329lect31 - 31 Periodic oscillations in lossless TL ckts V...

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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 source-free and co-sinusoidal voltage and current oscillations at a frequency of = 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 infinite and the oscillation waveforms are not restricted to co-sinusoidal 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 source-free 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.

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329lect31 - 31 Periodic oscillations in lossless TL ckts V...

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