RF Electronic Devices and Systems

Short-circuited half wavelength line resonator

Short-circuited half wavelength line resonator

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

Short-circuited half wavelength line resonator

This transmission line as characterised with characteristic impedance Z0, propagation constant β, attenuation constant α, length l=λ2, and resonant frequency ω=ω0.

Input impedance here Zin=Z0tanhαl+jtanβl1+jtanβl*tanhαl.

In practice it is usually used low-losses transmission lines where αl1, then tahnαlαl. Let’s consider the transmission line in the conditions close to resonant ω=ω0+ω, here ω is small.

Then βl=ω0lvp+ωlvpvp is a velocity of the transmission line. As we have half wavelength line resonator l=λ2=πvpω0.

For the condition of resonance we have ω=ω0βl=π+ωπω0, and ZinZ0(αl+jωπω0).  Input impedance can be presented as Zin=R+2jLω. So this transmission line can be presented as series RLC circuit, where R=Z0αlL=Z0π2ω0, and C=1ω02L.

Resonance in short-circuited transmission line also happens for l=nλ2, n=1,2... when ω=0.

Unloaded quality factor for this transmission line is Q0=β2α.

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