  A novel methodology to unleash electromagnetic coupling matrix information in
coupled-resonator microwave circuits has been recently proposed [1]. This
information is derived from Maxwell's equations and the natural language of
electromagnetics is employed. As a result, the coupling matrix coefficients
stand only for electromagnetics. In this work, we enhance this approach to
reveal valuable design information for microwave engineering, showing the
electromagnetic (EM) coupling among all EM resonators and ports. By the same
token, the similarities with the well-known classical coupling matrix theory
are addressed. We bridge this gap since the classical theory is the
preferential language among microwave engineers.
  Classical coupling matrix theory is a narrowband model for electromagnetics.
Thus, we carry out a narrowband approximation in the electromagnetic coupling
matrix. This makes it possible to describe the EM coupling coefficients in the
same framework as classical circuits. As a result, proper comparison between
both coupling matrices is allowed.
  Finally, both coupling matrix approaches have a common ground, namely, get
physical insight and valuable information for design purposes in
coupled-resonator microwave circuits. However, only the electromagnetic
coupling matrix details all EM behavior, including parasitic and leakage
couplings, and the higher-order mode influence in the microwave circuit.
Several microwave circuits, such as filters and diplexers, will show the
possibilities of this new technique and its relation to classical coupling
matrix theory.
