Smith Chart Angle of Transmission Coefficient

[Electronaut]

Question on the Smith Chart. The inner two circles on the traditional Smith Chart are supposed to measure "Angle of Reflection Coefficient in Degrees" (Gamma) and "Angle of Transmission Coefficient in Degrees" (T). It seems that the latter is rarely used, and the values on the chart don't make sense to me. Recall

Gamma = (Z1 - Z0)/(Z1 + Z0)
and
T = 1 + Gamma = 2 Z1/(Z1 + Z0)

The Angle of Reflection Coefficient should measure the phase angle of the complex number Gamma, and the Angle of Transmission Coefficient should
measure the phase angle of the complex number T (when both complex numbers are written in polar form).

But when I substitute test values for Z1 and Z0, and compute the corresponding phase angles for Gamma and T, they do not coincide with the
values on the inner two circles of the Smith Chart (where the ray from the origin through the point Z1/Z0 cuts the circles).

I have not been able to find documentation on the formula used to construct this part of the Smith Chart.

Any ideas or leads?

Thanks,
Electronaut

 

[unclejed613]

https://www.microwaves101.com/encyclopedias/smith-chart-basics#normalization
hope this helps