Can \$\Gamma\$ remain complex while calculating return loss?

Question

I'm working on adding .s1p touchstone export support to the xnec2c NEC2 antenna simulator for Linux. I would like the S11 dB values in the .s1p to have both real and imaginary values.

NEC2 calculates Gamma by breaking it into real-valued components \$Z_r\$ and \$Z_i\$. It then does a real (ie, not complex) calculation to obtain a real-valued Gamma:

\$\Gamma = \sqrt{ \frac{ (Z_r-Z_0)^2+Z_i^2 }{ (Z_r+Z_0)^2+Z_i^2 } }\$

S11 is then calculated as a real value using Gamma above as:

\$S_{11}\ = 20 log_{10}(\Gamma)\$

My question: Is it correct to calculate a complex Gamma from the real values \$Z_r\$ and \$Z_i\$ by creating a complex number \$Z = (Z_r+Z_i {\bf i})\$ and then calcule Gamma as shown in the Wikipedia article about reflection coefficient?

\$\Gamma = \frac{ Z-Z_0 }{ Z+Z_0 } \$

I ran the numbers and I get the same result for the real S11 value, but I have no way to validate the imaginary value. Is this a mathematically correct way to do this? Is it the "right" way?

If this works then I can split \$Z\$ into its real and imaginary parts for writing to the .s1p file's S11 real and imaginary columns. Here is the actual C code that generates the real-valued gamma and s11 variables:

zrpro2 = impedance_data.zreal[idx] + calc_data.zo;             
zrpro2 *= zrpro2;                                              
zrmro2 = impedance_data.zreal[idx] - calc_data.zo;             
zrmro2 *= zrmro2;                                              
zimag2 = impedance_data.zimag[idx] * impedance_data.zimag[idx];
gamma = sqrt( (zrmro2 + zimag2) / (zrpro2 + zimag2) );         

s11[idx] = 20*log10( gamma );

And here is my proposed change based on that:

double complex z_load = impedance_data.zreal[idx] + I*impedance_data.zimag[idx];
double complex cgamma = (z_load-Zo) / (z_load+Zo);                              

Answer 1

I'll keep it short:

Yes, the computation of: $$\Gamma=\frac{Z-Z_0}{Z+Z_0}$$ can be properly composed in the way you suggest.

It's also the case that your first equation isn't correctly written, though in contrast to that your actual C code is written correctly to calculate the result. (Comment is no longer true as this has now been corrected in the question.)

You can readily derive the real-number magnitude calculation from the above complex rational (by applying the complex magnitude or absolute value to the rational) as: $$\mid\, \Gamma\! \mid=\sqrt{\frac{\left(Z_\mathcal{R}-Z_0\right)^2+Z_\mathcal{I}^2}{\left(Z_\mathcal{R}+Z_0\right)^2+Z_\mathcal{I}^2}}$$

Note that this isn't the same as you wrote in your question.

However, you did get the right computation in C.

If you need help figuring out how to apply the absolute magnitude using the complex conjugate, I'll expand on that. But I think you should easily manage that much on your own.