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Journal of Microwaves, Optoelectronics and Electromagnetic Applications

On-line version ISSN 2179-1074

J. Microw. Optoelectron. Electromagn. Appl. vol.16 no.4 São Caetano do Sul Dec. 2017

https://doi.org/10.1590/2179-10742017v16i41077 

Article

A Direct Approach for Coupling Matrix Synthesis for Coupled Resonator Diplexers

Deeb Tubail1 

Talal Skaik2 

1Palestinian Technology Research Center, Palestine, Email: dtubail@gmail.com

2Electrical Engineering Department, IUG University, P.O. Box 108, Palestine, Email: talalskaik@gmail.com


Abstract

Diplexers are used to separate or combine two bands of frequencies and they are designed by combining channel filters using distribution circuit or alternatively designed based on coupled resonator circuits. Design parameters of coupled resonator circuits such as coupling coefficients, self-resonant frequencies and external couplings are done using optimization techniques. However, optimization techniques are not always efficient particularly for large set of parameters and hence more efficient approaches are required. In this paper, numerical solution instead of optimization is proposed to synthesize coupled resonator diplexers and obtain coupling coefficients directly from equations. The proposed direct approach achieves the desired characteristics for any given number of resonators and some numerical examples are given to demonstrate the validity of the proposed approach. One of the given examples is compared to a diplexer in literature synthesized using optimization technique and implemented using cavity resonators.

Index Terms coupling coefficients; coupling matrix; diplexer; resonators; T-topology

I. INTRODUCTION

Diplexers play an important role in communication systems where they are used to separate or combine different bands of frequencies. Usually diplexers are consisted of two filters connected with a junction. The design of coupled resonator filters is well known in filter theory and it is based on coupling matrix that represents coupling coefficients calculated using g-values [1, 2]. Another approach to synthesize filters based on optimization techniques is also presented in literature [3]-[5]. The theory of two-port coupled resonator filters is extended to multi-port coupled resonator structures that consist of only resonators without external junctions and thus forming miniaturized diplexers and multiplexers [6]-[18]. In [7], synthesis of coupled resonator diplexers is proposed. However, the synthesis method of coupling coefficients is done by optimization techniques that may not give a solution for relatively large structures with large set of optimization variables. Many coupled resonator diplexer components synthesized using coupling matrix optimizations have been presented in literature [7]-[11]. Furthermore, coupling matrix optimization approach has been extended to all resonator multiplexers [12]-[18]. The synthesis of such components becomes more complex for large structures and optimization algorithms may not converge to give a solution.

Fig. 1 exhibits the main coupled resonator diplexer topology presented in [7]-[10], where the coupling between the resonators, self-coupling and the external coupling coefficients are obtained using coupling matrix optimization. In this paper, we propose an alternative synthesis of coupling matrix for the topology in Fig. 1 directly from mathematical equations deduced empirically from optimization-based observations. The solution using the direct approach releases the need to use complex optimization algorithms that require defining a cost function, optimization variables and some constraints. The proposed direct synthesis achieves the desired characteristics for any number of resonators. Some numerical examples are presented in this paper for coupled resonator diplexers and an example is compared to a diplexer in literature synthesized using coupling matrix optimization.

Fig. 1 Diplexer Topology. 

II. DIRECT SYNTHESIS APPROACH

This section presents the proposed numerical solution to obtain the normalized coupling coefficients of the diplexer presented in Fig. 1 with pass-band ripple LAr=0.04321 dB. Considering a symmetrical channel diplexer with channel 1 extending from x1 to x2 (normalized frequencies) and channel 2 from –x2 to –x1, the normalized bandwidth Bw and the diplexer center frequency mc are given by

Bw=x2x1 (1a)
mv=0.5(x1+x2) (1b)
mc=x1x2 (1c)

The normalized coupling coefficient m12 is given by

a12=0.9756mv22.0877mv+1.4409 (2a)
b12=0.0382c12=0.0048m12=a12Bw2+b12Bw+c12+mc (2b)

where a12, b12 and c12 are the coefficients of the quadratic equation of the coupling coefficient m12.

The normalized coefficient m23 which equals to m2x is given by

a23=0.0209mv+0.6939 (3a)
b23=1.0001mv0.008 (3b)
m23=a23Bw+b23+mc (3c)

where a23 and b23 are the coefficients of the linear equation of the coupling coefficient m23. The normalized coupling coefficients between any other adjacent resonators mi,i+1 is given by

mi,i+1=0.5Bwgigi+1 (4)

The first two resonators R1 and R2 have no frequency offsets, and hence the self-coupling coefficients m11 and m22 are equal to zero. The self-coupling coefficient m33 which equals mxx is given by

a33=0.1177log10mv0.1349 (5a)
b33=0.0448mv0.1777 (5b)
c33=0.9936mv+0.0325 (5c)
m33=a33Bw2+b33Bw+c33+mvmc (5d)

The self-coupling coefficients mii of the other resonators in left arm Ri where i=4,…, x-1 are given by mii=mv in (1b), and similarly the other self-coupling coefficients mii of other resonators in the right arm Ri where i=x+1,…, N are equal to -mv in (1b).

The external quality factor at port one qe1, at port two qe2 and at port three qe3 are given from [7] as

qe2=qe3=2Bwqe±1 (6a)
qe1=qe22=qe32 (6b)

where qe±1 is the normalized external quality factor of the filter with edges of ±1, that can be calculated from the g-values as known in filter theory [1]. Finally, the scattering parameters are related to the coupling matrix as given in [7].

To illustrate how the proposed equations are deduced empirically from extensive study of optimization-based results, the following procedure is followed to give a brief insight on the foundation of the proposed method:

  1. The specifications of the diplexer are firstly defined which are the total number of resonators and the edges of the channels: x1 to x2 for channel 1 and –x2 to –x1 for channel 2.

  2. The center frequency for each channel mc is now directly calculated by (1c) and the bandwidth Bw by (1a).

  3. Optimization procedure is utilized to synthesize the coupling coefficients using cost function defined in [7].

  4. A comprehensive study is carried out on the effect of changing the channel bandwidth Bw on the values of coupling coefficients.

  5. Plots of optimized coupling coefficients (y-axis) versus channel bandwidth Bw (x-axis) are obtained.

  6. Step 5 is repeated for various channel positions mv as in (1b).

  7. The coupling coefficients between adjacent resonators other than those at the junction are observed from the optimization results and they are found to be directly related to the known g-values. Fig. 2 presents an example of the optimized values of the coupling coefficients (mi,i+1) between two adjacent resonators other than those at the junction. Those coupling coefficients linearly change with the bandwidth of the channels Bw and they are independent of the channel position mv and then (4) is deduced.

  8. The self-coupling coefficients of the resonators (except resonators at the junction: R1, R2, R3 and Rx) are displayed in Fig. 3 and they are found to be independent of the bandwidth Bw and their values are very close to the channel position mv where mii=mv is deduced.

  9. The graphs of the coupling coefficients at the junction: m12, m23 and m33 are presented in Figures 4, 5, 6 respectively. Curves in Fig. 4 represent quadratic equations for m12-mc versus channel bandwidth where each curve is given for a certain channel position mv. It can be noticed that the coupling coefficient m12 is dependent on channel bandwidth Bw, channel position mv and center frequency mc. Moreover, the curves representing m23-mc in Fig. 5 vary linearly with bandwidth and they are dependent on channel position mv. Furthermore, the coefficient m33 is depicted in Fig. 6 that exhibits quadratic equations for m33-mv+mc versus channel bandwidth Bw. It is evident from Fig. 6 that m33 depends on Bw, mv and mc.

  10. The equations of the coupling coefficients at the junction (m12, m23 and m33) are empirically deduced from the graphs in terms of the channel bandwidth Bw and channel position mv.

  11. The coefficients of the linear/quadratic equations which are labeled (a, b, c) in equations (2, 3, and 5) are also determined.

  12. For diplexers with different number of resonators, the previous steps are repeated. Very similar results for coupling coefficients have been obtained for diplexers with different number of resonators and general equations are deduced and proposed here.

Fig. 2 Optimized values of coupling coefficient mi,i+1

Fig. 3 Optimized values of coupling coefficient mii

Fig. 4 Optimized values of coupling coefficient m12

Fig. 5 Optimized values of coupling coefficient m23

Fig. 6 Optimized values of coupling coefficient m33

III. NUMERICAL EXAMPLES

This section presents numerical examples for coupled resonator diplexers synthesized using proposed equations in previous section. Five examples are given to prove the ability of the direct method to satisfy various diplexers' characteristics and requirements. These examples are different in number of resonators as well as different bandwidths. One of the examples is synthesized using direct method and compared with a diplexer in a literature synthesized using optimization and implemented using waveguide cavities.

Fig. 7 presents the topology of example one which consists of ten resonators. The normalized edges of the channels are x1=0.3, x2=1 and the normalized calculated coupling coefficients are m12=0.8127, m23=m27=0.3659, m34=m78=0.2226, m45=m89=0.2226 and m56=m9,10=0.3032. The self-coupling coefficients are m11=m22=0, m33=˗m77=0.5857, m44=m55=m66=˗m88=˗m99=˗m10,10=0.65. The normalized external quality factor of port two and port three is qe2=qe3=2.7754, while the normalized external quality factor of the common port is qe1=1.3877. The diplexer response is presented in Fig. 8 and it can be seen that the desired diplexer specification is achieved with five reflection zeros for each channel and with the specified channel edges and return loss.

Fig. 7 Topology of example one. 

Fig. 8 Theoretical response of example one. 

Topology in Fig. 9 presents the diplexer of example two of sixteen resonators. The normalized edges of the channels are x1=0.15, x2=1 and the normalized coupling coefficients are m12=0.8218, m23=m2,10=0.3838, m34=m10,11=0.2515, m45=m11,12=0.2354, m56=m12,13=0.2321, m67=m13,14=0.2354, m78=m14,15=0.2515 and m89=m15,16=0.3498. The self-coupling coefficients are m11=m22=0, m33m10,10= 0.5178, m44=m55=m66=m77=m88=m99= ˗m11,11= ˗m12,12 = ˗m13,13= ˗m14,14= ˗m15,15 = ˗m16,16 =0.5750. The normalized external quality factor of port two and port three is qe2=qe3=2.3932, while the normalized external quality factor of the common port is qe1=1.1966. The response of the diplexer is shown in Fig. 10 and it can be noticed that eight reflection zeros appear in each channel and that the specified channel bandwidth is satisfied.

Fig. 9 Topology of example two. 

Fig. 10 Theoretical response of example two. 

Example three is a diplexer formed of twenty coupled resonators and its topology is depicted in Fig. 11. The values of the boundaries are x1=0.12 and x2=1 and the calculated normalized external quality factors are qe2=qe3=2.3364 and qe1=1.1682. The calculated normalized coupling coefficients are m12=0.8226, m23=m2,12=0.3787, m34=m12,13=0.2570, m45=m13,14=0.2396, m56=m14,15=0.2342, m67=m15,16=0.2328, m78=m16,17=0.2342, m89=m17,18=0.2396, m9,10=m18,19=0.2570, and m10,11=m19,20=0.3580. The self-coupling coefficients are m11=m22=0, m33m12,12=0.5109, m44=m55=m66=m77=m88=m99=m10,10=m11,11m13,13m14,14m15,15m16,16m17,17m18,18m19,19m20,20 = 0.5600. The diplexer response is presented in Fig. 12 and it is evident that the specification of the diplexer is met which indicates that the proposed direct synthesis method is robust and convenient for large topologies. If this diplexer is synthesized using optimization techniques, it requires more than twenty variables with some constraints which may complicate the optimization problem and make the solution infeasible.

Fig. 11 Topology of example three. 

Fig. 12 Theoretical response of example three. 

Example four presents a diplexer with twelve coupled resonators and its topology is shown in Fig. 13. The specifications of this diplexer are the same as those in [9] that is practically implemented using waveguide cavity resonators at 10 GHz. The values of the boundaries are x1=0.30618 and x2=1 and the calculated normalized external quality factors are qe2=qe3=2.9806 and qe1=1.4903. A comparison between coupling coefficients for both methods is shown in Table I that clearly shows very close results for both methods. The response of diplexer four is depicted in Fig. 14 for both the direct and optimization techniques. It can be seen from the graph that the plots of S-parameters are close for both the proposed direct method and the optimization-based synthesis approach. The optimization has been performed on computer with Intel processor core i7, a CPU at 2.4 GHz and a RAM of 8 GB.

Fig. 13 Topology of example four. 

Table I NORMALIZED COUPLING COEFFICIENTS OF EXAMPLE FOUR 

Norm. Coupling Coef. Proposed Method Optimization Method
m12 0.8125 0.7963
m23=m28 0.3560 0.3466
m34=m89 0.2039 0.2101
m45=m9,10 0.1946 0.195
m56=m10,11 0.2039 0.2035
m67=m11,12 0.2814 0.2814
m11=m22 0 0
m33m88 0.6043 0.5942
m44m99 0.6665 0.655
m55m10,10 0.6665 0.6635
m66m11,11 0.6665 0.6652
m77m12,12 0.6665 0.6643
Time consumption Instant calculations 47.0782 second optimization time + time of writing optimization code.
Complexity -No complexity, direct calculations from equations -Complex cost function [see equation (3) in [9].
-Large number of optimization variables (17 variables).

Fig. 14 Theoretical response of example four. 

Example five presents a diplexer with fourteen coupled resonators and its topology is shown in Fig. 15. The values of the boundaries are x1=0.2 and x2=1 and the calculated normalized external quality factors are qe2=qe3=2.52 and qe1=1.26. The coupling coefficients synthesized by both direct and optimization methods are compared in Table II where it can be noticed that the results are close. Fig. 16 presents the response of the diplexer for both methods where small differences are observed.

Fig. 15 Topology of example five. 

Table II NORMALIZED COUPLING COEFFICIENTS OF EXAMPLE FIVE 

Norm. Coupling Coef. Proposed Method Optimization Method
m12 0.8183 0.7790
m23=m29 0.3842 0.3646
m34=m9,10 0.2396 0.2304
m45=m10,11 0.2255 0.2165
m56=m11,12 0.2255 0.2146
m67=m12,13 0.2396 0.2242
m78=m13,14 0.3324 0.3162
m11=m22 0 0
m33m99 0.5360 0.5188
m44m10,10 0.6000 0.5877
m55m11,11 0.6000 0.6022
m66m12,12 0.6000 0.5953
m77m13,13 0.6000 0.5928
m88m14,14 0.6000 0.5950
Time consumption Instant calculations 328.5 second optimization time + time of writing optimization code
Complexity -No complexity, direct calculations from equations -Complex cost function [see equation (3) in [9].
-Large number of optimization variables (20 variables).

Fig. 16 Theoretical response of example five. 

The results of those previous examples verify that the proposed numerical equations achieve the desired characteristics for different number of resonators and different channel bandwidths for coupled resonator diplexer with T-topology. Moreover, the proposed method is preferred to optimization techniques presented in literature for the given T-topology since the coupling coefficients are directly calculated from equations.

IV. CONCLUSION

This paper presents a novel direct approach for synthesis of coupled resonator diplexers with T-topology. The proposed method is based on synthesis of coupling coefficients directly from equations instead of optimization. The direct method achieves the desired diplexer characteristics for different number of resonators and various bandwidth requirements. An illustration on deducing the proposed equations from comprehensive optimization results is presented to give a good insight about the basis of the method. Five examples are given to prove the validity of the approach for relatively large number of resonators. Moreover, one numerical example is presented and compared to a diplexer synthesized and implemented in previous literature. The proposed method beats the optimization technique in reducing time consumption and complexity for structures with high number of resonators.

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Received: August 17, 2017; Revised: August 18, 2017; Accepted: October 19, 2017

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