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Area Transmission lines and waveguides / Standing-wave and impedance measurements

IEV ref 726-19-02

en
Z-Theta chart
graphical representation in polar coordinates of the amplitude reflection factor r, for a lossless uniform transmission line with characteristic impedance Z0:

r _ = Z _ Z 0 Z _ + Z 0 = Z _ / Z 0 1 Z _ / Z 0 +1 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbbjxAHXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeqabiGaciGacaqadmaadaqaaqaaaOqaaiqadkhagaqhaiabg2da9maalaaabaGabmOwayaaDaGaeyOeI0IaamOwamaaBaaaleaacaqGVbaabeaaaOqaaiqadQfagaqhaiabgUcaRiaadQfadaWgaaWcbaGaae4BaaqabaaaaOGaeyypa0ZaaSaaaeaadaWcgaqaaiqadQfagaqhaaqaaiaadQfadaWgaaWcbaGaae4BaaqabaGccqGHsislcaaIXaaaaaqaamaalyaabaGabmOwayaaDaaabaGaamOwamaaBaaaleaacaqGVbaabeaakiabgUcaRiaaigdaaaaaaaaa@49FC@

in terms of the complex impedance Z by two families of orthogonal circles on each of which either the modulus Z or the argument θ has a constant value, where Z = Z/θ is the complex impedance in the direction of propagation of the incident wave at the point at which the amplitude reflection factor is evaluated

Note 1 – The Z-Theta chart may be used with impedances Z, admittances Y _ = 1 Z _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbbjxAHXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeqabiGaciGacaqadmaadaqaaqaaaOqaaiqadMfagaqhaiabg2da9maalaaabaGaaGymaaqaaiqadQfagaqhaaaaaaa@3923@ , normalized impedances Z _ Z 0 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbbjxAHXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeqabiGaciGacaqadmaadaqaaqaaaOqaamaalaaabaGabmOwayaaDaaabaGaamOwamaaBaaaleaacaqGVbaabeaaaaaaaa@385D@ or normalized admittances Y _ Y 0 = Z 0 Z _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbbjxAHXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeqabiGaciGacaqadmaadaqaaqaaaOqaamaalaaabaGabmywayaaDaaabaGaamywamaaBaaaleaacaqGVbaabeaaaaGccqGH9aqpdaWcaaqaaiaadQfadaWgaaWcbaGaae4BaaqabaaakeaaceWGAbGba0baaaaaaa@3C85@ .

Note 2 – The Z-Theta chart is usually restricted to values of θ between π 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbbjxAHXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeqabiGaciGacaqadmaadaqaaqaaaOqaaiabgkHiTmaalaaaba accaGae8hWdahabaGaaGOmaaaaaaa@38CA@ and + π 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbbjxAHXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeqabiGaciGacaqadmaadaqaaqaaaOqaaiabgkHiTmaalaaaba accaGae8hWdahabaGaaGOmaaaaaaa@38CA@ corresponding to positive values of the real part of Z in which case it is bounded by an outer circle where the magnitude of the amplitude reflection factor is unity.

Note 3 – The Z-Theta chart has the same properties and applications as those of the Smith chart, but the complex impedance Z is represented with two families of orthogonal circles on each of which either the modulus Z either the argument θ has a constant value instead of the real and imaginary parts R and X of Z used for the Smith chart.


fr
abaque Z-Théta, m
représentation graphique en coordonnées polaires du facteur de réflexion complexe r, pour une ligne de transmission uniforme sans pertes, d'impédance caractéristique Z0:

r _ = Z _ Z 0 Z _ + Z 0 = Z _ / Z 0 1 Z _ / Z 0 +1 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbbjxAHXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeqabiGaciGacaqadmaadaqaaqaaaOqaaiqadkhagaqhaiabg2da9maalaaabaGabmOwayaaDaGaeyOeI0IaamOwamaaBaaaleaacaqGVbaabeaaaOqaaiqadQfagaqhaiabgUcaRiaadQfadaWgaaWcbaGaae4BaaqabaaaaOGaeyypa0ZaaSaaaeaadaWcgaqaaiqadQfagaqhaaqaaiaadQfadaWgaaWcbaGaae4BaaqabaGccqGHsislcaaIXaaaaaqaamaalyaabaGabmOwayaaDaaabaGaamOwamaaBaaaleaacaqGVbaabeaakiabgUcaRiaaigdaaaaaaaaa@49FC@

en fonction de l'impédance complexe Z à l'aide de deux familles de cercles orthogonaux sur chacun desquels soit le module Z soit l'argument θ a une valeur constante, Z = Z/θ étant l'impédance complexe dans la direction de propagation de l'onde incidente au point de détermination du facteur de réflexion complexe

Note 1 – L'abaque Z-Théta peut être employé avec des impédances Z, des admittances Y _ = 1 Z _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbbjxAHXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeqabiGaciGacaqadmaadaqaaqaaaOqaaiqadMfagaqhaiabg2da9maalaaabaGaaGymaaqaaiqadQfagaqhaaaaaaa@3923@ , des impédances normées Z _ Z 0 MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbbjxAHXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeqabiGaciGacaqadmaadaqaaqaaaOqaamaalaaabaGabmOwayaaDaaabaGaamOwamaaBaaaleaacaqGVbaabeaaaaaaaa@385D@ ou des admittances normées Y _ Y 0 = Z 0 Z _ MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbbjxAHXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaeqabiGaciGacaqadmaadaqaaqaaaOqaamaalaaabaGabmywayaaDaaabaGaamywamaaBaaaleaacaqGVbaabeaaaaGccqGH9aqpdaWcaaqaaiaadQfadaWgaaWcbaGaae4BaaqabaaakeaaceWGAbGba0baaaaaaa@3C85@ .

Note 2 – L'abaque Z-Théta est habituellement limité aux valeurs de θ comprises entre π 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbbjxAHXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeqabiGaciGacaqadmaadaqaaqaaaOqaaiabgkHiTmaalaaaba accaGae8hWdahabaGaaGOmaaaaaaa@38CA@ et + π 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbbjxAHXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeqabiGaciGacaqadmaadaqaaqaaaOqaaiabgkHiTmaalaaaba accaGae8hWdahabaGaaGOmaaaaaaa@38CA@ qui correspondent aux valeurs positives de la partie réelle de Z; tout l'abaque est alors compris à l'intérieur d'un cercle où le module du facteur de réflexion est égal à l'unité.

Note 3 – L'abaque Z-Théta a les mêmes propriétés et applications que l'abaque de Smith, mais l'impédance complexe Z y est représentée à l'aide de deux familles de cercles sur chacun desquels soit le module Z, soit l'argument θ a une valeur constante, au lieu des parties réelle R et imaginaire X de Z employées avec l'abaque de Smith.


ar
مخطط معاوقة (ثيتا)

de
Z-Theta-Diagramm, n

es
diagrama polar
diagrama Z-theta

fi
polaaridiagrammi

it
carta Z-θ

ko
Z- 세타 선도

ja
Z-シータ図表

pl
wykres biegunowy (impedancji lub admitancji)

pt
diagrama polar
diagrama Z-Teta

sv
impedansdiagram

Publication date: 1982
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