Could a copper RF cavity like this be reasonably expected to have a Q > 7000?

Question

The paper Measurement of Impulsive Thrust from a Closed Radio-Frequency Cavity in Vacuum (H. White et al, J. Propulsion & Power, November, 2016, http://dx.doi.org/10.2514/1.B36120) refers to an unusually shaped copper cavity with a resonance at about 1.94 GHz. This is described in the quoted section below. (further reading: https://space.stackexchange.com/questions/tagged/emdrive)

Fig. 4 suggests that the Q of this cavity is over 7,000 (7E+03). As far as I can tell there is no suggestion of an unusually conductive coating inside the copper.

My question is about the extremely high Q. I think among those with experience with ~GHz resonant copper cavities should be able to answer this based on experience, without it being too opinion based. Could a copper RF cavity like this be reasonably expected to have a Q > 7000?

I'm curious - with a drive of 50W, what would be the order of magnitude electric fields inside? kV/m? MV/m? I can break this off as a separate question if necessary.

An example of anything close in configuration and Q could be the basis of a "yes" and an example of anything close in configuration, highly optimized, and not even close in Q could be the basis of a "no" answer.

B. Test Article

The RF resonance test article is a copper frustum with an inner diameter of 27.9 cm on the big end, an inner diameter of 15.9 cm on the small end, and an axial length of 22.9 cm. The test article contains a 5.4-cm-thick disk of polyethylene with an outer diameter of 15.6 cm that is mounted to the inside face of the smaller diameter end of the frustum. A 13.5-mm-diam loop antenna drives the system in the TM212 mode at 1937 MHz. Because there are no analytical solutions for the resonant modes of a truncated cone, the use of the term TM212 describes a mode with two nodes in the axial direction and four nodes in the azimuthal direction. A small whip antenna provides feedback to the phase-locked loop (PLL) system. Figure 3 provides a block diagram of the test article’s major elements.

above: Figure 4 from here. Right click to open in a separate window to view clearly as full size, or view at the original link.

above: "Fig. 14 Forward thrust mounting configuration (heat sink is black finned item between test article and amplifier)." from here

above: "Fig. 17 Null thrust mounting configuration, b) view from side" from here

Answer 1

The trick to getting good microwave resonant-cavity Q is to have a good conductor, a smooth finish, precise alignment, light coupling of the input signal, and limited microphonic pickup.

The design in the picture looks like it might have been limited by microphonics, and then reworked to eliminate them. For example, it uses a large heatsink instead of a fan. It also looks like alignment would be a real chore!

The loaded Q specification for the Keysight Split Cylinder Resonator is >20,000 at 10 GHz. If you look into one of the resonator halves, you will see yourself in the mirror surface finish. The resonator is gold plated and precision diamond turned. The parts look so good that they used clear plastic for the instrument covers! Very unusual for Keysight gear.

Here is more background information about the Split Cylinder Resonator, in case anyone is interested:

The alignment is done with a kinematic mount, similar to how a telescope mirror is adjusted. The resonator halves can then be adjusted back and forth, while maintaining the alignment. A measurement sample is placed in the gap. The sample changes the Q and resonant frequency of the resonator. This, along with a Network Analyzer, enables measurement of the sample dielectric constant and loss. The accuracy of the dielectric measurement relies on having a high-Q resonator.

Here are the specifics on the surface finish from the datasheet: "Cylinders are precision diamond turned Al 6061-T6 plated with 0.5 μm Cu, 0.25 μm PdNi, and 2.0 μm Au."

Full disclosure: I am speaking for myself, not Keysight, even though I work there.

Answer 2

Yes, a quality factor of 7000 is not even near the upper limit for cavity resonators made of copper at that frequency. Microwave copper cavities with quality factors of \$10^6\$ are not uncommon. Exotic superconducting cavities can reach Q factors of \$10^{12}\$ (!!!).

Calculating the energy stored in a truncated conical cavity is nontrivial, and requires integrating the transverse magnetic and transverse electric fields, computed for a given geometry using Maxwell's equations. How to do so is beyond the scope of this question, but there is an excellent walkthrough and solution set of differential equations for a truncated spherical cone (not quite the same as this, but close enough) here. In fact, that entire page is just a wonderful write up on this topic and I heartily recommend it to anyone interested in getting dirty with the math.

Let's just do an easy one, a resonant cavity that is a simple cylinder. It's not a completely terrible substitute for a truncated cone, I think you'd agree.

The Q factor for such a cavity is:

$$Q = \frac{2\pi f\cdot \frac{\mu}{2}\int_{v}H^2dv}{\frac{R}{2}\oint_{s}H_{t}^{2}ds}$$

and I already have heartburn so I'm going to do what any engineer would do and use the much simpler approximation instead! One can show that a resonant cavity will have a Q that is in the order of magnitude of:

$$ Q\cong \frac{2}{\delta}\cdot \frac{V}{A} $$

where \$\delta\$ is the skin depth at the resonant frequency in question, and V and A is the volume and surface area of the cavity. In other words, the ratio of a cavity's volume to surface area is going to set a fairly narrow range of Q factors that a cavity, regardless of the exact geometry, can possibly have.

It should be apparent by now that creating a simple cylindrical cavity out of copper with a Q well above 7000, more like between 10,000 and 100,000. 7000 actually seems unusually low for a cavity shaped like the one in the photos. At the skin depth they're at, surface smoothness and imperfections become a concern, so if the surface quality inside is crappy, this could cause the Q to drop significantly.

Anyway, to answer the unasked question here, which is how does this thing produce thrust.... well, it's not at all anamolous. It seems to be exactly the right magnitude for the expected thrust due to uneven radiation of heat, as can be seen by the write up I linked earlier. This does produce thrust, and it will work in a vacuum. Unfortunately, relativity enforces a rather depressing limit on the thrust per power.

This drive will never produce more than micronewtons per killowatt. This makes it the most inefficient and impractical means of space propulsion available, reaction mass or no. And it will not get better. At least, that's the conclusion I've drawn, but I would love to be proven wrong.