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Short-circuited quarter wavelength wave resonator

This post tells about short-circuited quarter wavelength line resonator. Quarter wavelength line resonator is depicted below, it is short circuited at one end.

 

Input impedance for this resonator with length l is Zin=Z0tanh(α+jβ)l=Z01jtanhαlcotβltanhαljcotβl.

If l=λ4, ω=ω0, and ω=ω0+ω.

Here we can conclude that βl=π2+πω2ω0, then cotβlπω2ω0. And tanhαlαl, then ZinZ0αl+ω2ω0