S parameters for passive networks

Dear all

I need to know a theoretical point related to S parameters:

I am working with 3 port passive devices with no anisotropic materials. The S matrix for such a device is:

S=

|s11 s21 s31|
|s21 s22 s23|
|s31 s23 s33|

Now, I want to numerically find some results (gains and so on) when s23 varies and all the other parameters are known.

The point is how to restrict the s23 variation so as not to violate physical principles (such as energy conservation).

Thanks

cmunikat

As you know : there are some relationship between S parameters such as :

Σ(sij)^2=1 & Σ(sij)(skj)=Σ(sij)(sik)=0

and if your S parameters are specifided and invariant except one , it is possible to extract a relationship for desired Gain .

Thank you for your response.

In fact, what I need to do is to make a sweep on all possible values of s23 and calculate the resulting parameter. However, I think that the relationships you have brought back are not enough to solve the problem.

For example, consider the following device: NOT loseless with S matrix

|0 c c|
|c 0 x|
|c x 0|

c=.98/√2

So, I believe none of the relationships apply for x≠0. The point is how to get the sweep range for x so as not to violate physical principles.

cmunikat :sm28:

If [a] is the column matrix of input waves and [b] that of output waves then the total power delivered to the device is ?{[a*][a]-[b*]^[b]}and has to be ≥ 0 for passive devices. (* =conjugate ^=transpose)
[b]=[S][a] and thus:
[a*]^{I-[S*]^[S]}[a] ≥ 0 for any [a], ie the determinant of the {[S*]^ [S] ? I} matrix has to be smaller or equal zero.
Also Σ|sij|2 ≤ 1 for any j (results from the passivity condition when all a?s are 0 except aj )