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Im( )Re( )r = 0r r = 1r = 0.5 r = 5Transmission Lines Amanogawa, 2006 - Digital Maestro Series171Theresultfortheimaginarypartindicatesthatonthecomplex planewithcoordinates(Re(),Im())allthepossibleimpedances with a given normalized reactance xare found on a circle with 1 11,x x ` )Center = Radius =Asthenormalizedreactancexvariesfrom-to,weobtaina familyofarcscontainedinsidethedomainofthereflection coefficient | | 1.

Inordertoobtainuniversalcurves,weintroducetheconceptof normalized impedance () ()()01( )1nZ d dz dZ d+ = Transmission Lines Amanogawa, 2006 - Digital Maestro Series167ThenormalizedimpedanceisrepresentedontheSmithchartby usingfamiliesofcurvesthatidentifythenormalizedresistancer(real part) and the normalized reactance x (imaginary part) () ( ) ( )Re Imn n nz d z j z r jx = + = + Lets represent the reflection coefficient in terms of its coordinates () () ()Re Im d j = + Now we can write () ()() ()() () ()() ( ) ()2 2221 Re Im1 Re Im1 Re Im 2Im1 Re Imjr jxjj+ + + = + = + Transmission Lines Amanogawa, 2006 - Digital Maestro Series168The real part gives ( ) ( )( ) ( ) ( )( ) ( ) ( )( )( ) ( )( ) ( ) ( )( )( ) ( )( ) ( ) ( )( )( ) ( )( ) ( )( )2 22 22 2 2 22 2 222 222 221 Re Im1 Re Im1 1Re 1 Re 1 Im Im 01 11 1Re 1 Re 1 1 Im1 111 Re 2 Re 1 Im1 111Re Im1 1rr rr rr rr rr rr rr rrrr r = + + + + + =+ + + + + + =+ ++ + + + =+ ++ + =+ + = 0Add a quantity equal to zero Equation of a circle Transmission Lines Amanogawa, 2006 - Digital Maestro Series169The imaginary part gives ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )2 22 2 22 22 22 22 22222 Im1 Re Im1 Re Im 2 Im 1 1 02 1 11 Re Im Im2 1 11 Re Im Im1 1Re 1 Imxx xxx xxx xxx= + + + = + + = + + = + = 0Multiply by x and add a quantity equal to zeroEquation of a circleTransmission Lines Amanogawa, 2006 - Digital Maestro Series170The result for the real part indicates that on the complex plane with coordinates (Re(), Im()) all the possible impedances with a given normalized resistance rare found on a circle with 1, 01 1rr r `+ + )Center = Radius = Asthenormalizedresistancervariesfrom0to,weobtaina familyofcirclescompletelycontainedinsidethedomainofthe reflection coefficient | | 1.

Im( ) Re( )1 Transmission Lines Amanogawa, 2006 - Digital Maestro Series166The goal of the Smith chart is to identify all possible impedances on thedomainofexistenceofthereflectioncoefficient.Todoso,we start from the general definition of line impedance (which is equally applicable to a load impedance when d=0) ()()()()01( )1V d dZ d ZI d d+ = Thisprovidesthecomplexfunction( ) ( ) ( ) Re, Im Z d f = that we want to graph.It is obvious that the result would be applicable only to lines with exactly characteristic impedance Z0. Inthecaseofagenerallossyline,the reflectioncoefficientmighthave magnitudelargerthanone,duetothe complexcharacteristicimpedance, requiring and extended Smith chart. Thedomainofdefinitionofthereflection coefficient for a loss-less line is a circle of unitaryradiusinthecomplexplane.This is also the domain of the Smith chart.

Fromamathematicalpointofview,theSmithchartisa4-D representationofallpossiblecompleximpedanceswithrespectto coordinates defined by the complex reflection coefficient. Transmission Lines Amanogawa, 2006 - Digital Maestro Series165Smith Chart TheSmithchartisoneofthemostusefulgraphicaltoolsforhigh frequencycircuitapplications.Thechartprovidesacleverwayto visualizecomplexfunctionsanditcontinuestoendurepopularity, decades after its original conception.