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329lect28 - 28 Multi-line circuits In this lecture we will...

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28 Multi-line circuits In this lecture we will extend the bounce diagram technique to solve distributed circuit problems involving multiple transmission lines. One example of such a circuit is shown in the margin where two distinct TL’s of equal lengths have been joined directly at a distance l 2 away from the generator. + - 0 f ( t ) R g = 2 Z 1 R L = Z 2 z l Load Z 1 Z 2 = 2 Z 1 0 l 4 81 τ g = Γ g = Γ 12 = 1 3 τ 12 = 4 3 Γ L = 0 1 3 1 9 1 27 1 81 4 9 t l/ 2 v 2 = 2 v 1 v 1 The impulse response of the system can be found by first con- structing the bounce diagram for the TL system as shown in the margin. In this bounce diagram, z = l 2 happens to be the location of ad- ditional reflections as well as transmissions because of the sudden change of Z o from Z 1 to Z 2 = 2 Z 2 . These reflections and transmissions between line j and k — transmis- sion from j to k , and reflection from k back to j — can be computed with reflection coe ffi cient Γ jk = Z k - Z j Z k + Z j and transmission coe ffi cient τ jk = 1 + Γ jk that ensure the voltage and current continuity at the junction 1
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Z j is the characteristic impedance of the line of the incident pulse, while Z k is the impedance of the cascaded line into which the transmitted pulse is injected.
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