Short-circuited quarter wave transmission line

Short-circuited quarter wave transmission line

This post tells about short-circuited quarter wave transmission line. Quarter wave line resonator is depicted below, it is short-circuited at one end and opened at the other.  The length of this transmission line is λ4.

Short-circuited quarter wave transmission line

This transmission line as characterised with characteristic impedance Z0, propagation constant β, attenuation constant α, length l=λ4, and resonant frequency ω=ω0.

The input impedance of the short-circuited quarter wave transmission line is Zin=Z01j*tanhαl*cotβltanhαlj*cotβl.

In practice it is usually used low-losses transmission lines where αl1 , then tanhαlαl. If we have a transmission line with length l=λ4at resonance ω=ω0. Let’s consider a  close to resonance situation ω=ω0+ω.  Then β=π2+πω2ω0.

After simplification ZinZ0αl+jπω2ω0. This result can be interpreted like the impedance of parallel RLC circuit Zin=11R+2jωC. Resistance, capacitance and inductance here can be represented as:  R=Z0αl, C=π4ω0Z0, L=1ω02C.

Unloaded quality factor for this resonator is Q0=ω0RC=π4αl=β2α.

The application of quarter wave transmission line is impedance matching and impedance inversion in size relevant electronic structures, another application in RF/DC coupling in transistor amplifiers.

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