# 329lect29 - 29 Periodic oscillations in lossless TL ckts...

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29 Periodic oscillations in lossless TL ckts V ( t ) I ( t ) + - C L Lossless LC circuits (see margin) can support source-free and co-sinusoidal voltage and current oscillations at a frequency of ω = 1 LC known as LC resonance frequency . Lossless TL circuits can also support source-free voltage and current oscillations, but the number of resonance frequencies is infnite and the oscillation waveforms are not restricted to co-sinusoidal forms. Open λ 4 | I | | V ( z, t ) | z l 0 Open A TL segment open circuited at both ends can support voltage and current oscilla- tions such that the current waveform vanishes at both ends. Absolute values of a possible set of voltage and current waveforms satisfying this boundary condition are depicted above. Resonance frequencies of lossless TL’s are harmonically related , and therefore superpositions of resonant oscillations on TL’s can add up to arbitrary periodic waveforms as in Fourier series repre- sentation of periodic functions. In this lecture we will examine the periodic oscillations and resonances encountered in lossless and source-free TL circuits. Consider ±rst a TL segment of some length l having no electrical con- nection to any elements at either end as shown in the margin. 1

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Efectively, both ends oF the TL has been “open circuited”, and thus TL current I ( z, t ) needs to vanish at z =0 and z = l . Since I ( z, t )= V + ( t - z v ) Z o - V - ( t + z v ) Z o in general, these boundary conditions Open λ 4 | I | | V ( z, t ) | z l 0 Open A TL “stub” open circuited at both ends can support voltage and current oscilla- tions such that the current waveform vanishes at both ends. Absolute values of a possible set of voltage and current waveforms satisfying this boundary condition are depicted above. I (0 ,t V + ( t ) Z o - V - ( t ) Z o and I ( l, t V + ( t - l v ) Z o - V - ( t + l v ) Z o require that 1. V - ( t V + ( t ) , 2. V + ( t - l v V + ( t + l v ) V + ( t V + ( t + 2 l v ) . condition 1 says that Forward and backward going waveForms are the same, while condition 2 indicates that the waveForms are necessarily periodic with a 2
period of T = 2 l v , and fundamental frequency ω o = 2 π T = πv l .

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329lect29 - 29 Periodic oscillations in lossless TL ckts...

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