarly as the FD-Probing. Thus for meaningful results, each Point Source/Probe port should be placed between two metal elements separated by one FDTD cell (otherwise wire should be used). The S-Parameters system considers the port voltage (selected E-field integrated along the FDTD cell), and the current flowing into the embedding circuit (without the current component flowing into the FDTD cell). Reference impedance is taken as resistance of the Point Source/Probe port, unless this resistance is zero, in which case 50 ohm is taken.

Note that the differential method of S-parameters extraction converges slowly close to DC. This note is important for TEM transmission lines, where we may be interested in results close to DC. In such cases, software calculates threshold frequency f_T, depending on line dimensions, and in the frequency range up to f_T the reference impedance for S-parameters calculation is extracted from a linear approximation between the value calculated with differential method and characteristic impedance of TEM line, calculated during quasi-static template generation (at DC). The value of f_T frequency is reported in Simulation Info tab of Simulator Log window as f_threshold_QS.

The results of S-Parameters post-processing are available in the Results window.

The S-parameters calculation algorithm is supplemented with the calculation correction options S corrections assuming. When using the option Smn at reference planes (see below) we obtain the complete information about the circuit, which permits to compensate possible errors due to imperfect matching of the ports. Such complete information is not available when we use the option Sk1 at reference planes (see below). Thus some additional assumptions about the circuit are very helpful in the error compensating procedures. First of all, we use the information about the reciprocity. Since in principle all the calculated structures are reciprocal the option S corrections assuming: Reciprocal N-port should always be checked. Moreover, if we know that we are considering a lossless reciprocal two-port we should also check the reciprocal lossless 2-port option. It helps in accurate extraction of S11 andS21. More information about the correcting procedure can be found in ref. [9] [10] [44].

In QuickWave there are three options for S-parameters extraction:

• Sk1 at reference planes. In this case we excite the circuit only from port number 1 and calculate the incident (a_k) and reflected (b_k) waves at each port. This permits to calculate S_k1 = b_k/a₁. Note that:

-          The simple extraction produces results contaminated by numerical reflections from imperfect absorbing boundaries.

-          If the circuit is declared as reciprocal N-port, it is possible to correct the value of S₁₁, applying the expanded formula:

S₁₁=b₁/a₁-S₁₂a₂/a₁-S₁₃a₃/a₁-....

The software enforces S_k1=S_1k and calculates:

S₁₁=b₁/a₁-b₂a₂/a₁²-b₃a₃/a₁²-....

-          If the circuit is declared as reciprocal lossless 2-port, it is also possible to correct the value of S₂₁, applying the formula:

S₂₁=b₂/a₁-S₂₂a₂/a₁

The software enforces:

|S₂₂|=|S₁₁|

and

arg(S₁₁)+arg(S₂₂)-2arg(S₁₂)= ±p

-          The software will not verify your declaration regarding reciprocity and losses, but it will ignore the declaration of lossless 2-port if the number of ports is different.

• Smn at reference planes. This is a general case in which we perform N simulations exciting the structure from N different ports, and then compose the complete S-matrix from N sets of partial results. Each of these sets corresponds to excitation from port i when the software calculates the parameters S_ki in the same way as the parameters S_k1 are calculated in the Sk1 at reference planes post-processing. When all N sets are available, the software corrects all the S-matrix elements for imperfect absorbing at all ports (reciprocity is irrelevant at this stage). Thus the final results of S_mn calculation may somewhat differ from the N sets of partial results. There are two regimes of running the Smn at reference planes post-processing:

-          In the Sequential regime the software performs N consecutive simulations (with excitation from each of the N ports). Each simulation lasts for a predefined number of FDTD iterations, specified by the user in the Iterations per port box. Note that during one simulation we obtain S_k1 elements like in the Sk1 at reference planes post-processing, and these parameters (either uncorrected or partially corrected for imperfect absorbing boundaries) are being displayed. Only after completing the N simulations with excitation from N consecutive ports QW‑3D assembles the complete S-matrix, correcting all the S-matrix elements for imperfect absorbing at all port. Thus the final results of S_mn calculation may be somewhat different from the intermediate results displayed during calculation.

-          In the Multisimulator regime, the software runs N instances of internal simulator objects in the QW-Simulator application, with a different exciting port in each of them. An apparent disadvantage of this regime resides in the increased memory requirements. However, its important advantage is the possibility of the on-line monitoring of the full corrected S-matrix, calculated for the current number of FDTD iterations. This regime works on single-processor computers. More discussion concerning this regime is given in Multisimulator regime for Smn post-processing.

Attention: If one wants to analyse the results of S_k1 calculation for each elementary simulation it is important to know the indexes of the following ports. Let us assume that 4 port circuit is analysed with the port indexes given in the brackets:

Inp [1]

Out1 [2]

Out2 [3]

Out3 [4]

The above indexes are assigned in the first simulation. In the second simulation the following notation is applied:

Out1 [1]

Out2 [2]

Out3 [3]

Inp [4]

In the third simulation the notation is:

Out2 [1]

Out3 [2]

Inp [3]

Out1 [4]

In the fourth simulation we assign as follows:

Out3 [1]

Inp [2]

Out1 [3]

Out2 [4]

• Gamma K at ref. planes. It allows computing reflection coefficients G at several reference planes simultaneously during a single simulation run. It can be applicable when a multi-source network is considered with all the ports operating as sources simultaneously and, consequently, the scattering matrix cannot be computed.

Attention:

Only in the circuit, which is not declared to be reciprocal, the reflection coefficient is calculated independently of the outputs. In other cases it is corrected for the reflection on the output ports. In those cases any errors of analysis at the output (wrong port definition, wrong template and so on) may strongly influence also the S₁₁ calculation.

As mentioned above the S-parameters are calculated using the vector product of the simulated fields and the mode templates to filter out the influence of the modes other than the considered one. Nevertheless the presence of other modes may have some influence on the accuracy, especially when calculating wide-band characteristics of inhomogeneously filled lines.

Note that the ports which take part in S-matrix calculations must bear consecutive numbers. For example assigning to three available ports the numbers 1, 2 and 4 will result in computation error.

Note that the user can mark several ports as sources. In this case:

• If no S-Parameters post-processing is chosen, all these ports will be simultaneously excited with waveforms assigned to them in the port settings dialogue. In particular, the waveforms may have different amplitude and delay values.

• If Smn post-processing is chosen, each port will serve as a source in a different simulation, with other ones serving as loads.

• If Sk1 post-processing is chosen, all ports will act with waveforms assigned to them and port No.1 will be treated as a reference port. The results of such a post-processing will be inconsistent with standard understanding of S-matrices, but may be useful in some unusual studies.

It is worth noting at this point that the range of frequency in which we calculate the S‑parameters cannot exceed the spectrum of the exciting pulse defined in the port settings dialogue. If this is the case, the S-parameters may be calculated with significantly increased numerical errors and QW-Simulator gives an appropriate warning.

Formally we can choose an exciting pulse of the spectrum much wider than the spectrum needed for S-parameters analysis. This kind of choice will not change the final result to which the simulation converges. However, it may significantly prolong the computing time. The reason is that a wide-band pulse may excite high-Q resonance outside the band of interest. During the simulation process the energy accumulated in the circuit at these frequencies is dissipated slowly and thus the convergence of the S-parameters characteristics to their final shape may be much slower. Thus it is recommended as a standard approach to keep the frequency band of the exciting pulse equal to the band declared for S-parameter extraction.

Another case worth mentioning is the frequency step for S-Parameters post-processing, namely, number of frequency points that the S-parameters should be calculated at. It is a known advantage of the FDTD method that the number of frequency points has little effect on the computing time. This allows for example choosing the step small enough to detect narrow (high-Q) resonances. Nevertheless we suggest that some restraint be exercised in the choice of the frequency step. Usually a reasonable number of frequency points is supposed to be between 100 and 500 and in some cases should be increased to 1000 or so. A higher number of frequency points can slow down the Results display. It also affects the memory occupied during the software runtime and needed for the storage of the results on disk.

It has been already mentioned that in the case of transmission line ports the S-parameters are calculated at the reference planes. Note that a reference plane defined in the Input Interface remains at its user-defined position only if this position coincides with cell edges (i.e., a plane where tangential E-fields are available in QW-Simulator). If not, the reference plane is automatically shifted by the software to the nearest E-field plane (for ports oriented along the Z-axis), or to the nearest E-field plane in the positive X- or Y-direction (for ports oriented along the X- or Y-axis, respectively). The exact position of all reference planes whereat the S‑parameters are calculated may be checked in the Reference Location dialogue, which is active only for the S-Parameters post-processing. For the user convenience the Reference Location dialogue shows also the port location. The Reference Location dialogue offers additional functionality, which is a virtual shift of the reference plane. It is performed by defining a new location of the reference plane. This does not enforce an actual shift of the reference plane, however this newly defined location is taken for the S-parameters calculations and is remembered only in the Results window. Such virtual shift affects phases of S-parameters, but also amplitudes in the case of lossy transmission lines or evanescent modes. The virtual reference shift option can be used for multiple purposes but becomes especially useful when it is required to calculate the S-parameters at the plane which corresponds to the port plane, at which the reference plane must not be placed (see Transmission line port excitation).

For detailed introduction to the S-Parameters post-processing, refer to User Guide 3D: Waveguide-to-coax transition and to other subsections of User Guide 3D: S-parameter extraction problems.