How can we calculate the sqrt(2)*Zl of a rat race coupler?

[elektr0]

rat race coupler

Hello,

i guess the impedance sqrt(2)*Zl of the inner ring of a rate race coupler is because we want equal power split.

How can we calculate this sqrt(2)*Zl.
Assuming simple parallel connection of two lines does not give the right result.


elektr0

 

[Azulykit]

Re: rat race coupler

I have been looking at a similar question for a T junction. Pozar, "Microwave Engineering" notes that it is not possible to match all ports in a three port splitter (chapter eight). I have used the √2 used for rat races with good results. I was rushed and did not question why.

My current suspicion is that this choice reduces s22 and s33 at the expense of s11. I intend to look at the isolation and insertion loss interactions in more detail to see how they are effected by the input and output impedances. I am currently interested in an equal power split.