Even/odd mode decomposition for symmetrical network analysis; Application to simple RC circuits

Given an arbitrary but symmetrical 2 -port network:

Fig. 1

Even mode excitation: Odd mode excitation:

Fig. 2

Then, as demonstrated in Hong, Lancaster Chap. 2, pg 21:

(1)

For example, for a shunt C capacitance:

Fig. 3

whose symmetry is easily emphasized:

Fig. 4

Even mode: Odd mode:

Fig.5

For the even mode:

(2)

For the odd mode:

(3)

Then from (1):

(4)

Therefore the Scattering matrix for the shunt capacitance is:

(5)

For a shunt RC combination:

Fig. 6

the decomposition would provide:

Fig. 7

and the same approach can be applied. However it is convenient to emphasize its generality by not specifying the Y constitution before the end. Indeed, in general:

Fig. 8

and the decomposed network reflection coefficient gives:

(6)

therefore the combined S-parameters are:

(7)

Then going back to the case of the shunt RC resonator where Y=1/R+jωC the S-parameters become:

(8)

which reduces to (4) for infinite R. If |Y| is small, which is the case when R is in the tens of kΩ range and C is in the pF range then (7) shows that S₁₁≈ -YZ₀/2 and S₂₁≈1 .

For a shunt LC resonator, the admittance is

(9)

then:

(10)

.