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Unformatted text preview: 34 Line impedance, generalized reflection coef- ficient, Smith Chart • Consider a TL of an arbitrary length l terminated by an arbitrary load Z L = R L + jX L . as depicted in the margin. l Input port Z o- l z l d LOAD GENERATOR, Circuit Z L = R L + jX I ( d ) V ( d ) +- Voltage and current phasors are known to vary on the line as V ( d ) = V + e jβd + V- e- jβd and I ( d ) = V + e jβd- V- e- jβd Z o . In this lecture we will develop the general analysis tools needed to determine the unknowns of these phasors, namely V + and V- , in terms of source circuit specifications. • Our analysis starts at the load end of the TL where V (0) and I (0) stand for the load voltage and current, obeying Ohm’s law V (0) = Z L I (0) . Hence, using V (0) and I (0) from above, we have V + + V- = Z L V +- V- Z o ⇒ V- = Z L- Z o Z L + Z o V + . 1 – Define a load reflection coefficient “Load reflection coefficient” is a well justified name for Γ L since the forward travel- ing wave with phasor V + e jβd gets reflected from the load. Γ L ≡ Z L- Z o Z L + Z o and re-write the voltage and current phasors as V ( d ) = V + e jβd [1+Γ L e- j 2 βd ] and I ( d ) = V + e jβd [1- Γ L e- j 2 βd ] Z o . – Define a generalized reflection coefficient The term “generalized reflec- tion coefficient” is also well justified even if there is no reflection taking place at ar- bitrary d — the reason is, if the line were cut at location d and the stub with the load were replaced by a lumped load having a reflection co- efficient equal to Γ( d ) , then there would be no modifica- tion of the voltage and cur- rent variations on the line to- wards the generator. Γ( d ) ≡ Γ L e- j 2 βd and re-write the voltage and current phasors as V ( d ) = V + e jβd [1 + Γ( d )] and I ( d ) = V + e jβd [1- Γ( d )] Z o ....
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