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Unformatted text preview: 31 Periodic oscillations in lossless TL ckts V ( t ) I ( t ) + C L Lossless LC circuits (see margin) can support sourcefree and cosinusoidal voltage and current oscillations at a frequency of = 1 LC known as LC resonance frequency . Lossless TL circuits can also support sourcefree voltage and current oscillations, but the number of resonance frequencies is infinite and the oscillation waveforms are not restricted to cosinusoidal forms. Open 4  I   V ( z, t )  z l 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 TLs are harmonically related , and therefore superpositions of resonant oscillations on TLs 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 sourcefree TL circuits. Consider first a TL segment of some length l having no electrical con nection to any elements at either end as shown in the margin. 1 Effectively, 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 4  I   V ( z, t )  z l 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 = 0 and I ( l, t ) = V + ( t l v ) Z o V ( t + l v ) Z o = 0 require that 1. V ( t ) = V + ( t ) , 2. V + ( t l v ) = V + ( t + l v ) V + ( t ) = V + ( t + 2 l v ) ....
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This note was uploaded on 06/20/2011 for the course ECE 329 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Kim
 Frequency, Volt

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