Propagation constant of a microstrip transmission line on FR4 from ABCD-Parameters

[corsarosso]

Hi,

I am trying to calculate the propagation constant (γ=α+jβ) of a microstrip transmission line from the measured S-parameters (ABCD-parameters).
The relationship seems to be given by the equation:
γ=acosh (A)/d
where d is the length of the transmission line (0.15 m) and A is the complex A-parameter.

The measured S parameters (S11 & S21) at 3 frequencies were:
At 1 GHz, S11=-0.013+0.022i, S21=0.80+0.50i
At 1.5 GHz, S11=-0.0069+0.016i, S21=-0.88-0.23i
At 2 GHz, S11=-0.036+0.022i, S21=0.40+0.79i

The A parameter at 3 frequencies are as below (these are converted from the above S-parameters):
At 1 GHz, A=0.85-0.032i
At 1.5 GHz, A=-0.97+0.024i
At 2 GHz, A= 0.45-0.11i

According to the transmission line equation, γ at these frequencies are as below:
At 1 GHz, γ=0.40-3.7i
At 1.5 GHz, γ=0.62+19.19i
At 2 GHz, γ=0.82-7.39i

Since α and β never becomes negative, am I calculating wrong?
Or, is that from my measurement error?

Also, I would typically expect to observe the imaginary part of the propagation factor (β) increases with the frequency.

 

[FvM]

Without going into the calculation details, obviously the micro strip length is larger than lambda for some frequencies and calculating propagation constant from S21 phase is ambiguous.