Notes 16 Spring 2005

Notes 16 Spring 2005 - Notes 16 Spring 2005.doc Three...

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Notes 16 Spring 2005.doc Three limiting cases to note about : L Γ Short circuit the load : 0 Z L = 1 L = Γ Open circuit the load : L Z 1 L + = Γ Matched line ( ) o L Z Z = 0 V 0 L = = Γ So there is no reflection. We see that when the load is matched to the line, since 0 L = Γ , the standing wave ratio will be unity. We can define another impedance, , which is dimensionally the same as (has units of Ohms), but is the ratio of the voltage and current waves traveling in in Z 0 Z both directions (we use the full solution to the wave equation). This is the input impedance , , that we’ve seen before. in Z The input impedance is a function of position on the transmission line or, with the load ( ) arbitrarily placed at we let distances up the line be represented by L Z 0 z = l - to get i () l l l l l l l β + β + = β + β β + β = tan jZ Z tan jZ Z Z sin jZ cos Z sin jZ cos Z Z Z L 0 0 L 0 L 0 0 L 0 in This is equation 13.13 from your text. Clearly, the input impedance oscillates with position on the line. To graph the result it would be better expressed as l l l β + β + = tan jZ Z tan jZ Z Z Z L 0 0 L 0 in . Again we look at the same three limiting cases only now we look at the generalized impedance equation that we developed for arbitrary locations on the transmission line. Open circuit the load , Z L : ( ) L Z l l l l l β = β = β + β + = tan j Z tan jZ Z Z tan jZ Z tan jZ Z Z Z o L L o L o o L o You learned in circuit theory that impedance is generally composed of a real (resistive) component and an imaginary (reactive) component, i.e., jX R Z + = , and that the reactive component, X, can be either inductive or capacitive. In the present case of the open circuited load we can write jX cot jZ tan j Z Z o o = β = β = l l l 1
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Notes 16 Spring 2005.doc which is a purely reactive impedance, where ( ) l β = cot Z X o . When the load impedance is in between these limiting cases there is a resistive component to ( ) l Z as well. This lets us make a graph of the reactance vs. l β for the lossless line ( is real): 0 Z Reactance, X π 2 π 2 3 π π 2 Inductive Capacitive 4 λ 2 λ 4 3 λ λ 0 = β l l β
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This note was uploaded on 01/20/2012 for the course EE 4460 taught by Professor Czarnecki during the Fall '10 term at LSU.

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Notes 16 Spring 2005 - Notes 16 Spring 2005.doc Three...

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