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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL.XX, NO. XX, XXX XXXX 1

Coupled Mode Theory applied to Resonators in thePresence of Conductors

Sameh Y Elnaggar, Richard J. Tervo, and Saba M. Mattar

Abstract—Using the method of images, Energy Coupled ModeTheory (ECMT), a coupled mode equation in the frequency do-main, is extended to deal with important cases where resonatorsare in close proximity to conducting surfaces. Depending onthetype of conductors and the orientation of the resonators, themethod of images determines the relative phases of the images.Using the formed images, the coupled frequencies and fields canbe determined by applying ECMT. Two cases are studied. Inthe first case, it is shown that a dielectric resonator insertedin a cavity couples with both the mirror image and the cavity.The frequency behavior is described by the interaction withtheimage which counteracts that with the cavity. The second caseis that of resonators sandwiched between conducting plates. Itis shown that an infinite array of stacked images is formed. Thecoupling of the resonator with its images determines the coupledfrequencies and fields. In this context, the main advantage ofECMT-I is its ability to separate the effects of the walls fromthe uncoupled system. This means that the system parametersare independent of the separation distances and/or the typeofconductors, which renders the post processing analyses easierand predictable. Provided that the behaviors of the uncoupledresonators are known, ECMT-I is general and can be applied tomore complex systems.

Index Terms—Metamaterials, Perfect Magnetic Conductors,split ring resonators, dielectric resonators, coupled mode theory,coupled resonators, hybridization.

I. I NTRODUCTION

PRACTICAL situations arise where resonators are installedin the proximity of conducting surfaces. For example, itwas already shown that conducting plates affect the efficiencyof power transfer via inductive-resonance coupling [1], [2].Metamaterials can be realized by inserting split ring or di-electric resonators inside a waveguide operating below cut-off[3], [4]. Dielectric and loop gap resonators, used as probesin electron spin resonance spectroscopy, are usually housedin a shield or in a cavity [5]–[9]. In all these situations, thefrequencies and fields of the resonators are affected by thepresence of nearby conductors.

Taking the influence of the conducting walls on the res-onators was done byab-initio approaches. For example this

Manuscript received November 6, 2014; revised May 8, 2015. SMMacknowledges the Natural Sciences and Engineering Research Council ofCanada for financial support in the form of a discovery (operating) grant. SYNis grateful for the financial assistance from the Universityof New Brunswickin the form of pre-doctoral teaching and research assistantships.S Y. Elnaggar is currently with the School of Engineering andInformationTechnology, University of New South Wales, Canberra, ACT, Australia.Richard J. Tervo is with the Department of Electrical and Computer Engi-neering, University of New Brunswick, Fredericton, NB, Canada.Saba M. Mattar is with the Chemistry Department, Universityof NewBrunswick, NB, Canada.Corresponding author: Saba M. Mattar (email: [email protected] ).

was done through out the derivation of the eigenmodes oftwo stacked dielectric resonators [10]–[12] which was basedon the Cohn model [13]. For other resonators, such as loopgaps and split rings, analytical models are unavailable or verycomplex and one usually resorts to numerical techniques. Theeffect of the walls of a waveguide on a split ring resonatorwas numerically and experimentally investigated using thecoupling between the resonator and its mirror images. It wasnoted that a split ring resonator trapped inside a waveguidecouples with an infinite array of images [14].

To find the modes of an arbitrary number of coupledresonators, a Coupled Mode Theory in the form of an eigen-value problem was derived. The solution of the eigenvalueproblem determines the coupled frequencies and fields [15].Using the coupled mode equations, the eigenmodes of anelectron spin resonance probe, which consists of a dielectricresonator inserted in a cavity center were found [7]. Other fielddependent parameters, such as the quality factor, the fillingfactor and the resonator efficiency, were calculated [8], [9].The same approach was generalized to explain the behaviorof double split ring resonators. The approach was consideredto be a hybridization scheme which is the electromagneticanalog of Molecular Orbital Theory. The determined modesare proven to obey the energy conservation principle and hencethe eigenvalue equation was called Energy Coupled ModeTheory or ECMT for short [15].

The fields of the coupled system, obtained by ECMT, arethe linear superposition of those of the uncoupled modes.Therefore, an implicit assumption is that the tangential com-ponents of the electric (magnetic) fields of the uncoupledfields are negligibly small near Perfect Electric Conductors(Perfect Magnetic Conductors). Otherwise, the coupled modesseverely violates the boundary conditions. In the currentarticle, the method of images is combined with ECMT toextend its domain to cover situations where resonators are inthe proximity of conducting surfaces. Hereafter, the method isnamed ECMT-I. To show the validity of the method, two casesare theoretically and numerically studied. The first is thatofa dielectric resonator interacting with an enclosing cavity. Inprevious work, the interaction between the dielectric resonatorand the cavity was studied where the dielectric resonator wasinstalled in the cavity center and hence its field values aresmall near the walls [7], [8]. In the current work, the effectofthe walls on the coupled frequencies is taken into account.The second case discusses resonators sandwiched betweenparallel plate conductors. This broad class of problems coversthe presence of resonators inside waveguides which can findapplications in filters and metamaterial based devices. It also


proposes an analytical apparatus to study the behavior ofcoupled resonators, such as inductively coupled resonant coils,in the presence of conductors [1], [2].

This paper is partitioned as follows: Section II discussesthe method of images, the concept of theimage resonator andits integration with the coupled system. Expressions for thefrequency shifts for a dielectric and split ring resonatorsnear aconducting plane are derived. Similarly, the case of resonatorssandwiched between two conducting plates is discussed. Insection III two configurations are numerically solved and com-pared with finite element simulations and other independentanalytical models. Using ECMT-I, the numerical results areexplained based on the uncoupled parameters (frequencies andfields) which render the analysis simpler and more intuitive.Finally, the conclusions follow in Section IV.

II. T HEORY

When losses are negligibly small, the frequency and fieldsdistributions of the modes of a resonator can be obtained bysolving the eigenvalue problem

∇×1

µr(r )∇× E(r) =

ω2

c2ǫr(r)E(r), r ∈ D, (1)

over the domain boundary∂D, subject to the suitable boundaryconditions. HereE is the electric field,ǫr(r) is the relativepermittivity, µr(r ) is the relative permeability andω = 2πf isthe resonant angular frequency. The boundary can be dividedinto three main types: (1) Perfect Electric Conductor (PEC)where the tangentialE component vanishes, (2) Perfect Mag-netic Conductor (PMC) where the tangentialH componentvanishes and (3) Infinity (radiation) where the magnitude ofthe fields are zero (E, H → 0 wheneverr → ∞).

Consider the structure shown in Fig. 1(a), which representsa resonator in the proximity of a conducting plane (eitherPEC or PMC). If the upper half space is only considered, Fig.1(b) represents an equivalent problem. Both boundary valueproblems give the same results in the upper half space. Thisequivalency is used in full-wave solvers to reduce the problemspace. In this case, the original system is the one in Fig. 1(b)and the introduction of the suitable boundary (symmetry plane)cuts the space in half and hence reduces the computationaltime.

Energy Coupled Mode Theory is also in an eigenvalueproblem form. However unlike (1), the operator relies on thefields of the interacting resonators and hence the problemspace is limited to the number of resonators. Thus, one canstart from the configuration given in Fig. 1(b) to determinethe mode of interest of the structure depicted in Fig. 1(a).In general, the two modes of the resonators in Fig. 1(b)interact resulting in a symmetric and an anti-symmetric modes.Depending on the conductor type (PEC or PMC), each modewill represent a solution to the system in Fig. 1(a). Once themodes of a resonator/conducting plane system are determined,ECMT can be re-applied to study their interactions with othernearby resonators. It is crucial to note that the solution ofthestructure in Fig. 1(a) is a subset of the one shown in Fig. 1(b).The right solution is singled out by the theory of images.

z

xy

z

xy

(a) Original Structure

(b) Equivalent Structure

Magnetic

Moment

Magnetic

Moment

Conductor

Place of

Conductor

Fig. 1. A typical structure of a dielectric resonator interacting with aconducting plane. (a) shows the original system. (b) illustrates the equivalentstructure where the plane is replaced by an equivalent imageresonator.

The orientation of the resonator with respect to the con-ducting plane is crucial in determining the right coupledmode behavior. For example, consider a cylindrical dielectricresonator close to a PEC as shown in Fig. 1(a). The dielectricresonator’sTE01δ mode behaves like a magnetic dipole whereits direction is shown in Fig. 1 [13]. According to the ImageTheory, the real and image resonators have anti-parallel dipolemoments, which are both normal to the conducting plane [16].Therefore, the electric fields of both resonators are180◦ out ofphase which guarantees that the tangential component of theelectric field vanishes on the mid plane. Using Coupled ModeTheory, the coupling coefficientκ is found to be proportionalto∫

VDRE1 ·E2 dv, where the integration is carried over one of

the dielectric resonators volumeVDR [24] . SinceE1 andE2are anti-parallel,κ is negative. The frequency of the symmetricmode is [7]

f++ = f0√

1− κ(2s). (2)

Here f0 is the frequency of the dielectric resonator in freespace (hereafter called the uncoupled frequency) andκ(2s)is the coupling coefficient at double the distance between theresonator and the conducting plane. Thus, the effect of thePEC is to increase the frequency of the mode.

On the other hand, if a split ring resonator is broughtclose to a PEC wall as shown in Fig. 2, image theorydictates that the magnetic dipole moments of the resonatorand its image are parallel to each other and to the conductingplane as shown in Fig. 2 [16]. Again, the frequency of thesymmetric mode is given by (2). Nowκ is proportional to∫

V(µ0H1 · H2 − ǫ0E1 · E2) dv [24]. Because the magnetic

moments are parallel, the first term is positive. Moreover topreserve the parallelism of the magnetic dipoles, the electricfields, concentrated in the vicinity of the gaps are anti-parallel


(i.e,∫

VE1 · E2 dv ≤ 0) and henceκ is positive. Thus, the

resulting mode has a frequencyf++ which is lower than theuncoupled frequencyf0. This interpretation, based on coupledmode theory, explains why the frequency decreases when thegap of the split-ring gets close to the waveguide wall. Zhanget al. attributed this to the increase in gap capacitance [14].

Real

Image

Magnetic

Moment

Magnetic

Moment

Fig. 2. A split-ring resonator is coupled to its image in the symmetric mode.

As already mentioned, resonators sandwiched between twoparallel plates are not uncommon. For example, metamaterialscan be realized inside a waveguide by inserting dielectricor split ring resonators. Each resonator forms a unit cellwhich interacts with its neighbors and with the mirror images.The cylindrical cavity resonance technique, a method usedto measure the dielectric properties of a material, is anotherexample [17]. In this method, the dielectric sample is placedinside a cylindrical cavity which has a diameter at least doublethat of the sample. The unloadedQu, the central frequency, the3dB bandwidth and the transmission coefficient (S21) are usedto determine the sample’s loss tangent (tan δ). For sampleswhich have relatively low permittivity values, the walls ofthe cylindrical cavity can have a significant effect. Moreover,the cavity modes can interact with the dielectricTE01δ mode[7], [8]. The efficiency of Wireless Power Transfer via stronginductively-coupled resonant coils can also be affected bymetallic planes [1], [2]. To understand the behavior of suchcomplex systems, it is imperative to model the simplest cases:One and two resonator(s) installed between two parallel plates.

For the case of a resonator sandwiched between two con-ducting planes as shown in Fig. 3, the equivalent coupledmode structure is an array of infinite number of images[14]. Ignoring the coupling induced frequency shift terms,theeigenvalue problem of the coupled system can be written as[15].

ω20

1 −κ12 −κ13 · · · −κ1n · · ·−κ21 1 −κ23 · · · −κ2n · · ·

......

. . ....

......

−κm1 −κm2 · · · 1 −κmn · · ·−κn1 −κn2 · · · −κn(n−1) 1 · · ·

......

......

.... . .

a = ω2a

(3)

a = (a1 a2 · · · )⊤ .

Hereω0 is the angular frequency of the uncoupled resonator,

κij is the coupling coefficient between theith and jth res-onators,ai is the expansion coefficient of theith resonator andfinally ω is the frequency of the coupled system.κij can becalculated using the overlap integrals [7], [24] or the frequencysplitting formulaκij =

(

ω2+− − ω2++

)

/(

ω2+− + ω2++

)

, whereω++ andω+− are the angular frequencies of the symmetricand anti-symmetric modes, respectively. As will be shown,both methods were used in the current article. It is worthnoticing that although (3) has infinite number of eigenvectors,only one of them satisfies the boundary conditions at theposition of the conducting plane. Because the relative phasesof the images are taken into account in constructing the matrixin (3), this eigenvector is always the first one, where itscomponents have the same sign.

Fortunately, the coupling coefficient decreases sharply asthe distance between the resonators increases. Therefore,thecoupling coefficient between adjacent neighbors is consideredonly. Consequently, (3) simplifies to

ω20

1 −κ 0 · · · 0 · · ·−κ 1 −κ ... 0 · · ·...

.... . .

......

...0 · · · −κ 1 −κ · · ·0 0 · · · −κ 1 · · ·...

......

......

. . .

a = ω2a. (4)

s

Fig. 3. One resonator between two conducting planes. An equivalent system:Instead of the conducting planes, an infinite chain of image resonators can beplaced all separated by 2s.

In (4) the subscripts ofκij were dropped since all couplingcoefficients between nearest neighbors are equal. Since thematrix is symmetric and tri-diagonal, the first eigenvalueapproaches [18]

ω2 = ω20(1− 2κ). (5)

The corresponding eigenvector is [18]

am = limN→∞

sin(N −m+ 1)π

2N + 2, m = 0, 1, 2, ...N. (6)

m is the image order which, as shown in Fig. 4, is zero forthe original resonator. Image pairs are labelled accordingtotheir position from the original resonator.N is the number of


Fig. 4. The resonator sandwiched between two conducting planes. For PECwalls, the phases of the images are alternating. For the PMC walls, all theimages have the same phase of the original resonator. The left hand side showsthe magnitude of theam coefficients compared toa0 which is normalized tounity.

image pairs which tends to infinity when all pairs (orders) aretaken into account. In (6) the eigenvectors are normalized suchthat the coefficient of the original resonator (Zeroth order) a0is set to 1. It is worth noticing that the lower the image order(closer to the real resonator), the more its contribution tothesystem behaviour (higheram value).

In the current article, the following notation is used torepresent resonators sandwiched between conducting planes:The resonator is represented by ”R” and PEC wall as ”|”.Because of their resemblance with high impedance surfaces[19], ”❁” or ”❂” are used to represent a PMC wall. Forexample, a resonator sandwiched between two PMC walls isdenoted by ”❁ −R− ❂”. Two resonators sandwiched betweentwo PEC walls are represented by ”| − R −R− |”.

For a resonator sandwiched between two PEC walls (|−R− |), the magnetic dipole moment of any two consecutiveresonatorsp and p + 1 are anti-parallel i.e.κ < 0. Thisguarantees that the electric field vanishes on the walls. Onthe contrary, all moments are parallel (κ > 0) for ❁ −R− ❂.

The total electric fieldE is the vector sum of the fields,Emof all orders. Thus

E (r) = E0 (r) + limN→∞

N∑

m=1

amEm (r) , (7)

where

Em (r) = (−1)m+1 [E0 (r − ẑmd) + E0 (r + ẑmd)] (8)

for | −R− | and

Em = [E0 (r − ẑmd) + E0 (r + ẑmd)] (9)

for ❁ −R− ❂. d is distance between the two planes.

III. R ESULTS AND DISCUSSION

A. Dielectric-Cavity Interaction

The coupling between a dielectric resonator and a cavitywas previously studied [7], [20]. It was shown that inserting

a dielectric resonator in a cavity improves the signal to noiseratio of an electron spin resonance probe [9]. Adding anotherdielectric resonator, linearly changes the frequency of thedesired mode as a function of the separation distance betweenthe two dielectric resonators. Thisnice linear response wasused to tune the frequency over a wide range. However oncethey are close to the cavity walls, the benefit of using thedielectric resonators is lost and thus putting a limitationon themaximum separation distance and consequently the frequencyspanning range. [21]. The apparentloss of the dielectric effecton the probe performance was explained by using boundaryconditions arguments where the fields diminish due to thesimultaneous presence of perfect electric and perfect magneticwalls. In the analysis that was carried out in [7]–[9], only onedielectric resonator was used and it was installed at the cavitycenter. Hence, it was far away from the cavity walls and itsinteraction with the images was negligibly small. Therefore itwas justifiable to calculate the coupling coefficient betweenthe dielectric and the cavity using the dielectricTE01δ andthe cavityTE011 modes only [2]. The two modes couple togive two modes:symmetric and anti-symmetric. However, ifthe dielectric was brought close to one of the walls, as shownin Fig. 5, the coupling with the image becomes significant andcannot be ignored.

In this subsection, ECMT-I will be used to calculate theeffect of the cavity walls on the resonant frequencies of thesymmetric and anti-symmetric modes. A dielectric resonator(ǫr = 29.2, diameter = 6 mm, height = 2.65 mm) is insertedinto a cylindrical cavity (hiameter = height = 4.1 cm). Thesedimensions guarantee that the resonant frequencies of theTE01δ andTE011 modes are equal (9.7 GHz). For differentdistancess, the frequencies of both the symmetric and the anti-symmetric modes are calculated using ECMT [15], ECMT-Iand compared with HFSSR© (Ansys Corporation, Pittsburgh,PA, USA) eigenmodes simulations. Figs. 5(b) and 5(c) presenta conceptual view of the hybridization of the cavityTE011 andthe dielectric mode. In Fig. 5(b) when the effect of the imageisneglected, theTE011 mode couples with the dielectricTE01δmode. As a result, two coupled modes, the symmetric and anti-symmetric, arise. Fig. 5(c) shows a two stage hybridizationscheme. Here the dielectricTE01δ mode couples with itsown image, which is180◦ out of phase. Consequently, theresulting mode, theTE++01δ , then couples with the cavityTE011 mode yielding two coupled modes: symmetric andantisymmetric. The coupling between the cavity on one handand the dielectric or the dielectric-image pair on the otherisdetermined as in Ref. [7] using the overlap integrals of theelectric fields inside the dielectric material. Here, the fieldsexpressions of theTE01δ mode are obtained using the Cohnmodel which, as shown in Ref. [7], provides accurate fieldvalues inside the dielectric material. Because diffraction is nottaken into account, the accuracy of the electric field is lostoutside the dielectric material where the fields are evanescent[23]. Thus the Cohn model does not provide the accuracyneeded to compute the frequency of the dielectric-image pairwhich, unlike the coupling between the dielectric resonatorand cavity, strongly relies on the evanescent fields. Hence,HFSSR© eigenmode solver is used to determine the frequency


of the dielectric-image pair. This value is then plugged intothe ECMT eigenvalue problem together with the electric fieldexpression of the dielectric-image pair which is given byE′ = (E(x, y, z)− E(x, y, z − 2s)) /

√

(2), where E is theelectric field of theTE01δ mode.

(a)

(b)

(c)

Antisymmetric

Symmetric

Antisymmetric

Symmetric

Fig. 5. (a) A dielectric resonator allowed to move along its common axiswith the cavity. The holder is of low loss and low permittivity material. Itis therefore not shown. Conceptual view of the hybridization between themodes: (b) When the image is not taken into account. (c) When the image istaken into account.

Fig. 6. Coupled Frequencies calculated using ECMT, ECMT + Image andHFSSR©. (a): anti-symmetric mode. (b): Symmetric mode. In (a), theinsetsdepicts descriptive scenarios of the position of the dielectric resonator in thecavity .

Fig. 6 shows how the frequencies change as a function

of s. As the figure presents, incorporating the method ofimages gives more accurate results particularly for the anti-symmetric mode. This can be explained by noting that in orderto calculate the frequencies, ECMT uses theTE01δ mode,while ECMT-I uses theTE++01δ mode which, by referring to(2) and noting thatκ < 0, has a higher frequency value.When the dielectric is close to the wall, the electric fieldof the cavity’sTE011 mode is small (E ∝ cos πzd ) [8]. Thecoupling coefficientκ is the overlap of the electric fields ofthe cavity and the dielectric resonator and hence is small closeto the wall. This means that the modes tend to decouple. Thesymmetric mode tends to the cavityTE011 mode [6] and theanti-symmetric mode approaches the dielectricTE01δ mode(ECMT used) orTE++01δ mode (ECMT-I used). When thedielectric resonator is very close to the wall, the frequencyof the TE++01δ can be significantly higher than that of theTE01δ. This explains why ECMT completely fails to explainthe behavior of the anti-symmetric mode depicted in Fig. 6(b).

Similarly, the minimum of the anti-symmetric mode in Fig.6(b) can be explained by noting that it is the result of twoopposing actions:1- The decrease in the frequency due to thecoupling with the cavity and2- Increase in frequency due tothe coupling with the image. It is expected that the dielectricTE01δ mode couples with other cavity modes. However, thecontribution of their quantitative effect is very small [7]. It isalso expected that the discrepancy between HFSS and ECMT-I results will decrease if a more sophisticated model for thedielectric mode was used.

B. Resonator between two Parallel plates

Because an analytical model, based on the Cohn model,is already available, a dielectric resonator is considered. Re-gardless of the limitations of the Cohn model, it providesdeep insight into the coupling mechanism and allows thefinding of closed form expressions. However, the generalanalysis and findings are valid for any type of resonators.The coupled (coupling with the images) frequencies and fieldsare calculated and compared to the values obtained using theanalytical model. As mentioned in Section II, the resonatorsymmetrically couples with the stack of infinite images. Thecoupled, denoted byTE+++···01δ , has a frequency of

fTE+++···01δ

=√

1− 2κ(d)f0,

{

κ < 0 for | −R− |κ > 0 for ❁ −R− ❂

(10)Here d is the separation distance between the conductingplanes. The frequency given by (10) is calculated for differentseparations. The results are then compared with the valuesobtained using the analytical model [11]. For the sake ofcompleteness, the analytical models of both| −R− | and❁ −R− ❂ are presented in the appendix. The frequency val-ues are plotted in Fig. 7 for a dielectric resonator (ǫr = 29.2,diameter = 6 mm, height = 2.65 mm) inserted between twoplates which are separated by a distanced = 3.15−8.65 mm.

Although (10) is much simpler than the transcendental Eqs.(18) and (19), ECMT-I gives very accurate values whend >4 mm (Refer to Fig. 7). Unlike the analytical model, ECMT-I


(a)

(b)

Fig. 7. Frequency of the resonant mode for a dielectric resonator sandwichedbetween two conducting planes. (a): PEC. (b): PMC.

separates the effects of the walls from the resonator’s uncou-pled frequency (f0). The contribution of the walls is lumpedin κ. Ford < 4 mm, the discrepancy between the two methodscan be attributed to:1- The frequency expression (2) wasdetermined based on the nearest neighbour interactions only.2- When the resonators are close, Coupling Induced FrequencyShifts are not negligible. Coupling Induced Frequency Shiftschange the on-diagonal terms and hence shift the frequencies[15], [22]. This shift in frequency can be attributed to theproximity effects of the other resonators. The presence ofother resonators has the effect of changing the equivalent selfcapacitance and inductance and thus changing the on-diagonalterms. This behaviour is different from the coupling behaviourwhich introduces mutual capacitances and/or inductances.

In the following, expressions for the coupled electric fieldE will be derived and compared to the field expression (22).The electric field of theTE01δ mode, centered at the origin,can be written as [23]

E0 = −kρJ0(kρr){

cos k1z |z| <h2

cos k1h2 e

−k2(z−h

2) |z| > h2

φ̂. (11)

E0 is in the azimuthal direction̂φ. h is the dielectric height,kρ is the radial wave number,k2 is the axial free spaceattenuation factor andk1 is the propagation constant insidethe dielectric. Using ECMT-I, the total electric fieldE can bedetermined by (7) and (8) or (9). Equation (7) separates thefields of the uncoupled resonatorE0 from the contributionsof the conductors{Em}∞m=1. For the dielectric resonator, thefield of themth pair, Em, is found to be

Em = 2·(−1)m+1kρJ1(kρr)e

−k2(md−h/2) cos(k1h/2) cosh(k2z)φ̂(12)

for | −R− | and

Em = −2 · kρJ0(kρr)e−k2(md−h/2) cos(k1h/2) cosh(k2z)φ̂ (13)

for ❁ −R− ❂. The expressions (6), (12) and (13) are substi-tuted back into (7) and the field correctionÊ =

∑∞m=1 amEm

simplifies to

Ê = 2kρJ0(kρr) cos(k1h/2) ·

(

e−k2(d−h/2)

1 + e−k2d

)

· cosh(k2z)φ̂ (14)

for | −R− | and

Ê = −2kρJ0(kρr) cos(k1h/2)·

(

e−k2(d−h/2)

1− e−k2d

)

·cosh(k2z)φ̂ (15)

for ❁ −R− ❂.Expressions (14) and (15) are calculated overz ∈ [0, d/2]

and compared with the analytical model ((22)-(26)). The re-sults are plotted in Figs. (8) and (9). It is apparent that ECMT-I gives very accurate values ofE. Other than this, ECMT-I separates the conductors’ effects,̂E from the uncoupledfield, E0. Moreover, the parametersk1 and k2 depend onthe uncoupled resonance frequency and is independent of theseparation distanced. This means that̂E explicitly depends onthe uncoupled parameters. Its dependency ond is lumped inthe factors between brackets in (14) and (15). Separating thedifferent effects and contributions, renders the analysissimpler(compare to (22)-(26)).

Fig. 8. Dielectric resonator sandwiched between two PEC plates: Thenormalized electric field calculated overz ∈ [0, d

2]. The distance between

the two platesd is (a) 3.15 mm, (b) 8.65 mm.

Whend is very small (k2d≪ 1), the field of theTE++++...01δ

deviates significantly from that of theTE01δ mode. Based onECMT-I this can be explained by noticing that the smallerthe spacingd between the conductors, the larger the effect


Fig. 9. Dielectric resonator sandwiched between two PMC plates: Thenormalized electric field calculated overz ∈ [0, d

2]. The distance between

the two platesd is (a) 3.15 mm, (b) 8.65 mm.

of the mirror images. From Figs. 8 and 9, it is clear that ford = 3.15 mm the PMC walls have a more profound effect.This is due to the fact that, for PMC walls,all the images’modes are in phase with the uncoupled mode (Zeroth order).Thus, the effect of̂E is to enhanceE0. On the contrary, thepresence of the PEC walls, as shown in Fig. 4, alternates thephase between consecutive resonators, which makes the effectof the walls less profound. Quantitatively, the effect of thewalls appears in the terms inside the brackets of (14) and (15).For | −R− | this term approaches1/2, for smalld values. Onthe other hand, the term gets substantially large (approaches1/k2d) when PMC walls are present.

The last system examined here is that of two dielectricresonators sandwiched between two PEC walls (| −R−R− |)and is shown in Fig. 10. In general, the two resonators’ modescouple which result in symmetric and anti-symmetric modes.Because the two resonators are independent, the phase relationbetween the two real resonators modes (denoted by 1 and 2 inFig. 10(a)) can be arbitrarily chosen. In the following analysis,the two modes were chosen to be in phase. As in the caseof one resonator sandwiched between two conducting plates,the number of images is infinite. Due to the presence of tworesonators, the1st image order consists of four resonators. Thephase relations of the images are dependent on the walls type.The arrows in Fig. 10(b) show the relative phase when PEC

walls are used. Unlike the approximation used to determinethe modes of| −R− |, the full eigenvalue problem [15] issolved up to the1st order images only. This is a 6×6 systemresulting in 6 eigenvectors. However due to the presence oftwo independent resonators, only two eigenvectors correspondto the physical system depicted in Fig. 10(a). Dictated by themethod of images, the sign of thea coefficients of the imageresonators always follow the real ones. (By referring to Fig.10(b), the sign of thea coefficients of 1’ and 1” follow that of1. Similarly,the sign of thea coefficients of 2’ and 2” followthat of 2). Thus, one ends up with two eigenvectors whichhave the sign relations shown in Fig. 10(c). In Fig. 3d2, thedistance between the resonators and the PEC walls, is fixedto 3 mm andd1, the distance between the two resonators,changes from 1 to 6 mm. As shown by Fig. 11 shows thefrequency of the both the symmetric and the anti-symmetricmodes. The frequency is calculated using ECMT-I and theanalytical model obtained in [10]. Because the full eigenvalueproblem in Ref. [15] is solved, the coupling between all sixresonators and the Coupling Induced Frequency shifts termsare automatically taken into account. Although the first orderresonators are only used, ECMT-I can accurately calculate thefrequencies of both coupled modes. This is true even whend1is small. This suggests that taking into account the CouplingInduced Frequency shift terms and the coupling with relativelyfar resonators, improve the results.

It is worth noticing that provided that the fields and fre-quencies of the uncoupled resonators are known or estimated,ECMT-I can be applied to more complex systems where ananalytical model is not possible or very complex. For exampleinstead of using dielectric resonators, split-rings, loopgaps orcapacitively loaded coils can be used.

Fig. 10. (a): Two dielectric resonators separated by a distance of d1 mmapart. Both resonators ared2 away from PEC walls. (b): Equivalent coupledresonators images. (c): The phase shift of the symmetric andthe anti-symmetric modes.

IV. CONCLUSION

Using the method of images, the domain of ECMT wasextended to cover cases where resonators are in the proximityof conductors. Because ECMT is in the form of an eigenvalueproblem in the frequency domain, the extension can be easily


(a)

(b)

Fig. 11. Frequency of| −R−R− | whend2 = 3 mm. (a): the symmetricmode. (b): the anti-symmetric mode.

achieved. It was shown that the type of conductors (PEC orPMC) and the orientation of the resonators with respect tothem determine the phase of the images and consequently thesign of the coupling coefficientκ. This in turn determineswhether the resonant frequency is increased or decreased.To verify the validity of the new approach, two cases weretheoretically and numerically analyzed.

APPENDIX

For the| −R− | and❁ −R− ❂ structures, the modes canbe determined after solving the auxiliary Helmholtz’s equation

∇2ψ + k20ψ = 0, (16)

wherek0 = ωc , c is the speed of light. One is interested inthe azimuthal independent symmetric solution of (16). Thegeneral solution is

ψ = J0(kρr)

{

cos k1z |z| ≤h2

a sinhk2z + b coshk2z |z| >h2

(17)

Herekρ = 2.405rd , rd is the dielectric radius,k1 =√

ǫrk20 − k2ρ

andk2 =√

k2ρ − k20 .

For TE01δ like modes,Eρ = 0, Hφ = 0, Eφ =∂ψ∂ρ

andHρ = 1iωµ0∂2ψ∂ρ∂z

. The continuity of the fields across theboundaries can be written as(

cos(k1h/2) − cosh(k2h/2) − sinh(k2h/2)k1 sin(k1h/2) k2 sinh(k2h/2) k2 cosh(k2h/2)0 cosh(k2d/2) sinh(k2d/2)

)(

1ab

)

= 0

(18)for | −R− | and(

cos(k1h/2) − cosh(k2h/2) − sinh(k2h/2)k1 sin(k1h/2) k2 sinh(k2h/2) k2 cosh(k2h/2)0 sinh(k2d/2) cosh(k2d/2)

)(

1ab

)

= 0

(19)for ❁ −R− ❂. The resonant frequency is determined from(18) and (19) by solving the following transcendental equa-tions

tan(k1h/2) =k2k1

·1

tanh(k2(d− h)/2)(20)

for | −R− | and

tan(k1h/2) =k2k1

tanh(k2(d− h)/2) (21)

for ❁ −R− ❂.The electric field is found to be

E = −kρJ1(kρr){

cos k1z |z| ≤h2

a sinh k2z + b coshk2z |z| >h2

(22)For | −R− |

a =tanh(k2d/2)

tanh(k2d/2)− tanh(k2h/2)·cos(k1h/2)

cosh(k2h/2)(23)

and

b =1

tanh (k2d/2)− tanh(k2h/2)·cos(k1h/2)

cosh(k2h/2). (24)

For ❁ −R− ❂

a =cos(k1h/2)

cosh(k2h/2)− sinh(k2h/2) tanh(k2d/2)(25)

and

b =cos(k1h/2)

sinh(k2h/2)− cosh(k2h/2) coth(k2d/2). (26)

ACKNOWLEDGMENT

We would also like to acknowledge CMC Microsystems forproviding HFSSR© that facilitated this research.

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Sameh Y. Elnaggar has a B.Sc (Electrical Engi-neering) and M.Sc. (Electrical Engineering) fromAlexandria University and a Ph.D. (Electrical andComputer Engineering) from the University ofNew Brunswick in 2000, 2006 and 2013, respec-tively. His recent research interests are in com-putational electromagnetic, interferometry, magneticresonance, quasi-optics, metamaterials and wirelesspower transfer. He is currently with School of Engi-neering and Information Technology, University ofNew South Wales, Canberra.

Richard J. Tervo has a B.Sc. (Physics) and M.Sc.(Experimental Nuclear Physics) from McMasterUniversity and a Ph.D. (EE) from Laval Universityin 1980, 1982 and 1988, respectively. He joined theDepartment of Electrical and Computer Engineeringin 1986. His research interests include digital sys-tems, instrumentation, communications and signalprocessing.

Saba M. Mattar has a B.Sc. (Chemistry andPhysics) from Alexandria University, M.Sc. (Solidstate Science) from The American University inCairo and a Ph.D. (Chemistry) from McGill Uni-versity in 1968, 1974 and 1982, respectively. Hejoined the Chemistry Department, University of NewBrunswick in 1986. He is presently an HonoraryResearch Professor at UNB. His interests are the-oretical and experimental magnetic resonance, trap-ping of reactive intermediates, electronic structureof molecules, chemical physics, new metamaterials,

spectroscopy and instrumentation.

ieee transactions on microwave theory and ...samehelnaggar.ca/pdf/papers/tmtt2.pdftheory, the...

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