- #1
daudaudaudau
- 302
- 0
Hi. I have the following two equations
[tex]S_{21}=\frac{(1-\Gamma^2)z}{1-z^2\Gamma^2}[/tex]
[tex]S_{11}=\frac{(1-z^2)\Gamma}{1-z^2\Gamma^2}[/tex]
How would you go about solving these equations? I want to avoid square roots because they make the results ambiguous.
I myself have found that
[tex]z=\pm\sqrt{\frac{\Gamma-S_{11}}{\Gamma-S_{11}\Gamma^2}}[/tex]
but a better solution is
[tex]z=\frac{S_{21}}{1-S_{11}\Gamma}[/tex]
because it avoids the sign ambiguity. Yet another good solution is
[tex]z=\frac{(S_{11}+S_{21})-\Gamma}{1-(S_{11}+S_{21})\Gamma}[/tex]
but I have no clue how to arrive at these results. Any suggestions?
[tex]S_{21}=\frac{(1-\Gamma^2)z}{1-z^2\Gamma^2}[/tex]
[tex]S_{11}=\frac{(1-z^2)\Gamma}{1-z^2\Gamma^2}[/tex]
How would you go about solving these equations? I want to avoid square roots because they make the results ambiguous.
I myself have found that
[tex]z=\pm\sqrt{\frac{\Gamma-S_{11}}{\Gamma-S_{11}\Gamma^2}}[/tex]
but a better solution is
[tex]z=\frac{S_{21}}{1-S_{11}\Gamma}[/tex]
because it avoids the sign ambiguity. Yet another good solution is
[tex]z=\frac{(S_{11}+S_{21})-\Gamma}{1-(S_{11}+S_{21})\Gamma}[/tex]
but I have no clue how to arrive at these results. Any suggestions?