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Input files
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Resonator3D.MIN, Resonator3DSearch.ain, Resonator3D.ain,
Resonator3D.SCR, Resonator3DSearch.png, Resonator3DField.png
Download Resonator3D.zip
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Description
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This benchmark illustrates the procedure to find the resonant frequency and Q factor for the TM110 mode in a three-dimensional rectangular cavity using the Aether code. The cavity has widths 0.9" in the x and y dimensions and length 0.4" in z. The cavity is excited with a current source oriented in z at the center for preferential excitation of the TM110 mode. The calculation is performed in two stages: 1) a search to identify the mode frequency and 2) a field calculation at the identified frequency to determined stored energy and losses for the Q calculation.
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Results
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Analytic theory gives a predicted frequency of f = 9.273 GHz and a Q value of 7770 for a copper wall with conductivity sigma = 5.70E7 S/m. The Aether resonant search yields the frequency 9.279 GHz, a difference of 0.06%. At this frequency, the skin depth is less than a micron, much smaller than the width of elements representing the wall. A direct calculation of energy loss is not possible, so an alternate method can be applied. The time-averaged power loss per area in a high-Q resonator is given Ht^2*Rs/2 W/m2, where Ht is the magnitude of the mode magnetic field parallel to the copper wall and Rs is the surface resistance given by sqrt(pi*f*Mu0/Sigma) = 0.02535 ohm. The Analysis/Solution button of Aerial yields the following results:
Global and regional analysis of RF solution file
Time-averaged field energy and resistive power dissipation
Global electric field energy: 1.08400E-21 (J)
Global magnetic field energy: 1.08312E-21 (J)
Global total field energy: 2.16712E-21 (J)
Global power dissipation: 0.00000E+00 (W)
Time-averaged integral of Ht^2/2 over boundaries with metal regions
NReg HtIntegral (A^2) Area (m^2)
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2 6.41239E-13 1.97420E-03
The wall power loss is P = 0.02535*6.41239E-13 = 1.626E-14 W. The quality factor is Q = 2*pi*U/P = 7773, a difference of only 0.04%.
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Comments
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The geometry was based on a Comsol example for comparison. Note that this calculation uses a different coordinate axis convention and refers to the TE101 mode. The Comsol values were close to theory at the finest mesh resolution.
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