It is obvious that the result would be applicable only to lines with exactly characteristic impedance Z0. To do so, we start from the general definition of line impedance (which is equally applicable to a load impedance when d=0) ( ) ( ) ( ) ( )0 1 ( ) 1 V d d Z d Z I d d + Γ = − Γ This provides the complex function ( ) ( )( ) Re, ImZ d f= Γ Γ that we want to graph. Im(Γ ) Re(Γ ) 1 Transmission Lines © Amanogawa, 2006 - Digital Maestro Series 166 The goal of the Smith chart is to identify all possible impedances on the domain of existence of the reflection coefficient. In the case of a general lossy line, the reflection coefficient might have magnitude larger than one, due to the complex characteristic impedance, requiring and extended Smith chart. This is also the domain of the Smith chart. The domain of definition of the reflection coefficient for a loss-less line is a circle of unitary radius in the complex plane. From a mathematical point of view, the Smith chart is a 4-D representation of all possible complex impedances with respect to coordinates defined by the complex reflection coefficient. The chart provides a clever way to visualize complex functions and it continues to endure popularity, decades after its original conception. Where \$r\$ is your resistance, \$j\$ is your imaginary number \$\sqrt\$ and \$x\$ is your reactance which is your "resistance" of your capacitance or inductance, so to speak.Download Smith Chart Notes - Lines, Fields, Waves | ECE 450 and more Electrical and Electronics Engineering Study notes in PDF only on Docsity!Transmission Lines © Amanogawa, 2006 - Digital Maestro Series 165 Smith Chart The Smith chart is one of the most useful graphical tools for high frequency circuit applications. Remember that your formula for impedance is: If the impedance was only real and not complex, it would mean your transmission line would be purely resistive with no indication of induction or capacitance. I'm not sure what you mean by "assuming the TL impedance is real". Otherwise if the reflection coefficient, \$\Gamma=-j\$, it would indicate a purely capacitive load. Henceforth, using the picture above, if the reflection coefficient, \$\Gamma=j\$, it would mean that the transmission has a purely inductive load. If you had to cut horizontal line across the middle of this circle in half, you would see that top half would be a more inductive load and the bottom half being a more capacitive load. real axis, you can basically determine real and imaginary components of the impedance. Wikipedia has a very good image of how the Smith Chart is organized (for impedance): Forgive me for my knowledge of transmission lines and microwave circuits is very minuscule.
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