Coupling coefficient of resonators

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. 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. 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. 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 k {displaystyle k} for the case of two equal resonators is defined by formula k = | f o − f e | / f 0 , {displaystyle k=|f_{o}-f_{e}|/f_{0},} (1) where f e , {displaystyle f_{e},} f o {displaystyle f_{o}} are the frequencies of even and odd coupled oscillations of unloaded pair of the resonators and f 0 = f e f o . {displaystyle f_{0}={sqrt {f_{e}f_{o}}}.} It is obvious that the coupling coefficient defined by formula (2) is a positive constant that characterizes interaction of resonators at the resonant frequency f 0 . {displaystyle f_{0}.} 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 k {displaystyle k} is defined by the formula k = K 12 x 1 x 2 {displaystyle k={frac {K_{12}}{sqrt {x_{1}x_{2}}}}} (2)