In order to adjust the frequency response of the filter it is sufficient to optimize only these generalized parameters.
Meaning of this term has been improved many times with progress in theory of coupled resonators and filters.
Earlier well-known definitions of the coupling coefficient of resonators are given in monograph by G. Matthaei et al.[2] Note that these definitions are approximate because they were formulated in the assumption that the coupling between resonators is sufficiently small.
In accordance with new definition (6), the value of the inductive coupling coefficient of resonant LC-circuits
In accordance with (6), the formula (5) for the capacitive coupling coefficient of resonant circuits takes a different form
Frequency response of the filter will not change if signs of all the coupling coefficients would be simultaneously alternated.
Therefore it would be more accurate to characterize interaction of resonators by a continuous function of forced-oscillation frequency
where transmission of high frequency power from one resonator to another one is absent, i.e. must meet the second condition
(10) The transmission zero arises in particularly in resonant circuits with mixed inductive-capacitive coupling when
that generalizes formula (6) and meets the conditions (9) and (10) was stated on energy-based approach in.
[6] This function is expressed by formula (8) through frequency-dependent inductive and capacitive coupling coefficients
denotes energy of high frequency electromagnetic field stored by both resonators.
denotes magnetic part of high frequency energy, and subscript
Theory of microwave narrow-band bandpass filters that have Chebyshev frequency response is stated in monograph.
Frequency response of the low-pass prototype filters is characterized by Chebyshev function of the first kind.
Exact expressions for the coupling coefficients in prototype filter were obtained in.
Inaccuracy of formulas (16) and their refined version is caused by the frequency dispersion of the coupling coefficients that may varies in a great degree for different structures of resonators and filters.
Therefore, the formulas (16) may be used to determine initial values of the coupling coefficients before optimization of the filter.
The approximate formulas (16) allow also to ascertain a number of universal regularities concerning filters with inline coupling topology.
For example, widening of current filter passband requires approximately proportional increment of all the coupling coefficients
Real microwave filters with inline coupling topology as opposed to their prototypes may have transmission zeroes in stopbands.
Amplitudes of waves transmitted through different paths may compensate themselves at some separate frequencies while summing at the output port.
in a tuned filter are equal to zero because a susceptance vanishes at the resonant frequency.
is the fact that it allows to directly compute the frequency response of the equivalent network having the inductively coupled resonant circuits,.
[13] Utilization of a coarse model allows to quicken filter optimization manyfold because of computation of the frequency response for the coarse model does not consume CPU time with respect to computation for the real filter.
Because the coupling coefficient is a function of both the mutual inductance and capacitance, it can also be expressed in terms of the vector fields
Hong proposed that the coupling coefficient is the sum of the normalized overlap integrals [14][15]
(21) On the contrary, based on a coupled mode formalism, Awai and Zhang derived expressions for
Using Lagrange’s equation of motion, it was demonstrated that the interaction between two split-ring resonators, which form a meta-dimer, depends on the difference between the two terms.
In this case, the coupled energy was expressed in terms of the surface charge and current densities.