Calculating the resonance frequency of the TDO is difficult if you would like to include the transmission line in the calculation. The calculation below is based on finding the impedance of the resonator in this circuit, everything to the right of the parasitic resistor, and solving for when the imaginary part of the impedance goes to zero. The resulting expression is very ominous. Thanks to Lily Philips of Smith College (now at Yale U.) for doing the symbolic math to rationalize this expression, seen in its full glory below.
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Result:
The full expression is shown here. Because it is so long, it is broken into parts. The full expression is z_0(A+B)/C, where A, B and C are in order below.
These are parts A, B, and C of the expression for resonant frequencies of a tank circuit connected to a transmission line. The full expression is z_0(A+B)/C, where A, B and C are in order above this caption.
