Microwave Breakdown Calculator
In ordinary experience, air is transparent to electromagnetic waves at microwave frequencies. This is because field strengths are low in most situations. Home microwaves occasionally illustrate what happens at higher field strengths: Air breaks down into a plasma that absorbs and reflects microwaves.
In clean dry air that is far from surfaces, flames, shockwaves, radioactive decay or UV light, the free electrons that seed breakdown are generated by cosmic rays that continually shower the Earth from space. Though free electrons only last nanoseconds to microseconds in the lower atmosphere (< 36 km), cosmic rays generate them often enough to initiate breakdown almost as soon as an electromagnetic beam reaches a critical intensity. At this critical intensity and beyond it, the electric field of the beam accelerates the free electrons, enough that each collision with an air molecule results in (statistically) enough free electrons to replace those that recombine with oppositely charged ions, or attach to neutral molecules to form negative ions. The ensuing avalanche ionizes neutral air into microwave-absorbing plasma within a microsecond.
This model implements the equations of Guo-Zhi, Liu, et al. “A study of high power microwave air breakdown.” Chinese Physics 9.10 (2000): 757 to estimate the intensity at which clean dry air begins to break down into a plasma, and also the intensity at which it breaks down no further (because the plasma begins to reflect the incident beam as its ionization increases, thus limiting the incident beam’s intensity). MacDonald (1966) gives a more general account of microwave breakdown in gases.
Example: Breakdown at home microwave oven conditions
- Assert a microwave beam frequency of 2.45 GHz at an altitude of 0 km by typing:
f = 2.45 «enter» h = 0 «enter»
The model infers what it can from this and populates the answer in its data structure: Typing 0 «enter» shows the estimate for the breakdown threshold, which the model calculates by assuming that the electron number density is that generated by cosmic rays (ne=ne0). Note that the function ne0(h) is a gross approximation, especially above 36 km altitude, as it actually varies with space weather, day/night cycle and other factors, and these will somewhat alter the minimal breakdown threshold. Given this approximation, the model estimates that breakdown initiates at an effective electric field strength (Ee) of 3.1 megavolts per meter in this case, which is similar to the often-quoted 3 MV/m dielectric breakdown strength of air. Despite being resonant cavities (that multiply field strength in proportion to the quality of their resonance), home microwaves do not actually generate high enough fields to break down clean dry air, and it is additional ionization caused by a spark or flame that lowers the breakdown threshold.
Example: 10 GHz microwave thermal rocket
- Assert a 10 GHz microwave beam frequency at 43 km altitude by typing:
f = 10 «enter» h = 43 «enter»
The model infers what it can from this and populates the answer in its data structure: Typing 0 «enter» shows 6.8 MW/m² estimated breakdown intensity (I), which is just enough to sustain a plasma. Typing 1 «enter» shows the conditions at which the plasma starts to reflect the incident beam, which the model calculates by asserting that the plasma frequency has risen to equal the incident beam frequency. This condition is reached at 7.0 MW/m² intensity (I) and represents the (estimated) upper limit of microwave power than can be focused onto a microwave thermal rocket at 43 km. Experimenting by changing the altitude (h) shows that this intensity limit is minimum at this altitude (it is larger at both lower and higher altitude). A microwave thermal rocket that is designed to operate at greater than 7.0 MW/m² illumination has the option of operating at reduced thrust as it ascends through 43 km altitude or switching to higher frequency (which raises the intensity limit, as can be seen by increasing the model’s frequency f).
- We see that there is only a 3% intensity difference between no breakdown and a plasma that reflects the 10 GHz incident beam (I₁/I₀=1.03). To calculate the plasma frequency (and other conditions) corresponding to somewhere between the minimum (₀) and maximum (₁) beam intensity, let us choose I/I₀=1.01 by typing:
I/= 1.01 «enter»
The resulting plasma frequency (plasma.f) is 5.4 GHz, so this plasma is expected to (mostly) reflect incident microwaves whose frequency is less than 5.4 GHz.
Example: Breakdown over a range of altitudes and frequencies
The model can be used to generate plots of how breakdown and reflection thresholds vary with altitude and frequency.
- Plotting is started by specifying how many plots to calculate on each axis: To specify 100 points, type
plo.n = 100 «enter»
- Once calculations have finished (this may take a few seconds), press «enter» again to show the top-level variables. Plot(f,h) is the plot object, and its description should now be a hyperlink to an SVG file. Clicking this hyperlink opens a new browser window containing the plots that were just generated. If any relevant changes are made to the input values, then the plots are automatically recalculated (pressing the browser’s refresh button in the plots’ window updates the plots).
- Elements of the plots can be moved around to improve appearance. Clicking the ‘save’ button in the upper right corner of the browser’s window saves the plots as an SVG file.