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Coupling Matrices				The method used here for synthesis of general coupling matrices is based on the work by Dr. R.J. Cameron as described in the papers [1], [2] and by A.E Atia and A.E. Williams as described in [3].

				Coupling Matrix
				A coupling matrix of the fully-canonical folded form is shown in figure 1.

				Figure 1. Folded canonical network coupling matrix form - fifth degree example. (a) Folded coupling matrix form. "s" and "xa" couplings are zero for symmetric characteristics. (b) Coupling and routing schematic [1]^*.

				As described in [1] the above folded canonical network coupling matrix has the following characteristics/advantages: Multiple input/output couplings may be accommodated, i.e. couplings may be made directly from the source and/or to the load to internal resonators, in addition to the main input/output couplings to the first and last resonator in the filter circuit. Fully canonical filtering functions (i.e., Nth-degree characteristics with N finite-position transmission zeroes) may be synthesized.
				Normalization
				The coupling coefficients presented on this site are normalized with respect to ripple bandwidth and center frequency. The normalized coupling coefficient between resonators i and j is denoted Mij and is related to the 'normal' or actual coupling coefficients by multiplication with the relative bandwidth, i.e. ripple bandwidth divided by center frequency, *Kij=wMij** where w = bw/f0. If coupling bandwidths are desired the entries of the coupling matrix need only to be multiplied by the ripple bandwidth 'bw'. In this case the coefficients will have dimension of frequency (e.g. MHz). The external couplings are found from: Qe,S=1/(wMS1²) Qe,L=1/(wMNL²) MS1 and MNL are often also denoted M01 and MNN+1 respectively. N is the filter order. 'MS1' refers to the input coupling value in row 'S' and column 1. The normalized source and load terminations- RS and RL - are unity due to the transformation by MS1and MNL, respectively.

				Transformation to other topologies than the folded.
				Even though the folded form is an often used topology in real microwave filters other forms may be more practical for a given application. The folded form can be modified by applying a series of "plane rotations" to the folded coupling matrix whereby other filter configurations may be obtained. A general and very flexible method for this purpose has been presented by Atia [3]. Here the wanted topology is described by a "topology matrix" which is a NxN matrix of zeroes and ones, depending on whether the corresponding element in the final coupling matrix has to be reduced to zero or not. With the topology matrix as input, successive plane rotations are applied to the coupling matrix. The angle of rotation is determined so as to minimize the elements that are affected by the plane rotation and that have to be reduced to zero. The reduction procedure continues until all elements that have to be reduced to zero become smaller than a prescribed error. Any topology can - in principle - be realized by the method. If, however, the desired filter characteristic is not realizable with the wanted topology, the method will not converge and another - and more suitable - topology must be defined, e.g. with additional non-zero entries in the topology matrix.

				On this site the default topology is the folded form, however, Atia and Williams' general method is implemented and any topology can in principle be specified/synthesized.

				References:
				[1] R.J. Cameron, "Advanced coupling matrix synthesis techniques for microwave filters", IEEE Trans. Microwave Theory Tech., vol 51, pp. 1-10, Jan 2003.

				[2] R.J. Cameron, "General coupling matrix synthesis methods for Chebyshev filtering functions", IEEE Trans. Microwave Theory Tech., vol 47, pp. 433-442, Apr 1999.

				[3] A.E Atia and A.E. Williams, "New types of bandpass filters for satellite transponders", "COMSAT Tech. Rev., vol 1, no. 1, pp. 21-43, Fall 1971.

				* Figure reprinted from [1] with permission from Dr. Cameron.