Im(Γ ) Re(Γ ) r = 0 r →∞ r = 1 r = 0.5 r = 5 Transmission Lines © Amanogawa, 2006 - Digital Maestro Series 171 The result for the imaginary part indicates that on the complex plane with coordinates (Re(Γ), Im(Γ)) all the possible impedances with a given normalized reactance x are found on a circle with 1 11, x x Center = Radius = As the normalized reactance x varies from -∞ to ∞, we obtain a family of arcs contained inside the domain of the reflection coefficient | Γ | ≤ 1. In order to obtain universal curves, we introduce the concept of normalized impedance ( ) ( ) ( )0 1( ) 1n Z d dz d Z d + Γ = − Γ Transmission Lines © Amanogawa, 2006 - Digital Maestro Series 169 The imaginary part gives ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 Im 1 Re Im 1 Re Im 2 Im 1 1 0 2 1 1 1 Re Im Im 2 1 1 1 Re Im Im 1 1 Re 1 Im x x x x x x x x x x x Γ = − Γ + Γ − Γ + Γ − Γ + − = − Γ + Γ − Γ + = − Γ + Γ − Γ + = ⇒ Γ − + Γ − = = 0 Multiply by x and add a quantity equal to zero Equation of a circle Transmission Lines © Amanogawa, 2006 - Digital Maestro Series 170 The result for the real part indicates that on the complex plane with coordinates (Re(Γ), Im(Γ)) all the possible impedances with a given normalized resistance r are found on a circle with 1, 0 1 1 r r r + + Center = Radius = As the normalized resistance r varies from 0 to ∞, we obtain a family of circles completely contained inside the domain of the reflection coefficient | Γ | ≤ 1. 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. 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.
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