even and odd mode impedances

Hello,

studying symmetric coupled microstrip lines, we identifiy to modes of propagation (common mode and differential mode).

Whereas the even and odd mode impedances, commonly used for microwave design purposes, are defined for half of the structure.
Zeven = V/I, if a perfect magnetic wall is placed between the strips.
Zodd = V/I, if a perfect electric wall is placed between the strips.

Lets say Zcommon = 20 Ohm, Zdifferential = 60 Ohm.
It appears Zeven=2*Zcommon=40 Ohm, Zodd=0.5*Zdifferential = 30 Ohm.
In this sense, Zcommon is the series connection of Zeven (first line) and Zeven (second line),
while Zdifferential is the parallel connection of Zodd (first line) and Zodd (second line).

Zcommon = Zeven1+Zeven2
Zdifferential = (Zodd1*Zodd2)/(Zodd1+Zodd2)

If we know think of an asymmetric configuration.
Left strip width differs from right strip width.
Is it still possible to use the above equations.
Series connection of two different Zeven values and parallel connection of two different Zodd values. ?

Thanks for your comments.

-e

I think one problem is how you define Zeven1 Zeven2 Zodd1 and Zodd2.

Assume the situation is that you have two signal lines S1 and S2 and one shared large ground line G. In the symmetric case, S1 and S2 has exactly the same dimensions, so you place PEW or PMW in the middle that does not alter field distribution in the anti-symmetric (differential) and symmetric mode (common), this feature is the key to use the equation to calculate Zcommon, Zdiff from Zeven and Zodd.

For asymmetric case, it'll be almost impossible to find a surface to place PEW and PMW (they won't be a plane anymore since S1 and S2 has different dimensions) without disturbing the field distribution in common and diff mode. Alternatively, you could sove the Zeven1 and Zodd1 in a configuration using two S1 and one G and Zeven2 and Zodd2 in a configuration using two S2 and one G, but I think you cannot combine them (two different physical situation) to generate the Zcommon and Zdiff for one S1, one S2 and one G situation.