IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 12, DECEMBER 2004 2645

A Direct Method to Compute the CouplingBetween Nonidentical Microwave Cavities

Ayyangar R. Harish, Senior Member, IEEE, and Joseph Sahaya Kulandai Raj

Abstract—A new technique to compute the coupling, as a func-tion of frequency, between nonidentical resonators is presented inthis paper. The structure is separated by introducing electric andmagnetic wall boundary conditions on the plane of separation, andcoupling coefficient is computed directly from the eigenvalues ofthe four individual structures. The proposed technique has beenused to compute the coupling between coaxial combline resonators.It is found to be computationally much faster than the iterativetechnique. The computed results are in good agreement with themeasured results.

Index Terms—Coaxial resonators, microwave filters, microwaveresonators.

I. INTRODUCTION

BANDPASS filter, an important block in a microwavecommunication system, can be realized using coupled

resonators [1]. The quality factor of a resonator and, hence, itsinsertion loss, depends on the size of the resonator. Using largerresonators having a high quality factor it is possible to realizefilters with lower insertion loss. For simplicity, it is desirableto realize a bandpass filter with identical resonators. It may, attimes be possible to use resonators of different sizes and shapesso that the available space is efficiently utilized to achieve anoptimum solution. In such cases, it is necessary to compute thecoupling between nonidentical resonators.

In order to compute the coupling coefficient between twoidentical cavities Zaki and Chen [2] introduced symmetry wallsbetween the resonators and calculated the eigenfrequencies ofone-half of the structure, once with an electric wall and then witha magnetic wall boundary condition at the symmetry plane. Thecoupling coefficient can be calculated from these two eigenfre-quencies. It is also possible to compute the eigenfrequencies ofthe entire structure and, hence, the coupling coefficient [3].

Computationof thecouplingcoefficientbetween twononiden-tical cavities is not quite straightforward. To achieve synchronoustuning when the two cavities are not identical, the electrical pa-rameter, e.g., capacitance, of one of the cavities is varied by ad-justing the tuning screw of the resonator until the eigenfrequencyseparation becomes minimum. This iterative technique is quitesimple to use on a test bench to measure the coupling betweentheresonators[4];however,computationally, it isvery inefficient.Direct techniques [5], [6] have been used to compute the cou-

Manuscript received April 20, 2004; revised June 15, 2004A. R. Harish is with the Department of Electrical Engineering, Indian Institute

of Technology, Kanpur 208 016, India (e-mail: arh@iitk.ac.in).J. S. K. Ray was with the Department of Electrical Engineering, Indian

Institute of Technology, Kanpur 208 016, India. He is now with the Departmentof Electronics and Communications Engineering, Bannari Amman Institute ofTechnology, Sathyamangalam 638 401, India.

Digital Object Identifier 10.1109/TMTT.2004.837311

pling coefficient between two dielectric resonators operating intwo different modes. Liang et al. [5] show that the coupling be-tween two dissimilar cavities (say, A and B) is equal to the geo-metricmeanof thecouplingbetweentwoidenticalcavitiesof typeA and the coupling between two identical cavities of type B. In[7], this method has been extended to include the reference fre-quency for the coupling coefficient and applied to calculate thecoupling between two nonidentical microwave resonators cou-pled magnetically, and it has been further extended in this paperto the case of electric coupling. The coupling coefficient betweendissimilar resonators has been computed using the iterative tech-nique, as well as the proposed direct technique, and the results arecompared with the measured values over a band of frequencies. Itis foundthat theaccuracyof theproposedtechniqueiscomparablewith that of the iterative technique and is within approximately2%of themeasuredvalues.Theproposedtechnique isalsoused tocompute the coupling coefficient between symmetric resonatorscoupled by asymmetric coupling elements.

II. FORMULATION

A. Magnetic Coupling

Consider two microwave resonators of different sizes thatare coupled magnetically. Following the work of Hong [8], alumped-element equivalent circuit for the system is shown inFig. 1(a). This can be rewritten as a T-network [see Fig. 1(b)]. InFig. 1, and ( ) are the self-inductance and self-ca-pacitance of the cavity , respectively, and is the mutual in-ductance between the two cavities. The eigenfrequencies of thefour individual structures obtained by introducing electric andmagnetic walls on the plane of separation are given by (1). Thesubscripts 1 and 2 refer to cavities 1 and 2, and and ofrefer to the magnetic and electric wall boundaries, respectively,

(1)

The coupling coefficient is defined by the following equation[8]:

(2)

0018-9480/04$20.00 © 2004 IEEE

2646 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 12, DECEMBER 2004

Fig. 1. Lumped-element equivalent circuits for the magnetically coupledresonators.

Solving (1) simultaneously for products , , ,and , and substituting in (2), we get the following expres-sion for the coupling coefficient:

(3)

If and are the coupling coefficient and resonant frequencyof two identical cavities of type 1 coupled by , and and

are the coupling coefficient and resonant frequency of twoidentical cavities of type 2 coupled by , the coupling coeffi-cient is a geometric mean of and , as given in [5].

If the two resonators are identical and synchronously tuned,and . Thus, the coupling

coefficient reduces to

(4)

It can be shown that, for a synchronously tuned system, the cou-pling coefficient is referenced to the resonant frequency givenby the following expression:

(5)

For a general case, when the two cavities are nonidentical andthe individual resonant frequencies are different, the couplingcoefficient given by (3) is referenced to an average resonant fre-quency given by the following equation [7]:

(6)

This can be recognized as a geometric mean of the resonantfrequencies and . This approximation gives best resultswhen the resonant frequencies are close to each other. As the twofrequencies drift apart due to dissimilarities in the two cavities,the accuracy of the coupling coefficient referenced to givenby (6) also degrades.

Fig. 2. Lumped-element equivalent circuit for the electrically coupled cavities.

The eigenfrequencies used in (3) can be computed by usinga numerical technique such as mode matching, finite-elementmethod, or finite-difference technique, etc., on two individualcavities with appropriate boundary walls. Therefore, using theproposed technique, only four independent simulations are re-quired to compute the coupling coefficient. If the two resonatorsare identical, the number of simulations reduces to two.

B. Electric Coupling

The equivalent circuit for two nonidentical cavities coupledelectrically is shown in Fig. 2. The elements and (

) are the self-inductance and self-capacitance of the cavity ,respectively, and is the mutual capacitance between the twocavities.

The geometry is separated into four individual structures byintroducing electric and magnetic walls at the plane of separa-tion. The eigenfrequencies of the four individual structures aregiven by

(7)

The electric coupling coefficient is defined by the followingequation [6]:

(8)

Solving (7) simultaneously for products , , ,and , and substituting in (8), we get the following expres-sion for the coupling coefficient:

(9)

This coupling coefficient is referenced to an average resonantfrequency given by (6) with replaced by . If the two res-onators are identical and synchronously tuned, the coupling co-efficient reduces to

(10)

HARISH AND RAJ: DIRECT METHOD TO COMPUTE COUPLING BETWEEN NONIDENTICAL MICROWAVE CAVITIES 2647

C. Iterative Method

The eigenfrequencies of the magnetically coupled struc-ture shown in Fig. 1(a) and of the electrically coupled struc-ture shown in Fig. 2 are given by (11) and (12), shown at thebottom of this page [8]. The difference between the two eigen-frequencies ( or ) willbe minimum when the two cavities are synchronously tuned

and the coupling coefficient under this con-dition is given by

or (13)

In (13), the resonant frequencies and are given by

and (14)

The eigenfrequencies that are required for computing the cou-pling coefficient can be calculated using full-wave electromag-netic (EM) simulation of the entire structure. The synchronouslytuned condition is achieved by keeping one of the capacitancesfixed and varying the other until the difference in the two eigen-frequencies is minimum. To achieve this minimum condition, itmay take several independent simulations of the entire structure.

III. NUMERICAL EXAMPLES

The usefulness of the proposed technique is demonstrated byconsidering several examples.

A. Iris Coupling

Let us consider two combline resonators of different sizeswith identical resonator rods coupled magnetically by an iris inthe common wall (Fig. 3). Therefore, setting the tuner heightsequal to each other does not yield a synchronous resonance con-dition. It is possible to achieve a synchronously tuned conditionby keeping one of the tuning screw heights (say, ) fixed andadjusting the other tuning screw height (say, ) iterativelyuntil the difference in the two eigenfrequencies of the completestructure reaches a minimum value.

Fig. 4 shows the difference between the two eigenfrequenciesas a function of the tuning screw height for a fixed

of 5 mm. In order to achieve a reasonably accurate valueof minimum , several computational runs are carried out.The simulations are performed in FEMLAB [9], a commercialEM simulator, using 50 000 edge elements. The coupling coef-ficient is computed using (13) and the corresponding resonantfrequency is given by (14). The entire procedure is repeated by

Fig. 3. Geometry of the combline resonator cavities magnetically coupledthrough an iris in the common wall (different cavity sizes and identicalresonators). All dimensions are in millimeters.

Fig. 4. Difference between the two eigenfrequencies as a function of tunerheight of cavity 2 (Th2) for the structure shown in Fig. 3.

setting a different value of to compute the coupling coeffi-cient at any other frequency.

In order to use the proposed technique, the structure is sep-arated into two cavities at the plane of separation (as shown inFig. 3) and electric and magnetic walls are introduced at thisplane resulting in four individual structures. FEMLAB is againused to compute the eigenfrequencies of the four individualstructures. The coupling coefficient and the corresponding res-onant frequency are computed using (3) and (6), respectively.

(11)

and

(12)

2648 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 12, DECEMBER 2004

Fig. 5. Coupling coefficient of combline resonator cavities (shown in Fig. 3)magnetically coupled through an iris in the common wall.

Fig. 6. Geometry of combline resonator cavities magnetically coupled throughan iris in the common wall (different cavity and resonator sizes). All dimensionsare in millimeters.

Once again, the tuner heights are changed to calculate the cou-pling coefficient referenced to any other frequency. Using theproposed approach, only four single cavity simulations are re-quired to calculate the coupling coefficient at any one frequency.

The coupling coefficients, as a function of frequency, of thestructure shown in Fig. 3 have been computed using the iterativemethod, as well as the proposed technique, and compared withthe measured results (Fig. 5). The coupling coefficients com-puted using the proposed method is in close agreement (within1%) with those computed using the iterative technique. Thecomputed results using both the techniques are in close agree-ment (within 2%) with the measured results.

Another example of iris-coupled combline resonators with adifferent cavity, as well as resonator rod sizes, has been consid-ered (Fig. 6) and the coupling coefficients have been computed

Fig. 7. Coupling coefficient of combline resonator cavities (shown in Fig. 6)magnetically coupled through an iris in the common wall.

Fig. 8. Coupling coefficient of two identical combline resonator cavitiesmagnetically coupled through an iris in the common wall.

using the iterative and proposed technique. The results are com-pared with the measured results in Fig. 7. Once again, a closeagreement is found between the measured and computed cou-pling coefficients.

In order to understand the issues involved with the accuracyof (6), let us consider two identical combline resonator cavi-ties coupled by an iris in the common wall. We can use thetechnique given in [2] to compute the coupling coefficient be-tween the two cavities. The coupling coefficient as a functionof frequency is calculated by changing the heights of the tuningscrews in both the cavities simultaneously so that the two cav-ities are always synchronously tuned (Fig. 8). Now the heightof the tuning screw in one of the cavities is kept fixed, and theheight of the tuning screw in the other cavity is changed. Thisconstitutes nonidentical cavities and the proposed method hasbeen used to compute the coupling coefficient as a function offrequency and plotted in Fig. 8. These results clearly indicatethat the two curves get closer to each other as the difference be-tween the resonant frequencies and gets smaller.

HARISH AND RAJ: DIRECT METHOD TO COMPUTE COUPLING BETWEEN NONIDENTICAL MICROWAVE CAVITIES 2649

Fig. 9. Geometry of two identical combline resonator cavities without anycoupling elements. All dimensions are in millimeters.

Fig. 10. Geometry of two identical combline resonator cavities coupled by anasymmetric lid loop.

Fig. 11. Geometry of two identical combline resonator cavities coupled by alid-to-floor loop.

B. Loop Coupling

Let us consider two identical cavities with a small iris sym-metrically placed in the common wall (Fig. 9). The iris in thecommon wall is so small that its contribution to the couplingcan be neglected. The coupling between the two cavities can berealized by an asymmetrically placed loop element (Fig. 10).The loop is constructed by attaching the two ends of a wire ofdiameter to the lid in either cavity. The vertical dimension ( )of the loop is measured from the bottom surface of the lid to thecenter of the wire. The horizontal dimensions ( and ) aremeasured from the central plane of the cavities to the center ofthe wires on either side. This is called a lid loop. The couplingcoefficient of the loop and the iris has the same sense (say, posi-tive). It is possible to change the sense of the coupling (i.e., makeit negative) by changing the orientation of the loop, as shown inFig. 11. The proposed structure is called a lid-to-floor loop. Thecoupling coefficient of both the structures as a function of fre-quency has been presented in Fig. 12. From the transmissionresponse of a four-cavity filter, with irises realizing the main

Fig. 12. Coupling coefficient of two identical combline resonator cavitiescoupled by: (a) an asymmetric loop (legend: �: L1 = 9:2, L2 = 7:2,H = 12:65, d = 1:3), (b) a lid-to-floor loop (legend: +: L1 = L2 = 9:2,H = 12:65, d = 1:3, and (c) a capacitive probe (legend: �: L1 = 13:65,L2 = 12:35, H = 10, d = 2, T = 3, D = 8, �: measured). Solid linesindicating coupling coefficient computed by the proposed method and dottedlines by the iterative method. All dimensions are in millimeters.

Fig. 13. Geometry of two identical combline resonator cavities coupled by anasymmetric capacitive probe.

line couplings, and a lid-to-floor loop realizing the cross-cou-pling between resonators 1 and 4, it is possible to show that thelid-to-floor loop gives negative coupling.

C. Probe Coupling

Negative coupling can also be realized with the help of a ca-pacitive probe of diameter . In order to realize a stronger cou-pling, the probe ends are loaded with metallic discs of thickness

and diameter . The coupling element extends into each ofthe cavity by and , respectively (Fig. 13). The probe isheld in place by a polytetrafluoroethylene (PTFE) bush fittedinto the iris in the common wall. Fig. 11 also shows the cou-pling coefficient of the capacitive probe as a function of fre-quency computed by both iterative and proposed techniques incomparison with the measured results.

IV. DISCUSSION

The computational advantage of the proposed method can beillustrated by considering the computer time required to calcu-late the coupling coefficient at one frequency point. For the ex-amples considered in this paper, the iterative method required8–12 simulations of the entire structure to achieve minimum fre-quency difference condition (synchronously tuned condition). A

2650 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 12, DECEMBER 2004

Fig. 14. Coupling coefficient of two identical combline resonator cavitiescoupled by a symmetric capacitive probe (T = 5, D = 10, H = 12:65,d = 1:3) and a lid-to-floor loop (H = 12:65, d = 1:3). All dimensions are inmillimeters.

total of 50 000 elements were used to represent the entire struc-ture and it took 15 min to complete one simulation. In total, tocalculate the coupling coefficient at one frequency point, the it-erative method required approximately 2–3 h. In the proposedmethod, the individual cavities were modeled using 30 000 ele-ments and it took 5 min to compute the eigenfrequency of onecavity with an appropriate boundary condition. Therefore, theproposed method required only 20 min to compute the couplingcoefficient at one frequency, a significant saving in the compu-tational effort compared to the iterative method.

The lid-to-floor loop, as well as the capacitive probe, gen-erate negative (with respect to the iris coupling) coupling. Theresults indicate that the lid-to-floor loop has two distinct advan-tages compared to the capacitive probe, which are: 1) the vari-ation of coupling coefficient with frequency (dispersion) of thelid-to-floor loop is very similar to that of the iris coupling [on theother hand, the capacitive probe is highly dispersive (Fig. 12)]and 2) the coupling coefficient of the lid-to-floor loop is less sen-sitive to dimensional changes compared to the capacitive probe(Fig. 14).

V. CONCLUSION

In this paper, a noniterative method has been proposed tocompute the coupling, as a function of frequency, between syn-chronously tuned nonidentical microwave cavities. It has beenshown with several examples that the results obtained using theproposed method is in close agreement with those computedusing the iterative method, as well as measurements. The pro-posed method takes significantly less time to compute the cou-pling coefficient compared to the iterative method.

The method has also been used to compute the coupling be-tween identical cavities coupled by asymmetric coupling struc-tures. It is found that one such element, the lid-to-floor loop, hasbetter performance compared to the capacitive probe.

ACKNOWLEDGMENT

The authors wish to acknowledge G. Dobbs and K. Hashmi,both of Mitec Telecom Ltd., Dunstable, U.K., for providingsome of the measured data, the reviewers for their comments,which helped in clarifying certain aspects, and especially onereviewer for pointing out [5] and [6].

REFERENCES

[1] A. E. Atia and A. E. Williams, “Narrow-bandpass waveguide filters,”IEEE Trans. Microwave Theory Tech., vol. MTT-20, pp. 258–265, Apr.1972.

[2] K. A. Zaki and C. Chen, “Coupling of nonaxially symmetric hybridmodes in dielectric resonators,” IEEE Trans. Microwave Theory Tech.,vol. MTT-35, pp. 1136–1142, Dec. 1987.

[3] J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/MicrowaveApplications. New York: Wiley, 2001.

[4] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters,Impedance-Matching Networks, and Coupling Structures. New York:McGraw-Hill, 1964.

[5] J. F. Liang, K. A. Zaki, and A. E. Atia, “Mixed modes dielectric resonatorfilters,” IEEE Trans. Microwave Theory Tech., vol. 42, pp. 2449–2454,Dec. 1994.

[6] C. Wang, H. W. Yao, K. A. Zaki, and R. R. Mansour, “Mixed modescylindrical planar dielectric resonator filters with rectangular enclosure,”IEEE Trans. Microwave Theory Tech., vol. 43, pp. 2817–2823, Dec.1995.

[7] A. R. Harish and J. S. K. Raj, “A direct method to compute the cou-pling between nonidentical microwave cavities,” in IEEE MTT-S Int.Microwave Symp. Dig., June 2004, pp. 1089–1092.

[8] J.-S. Hong, “Couplings of asynchronously tuned coupled microwaveresonators,” Proc. Inst. Elect. Eng., pt. H, vol. 147, no. 5, pp. 354–358,Oct. 2000.

[9] FEMLAB Ver. 2.3 User Manual, Comsol AB, Stockholm, Sweden,2002.

Ayyangar R. Harish (M’00–SM’03) received thePh.D. degree in electrical engineering from theIndian Institute of Technology, Kanpur, India, in1997.

From 1997 to 2000, he was a Senior Engineer, andfrom 2000 to 2002, he was the Chief RF Passive En-gineer with COM DEV Wireless, Dunstable, U.K.From March 2002 to July 2002, he was the ChiefRF Passive Engineer with Mitec Telecom Ltd., Dun-stable, U.K. He is currently an Assistant Professorwith the Department of Electrical Engineering, In-

dian Institute of Technology. His research interests are the analysis and syn-thesis of microwave filters, computational electromagnetics, fractal structures,and neural networks.

Joseph Sahaya Kulandai Raj was born in Madurai,India, in 1979. He received the B.E. degree inelectronics and communication engineering from theUniversity of Madras, Madras, India, in 2000, andthe M. Tech. degree in electrical engineering fromthe Indian Institute of Technology, Kanpur, India, in2004.

He is currently with the Department of Electronicsand Communications Engineering, Bannari AmmanInstitute of Technology, Sathyamangalam, India. Hisresearch interests include computational electromag-

netics, antenna theory, and modeling.