IEVref:726-19-02ID:
Language:enStatus: Standard
Term: <i>Z</i>-Theta chart
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Definition: 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.


Publication date:1982
Source
Replaces:
Internal notes:2015-12-04: <i>Z</i><sub>o</sub> corrected to <i>Z</i><sub>0</sub>. JGO
2016-06-21: amplitude reflection factor <i>r</i> corrected to <i><u>r</u></i> and <i>Z</i> corrected to <i><u>Z</u></i> in Note 3. <i>Z</i><sub>o</sub> corrected to <i>Z</i><sub>0</sub> in equations. JGO

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