Naji, Adham

Warr, Paul

Adham Naji

Paul A. Warr

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IEEE Microwave and Wireless Components Letters

Proof of Perturbational Duality Between Classical Cavities and Planar Resonators With Magnetic Side-Walls

In this note a formal proof is given to an observation that the perturbational response of a planar resonator, which can be modelled as a cavity with magnetic side-walls, is the dual of the behaviour of a classical cavity with electric walls. This perturbational duality is observed even though the respective boundary types are not exactly dual. This observation is important as it provides the designer of planar resonators, such as filters and antennas, with a new ‘intuition’ to the pertubational reaction of such structures, and how it opposes that of classical cavities during perturbation. The formalism of the proof follows clear steps that parallel those found in the classic literature, and the observation is confirmed using numerical methods and measurements

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                          Naji, A., & Warr, P. (2016). Proof of Perturbational Duality BetweenClassical Cavities and Planar Resonators with Magnetic Side-walls. IEEEMicrowave and Wireless Components Letters, 26(7). DOI:10.1109/LMWC.2016.2574817Peer reviewed versionLink to published version (if available):10.1109/LMWC.2016.2574817Link to publication record in Explore Bristol ResearchPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia IEEE at http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7490405. Please refer to any applicableterms of use of the publisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.html1Proof of Perturbational Duality Between ClassicalCavities and Planar Resonators with MagneticSide-wallsAdham Naji, Member, IEEE, and Paul WarrAbstract—In this note a formal proof is given to an obser-vation that the perturbational response of a planar resonator,which can be modelled as a cavity with magnetic side-walls,is the dual of the behaviour of a classical cavity with electricwalls. This perturbational duality is observed even though therespective boundary types are not exactly dual. This observationis important as it provides the designer of planar resonators, suchas filters and antennas, with a new ‘intuition’ to the pertubationalreaction of such structures, and how it opposes that of classicalcavities during perturbation. The formalism of the proof followsclear steps that parallel those found in the classic literature,and the observation is confirmed using numerical methods andmeasurements.Index Terms—Perturbation theory, eigenvalue problem, planarresonator, boundary perturbation, microstrip resonator.I. INTRODUCTIONPERTURBATION calculations are of great importance inthe study of classical cavities, as they provide a quickapproximate tool for the evaluation of a cavity’s resonantfrequency shift in response to a relatively small disturbancein its geometric boundaries or its material [1]–[3]. Herewe investigate perturbations to the geometric boundaries. Inclassical cavities, which are made ideally of perfect conductors(electrical-walls), as in Figure 1a, it is known that the fre-quency shift (relative to the original frequency ω0) in responseto a perturbation is given byω − ω0ω0≈ ∆Wm −∆WeW0(1)where ∆Wm and ∆We are the energies removed by theperturbance from the original resonator volume, and W0 isthe total energy (magnetic and electric) stored in the originalresonator before any perturbation.In planar resonators, which can be modelled as cavities withmagnetic side-walls [1]–[3], however, it is observed that theshift direction is the reverse of that dictated by equation (1). Ineffect, the planar resonator is observed to have a perturbationalresponse that is the dual of that of the classical cavity. In thisnote, we prove this observation rigorously, and it becomesa new ‘intuition’ for the designer of planar resonators. Suchresonators are extremely common in today’s technology, suchas microstrip, stripline, and substrate-based filters and patchA. Naji is with the British University in Egypt, Egypt, and also with the Fac-ulty of Engineering, University of Bristol, England. P. Warr is with the Facultyof Engineering, University of Bristol, England; e-mails: adham.naji@ieee.organd paul.a.warr@bristol.ac.uk.Manuscript received September, 2015.t,bsurfaceenclosing τ,Svolume  τSSS surface ofside wallsSt,b surface of top andbottom wallsremoved volumeand surface(Δτ, ΔS)removed volumeand surface(Δτ, ΔS)(a) Conductor-enclosed       (classical) cavity(c) Resonator with       outer perturbance(d) Resonator with inner      (island) perturbancesurfaceenclosing τ',S' remaining volumeτ' h h(b) Planar resonator or cavity       with magnetic side-wallsremainingvolumeτ' surfaceenclosing τ',S' t,bS St,bS surfaceenclosing τvolume  τzx yFig. 1: Illustrations of a classical cavity and a planar resonator,with and without perturbations: (a) the classical conductor-enclosed cavity, (b) the planar resonator or the cavity modelwith magnetic side-walls and conductive walls (electric-walls)at the top and bottom planes, (c) the planar resonator after anouter perturbation, and (d) the planar resonator after an innerperturbation. The substrate dielectric material is shown asdark-hatched regions, while conductors are shown as white re-gions. h is the substrate height, τ the original volume, enclosedby surface S, ∆τ the perturbation size (removed), ∆S thesurface enclosing ∆τ , and τ ′ the remaining volume enclosedby the remaining surface S′, with τ ′=τ−∆τ , S′=S−∆S,S=Ss+St,b, S′=S′s+S′t,b, and ∆S=∆Ss+∆St,b.antennas. This result can also be used in perturbation effectcalculations in reconfigurable planar structures, where theinsertion of tuning elements constitutes a form of geometricperturbation (e.g. [4]).It is important to note here that the term planar impliesno field variations along substrate depth (along z, say),which is much smaller a dimension compared to the resonantwavelength. The latter is in the order of the planar surfacedimensions (in the xy plane) of the resonator. Dominant2(lowest-frequency) modes in such structures are assumed tobe TMzmn0 type modes. In other words, a planar resonatoris effectively two-dimensional (whereas a transmission lineresonator, for example, is effectively one-dimensional).II. PROOF FORMALISMThe problem of finding the resonant (natural) frequencyshift in a planar resonator (or a cavity with magnetic side-walls) due to a perturbation in the geometry can follow thesame initial steps as those used in perturbation calculationsof classic conductor-enclosed cavities (e.g. [1]). However,once the calculation steps reach the stage where the boundaryconditions are to be incorporated, the treatment diverges fromthe classic case, as shown below, due to the mixture of surfaceboundary conditions. It is shown that the actual perturbationalresponses (frequency behaviour) of the two cavity types areexactly dual.With reference to Figure 1, and assuming that the cavity issource-free (i.e. an eigenvalue problem), we can write the fieldequations before and after perturbation, then attempt to relatethem together, to deduce the frequency shift. As in classicliterature, we denote quantities before perturbation with a 0-subscript, such as E¯0, H¯0, ω0, k0, which denote the electricfield intensity, magnetic field intensity, angular frequency, andwavenumber in the original resonator. Symbols without suchsubscript denote the new quantities after perturbation, such asE¯, H¯, ω, k. Barred symbols (e.g. A¯) denote vectors, starredsymbols (e.g. A∗) denote complex conjugates, j =√−1, is the permittivity, and µ is the permeability. Amplitudes aretaken as effective (rms) values, and time dependence as ejωt.In the original resonator (Figure 1b), the field equations arederived from Maxwell’s equations as−∇× E¯0 = jω0µH¯0, (2)+∇× H¯0 = jω0E¯0. (3)Similarly, after perturbation (Figures 1c,d) the equations are−∇× E¯ = jωµH¯, (4)+∇× H¯ = jωE¯. (5)After some inter-manipulation [1], integrating across theperturbed cavity volume (τ ′) and applying the DivergenceTheorem, equations (2)–(5) are combined and reduced to [1]:j(ω − ω0)∫∫∫τ ′(E¯E¯∗0 + µH¯H¯∗0 )dτ=∫∫∫τ ′[∇ · (H¯ × E¯∗0 +∇ · (H¯∗0 × E¯))] dτ=∫∫∫τ ′∇ · (H¯×E¯∗0+H¯∗0×E¯)dτ=∮S′(H¯×E¯∗0+H¯∗0×E¯)ds¯.(6)Up to this point, the derivation is standard. Now we examinehow the magnetic-wall condition on the side walls and electric-wall condition on the top/bottom walls affect the result. It iseasier to start by considering the scenario of a disturbanceeffected in the outer boundary of the cavity, as in Figure 1c.In this scenario, we may decompose the surface integral of (6)to components on the top/bottom surfaces (S′t,b) and on theside-walls (S′s) of the perturbed resonator, giving∮S′(H¯×E¯∗0+H¯∗0×E¯)ds¯=∮S′(H¯×E¯∗0 )nˆds+∮S′(H¯∗0×E¯)nˆds=∫∫S′t,b(E-wall)(H¯ × E¯∗0 )nˆt,bds+∫∫S′s(M-wall)(H¯ × E¯∗0 )nˆsds+∫∫S′t,b(E-wall)(H¯∗0 × E¯)nˆt,bds+∫∫S′s(M-wall)(H¯∗0 × E¯)nˆsds. (7)But such planar structures will always have the conductorsconfined to their top or bottom (ground) planes, and thedominant fields will have the electric field direction transverseto them, i.e. TMzmn0, and S¯t,b ‖ S¯′t,b ‖ ∆¯St,b ‖ nˆt,b ‖ E¯ ‖ E¯0(symbol ‖ denotes parallel vectors), we have nˆt,b × E¯0 = 0(before perturbation) and nˆt,b × E¯ = 0 (after perturbation).On the other hand, on the magnetic-wall of the perturbedresonator, the magnetic field has re-adjusted itself to remainorthogonal to the magnetic-walls on the sides, which will givenˆs × H¯ = 0. Thus, the only term remaining from (7) is thefourth (H¯0 is generally not orthogonal to the S′s side walls):∮S′(H¯ × E¯∗0 + H¯∗0 × E¯)ds¯ =∫∫S′s(M-wall)(H¯∗0 × E¯)nˆsds. (8)We can re-write this as a closed integral by noting that wecan safely take the right hand side as∮S′(H¯∗0 × E¯)ds¯, since∫∫S′t,b(E-wall)(H¯∗0 × E¯)nˆt,bds = 0 always.Since either H¯0 or E¯ is orthogonal to the different surfacesthat make up S, then∮S(H¯∗0×E¯)ds¯=0. We note also that,by keeping the convention of the surface normal vectors nˆpointing outwards, we can reduce the integral domains asfollows:∮S′ ≡∮S−∆S = 0−∮∆S, giving∮S′(H¯∗0 × E¯)ds¯ = −∮∆S(H¯∗0 × E¯)ds¯. (9)We now make the assumption central to perturbation theory,that small disturbances (with smooth and shallow surfacedistortion) will allow us to consider the E¯ and H¯ fieldfunctions to be unchanged under perturbation, and that thevolume integrals over τ ′ are approximately equal to those overτ [1]–[3]. Poynting’s complex power balance gives:∮∆S(E¯0 × H¯∗0 )ds¯ = −jω0∫∫∫∆τ(µ∣∣H¯0∣∣2 −  ∣∣E¯0∣∣2) dτ= −2jω0(Wm −We). (10)Combining equations (9), (6), and (10) together and solvingfor (ω − ω0)/ω gives the final result:ω−ω0ω0≈∫∫∫∆τ(∣∣E¯0∣∣2−µ ∣∣H¯0∣∣2) dτ∫∫∫τ(∣∣E¯0∣∣2 +µ ∣∣H¯0∣∣2) dτ≈∆We−∆WmW0. (11)3feedfeedxyxyx1=½aaay1=a–½bybxbyzgaagy1bxbyx1=½a(a) (b) zFig. 2: Two simple perturbation scenarios to validate thetheoretical observation, using square stripline planar resonatorson r = 2.17 and h = 1.5 mm substrate (each side) with 17.5µm copper cladding. One has an E-cut (a) while another hasan M-cut (b), with g = 0.4, a = 20 and bx variable in bothscenarios, and by = 2 mm in (a) and 1 mm in (b).0 1 2 3 4 5 6 7 8 9 10-0.12-0.1-0.08-0.06-0.04-0.0200.020.040.06Gap size (bx), mm  f / f0Measurements, FEM results, and perturbation approximation M-cut, Numerical FEMM-cut, Perturbed Approx.E-cut, Numerical FEME-cut, Perturbed Approx.M-cut, MeasuredE-cut, MeasuredUnpertubed (reference)M-cut typeE-cut typeFig. 3: Results of measurements, FEM, and perturbationapproximation, all confirming the predicted behaviour.Note that the numerator is the inverse of that of the classicconductor-enclosed cavity case expressed in (1), which makesthe planar resonator’s cavity model the perturbational dual ofthe classic cavity.The second case of Figure 1d, where an inner perturbance ismade (resembling an island), gives the same result. This canbe shown by noticing that the first, second, and third termsin the right hand side of equation (7) still vanish because thenew magnetic field H¯ will position itself to be orthogonal tothe gap’s inner magnetic-walls as well as the resonator’s outerside walls, while the electric field E¯ remains transverse to theresonator’s plane and is discouraged from occupying the gap(except for a small fringing effect).Experimental and numerical validation of this observationcan be readily carried out by considering a simple structurethat is perturbed in two different scenarios, each of whichdesigned in such a way as to cause more disturbance toeither the E or the H field energies. To prove the conceptit is preferred to use simple structures that have well-definedmodes with clear domains for E and H energies and simpleperturbation shapes (cuts), which do not cause complex phe-nomena such as mode-rotation and mode-splitting. We choosethe simple square resonator shown in Figure 2. The two typesof cuts shown remove mostly E-energy or mostly H-energywithout causing mode rotation, since they do not break theshape symmetry with respect to the mode concerned. Thedominant mode TMz01 is expressed as H¯0 = xˆA sin ky andE¯0 = zˆjAη cos ky, where A is an arbitrary amplitude constantand η =√µ/ is the wave impedance.Substituting these equations into equation (11), after inte-gration (variables are defined in Figure 2), gives∆ff0= bx cos 2ky1 sin kby/(ka2), (12)which is an approximate model accurate to the first order only(linear with bx for given by, a and y1), where the perturbationsize is small, as expected from a perturbation technique.Measurement results, alongside the predictions given by thisapproximate perturbation model and by the numerical FiniteElement Method (FEM), are shown in Figure 3. We clearly ob-serve how frequency is confirmed to increase with increasingE-cut sizes and decrease with increasing H-cut sizes.We finally note that a previous general analysis by [5]gave a similar theoretical observation for perturbations that‘pushed-in’ outer walls, as in Figures 1c and 2a, whoseperturbed surfaces represent simply-connected regions, but didnot explicitly include the case of multiply-connected perturbedsurfaces (inner cut), such as that in Figures 1d and 2b.III. CONCLUSIONIn this note it was proved that the geometric perturbationalbehaviour of a a classic cavity is the dual of that of aplanar resonator (or a cavity with magnetic side-walls). Thishas important implications on approximate (perturbational)calculations of many popular resonating structures and systemsof the planar type, such as microstrip, stripline, and substrate-based patches in filters, antennas or reconfigurable structures.More importantly, this simple result provides the designer withan intuition of how the frequency of resonance would shift ina planar resonator, depending on whether the perturbation haslargely affected stored electric energy or magnetic energy, andthat this shift is opposite to that seen in classic cavities.ACKNOWLEDGMENTThis work was supported by ITAC project CFP99, Egypt.REFERENCES[1] R. F. Harrington, Time-Harmonic Electromagnetic Fields. McGraw-Hill, 1961.[2] D. M. Pozar, Microwave Engineering, 3rd ed. John Wiley & Sons,2004.[3] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. WileyBlackwell, 2000.[4] A. Naji, P. Warr, M. Beach, and K. Morris, “A fundamental limiton the performance of geometrically-tuned planar resonators,” IEEETransactions on Microwave Theory and Techniques, vol. 59, no. 6, pp.1491–1499, Jun. 2011.[5] P. H. Ladbrooke, “ Some Effects of Field Perturbation Upon Cavity-Resonance and Dispersion Measurements on MIC Dielectrics,” IEEETransactions on Microwave Theory and Techniques, vol. 25, no. 11, pp.892–903, Nov. 1977.

Proof of Perturbational Duality Between Classical Cavities and Planar Resonators with Magnetic Side-walls

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Proof of Perturbational Duality Between Classical Cavities and Planar Resonators with Magnetic Side-walls

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