geometry with K�(0) � 1.1 is quite appropriate. As for the next

following fundamental harmonic of the periodic cavity (s � 14),the general trend is a decrease of �AN�q/Aq� with increasing thedepth of the irregular resonator(s) and, hence, with increasing W.

One may use the sine modes whose frequencies are the nearestto those of the modes c3 and c4, because the competition withthese cosine modes should be rather weak due to their amplitudecomposition destruction, which resulted from the strong-modemixing. If the cavity is employed in the millimeter-wave magne-tron [2], the power output for these modes is carried out from oneof the resonators with odd r, which is regular in the frame of thepresent theory. However, at typical values of the loaded Q-factor(one to several hundred), the frequency shift should be less than1%, so that variations in the amplitudes should not be stronger, ascompared to those observed for the cosine modes at r� � 1%.Hence, these sine modes can be employed.

As an example of the effect of the geometry of the irregularresonator on the amplitudes of one of the cosine modes, whosenatural frequency is close to that of such a sine mode, Figure 6shows the fragment of the amplitude composition for the mode c4.Since the sine mode is the operating one, the competing cosinemodes must be mixed as strongly as possible. From the datashown, one can see that the strongest mixing is observed in thecase of K�

(0) � 1 and K�(N/ 2) � 1. This geometry is the most

appropriate among those which have been studied.Another cosine mode, which can be employed, is the c8 (�

mode). In this case, the difference in depth of the irregular reso-nators should be rather large, while either K�

(0) or K�(N/ 2) should be

less than 1. If K�(0) � 1 and K�

(N/ 2) � 1, r� is not large enoughfor this mode. From the obtained results it follows that among themodes, which have been studied, the modes c1, c2, s3, s4, and c8are the most appropriate to be employed.

4. CONCLUSION

In this paper, we have studied different manifestations of themixing of the eigenmodes of a periodic azimuthally corrugatedcavity, which occurs for a non-periodic cavity with a relativelysmall depth of resonators when the first passband is rather wide.Based on the results obtained, that exert the main effect on themixing the factors have been revealed. These factors are thedifferences in the natural frequency between a cosine and thenearest sine modes, the azimuthal mode index, and the relativedepth of the irregular resonators.

In line with the obtained results, the design strategy for thecosine modes should be based on the sequential choosing of theoperating mode from the frequency range required and the geom-etry whose rarefaction factor is realized within a minimum appro-priate value (2–3%), while the difference in the geometry of theregular and irregular resonators should be as small as possible. Forthe sine modes, r� must be maximized in order to provide thestrongest mixing for the cosine modes that are the main compet-itors in this case, while the regular and irregular resonators maystrongly differ in their geometry.

ACKNOWLEDGMENTS

The author would like to thank Prof. D. M. Vavriv for suggestingthe topic for this study, Prof. A. I. Nosich for discussions, and Mr.S. Yu. Puzirkov for aid with the illustrations.

REFERENCES

1. A. Serebryannikov and K. Schuenemann, Enhancing the natural fre-quency doublet splitting in almost periodic azimuthally corrugatedcavities, IEEE Microwave Wireless Compon Lett 12 (2002), 54–56.

2. I.D. Truten, Pulsed millimeter-wave magnetrons, in A.Y. Usikov (Ed.),Electronics and radio physics of millimeter and submillimeter waves,Naukova Dumka Publ., Kiev, 1986 (in Russian), pp. 7–21.

3. U. Balaji and R. Vahldieck, Radial mode matching analysis of ridgedcircular waveguides, IEEE Trans Microwave Theory Techn 44 (1996),1183–1186.

4. K. Schuenemann, A.E. Serebryannikov, and D.M. Vavriv, Analysis ofthe complex natural frequency spectrum of the azimuthally-periodiccoaxial cavity, Microwave Opt Technol Lett 17 (1998), 308–313.

© 2003 Wiley Periodicals, Inc.

WIDTH AND THICKNESS EFFECTS OFDIELECTRIC OVERLAY ON EVEN ANDODD MODE RESONANCE FREQUENCYOF MICROSTRIP RING RESONATOR

Julie Jacob, Chote Nuangnum, and R. C. AiyerDepartment of PhysicsUniversity of PunePune-411007, India

Received 29 November 2002

ABSTRACT: The effect of perturbation on even- and odd-mode reso-nance frequencies of a microstrip ring resonator (MRR) is studied.Overlay of alumina strips with variable width (1 mm to 4 mm) andthickness (0.635 mm to 3.175 mm) are used to perturb the MRR. Thechange in even- and odd-mode resonance frequencies, due to width ofthe overlay, is found to be nonlinear. Odd-mode frequency is observedto be independent of perturbation due to the thickness of the overlay.Even-mode frequency variations tend to saturate for thickness greaterthan about four times the substrate height. Based on the experimentalresults, empirical equations for even- and odd-mode resonance frequen-cies are reported. © 2003 Wiley Periodicals, Inc. Microwave OptTechnol Lett 37: 465–470, 2003; Published online in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/mop.10952

Key words: MRR; perturbation; even mode; odd mode

1. INTRODUCTION

The microstrip ring resonator is a simple transmission line formedin a closed loop. It has been widely used for various applicationsat microwave and millimeter-wave frequencies based on the prin-

Figure 6 Amplitude composition for the mode c4, for several APCswith �0/�1 � 0.5, �2

(1)/�1 � 1.25, � � 0.55, and N � 16. Numbers at amarking sign mean K�

(0), K�(N/ 2), and rC4, respectively

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 37, No. 6, June 20 2003 465

ciple that the ring is resonant when its mean circumference is equalto an integral multiple of guided wavelength, 2�r � n�g where ris the mean radius of the ring, �g is the guided wavelength, and nis the mode number [1]. Using the magnetic wall model it can beshown that the microstrip ring resonator actually supports twodegenerate modes [2]. Two field equations that are related to thedegenerate modes are given by [3]:

Ez � �AJn�kr� � BNn�kr�cos�n� (1a)

Hr � n/j�o�AJn�kr� � BNn�kr�sin�n� (1b)

H � k/j�o�AJn�kr� � BNn�kr�cos�n� (1c)

Ez � �AJn�kr� � BNn�kr�sin�n� (2a)

Hr � �n/j�o�AJn�kr� � BNn�kr�cos�n� (2b)

H � k/j�o�AJn�kr� � BNn�kr�sin�n� (2c)

where A and B are constants, k is the wave number, is theangular frequency, Jn(kr) is the Bessel function of first kind oforder n, Nn(kr) is the Bessel function of second kind of order n.Jn(kr) and Nn(kr) are the derivatives of the Bessel functions withrespect to the argument (kr). The boundary conditions to beapplied are H � 0 at r � ro and H � 0 at r � ri, where ro andri are the outer and inner radii of the ring, respectively. Applicationof boundary condition leads to the eigen value equation

Jn�kro�Nn�kri� � Jn�kri�Nn�kro� � 0 (3)

where k � ( o r�o)1/2. The difference between the field com-ponents of Eqs. (1a–c) and (2a–c) is that cosine as well as sinefunctions are solutions to the field dependence in the azimuthaldirection . Both sets of solutions also have the same eigen valueof Eq. (3). This means that two degenerate modes can exist atresonance frequency. Because the modes are orthogonal, there isno coupling between them. These two solutions could be inter-preted as two waves—one travelling clockwise and the otheranti-clockwise. If the paths traversed by these waves before ex-traction are of equal lengths, then the waves are orthogonal. Thus,in a symmetric ring resonator, the paths traversed by these wavesare of equal lengths and hence degeneracy is maintained [1]. Onlyeven mode gets excited in a symmetric ring resonator [4]. If thesymmetry of the ring is disturbed, coupling between the twodegenerate modes takes place and the two modes resonate at aslightly different frequency [2], which can be detected by thesplitting of the resonance curve. Degrees of splitting, modes thatsplit, and insertion loss are also studied using a distributed trans-mission model [5].

Asymmetries are usually introduced by making an asymmetricarrangement of end-coupled microstrip transmission line [2] con-necting some impedance, for example, by drilling a hole in the ringand filling it with mercury [6] or by introducing a notch [5] or gap[4]. It is shown that the resonance splitting has a strong functionaldependence on the mode number and position of perturbation [5].A nondestructive method of disturbing the symmetry of ring bykeeping overlay of alumna strips is also reported [7]. This methodhas the flexibility of changing the position of perturbation byplacing alumna strips at different azimuthal angles . Moreover at � n�/ 2, (where n is an even integer) splitting of resonancecurve is not observed and for all other intermediate angles splittingis observed [7].

This paper reports the width and thickness effects of dielectricoverlay on frequency characteristics of a flexibly perturbed MRR.

The reported empirical equations for even- and odd-mode fre-quency variations is also modified based on the experimentalresults.

2. EXPERIMENTAL

The experiment consist of two parts: i) Design and fabrication ofmicrostrip ring resonator (MRR), and ii) Characterization of MRRunder different perturbation conditions.

In i), an MRR with a resonating frequency of 10 GHz, 1%bandwidth, and characteristic impedance 50� is designed andfabricated on alumna substrate (height h � 0.635 mm, r � 9.8)as reported in [8, 9]. The fabricated MRR is characterized using ascalar network analyzer (HP8757C) and it shows a resonatingfrequency of 10.11 GHz with bandwidth of 124 MHz and insertionloss of �17.87 dB.

In ii), an MRR is perturbed by keeping crossed partial overlaysof alumna strips ( r � 9.8, length � 22 cm, height � 0.635 mm,and width of the strips � 1, 1.3, 1.5, 1.8, 2, 2.5, 3, 3.5, and 4 mm)at various azimuth angles . The height of the overlay is increasedby keeping alumina strips of the same width one over the other(with vacuum grease in between to avoid air gap). The azimuthangle is varied from 0° to 360° in steps of 22.5° in all the fourquadrants (Fig. 1). For each position of the overlay even/odd moderesonant frequencies, insertion loss and bandwidth are measured.The maximum width that is used to perturb the MRR is 4 mmbecause, at higher width, the angular positions of the overlaycannot be specified due to the small circumference of the ring usedand also because the repeatability of measurements tends to de-grade at higher width.

Figure 1 Overlay orientation on a microstrip ring resonator (MRR)

466 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 37, No. 6, June 20 2003

3. RESULTS AND DISCUSSION

It is observed that the resonance curve splits up to give even- andodd-mode frequencies for the azimuthal angles other than �n�/ 2 (where n is an integer). The variation in frequency for boththe modes as a function of width and thickness of the overlay isdiscussed below.

3.1 Even ModeFigure 2 shows variation in resonance frequency (at 0°) withdifferent width and thickness of the overlays. Here the width(wover) is normalized with respect to guided wavelength �g andthickness (tover) with respect to substrate height (tsub). The reso-nance frequency decreases with an increase in thickness of theoverlay and tends to saturate for thickness greater than about fourtimes the substrate thickness. The decrease in resonance frequencywith width and thickness of overlay can be attributed to a rise inthe effective dielectric constant due to change of mode fromquasi-TEM to TEM [10]. Variation in even-mode frequency withoverlay angle has (1 � cos 2) dependence as shown in Figure3. The reported [7] empirical equation for even-mode resonancefrequency variation due to flexible perturbation is as follows:

fre � fr � k�p� � 0.067�1 � cos 2� (4)

where n�/ 2, n is odd for fre variations and fr is the originalresonance frequency without overlay. Here the perturbation factork( p) is claimed to be a linear function of its cause (widths of thestrips) and the second term represents the angle dependent pertur-bation constant, are found from intuitive fits. In the present work,Eq. (4) is written in a general form as

fre � fr � k�p�e � a�p�e�1 � cos 2� (5)

where a( p)e is the angle-dependent perturbation factor. The per-turbation factor k( p)e and the factor a( p)e due to various widthand thickness of the overlay are obtained from intercepts andslopes of the linear equations, which are fitted to the graphs of ( fre

� fr) versus (1 � cos 2).Figure 4 depicts the variation of k( p)e with width of the

overlay. From the graph it is obvious that k( p)e is linear only upto lower widths (1.8 mm), higher than that it is a nonlinear functionof the width of the overlay. Further, it is observed that a( p)e is alsoa variable depending on the width of the overlay as shown inFigure 5. Hence, the second-order equations fitted to k( p)e anda( p)e variations with width of the overlay are as follows:

k� p�e � 4.1417�wover/�g� � 4.90858�wover/�g�2 (6a)

a� p�e � 0.21671�wover/�g� � 0.21306�wover/�g�2 (6b)

Figure 6 shows variation of k( p)e with thickness t of theoverlay. The factor a( p)e is found to be independent of thicknessvariation. Hence, taking into account width and thickness of theoverlay, the reported equation for even-mode frequency variationof a flexibly perturbed MRR can be rewritten as

fre � fr � k�p�e � a�p�e�1 � cos 2�, (7)

where k( p)e � 5.65(1 � exp(�1.26(tover/hsub)))(wover/�g) �6.35(1 � exp(1.28(tover/hsub)))(wover/�g)2, and a( p)e �0.21671(wover/�g) � 0.21306(wover/�g)2 where fr is in GHz andtover, hsub, wover, and �g are in mm.

Figure 3 Variation of even mode frequency fre with overlay angle forvarious widths. ■: wover � 1 mm; F: wover � 1.3 mm; Œ: wover � 1.5 mm;�: wover � 1.8 mm; }: wover � 2 mm; E: wover � 2.5 mm; �: wover � 3mm; ƒ: wover � 3.5 mm; �: wover � 4 mm

Figure 2 Variation of even-mode frequency fre with width (wover/�g) ofthe overlay (at 0°) for representative thickness (tover/hsub). ■: tover/hsub �1; F: tover/hsub � 2; Œ: tover/hsub � 3; �: tover/hsub � 4; }: tover/hsub � 5

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 37, No. 6, June 20 2003 467

Figure 7 Variation of odd mode resonance frequency fro (at 90°) withwidth (wover/�g) of the overlay

Figure 4 Variation of perturbation factor k( p)e with width (wover/�g) ofthe overlay [Eq. (6a)]. [Color figure can be viewed in the online issue,which is available at www.interscience.wiley.com.]

Figure 5 Variation of angle dependent perturbation factor a( p)e withwidth (wover/�g) of the overlay [Eq. (6b)]. [Color figure can be viewed inthe online issue, which is available at www.interscience.wiley.com.]

Figure 6 Variation of perturbation factor k( p)e with thickness (tover/hsub) of the overlay

468 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 37, No. 6, June 20 2003

3.2 Odd ModeThe variation in odd-mode resonance frequency (at 90°) withwidth of the overlay is as shown in Figure 7. The reported [7]empirical equation for odd-mode resonance frequency is

fro � fr � a�p��1 � cos 2�, (8)

where fro is claimed to be independent of strength of perturbation.From Figure 7 it can be seen that fro is more or less independent onlyat lower width but pronounced change is seen at higher widths of theoverlay. Further, it is observed to be independent of the thickness ofthe overlay. Hence, based on the experimental results, the odd-modefrequency variation due to perturbation is modified as

fro � fr � k�p�o � a�p�o�1 � cos 2�, (9)

where k(p)o � 0.38373(wover/�g) � 5.57972(wover/�g)2 and a(p)o �0.013 is obtained from the slope of the graph ( fo � fr) versus (1 � cos2) for various widths (1 to 4 mm as mentioned above).

Here fro is greater than original resonance frequency at thelowest width of the overlay (1 mm) and then continues to decreasewith strength of perturbation (width of overlay). The proposedequations for even- and odd-mode resonance frequencies are ver-ified by overlaying alumna strips on MRR with resonating fre-quency 10.27 GHz and bandwidth 92.29 MHz. The curves gener-ated using the above equation and experimentally observed datafor representative width and thickness are shown in Figure 8(a)–(c)and Figure 9. The shifts in resonance frequencies are viewed ascapacitive loading at the overlaid portion. Dielectric overlaychanges the effective dielectric constant of the system effectivelychanging the capacitance per unit length at the overlaid portion.

4. CONCLUSION

Overlay of alumina strips at various azimuthal angles excited thedegenerate modes (even and odd) of a microstrip ring resonator.

Figure 8 Validation of Eq. (7) for (a) wover � 1 mm, (b) wover � 1.3mm, and (c) wover � 1.5 mm, respectively. -: represent curves generatedusing Eq. (7). Symbols which represent experimentally observed values forvarious tover/hsub have the same meaning as in Fig. (2). [Color figure canbe viewed in the online issue, which is available at www.interscience.wiley.com.]

Figure 9 Validation of Eq. (9) for various widths (wover/�g) of theoverlay. -: represent curves generated using Eq. (9). Symbols representexperimentally observed values. ■: wover � 1 mm; Œ: wover � 1.3 mm; �:wover � 1.5 mm; F: wover � 1.8 mm. [Color figure can be viewed in theonline issue, which is available at www.interscience.wiley.com.]

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 37, No. 6, June 20 2003 469

The change in even-mode frequency with width and thickness isnonlinear. Odd-mode frequency was found to vary nonlinearlywith strength of perturbation. Empirical equations accounting forthis nonlinearity were reported.

REFERENCES

1. K. Chang, Microwave ring circuits and antennas, Wiley, New York,1996, Ch. 2 and 3.

2. I. Wolff, Microstrip band-pass filter using degenerate modes of amicrostrip ring resonator, Electron Lett 8 (1972), 302–303.

3. I. Wolff and N. Koppik, Microstrip ring resonator and dispersionmeasurement on microstrip lines, Electron Lett 7 (1971), 779–781.

4. V.K. Tripathi and I. Wolff, Perturbation analysis and design equationsfor open and closed ring microstrip resonators, IEE Trans MTT-32(1984), 405–410.

5. G.K. Gopalkrishnan and K. Chang, Band pass characteristic of splitmodes in asymmetric ring resonator, Electron Lett 26 (1990), 774–775.

6. W.J.R. Hoefer and A. Chattopadhyay, Evaluation of equivalent circuitparameters of microstrip discontinuities through perturbation of aresonant ring, IEE Trans MTT-23 (1975), 1067–1071.

7. R.A. Yogi, S.A. Gangal, R.C. Aiyer, and R.N. Karekar, Split modes inasymmetric microstrip ring resonator structure by flexible perturba-tion, Microwave Opt Technol Lett 19 (1998), 168–172.

8. M.P. Abegaonkar, R.N. Karekar, and R.C. Aiyer, A miniaturizednon-destructive microwave sensors for chickpea moisture measure-ment, Review of Sci Instrum 70 (1999), 3145–3148.

9. M.P. Abegoankar, R.N. Karekar, and R.C. Aiyer, Microwave micros-trip ring resonator as a moisture sensor for bio-materials: Applicationto wheat grain, Meas Sci Tech 10 (1999), 195–200.

10. K.K. Joshi, R.D. Pollard, and V. Postoyalko, Microstrip with dielectricoverlay: Variational analysis and validation, IEE Proc MicrowavesAntennas Propagation 141 (1994), 138–140.

© 2003 Wiley Periodicals, Inc.

A NOVEL PHOTONIC BAND-GAPMICROSTRIP STRUCTURES FOR LOW-PASS FILTERS OF WIDE STOP-BAND

Haiwen Liu,1 Zhengfan Li,1 Xiaowei Sun,2 and Junfa Mao1

1 Department of Electronic EngineeringShanghai Jiaotong UniversityShanghai 200030, P. R. China2 Shanghai Institute of Microsystem and Information TechnologyChinese Academy SciencesShanghai 200050, P. R. China

Received 29 November 2002

ABSTRACT: Two novel low-pass filters (LPFs) of wide stop-bandare proposed in this paper. One filter is implemented by a photonicband-gap (PBG) microstrip structure with T-junction opened stubsand its frequency characteristics are discussed. The other filter isimplemented by a PBG microstrip structure with cross-junctionopened stubs. The new wide-band harmonic rejection LPFs are easyto design and fabricate. The simulated and experimental results ver-ify the validity of the LPFs. © 2003 Wiley Periodicals, Inc.Microwave Opt Technol Lett 37: 470 – 472, 2003; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10953

Key words: low-pass filter; photonic band-gap (PBG); microstrip;finite-difference time-domain

1. INTRODUCTION

Filtering of undesired frequencies in microstrip designs can beimplemented with shunt stubs or stepped-impedance lines, butthese techniques provide typically narrow-band and spurious pass-band in stop-band and occupy valuable circuit layout area. Pho-tonic band-gap (PBG) structures have been considered as an al-ternate to solve this problem in microwave and millimeter-wavecircuit applications recently [1–8]. (PBG structures are periodicstructures which exhibit a band gap within a certain band ofelectromagnetic propagation is prohibited [9, 10].) But most of thePBG structures show a periodic frequency stop-band, of which thehigher-order stop-band is termed harmonic. In order to achievewide stop-band characteristics, several techniques have been pro-posed, which use serial or parallel PBG structures to meet this goal[11–13]. They provide an effective method to suppress high-orderharmonics in active circuits.

However, most of the work has focused on PBG structures witha straight microstrip line. In this paper, the low pass-band filters(LPFs) are fabricated by a PBG microstrip structure with T-junction opened stubs and cross-junction opened stubs, respec-tively. The first section describes how the LPFs with the PBGstructure were designed. This is followed by a section describingfrequency characteristics of the proposed filter with T-junctionopened stubs. The finite-difference time-domain (FDTD) simu-lated and experimental results are discussed finally.

2. DESIGN OF THE LPFs WITH THE PBG STRUCTURES

Figure 1 shows a schematic of the proposed filters. The openedstubs are realized with T-junction structures and cross-junctionstructures, respectively. And the length l is 10.9 mm, which isequal to 4/�g (�g is the guided wavelength of the standard micros-trip line). The PBG structure used in the following studies wasimplemented by etching rectangle holes on the backside metallicground plane. The periodic distance d is 21.8 mm, which isdetermined by �g/2. The side length a of the square hole is 10.9mm. The dielectric constant of material is 2.22 with a thickness of0.254 mm. The width of the microstrip line w is 0.78 mm,corresponding to a 50� characteristic impedance.

3. FREQUENCY CHARACTERISTICS OF THE PROPOSEDPBG MICROSTRIP STRUCTURE

To investigate the frequency characteristics of the proposed PBGmicrostrip structure, we simulated the proposed PBG microstripstructure with T-junction stubs using FDTD.

3.1. Influence of the Length of the StubThree PBG microstrip structures with T-junction stubs are simu-lated with the different length l of the stub. The simulated resultsare illustrated in Figure 2. From Figure 2, the filter’s cutoff

Figure 1 Schematics of the proposed filters: microstrip line with (a)T-junction opened stub and (b) cross-junction opened stub

470 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 37, No. 6, June 20 2003