Circuit-slide-4

# Circuit-slide-4 - 29 1 j z j z j z j z V z V e V e I z V e...

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Unformatted text preview: ( 29 ( ) 1 ( ) j z j z j z j z V z V e V e I z V e V e z β β β β + - - + - - = + = - ( 29 ( ) 1 ( ) z z z z V z V e V e I z V e V e z γ γ γ γ + - - + - - = + = - We have obtained the following solutions for the steady-state voltage and current phasors in a transmission line: Loss-less line Lossy line Since V (z) and I (z) are the solutions of second order differential (wave) equations , we must determine two unknowns , V + and V - , which represent the amplitudes of steady-state voltage waves, travelling in the positive and in the negative direction, respectively. Therefore, we need two boundary conditions to determine these unknowns, by considering the effect of the load and of the generator connected to the transmission line. Before we consider the boundary conditions, it is very convenient to shift the reference of the space coordinate so that the zero reference is at the location of the load instead of the generator. Since the analysis of the transmission line normally starts from the load itself, this will simplify considerably the problem later. We will also change the positive direction of the space coordinate, so that it increases when moving from load to generator along the transmission line. ( 29 ( ) 1 ( ) j d j d j d j d V d V e V e I d V e V e z β β β β + - - + - - = + = - ( 29 ( ) 1 ( ) d d d d V d V e V e I d V e V e z γ γ γ γ + - - + - - = + = - (0) V V V + - = + ( 29 1 (0) I V V z +- = - We adopt a new coordinate d = - z , with zero reference at the load location. The new equations for voltage and current along the lossy transmission line are Loss-less line Lossy line At the load (d = 0) we have, for both cases, (0) (0) R V Z I = ( 29 R Z V V V V Z +- +- + = - R R R Z Z V V Z Z- +- Γ = = + For a given load impedance Z R , the load boundary condition is Therefore, we have from which we obtain the voltage load reflection coefficient ( 29 ( 29 2 2 ( ) 1 ( ) 1 j d j d R j d j d R V d V e e V e I d e z β β β β + - +- = + Γ = -Γ ( 29 ( 29 2 2 ( ) 1 ( ) 1 d d R d d R V d V e e V e I d e z γ γ γ γ + - +- = +Γ = - Γ 2 ( ) j d R d e β- Γ = Γ 2 ( ) d R d e γ- Γ = Γ ( 29 ( 29 ( ) 1 ( ) ( ) 1 ( ) j d j d V d V e d V e I d d z β β + + = + Γ = - Γ ( 29 ( 29 ( ) 1 ( ) ( ) 1 ( ) d d V d V e d V e I d d z γ γ + + = +Γ = - Γ We can introduce this result into the transmission line equations as Loss-less line Lossy line At each line location we define a Generalized Reflection Coefficient and the line equations become ( ) 1 ( ) ( ) ( ) 1 ( ) V d d Z d z I d d + Γ = =-Γ We define the line impedance as A simple circuit diagram can illustrate the significance of line impedance and generalized reflection coefficient: R e 0 R e R e 0 ( ) q q q Z Z d Z Z- Γ = Γ = + ( ) 1 ( ) ( ) 1 ( ) in in in V V L L z z I I L L + Γ = = =- Γ If you imagine to cut the line at location d , the input impedance of the portion of line terminated by the load is the same as the line impedance at that location “ before the cut ”. The behavior of the line”....
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## This note was uploaded on 03/09/2012 for the course ECE 450 taught by Professor E.kudeki during the Spring '12 term at Illinois College.

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Circuit-slide-4 - 29 1 j z j z j z j z V z V e V e I z V e...

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