The coupling coefficient of resonators is a dimensionless value that characterizes interaction of two resonators. Coupling coefficients are used in resonator filter theory. Resonators may be both electromagnetic and acoustic. Coupling coefficients together with resonant frequencies and external quality factors of resonators are the generalized parameters of filters. In order to adjust the frequency response of the filter it is sufficient to optimize only these generalized parameters.
Contents
- Evolution of the term
- Coupling coefficient considered as a positive constant
- Coupling coefficient considered as a constant having a sign
- Coupling coefficient considered as a function of the forced oscillation frequency
- Bandpass filters with inline coupling topology
- Bandpass filters with cross couplings
- Coupling Coefficient in terms of the Vector Fields
- References
Evolution of the term
This term was first introduced in filter theory by M Dishal . In some degree it is an analog of coupling coefficient of coupled inductors. Meaning of this term has been improved many times with progress in theory of coupled resonators and filters. Later definitions of the coupling coefficient are generalizations or refinements of preceding definitions.
Coupling coefficient considered as a positive constant
Earlier well-known definitions of the coupling coefficient of resonators are given in monograph by G. Matthaei et al. Note that these definitions are approximate because they were formulated in the assumption that the coupling between resonators is sufficiently small. The coupling coefficient
where
In case when an appropriate equivalent network having an impedance or admittance inverter loaded at both ports with resonant one-port networks may be matched with the pair of coupled resonators with equal resonant frequencies, the coupling coefficient
for series-type resonators and by the formula
for parallel-type resonators. Here
When the resonators are resonant LC-circuits the coupling coefficient in accordance with (2) and (3) takes the value
for the circuits with inductive coupling and the value
for the circuits with capacitive coupling. Here
Coupling coefficient considered as a constant having a sign
Refinement of the approximate formula (1) was fulfilled in. Exact formula has a form
Formulae (4) and (5) were used while deriving this expression. Now formula (6) is universally recognized. It is given in highly cited monograph by J-S. Hong. It is seen that the coupling coefficient
In accordance with new definition (6), the value of the inductive coupling coefficient of resonant LC-circuits
Whereas the value of the capacitive coupling coefficient of resonant LC-circuits
Coupling between electromagnetic resonators may be realized both by magnetic or electric field. Coupling by magnetic field is characterized by the inductive coupling coefficient
Summation of the inductive and capacitive coupling coefficients is performed by formula
This formula is derived from the definition (6) and formulas (4) and (7).
Note that the sign of the coupling coefficient
Coupling coefficient considered as a function of the forced oscillation frequency
Two coupled resonators may interact not only at the resonant frequencies. That is supported by ability to transfer energy of forced oscillations from one resonator to the other resonator. Therefore it would be more accurate to characterize interaction of resonators by a continuous function of forced-oscillation frequency
It is obvious that the function
Besides, the function
The transmission zero arises in particularly in resonant circuits with mixed inductive-capacitive coupling when
The definition of the function
Here
Explicit functions of the frequency-dependent inductive and capacitive couplings for pair of coupled resonant circuits obtained from (12) and (13) have forms
where
Bandpass filters with inline coupling topology
Theory of microwave narrow-band bandpass filters that have Chebyshev frequency response is stated in monograph. In these filters the resonant frequencies of all the resonators are tuned to the passband center frequency
Derivation of approximate formulas for the values of the coupling coefficients of neighbor resonators in filters with inline coupling topology
where
Prototype element values
Here the next notations were used
where
Formulas (16) are approximate not only because of the approximate definitions (2) and (3) for coupling coefficients were used. Exact expressions for the coupling coefficients in prototype filter were obtained in. However both former and refined formulae remain approximate in designing practical filters. The accuracy depends on both filter structure and resonator structure. The accuracy improves when the fractional bandwidth narrows.
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. In other words, the optimal values of the coupling coefficients
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. Transmission zeroes considerably improve filter selectivity. One of the reasons why zeroes arise is frequency dispersion of coupling coefficients
Bandpass filters with cross couplings
In order to generate transmission zeroes in stopbands for the purpose to improve filter selectivity, a number of supplementary couplings besides the nearest couplings are often made in the filters. They are called cross couplings. These couplings bring to foundation of several wave paths from the input port to the output port. Amplitudes of waves transmitted through different paths may compensate themselves at some separate frequencies while summing at the output port. Such the compensation results in transmission zeroes.
In filters with cross couplings, it is convenient to characterize all filter couplings as a whole using a coupling matrix
Important merit of the matrix
Coupling Coefficient in terms of the Vector Fields
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
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.
Recently, based on Energy Coupled Mode Theory (ECMT), a coupled mode formalism in the form of an eigenvalue problem, it was shown that the coupling coefficient is indeed the difference between the magnetic and electric components