Bonjour Francois:
To your first point, your expanded expression in the seventh paragraph for gamma in terms of A,B, and Zo is mathematically correct.
But I will say it is not a form I have ever seen used, and while correct, I find, for my purposes, not particularly useful.
As far as the basic question on the fifth paragraph of your post, the question indeed is answered in the attachment. (For your convenience, I attach the link to Pilloud¡¯s description, written en Francaise)
Typical steps in finding S11 from impedances follow below. But note finding S11 is typically one step along the way towards calculating VSWR and Reflection Coefficient.
General steps are:
1. Measure Zs = Rs + Zs.
Zs will be a vector in the form a + jb
2. Calculate ABS Zs = I Zs |
In terms of SQRT (Rs*Rs + Xs*Xs)
3. Calculate Impedance Phase
Angle Zs = arctan (Xs/Rs)
4. Note Zo
5. Calculate Gamma = (Zs-Zo)/(Zs +Zo)
in vector form using the Excel complex division feature, or by hand is so inclined.
6. The resultant vector will indeed also be in the form Gamma= a + jb
7. Gamma vector will have magnitude rho and Gamma angle, often called Theta, when converted to polar form using the same technique you used with impedance, above.
9. Note Gamma and S11 are the same, and are sometimes referred to as the Reflection Coefficient, a complex quantity, usually discussed in terms of rho and Theta.
8. Once all this is in hand, you can calculate
VSWR = ( 1+ Mag Gamma)/(1-Mag Gamma)
8 As noted above, Mag Gamma is referred to as rho, so VSWR = (1+rho)/(1-rho)
9. You can calculate Return Loss (RL)
using RL = -20 log(rho) in terms of dB.
You will find in this site various Excel-based programs that will allow you to derive Z from S11 and S21, as well as starting with Zs in the form Rs and Xs to derive Gamma and all the other parameters of interest.
My iPhone probably has means to express Gamma, rho, Theta in symbols but I¡¯ve not yet found the appropriate app.
Study Pilloud. You have no excuse! It¡¯s written in French
Ed McCann
AG6CX
