Coupling coefficient of resonators

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 k for the case of two equal resonators is defined by formula k=|f_o-f_e|/f_0, (1) where f_e, f_o are the frequencies of even and odd coupled oscillations of unloaded pair of the resonators and f_0=\sqrt{f_ef_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. 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 is defined by the formula k=\frac{K_{12}}{\sqrt{x_1x_2}} (2) for series-type resonators and by the formula k=\frac{J_{12}}{\sqrt{b_1b_2}} (3) for parallel-type resonators. Here K_{12}, J_{12} are impedance-inverter and admittance-inverter parameters, x_1, x_2 are reactance slope parameters of the first and the second resonant series-type networks at resonant frequency f_0, and b_1, b_2 are the susceptance slope parameters of the first and the second resonant parallel-type networks. When the resonators are resonant LC-circuits the coupling coefficient in accordance with (2) and (3) takes the value k_L=\frac{L_m}{\sqrt{L_1L_2}} (4) for the circuits with inductive coupling and the value k_C=\frac{C_m}{\sqrt{(C_1+C_m)(C_2+C_m)}}. (5) for the circuits with capacitive coupling. Here L_1, C_1 are the inductance and the capacitance of the first circuit, L_2, C_2 are the inductance and the capacitance of the second circuit, and L_m, C_m are mutual inductance and mutual capacitance. Formulas (4) and (5) are known for a long time in theory of electrical networks. They represent values of inductive and capacitive coupling coefficients of the coupled resonant LC-circuits. Coupling coefficient considered as a constant having a sign Refinement of the approximate formula (1) was fulfilled in. Exact formula has a form k=\frac{f_o^2-f_e^2}{f_o^2+f_e^2}. (6) 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 k has a negative value if f_o In accordance with new definition (6), the value of the inductive coupling coefficient of resonant LC-circuits k_L is expressed by formula (4) as before. It has a positive value when L_m>0 and a negative value when L_m Whereas the value of the capacitive coupling coefficient of resonant LC-circuits k_C is always negative. In accordance with (6), the formula (5) for the capacitive coupling coefficient of resonant circuits takes a different form k_C=\frac{-C_m}{\sqrt{(C_1+C_m)(C_2+C_m)}}. (7) Coupling between electromagnetic resonators may be realized both by magnetic or electric field. Coupling by magnetic field is characterized by the inductive coupling coefficient k_L and coupling by electric field is characterized by the capacitive coupling coefficient k_C. Usually absolute values of k_L and k_C monotonically decay when the distance between the resonators increases. Their decay rates may be different. However absolute value of their sum may both decay all over distance range and grow over some distance range. Summation of the inductive and capacitive coupling coefficients is performed by formula f_z=\frac{1}{2\pi}\sqrt{\frac{L_m}{(L_1L_2-L_m^2)C_m}}.(11) The definition of the function k(f) that generalizes formula (6) and meets the conditions (9) and (10) was stated on energy-based approach in. This function is expressed by formula (8) through frequency-dependent inductive and capacitive coupling coefficients k_L(f) and k_C(f) defined by formulas k_L(f)=\frac{\dot{W}_{12L}(f)}{\sqrt{[\bar{W}_{11L}(f)+\bar{W}_{11C}(f)][\bar{W}_{22L}(f)+\bar{W}_{22C}(f)]}}, (12) k_C(f)=\frac{\dot{W}_{12C}(f)}{\sqrt{[\bar{W}_{11L}(f)+\bar{W}_{11C}(f)][\bar{W}_{22L}(f)+\bar{W}_{22C}(f)]}}. (13) Here W denotes energy of high frequency electromagnetic field stored by both resonators. Bar over W denotes static component of high frequency energy, and dot denotes amplitude of oscillating component of high frequency energy. Subscript L denotes magnetic part of high frequency energy, and subscript C denotes electric part of high frequency energy. Subscripts 11, 12 and 22 denote parts of stored energy that are proportional to |U_1|^2, |U_1||U_2| and |U_2|^2 where U_1 is complex amplitude of high frequency voltage at the first resonator port and U_2 is complex amplitude of voltage at the second resonator port. Explicit functions of the frequency-dependent inductive and capacitive couplings for pair of coupled resonant circuits obtained from (12) and (13) have forms k_L(f)=\frac{L_m}{\sqrt{L_1L_2}}\frac{2}{\sqrt{(1+f_1^{-2}f^2)(1+f_2^{-2}f^2)}}, (14) k_C(f)=\frac{-C_m}{\sqrt{(C_1+C_m)(C_2+C_m)}}\frac{2}{\sqrt{(1+f_1^2f^{-2})(1+f_2^2f^{-2})}} (15) where f_1, f_2 are resonant frequencies of the first and the second circuit disturbed by couplings. It is seen that values of these functions at f=f_1=f_2 coincide with constants k_L and k_C defined by formulas (14) and (15). Besides, function k(f) computed by formulae (8), (14) and (15) becomes zero at f_z defined by formula (11). == Coupling coefficients in filter theory ==

Bandpass filters with inline coupling topology Theory of microwave narrow-band bandpass filters that have Chebyshev frequency response is stated in monograph. They have a form k_{i,i+1}=\frac{f_2-f_1}{\sqrt{f_1f_2g_ig_{i+1}}}, (16) where g_i (i=0,1,2...n) are normalized prototype element values, n is order of the Chebyshev function which is equal to the number of the resonators, f_1, f_2 are the band-edge frequencies. Prototype element values g_i for a specified bandpass of the filter are computed by formulas g_0=1, g_1=2a_1/\gamma, g_i=\frac{4a_{i-1}a_i}{b_{i-1}g_{i-1}}, (i=2,3,...n), (17) g_{n+1}=1, if n is even, g_{n+1}=\mathrm{coth}^2(\beta/4), if n is odd. Here the next notations were used \beta=2\mathrm{artanh}\sqrt{10^{-\Delta L/10}}, \gamma=\mathrm{sh}(\frac{\beta}{2n}), (18) a_i=\mathrm{sin}\frac{(2i-1)\pi}{2n}, b_i=\gamma^2+\mathrm{sin}^2(\frac{i\pi}{n}), (i=1,2,...n), where \Delta L is the required passband ripple in dB. 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 k_{i,i+1} at frequency f_0 depend on both specifications of the required passband and values of the derivatives dk_{i,i+1}/df|_{f=f_0}. That means the exact values of the coefficients k_{i,i+1} ensuring the required passband can not be known beforehand. They may be established only after filter optimization. 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 k_{i,i+1}. The coefficients k_{i,i+1} are symmetrical with respect to the central resonator or the central pair of resonators even in filters having unequal characteristic impedances of transmission lines in the input and output ports. Value of the coefficient k_{i,i+1} monotonically decreases with moving from the external pairs of resonators to the central pair. 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 k_{i,i+1} for one or more pairs of resonators expressing in their vanishing at frequencies of transmission zeroes. 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 \mathbf M of dimension n\times n,. It is symmetrical. Every its off-diagonal element M_{ij} is the coupling coefficient of ith and jth resonators k_{ij}. Every diagonal element M_{ii} is the normalized susceptance of the ith resonator. All diagonal elements M_{ii} in a tuned filter are equal to zero because a susceptance vanishes at the resonant frequency. Important merit of the matrix \mathbf M is the fact that it allows to directly compute the frequency response of the equivalent network having the inductively coupled resonant circuits,. 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. == 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 \mathbf E and \mathbf H . Hong proposed that the coupling coefficient is the sum of the normalized overlap integrals \kappa = \kappa_E+\kappa_M, (19) where \kappa_E=\frac{\int_V \epsilon \mathbf{E}_1\dot\mathbf{E}_2 dv}{\sqrt{\int_V \epsilon|\mathbf{E}_1|^2 dv\times\int_V \epsilon |\mathbf{E}_2|^2 dv}}(20) and \kappa_M=\frac{\int_V \mu \mathbf{H}_1\dot\mathbf{H}_2 dv}{\sqrt{\int_V \epsilon|\mathbf{E}_1|^2 dv\times\int_V \epsilon |\mathbf{E}_2|^2 dv}}.(21) On the contrary, based on a coupled mode formalism, Awai and Zhang derived expressions for \kappa which is in favor of using the negative sign i.e., \kappa=\kappa_M-\kappa_E. (22) Formulae (19) and (22) are approximate. They match the exact formula (8) only in case of a weak coupling. Formulae (20) and (21) in contrast to formulas (12) and (13) are approximate too because they do not describe a frequency dispersion which may often manifest itself in a form of transmission zeros in frequency response of a multi-resonator bandpass filter. 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 \kappa_M and \kappa_E . Using Poynting's theorem in its microscopic form, it was shown that \kappa can be expressed in terms of the interaction energy between the resonators' modes. == References ==