Formal Expression of Sensitivity and Energy Relationship in the Context of the Coupling Matrix

Precise formulas to express the formal relationship between the time average stored energy in the resonators of a low-pass filter network and the sensitivity of the reflection
S
-parameter with respect to the coupling matrix terms are demonstrated in this paper, considering the normalized frequency axis. These relationships are found in the modern context of the
N
+2 coupling matrix, and for both diagonal and general nondiagonal coupling elements of the matrix. The results are valid for any type of coupling topology represented by the
N
+2 coupling matrix. Different examples are included as validation. Furthermore, important implications and applications derived from the new relationships are highlighted.