S-Matrix Simulation

Problem Description

A common measurement made on a 2-port rf device is reflection and transmission of an rf signal, for either a single frequency, or for a range of frequencies. This measurement results in the Scattering-Matrix, or S-Matrix, whose elements S11 and S21 are the reflected and transmitted signal for unit input at Port 1. VORPAL provides the capability to simulate these S-Matrix parameters for arbitrarily complex devices connected to waveguides propagating TE, TM, and TEM modes. To demonstrate this capability, we show in this example how to measure S11 and S21 in a dual-mode cavity filter, connected to WR-90 waveguide, with the narrow-band band-pass tuned to pass frequencies between 9.95 and 10.05 GHz.

Note: All visualization for this example is done using VorpalView. To do the visualization, navigate to where the data is saved and open VorpalView.

Input File Features

The simulation geometry consists of a standard WR-90 rectangular waveguide with the filter cavity (also referred to as the Device-Under-Test (DUT) in this writeup) in the center. A planar antenna in the waveguide, near the DUT, launches the incident wave while allowing reflected signals to pass through into the waveguide behind it. The waveguide ends are terminated in gradual absorbing layers with negligible reflection, and the reflected and transmitted signals are measured just in front of these absorbers.

A main feature of this input file is that the waveguide description and the DUT description are short compact sections of input, that are easily substituted. Thus this example is effectively a template for an S-Matrix simulation of any device. The time histories of voltage signals used to measure S11 and S21 are also built in and automated for easy substitution. Furthermore, these signals are easily turned into S11 and S21 values, or frequency variation curves, using the standard “Amplitude” and “FFT” capabilities in the VorpalView program’s History Tab.

Running the Simulation

Start VorpalComposer and select File -> Clone Example. Highlight Solving Classical Physics Problems and then select Next. Highlight sMatrix and then select Choose. Create a new folder and then select Choose.

Alternatively, save the VORPAL input file, sMatrix.pre, and open in VorpalComposer.

The file should be displayed in the right pane of the Setup window. Click on the Save and Process Setup button in the lower right corner. Proceed to the run window as instructed. To run the file, click on the Run button in the lower left corner of the window. You can see the real time output of the run in the right pane.

This example is more sophisticated than some of the others, in that successful determination of S-Matrix parameters is not the result of a single run, but rather a result of a procedure involving several runs. This includes at least one Calibration Run, and at least one Data Run to determine S11 and S21, and then a repetition with input switched to Port 2, in order to determine S22 and S12. All runs use the same input file and it is a set of control variables which determine the action of a particular run. Below we discuss in detail some of the features of this example.

Frequency Band vs. Single Frequency

The user may choose whether to compute a single-frequency value of the S-Matrix parameters, or to compute the variation of the parameters as a function of frequency across a user specified frequency band. The variable, FREQ_CENTER, specifies either the single frequency or the center frequency of the band. The variable, FREQ_BANDWIDTH, provides the bandwidth or is set to 0 if a single-frequency simulation is desired.

With a single frequency simulation, the variable, NUMBER_OF_CYCLES_TO_DRIVE, should be large enough to ensure that the S11 and S21 histories reach a steady amplitude. The “Amplitude” button on VorpalViews’s History Tab is used to give the amplitude, which is the S-Matrix value.

With a finite frequency band, the same variable, NUMBER_OF_CYCLES_TO_DRIVE, can be adjusted upward to increase the detail and sharpness of the S-Matrix variation with frequency. The variable, NUMBER_OF_CYCLES_TO_COAST, may also need to be adjusted upward if the DUT contains internal mode oscillation of large Q (quality) factor. This variable needs to be large enough so that the signal histories have decreased to a negligible value (10^-4, relative to maximum) by the end of the simulation. The “FFT” button on VorpalView’s History Tab is then used to give the S-Matrix variation with frequency, with the plot’s Y-axis units being dB. Be aware that it is usually necessary to zoom in significantly on this plot in order to see the frequency band of interest.

Finally, in both these cases, only the amplitude of the complex-valued S-Matrix parameters is given by the VorpalView program. More sophisticated post-processing (not covered in this example) is needed in order to get the phase information.

Calibration Run

The Calibration Run is done first, and the user must ensure that the variable, CALIBRATION_RUN=1, is set. In the Calibration Run, the DUT is automatically omitted and replaced with a continuation of the waveguide, so that this is a near trivial simulation of a straight length of waveguide that should have effectively 100% transmission. The calibration run serves two very important purposes:

a. To insure that there is negligible (below 1% amplitude, -40 dB) reflected voltage (S11). If the reflection is too high it indicates that either the absorbing boundaries are not working well enough, or that the waveguide’s “modeProfile” description is not accurate enough, and/or that there is not enough grid resolution.

b. To adjust the variable DRIVE_NORMALIZATION, which runs in proportion to observed transmitted voltage (S21), so that the next time the calibration run is done, the transmitted voltage (S21) will be exactly unit amplitude (single frequency) or zero dB (across frequency band). For example, if the first Calibration Run shows an amplitude of 0.667 for S21, change the variable DRIVE_NORMALIZATION to 1.5 times its present value for the next Calibration Run, since 1/0.667 = 1.5.

Changing center frequency, or any waveguide parameter, or even the nominal cell size, will require re-calibration. If not sure, always recalibrate, when changing a parameter.

Data Run

Once the Calibration Run is successful at achieving unit transmission with negligible reflection, the Data Run is then done. The user should ensure that the variable, CALIBRATION_RUN=0, is set. The S-Matrix results are then read from the VorpalView History Tab. An example run, for a frequency band simulation from 8 to 12 GHz, is shown below in the figure. The left part of the figure shows VorpalView’s History Tab, with the FFT’s of the “S11_Voltage” history and the “S21_Voltage” history shown as the top and middle plots, and the actual time-history (no FFT) data shown as the bottom plot. Screen captures, and simple cut-and-paste edits in a Paint program were used to place the two curves on the same plot, with one of them changed in color, as shown in the right part of the figure.

Frequency variation of the S-Matrix Parameters for the dual-mode cavity fielter DUT.

The control variable, SWAP_ENDS, can be set equal to 1 to move the source plane from Port 1 to Port 2 so that the S-Matrix parameters S22 and S12 can be found. However, due to symmetry of this example’s dual-mode filter cavity, these will be the same as S11 and S21.

Dual Mode Cavity Filter

The X coordinate is always the direction aligned with the length of the waveguide, and the input and output waveguide is specified with 10 variables that provide YZ bounding coordinates of the waveguide, YZ coordinates for the voltage measurement, a boolean specifying whether the excited mode is TM or not, and the waveguide cutoff frequency for the mode. Three additional analytic functions provide the interior of the waveguide’s YZ cross-section and the Y and Z components of the mode’s electric field pattern. For this simulation, these parameters all derive from WR90 dimensions, with the waveguide cross-section centered at the YZ origin. The excited mode is the standard lowest mode, TE01, and in particular note that for this mode, the Ez component of the field is zero.

The Device-Under-Test, in this case the dual mode cavity filter, is specified by 6 variables that provide the XYZ bounding coordinates and one macro that constructs the DUT. These all derive from the dimension parameters for the dual mode cavity filter, which are height, width, and length of the rectangular cavity, and the size of the apertures connecting to the waveguide. The figure illustrates these dimension parameters, and the user is encouraged to experiment with changing them.

Dimension parameters of the Dual Mode Cavity Filter

The Dual Mode Cavity Filter operates by coupling the TE01 waveguide mode into the two nearly degenerate TE102 and TE201 modes of the cavity, since CAVITY_LENGTH is very close to CAVITY_WIDTH, in value. The differences in these values, along with the symmetry breaking along the waveguide axis, determine the frequency separation of the two modes. This separation is what gives the filter finite-bandwidth since frequencies between these modes are passed, and frequencies above or below the modes are rejected. A pole in the transmitted signal just below the band contributes to sharpness of the band’s lower edge, but this pole moves easily to the upper frequency edge with small adjustments to the cavity dimension parameters, and the user is encouraged to experiment in finding optimal placement of this pole. The figure shows field patterns from single frequency excitations at three important locations, the pre-band transmission pole, the lower band edge transmission resonance, and a point just above the upper band edge where reflection and transmission are approximately equal. These figures come from VorpalView’s 3D Tab, with the “Paint Field” button activated, after having first selected magnetic field on the Fields Tab. These figures also show the location of the antenna driving the incident wave with respect to the DUT, and the TE102 and TE201 mode mixing within the cavity.

Field patterns from three single-frequency runs, selected at interesting points along the S-Matrix frequency variation curves.

References

This example’s Dual Mode Cavity Filter DUT is patterned after a device detailed at the website:

http://www.mel.diei.unipg.it/limbo/index.php?ption=content&task=view&id=81&catid=34&itemid=55