# Shawyer's Cavity Revisited

The assumption, up until now, was that Shawyer's cavity used a probe to inject the signal into the cavity (see figure 5 here), however, upon closer examination, it is clear that an aperture was used. It is difficult to see, but figure 1 below shows the waveguide goes up, over the cavity, back down the other side and into the resonator. Using an aperture is also consistent with the previous "Experimental Thruster" cavity (see figure 2 here).

Using the known waveguide width ("a") of WR340 at 8.6cm, the size of Shawyer's cavity is estimated to be four times that at 34.4cm or 344mm. Another method to estimate the length was to use the known diameter[1] of the large end. By measuring the size of the large end in the picture of the unmounted cavity (see figure 5 here) and the length, we get a ratio of length to diameter of 1.39, which means the length is $280*1.39=390$. Taking a rough average of the two estimation methods means a length of 370mm. The diameter of the narrow end was then estimated based on a profile comparison as shown in figure 4, between the modeled cavity and the actual cavity.

With the diameter of the large end known and the length estimated at 370mm, a facsimile was built in CST.

After simulation, the calculated Q of the simulated cavity was 30K which is lower then the reported 45K[2], however, using the maximum estimated length of 390mm, the calculated Q was 45K.

Figure four below shows a comparison of the cavity shape to the simulated cavity and figure five shows the mode.

In email correspondence with Shawyer, he states the mode used was TE0,1 [3], however the mode shown in the simulation (Fig. 5) is TE3,1 given there are three wavelengths around the perimeter. After changing the length to 390mm and the diameter of the small end to 166mm, the TE0,1,4 was generated as shown in figure 6. The TE0,1,4 was very sensitive to the diameter of the narrow end with only a 1mm difference in either direction creating a different mode. The length of the cavity from front to back was not as sensitive, but changes of 5 to 10mm would distort and eventually change the mode (usually an impure TE1,1,n mode).

Once the cavity is properly tuned for a TE0,1,4 mode, it has a significantly higher Q because of low field strengths along the walls where losses occur. It should be possible to tune the cavity with a large movable circular plate as the short at the narrow end because the mode is TE0,1,n and the field strength around the edges are small. As shown in figure 7, for square tuners, it is clear when field strengths are low, as on the sides, that fairly large spaces can be used. In the middle of the square waveguide where the field forces are the highest, sliding contacts are necessary.

A simple circular cavity at 152mm in diameter and 260mm in length was simulated with a loop instead of a probe to properly generate a TE0,1,1 mode (see figure 8). Fabrication of a loop probe maybe difficult considering it has to terminate in an N connector and would have to be inserted from the inside after the cavity has been built.

# Demonstration Engine Simulation Models

Figure 9: Demonstration Engine simulation with port, simulated with 541K tetrahedrons, Z and Y cut planes shown
The vacuum-surrounded-by-PEC design (shown in figure 6, 6a) was converted to a thin copper shell cavity to more closely simulate the final real world design. Two thin-shell cavities were simulated, one with a single waveguide port (figure 9 and the download link above) and the other with probes, both of which had a large tuning plate at the back of the narrow end. The tuning plate had a gap of 1mm between its edges and the side of the cavity.

With the first single port model, by using the optimization method discussed below, it was determined that the TE0,1,4 mode had moved. In the vacuum only design shown in figure 6(a), the cavity resonated in a pure TE0,1,4 mode with the narrow end at 166mm, however, with a copper shell, the narrow end diameter had to be adjusted to 170mm for the same mode (figure 9).

For the second cavity, two probes were added, one in the WR-340 sized waveguide to convert the TEM mode in the coax to a TE mode in the waveguide before being launched through an aperture into the cavity. A second smaller loop probe was inserted near the large end of the cavity to be used to measure Q. The dimensions of the SMA probe were modeled after a real world example, not including the loop.

Optimizations took considerable time because the size of the simulations frequently went over a quarter-million tetrahedrons. All the simulations were done with the more accurate, but slower frequency domain solver. A typical optimization took the form of changing the diameter of the narrow end a few millimetres, then running a simulation until the field results could be checked for the TE0,1,n mode. Because the TE0,1 mode was very sensitive to the diameter, it was necessary to change the diameter only a few millimetres before rerunning the simulation.

There are two factors important when designing a Shawyer cavity, the first is injecting the signal correctly such that a TE0,1,n mode is the result and getting as high a quality factor or Q as possible. The "purity" of the TE0,1 mode directly affects the Q, for example, as shown in figure 9, when a perfect TE0,1,n mode is generated with a "perfect" waveguide port, the Q can be as high as 81K.

Because our experiment requires the cavity to be hung from a pendulum, a probe is necessary, either in a waveguide attached to the cavity or directly into the cavity itself. In order to experimentally measure the Q and S1,1 parameters of the physical cavity, it is necessary to include two probes or ports, even if one is left open (or shorted) when power is applied to the other during the experiment.

## Matching a Single Port to Closed Resonating Cavities

The results shown in Figure 9 represent the best possible condition with a Q of 81K, from which the cavity with probes was derived. In order to change only one variable at a time and minimize the number of tetrahedrons, first one probe was added and matched, then a second added. Working with closed resonator cavities with only one port introduces difficulties in using the usual scattering parameters or S-parameters to check for matching. The problem is that at steady state, all the power is reflected and the S1,1 is always zero.

The method used for matching a single probe to a closed resonating cavity then consisted of maximizing the field strength inside the cavity. In practice, a maximum field strength consisted of a high V/m during the peak cycle of the standing wave. When the maximum field strength of the E field in the narrow end during the peak cycle was close to those found in the ideal case shown in figure 9, the probe was considered matched.

During the process of designing the "measurement" SMA-to-loop probe, it was found that the calculated Q was 61K. The maximum Q from an aperture launch was 45K, similar to what was reported by Shawyer, which is significantly lower. The loop probe also has the benefit of being easier to mount on the cavity than a waveguide. Connecting a square waveguide to a circular cavity with sloping sides means cutting at an angle and creating curving edges. A probe in the center and top of the cavity is also easier to hang from a coax cable unlike the off-center waveguide.

As an aside, when attempting to use a probe-to-waveguide as the backing for the aperture window, it was found that the location of the aperture was important in generating a pure TE0,1,n mode. Over about a range of 10 cm, the Q, a measurement of the purity of the TE0,1,n mode, could differ by 4 or 5K. Locating the aperture correctly makes sense because the waveform needs to enter in such a manner that it lines up with the circular standing wave inside the cavity. The minimum E field must reside at the aperture opening in order for the maximum E field to occur at the location necessary to create the TE0,1 mode (i.e. in the circular E field). At first, attempts were made to use the length of the waveguide leading up to the aperture to control the minimum E field at the aperture window. However, the standing wave in the cavity determined the mode inside the waveguide. In practice this meant that the length of the waveguide leading up to the aperture had no affect on the phase at the aperture window, a counter-intuitive result. The proper design methodology, discovered in reverse, was to first locate the aperture along the side of the cavity such that it generated the cleanest highest-Q TE0,1 mode. Then the waveguide should be just long enough that a probe can be inserted at the maximum E field strength position relative to the short.

## Two Loop Probes Matched

Figure 10: Demonstration Engine simulation with loop probes
As shown in figure 10, two probes were added, a "measurement" loop probe and a second nearly identical "power" loop probe. The "power" loop probe was modeled with the larger N-type connector in mind, which has a larger diameter dielectric and probe but with the same impedance of 50ohms.
Figure 11: S-parameters with loop probes

The success of the matching can be seen in figure 11, which shows the S-parameters at 2.45Ghz. The S1,1 is -14dB and is indicative of a very pure TE0,1,4 with a maximum E field in the narrow end of 6378 V/m. With a correctly sized cavity, the next most important criteria in determining the match between the loop probe and the cavity was the diameter of the loop.

Figure 12: Multiple resonances
There is a method used to make sure the cavity is resonating in the right mode, TE0,1,n at the right frequency, 2.45Ghz. Typically when the cavity is out of tune, it means that the TE0,1,n mode is resonating at a frequency other then the desired 2.45Ghz. The problem is that there are a lot of resonant frequencies, as shown by each dip in figure 12 and it is first necessary to determine which one is the TE0,1 mode before moving the tuning plate. The method is simple. First, with an S-parameter plot, locate the frequency of the nearby dips and add a "field monitor" at the frequency of interest in the simulator, then rerun the simulator which will calculate fields for those frequencies. After looking at the field configuration and determining which one is the TE0,1,n mode, then determine if it higher or lower then the desired 2.45Ghz. If it is higher, then make the chamber longer, if it is lower then make the chamber shorter. It make take a few iterations before the appropriate mode is resonating at 2.45Ghz.

On the lab bench, it will be critical to setup the cavity tuning as closely as possible to the simulated cavity, and then move it just enough to bring the resonance into focus. The problem with a real cavity is that there is no method to tell which mode the cavity is resonating in.

This iteration of the model had a proper 1mm space around the tuning plate at the narrow end of the cavity. Further tests included:

• The distance between the probe's loop connectors, i.e the one from the source and the grounding side, was large enough to avoid arcing.
• A sensitivity analysis was carried out to see how much the S-parameters dropped for a given loop diameter and changes of 1mm typical dropped the match by upwards of 10dB.

Because the cavity will be tested inside a vacuum chamber, it is necessary to have hole(s) in the cavity to the let the air out. Otherwise, as the pressure in the vacuum chamber is reduced, the cavity will bulge.

# References

1. "Microwave Propulsion - Progress in the EMDrive Programme" (PDF), Roger Shawyer, SPR Ltd., United Kingdom, International Astronautical Congress 2008 states "280mm"
2. "Microwave Propulsion - Progress in the EMDrive Programme" (PDF), Roger Shawyer, SPR Ltd., United Kingdom, International Astronautical Congress 2008, Section 7 "Demonstrator Engine"
3. "Re: EMDrive is Going Backwards - Is Fg1 really > then Fg2?", Private email correspondence, Roger Shawyer, July 25th, 2007