329lect32 - 32 Input impedance and microwave resonators...

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32 Input impedance and microwave resonators Series Parallel C L LC The input impedance and admittance of the series and parallel LC resonators shown in the margin are, respectively, Z s = j ( ωL - 1 ωC ) and Y p = j ( - 1 ) , both of which vanish at the common resonance frequency of these net- works, namely ω = 1 LC ω o . Recall that LC resonators play an important role in the design of Flter and tuning circuits. In this lecture we will examine the input impedance of microwave resonators consisting of open or short circuited TL stubs. ±irst let us consider a shorted stub of some length l as a two-terminal circuit element as depicted in the margin. Short l Input port Z o If the circuit containing this element is in sinusoidal steady-state at a source frequency ω matching one of the resonant fre- quencies ω n =(2 n +1) πv 2 l ,n 0 , of the shorted stub — the resonances corresponding to an odd number of quarter wavelengths — then the element will draw no 1
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current from the source circuit in analogy with the parallel LC network ,andconsequentlythe input admittance of the element will be zero ( open condition), which we will indicate by Y ( l )=0 . This type of resonant behavior of the shorted stub will be called parallel resonance because of the analogy just mentioned. Short l Open input Z o Y ( l Odd # of quarter wavelengths Short l Short input Z o Z ( l Integer # of half wavelengths Parallel resonance Series resonance By contrast, if the circuit containing the element is in sinusoidal steady-state at a source frequency ω matching one of the other resonant frequencies ω n = n πv l ,n 1 , of the shorted stub — the resonances corresponding to integer mul- tiples of half wavelengths — then the voltage across the element terminals will be identically zero in analogy with the series LC network input impedance of the element will be zero ( short condition), which we will indicate by Z ( l . This alternate type of resonant behavior of the shorted stub will be called series resonance . We will next examine how Z ( l ) and Y ( l ) 1 Z ( l ) varies with ω for frequencies not matching any of the resonant frequencies of the shorted stub. 2
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If the stub is a part of a circuit in sinusoidal steady-state at an arbitrary frequency ω ,thend ’A lembe r tso lu t ion so fvo l tageandcu r ren tonthe line are V ( z,t )= Re { V + e ( t - z v ) } + Re { V - e ( t + z v ) }⇔ ˜ V ( z V + e - jβz + V - e and I ( Re { V + e ( t - z
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329lect32 - 32 Input impedance and microwave resonators...

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