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Coupling Through Tortuous Path Narrow SlotApertures Into Complex Cavities

Russell P. Jedlickal, Steven P. Castillol, Larry K. Warne2

Klipsch School of Electrical and Computer EngineeringNew Mexico State University

Las Crgces, NM 88003

Sandia National Laboratories2Albuquerque, NM 8’i185

July 9, 1999

Abstract

A hybrid l?EM/MoIvI model has been implemented to computethe coupling of fields into a cavity through narrow slot apertures hav-ing depth. The model utilizes the slot model of Warne and Chen[23] - 129]which takes into account the depth of the slot, wall losses,and imhonogeneous dielectrics in the slot region. The cavity interi-or is modeled ~vith the mixed-order, covariant-projection hexahedral’elements of Crowley [32]. Results are given sho~~ing the accuracyand generality of the method for modeling geometrically complex slot-cavity combinations.

DISCLAIMER

This report was prepared as an account of work sponsoredby an agency of the United States Government. Neither theUnited States Government nor any agency thereof, nor anyof their employees, make any warranty, express or implied,or assumes any legal liability or responsibility for theaccuracy, completeness, or usefulness of any information,apparatus, product, or process disclosed, or represents thatits use would not infringe privately owned rights. Referenceherein to any specific commercial product, process, orservice by trade name, trademark, manufacturer, orotherwise does not necessarily constitute or imply itsendorsement, recommendation, or favoring by the UnitedStates Government or any agency thereof. The views andopinions of authors expressed herein do not necessarilystate or reflect those of the United States Government orany agency thereof.

DISCLAIMER

Portions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.

I

t

1 Introduction

Electromagnetic coupling can adversely impact a multitude of applicationsranging from modern telecommunication systems to sophisticated electronicwarfare equipment. The primary coupling issues are electromagnet ic compat-

ibility and interference, either intentional or unintentional. The functionalityof the systems involved, which typically consist of numerous subsystems op-erating concurrently, can be characterized in terms of their susceptibility,vulnerability and survivability in the electromagnetic environment in which

they are expected to operate.“Todesign effectively hardened, complex operational systems, it is critical

to characterize the worst case electromagnetic coupling, that is, the upper

bound of the field penetrating a system and the distribution of the energywithin that complicated system. System performance may be compromised

by a variety of penetration mechanisms. In this study, it is assumed thatthe shield is constructed from good electrical conductors so diffusion can beignored and isolated so conductive effects are negligible. Back-door couplingoccurs through inadvertent cracks and gaps, createci by warping or bowing,at the mechanical interfaces between pieces of a shielded enclosure. Theseback-door phenomena are much less predictable or controllable, and so meritparticular interest and study.

Coupling through unforeseen apertures such as the tortuous-path, lapped

seam depicted in Figure 1 into conducting cavities is the major thrust of thiswork. To model realistic coupling problems, it is necessary to develop a model

which includes a three-dimensional representation of the slot/cavity config-

uration. Specifically. such a model should incorporate narrow slot apertureshaving depth. loss and gaskets. backed by arbitrarily-shaped cavities, filledwith inhomogeneous, lossy dielectrics. The model should also incorporate

the effect of the finite conductivity of the enclosure walls.The conceptual formulation for the cavity-backed aperture has been pre-

sented in detail by Barrington and Mautz [1] as well as Butler, Ramat-Samiiand J[ittra {2]. The generalized network formulation presentation in the for-mer is an extension of the, work published by Harrirgton in a 1968 book [3].In both papers, the problem is segmented into an interior and an exterior

region. Equivalent problems are formulated for the two regions and coupledthrough the boundary conditions in the aperture. In these formulations: theinterior and exterior are generally considered to be homogeneous and, ex-cept for the aperture, the regions are isolated by good electric conductors.

<‘

l~:ang, Barrington ant! IIa.utz considered not only Cransmlsslon into, but al.

so scatterin: from, conducting bodies ~vith arbitrary apertures [1]. In thiswork, they formulateci the three-dimensional problem for arbitrarily shaped

bodies and apertures: however, only zero thickness slots were considered, Inaddition, the interior region did not include inhomogeneous media since an

integral equation method was used.In 1988 Taflove et al. applied a FDTD technique to the analysis of narrow

slots and lapped joints in a two-dimensional conducting screen [.5]. This paperrecognized the importance of narrow slot apertures, in particular the tortuouspaths associated with lapped joints.

Following the work of Merewether [6], Riley and Turner applied the FDTD

technique to the analysis of a cavity-backed aperture loaded by boxes andterminated wires. Their hybrid thin slot algorithm [7] advances the earlierwork of Gilbert and Holland [8]. Later in 1990, they incorporated the model

of Warne and Chen into an FDTD analysis of a narrow slot aperture having

depth and losses into a regularly shaped cavity [9].The demand for solutions of aperture/cavity problems has brought progress

in numerical techniques and their application. Historically, finite-differencetechniques have received greater attention in e!ectromagnetics; the use of the

finite-element method has been more recent. Subsequently, hybrid techniquesusing boundary inteb~al/modal expansion techniques, method of moments(MOM) /finite-difference/time-domain methods as well as finite-element/boundaryintegral formulations have become commonplace [10], (20]. Non-zero thick-ness was incorporated by using finite-difference (FD) techniques or finite-element methods (FIN) to solve the interior problem increasing the compu-

tational size of the problem considerably. In recent years, the finite-elementmethod has proven to be a useful ENIC prediction tool [21], [22].

The research performed to date demonstrates the maturity of hybrid nu-merical methods in eiectromagnetics. The only slot/cavity work that in-

corporated a practical siot model was the effort of Riley and Turner [7].However. their finite-difference implementation was not conducive to curvedsurfaces and neither included wall loss nor dielectric inhornogeneities. Thus,another formulation and solution are required to include additional featuresin the model of a real world system.

In this paper, we present a hybrid numerical technique which can be usedto calculate the amount of ener~q penetrating a system through a narrowslot aperture with depth and? at the same time, to determine its disposi-tion; that is, of the power available in an impinging wave, how much is lost

3

during transmission through the aperture and dissipated in the walls and

dielectrics occupy-in: the interior, and how much is delivered to loads, or

Components. ~vitlill the configuration. Specifically. t his involves comput in:the electromagnetic coupling through narrow slot apertures into complexcavities (arbitrarily shaped, with an inhomogeneous dielectric loading). The

slot models of Warne and Chen [23] - [29] are coupled to a three-dimensionalvector FEM formulation for the interior cavity region. W_arne and Chen’s

slot model avoids tile need to resolve a tortuous path slot with a very finefinite-element or finite-difference mesh while being general enough to includedielectrics in the slot interior, slot wall losses, and bolt loading across the slot

[30]. In addition, the slot model can be extended to include very deep slotsin which a resonance is excited in the depth direction. A comparison is made

betw-een experimental and numerical results for several different slot-cavitycombinations to validate the model.

2 Coupled Slot-Cavity Model

Consider the slot geometry in Figure 2. The depth through the slot may bea tortuous path more characteristic of overlapped joints. Warne and Chen’smodel assumes that

l,~Ot,~ >> pO > d

‘The slot N-ails are assumed to have a large but finitethey can be characterized by the usual perturbation:

)

(1)

conductivity so that

(2)

Consider Figure 3. The governing integro-differential equat.icm is given as

1 d21,n ~ &=_ Hy ’(z)= -H? (~) (3)Hz- (P;!z) + -@-~ – Po

4 ~

where

*

ancl

The half length of the slot is h, and pOis the radial distance on the illuminated

side. The transmission-line coefficient terms are

The parametersrameters for the

Zpo = Zi,tt + j~~p. (6)

Zint = R + jtiLint

Ypo= G + juCpo (7)

Lpo and CPO are the per-unit-length transmission line pa-

slot wit h lossless walls and no gasket. The parameters R,Lint, and G represent the per-unit-length transmission-line parameters for

the siot when the walls are lossy and a iossy gasket material is contained inthe slot. Note that Equations (3) - (7) are dependent on the radial distance,

PO,which is arbitrary.Using the equivalent antenna radius derived by W-arne and Chen [24], [25],

and assuming that. the field in the unilluminated region, 11~, is in an infinitehalf-space: the slot equations, evaluated at pO= a, become

AY-~ d21m (-z)Hz- ((2, 2) -i —

AYC– #m (z) = –H~c (z)

4 d.zz

where.

H; (U,z) = [(3)(i$+k’)])’n(’dz’-h

(8)

(9)

and

AYC = G’ + jsCpo – jtiC~o (lo)

(11)

‘The terms. AYC and AY~ represent the excess capacitance and inductanceper-unit-length of the slot due to wall losses and the presence of a 10SSYgasket

.5

I Quantity I Description 1~G ‘ Accounts for losses in the gasket materialj : in the interior of the slot

Local external* capacitance per-unit-Iength without the gasket present (walls

may have finite conductivity)

Local external* capacitance per-unit- ~length with the gasket present (walls may \

have finite conductivity) ILocal external’ inducta~ce per-unit-length Iwithout the gasket present (assuming per- 1fectly conducting walls) ILoc~l external* inductance per-unit-length \with the gasket present (assuming perfect- 1ly conducting walls) IInternal impedance’ per-unit-length which Iaccounts for losses wit hin the walls and in-cludes changes in the inductance due tovariations of the magnetic field outside thewalls due to finite internal** conductivity I

1

* External to the conductirw walls

** Internal to the conducting wallsI

Table 1: Gasket and Slot Parameters

#

not taken into account by the equivalent antenna radius, a. Table 1 gives a

summary of the t ems included in A Yc and AY~.We now want to add a shielded enclosure in the unilluminated region.

Consider the cavity-backed narrolv slot aperture shown in Figure 4 with aplane wave impinging from the left half-space, z < 0. The first term in

(3) represents the total field on the unilluminated. or cavity side. In thehalf-space case, this term is combined with the scattered field expression onthe illuminated side since the Green’s function is the same for both region-s. Removal of the pO dependence in the half-space case was achieved by

expanding the integral for the scattered field and canceling terms of the lo-cal transmission-line parameters, ZPOand YPO.With an enclosure, the same

approach is taken except that the field on the unilluminated side will bewritten as a sum of a half-space scattered field identical to that used in the

pure half-space case and a difference field:

H: (00, z) = H:s (po, z) + /!MZ(p., z)

= –Hz- (po, 2) + Mlz (p., z)

where as before

H:s (p., z) =E) (s+ ’’)])’.(’’)=’”

-h

(12)

(13)

= –H; (/2.,.2’)

(14)

Therefore, the governing integral equation becomes

[

1 d21m–Mr. (p., 2) +“ 2HZ-(po, z) + ~~ – ypo+

-1=–H;c(z)(15)PO

If we use the procedure of Warne and Chen, the p. dependence in the secondterm of (15) is eliminated yielding

L3YLCPI,. (z) AY~–AHZ (po, -z) + 2HZ-(a, 2) + ~

dzz– ~I,n (Z) = –H:c (16)

Note that the difference field is still dependent on the arbitrary distance+?parameter, PO. ,

For the unilluminated side, the governing equation is the vector waveequation:

7 x h x 7 – /.L.k@-= –m (17)c.

7

‘

Ivhere the 2 component of ~y is given by (12). To solve the slot/cavitycoupling problem, we determine the z component of the difference fieid at a

radius p. from the aperture wall. This is accomplished by either: 1) solving(16), with ~ specified in terms of 1,., and subtracting the half-space field,or 2) computing the difference field directly. In either case, the boundary

conditions on the total field must be enforced. At the surface of a perfectelectric conductor we have:

ficxvxm=o (18)

where the unit normal points outward from the cavity volume. At the surface

of a good conductor, we relate the electric and magnetic field via the surfaceimpedance:

K. ‘Zs (’w ‘R+)(19)

or more. conveniently as

+ —+ficx’Vx H =juEz,ficxfi.!x H (20)

where fiWpoints from the conducting wall into the cavity volume.

Solution of (17) is simpler in terms of enforcing the boundary conditions,but involves added complication because the equivalent magnetic current ~

must “drive” the cavity. Although possible, this can lead to numerical prob-lems because the mesh density must be fine enough to adequately representthe magnetic current along the slot. This somewhat defeats the purpose ofusing the local transmission-line model. Here, we take the second approach

and solve for the difference field directly. Instead of a forced problem wesolve the homogeneous wave equation, wit h ~ = O:

vx&7xm–p,k:E+=o (21)e~

Since the half-space field, ~Rs, is a solution of the homogeneous wave equa-tion we obtain the following result upon substitution of (12) into (21):

Vx&xA~-prk;A~=O (22)tr

‘The forcing function for this problem becomesvia Equations (20) and (12).

~

the incident. half-space field

I

1

i . .

I

3 Hybrid Numerical Model

Here, w-ediscuss the numerical solution of the governing equations previously

formulated. l\~e shall first discuss the discretization of the integral equation

(16). Then the finite element method solution of the three-dimensional vectorwave equation is given. The hybrid technique utilized in the solution ofslot/cavity problems will then be presented.

3.1 Method of Moments

The unknown is the equivalent magnetic current, 1,.. Just as in the thin-wiredipole problem, it is convenient to utilize piecewise sinusoids. The magneticcurrent is approximated by

lv~lot

in = ~ A,Lb,,(2) (23)71=1

where A,, is a complex constant to be determined and b,, (z) are the piecewise

sinusoidal. Each basis function is described mathematically as

{

sin[k((2,t:Z)+~z)l v ~n < ~ < 2 + 1

sin[kflz] —— n

bn (Z) =sin[k((z–z,,)+lu)j v

zTL_l < z < ZTLsin[kk]

(24)

o otherwise

tvhere

Xz=zn+l–zn V n=l,2, . . .. NS10tA’? = z,, – .z,,_~ ‘u’ n=l,2, . . .. NS10t

(25)

Using the Galerkin method to discretize (16) yields the N x N system ofequations:

[[n]-!- [m’:] - p:] - [cc]]{.4} = - {f}

where the entries for each matrix are

(26)

(27)

9

,

~y: - ~’~I:,,,+Lk

2bi (z) b,, (z) dZ

t>,&z,,, -,

(29)

z,,,+Lkc

f =!z bi (z) H;cck (30)Z,?,—AZ

!Z,,,+AZ

c: = bi (Z) L@z (po: z) dz (31)Z,,t –AZ

Note that there are t.lvo unknowns in (26): the eoeficients, A,, of the slot

magnet ic current and the difference field, Afi. (p., z). The next section will

discuss the numerical formulation for the unkno~vn carity difference field.

3.2 Finite-Element Discretization

‘The finite-element method is chosen so that irregularly shaped cavities con-taining highly inhomogeneous materials can be modeled. Due to the finiteelement representations selected, the shapes are limited by bounding sur-

faces which can be approximated by second order polynomial functions. Inaddition, the implementation allows varying dielectrics to be placed withinthe cavity as long as each finite element of the geometric mesh contains ahomogeneous medium. As with many real cavities, the Q of the systemsmodeled, as well as measured, in this investigation are in many cases ratherlarge. As has often been encountered in the three-dimensional finite-elementmodeling of electromagnetic systems, spurious modes may result from the

choice of basis functions and their numerical implementation [31]. These aresimply artifacts of the numericai implementation and are non-physical. Toovercome these pro blems, this investigate ion utilized the mixed-order elements

of Crowley (32]. These are-similar in form to the edge-based functions usedby others but, as implemented here, are nodal-based.

We begin by testing (22) with a set of testing, or weighting, functions, asyet unspecified. to oh t.ain the w-eak form of the vector ~vave equation.

[// [q . ‘ii x -h x AR – prk;mld“ = 0,‘dp (32)

e,. . .~c

10

The index on1,2, ... . .\”[.-~,l(.

the set of independent weighting functions varies as p =:\”~&J[is the number of unkno~vns in the finite-element prob-

lem. AR is the desired field c~uant.ity for ~~hich lye ~~-ishto find an approx-imate solution. It is a function of the unknown magnetic current along the

slot, 1,. (z). Using Green’s theorem (32) can be rewritten as

Vc

+//Tp.ti=x (:vxAn)ds=o:’dp (33)

s’

The unit normal which points out of the cavity volume, V’, is fiC.A major contributor to the reduction of Q in most cavities is wall loss

which must be included in our model. For a good, but not perfect, conductor

we substitute the expression for the total field, the vector form of (12), into

(20) and use llamvell”s equation from Ampere’s law- to find:

(:vxAE)-’’Tk’T~ ”AH’v

//+ 77Po ii. x iiwx jueoz,mds.,

=— II~p.iicx iiwx jLXOZ,H–Hscls (34)

s;

(35)

Arote that in the case of perfectly conducting walls, the surface impedance

approaches zero, 2s = 0, and -weobtain (33).Xom- N-eaddress the implementation of the finite-element method used to

compute the solution. ll~e ~vish to find tin approximation to AR. call it 111~.Discretizing the domain into finite elements. ~shere the elements are assumed

[0 be holnogeneous. [35) becomes

(36)

‘Tile finite element cliscretization of Crowley is used here. The finite ele-

ments are hexahedral. with the vector basis being a set of fifty-four differentpolynomial functions which, for each scalar direction, utilize a set of eigh-teen independent shape functions; that is, there are :Vc = 18 unknowns per

component for each finite element subdomain. In compact matrix notation

(36) on a term-by-term basis becomes

Note that the boundary terms cancel for internal elemental surfaces so that

the elemental matrices, [GMq.], {glc}, and {~zc} must only be evaluatedfor elements bordering a conductor. The terms {gl. } and {gzc} are vectors

computed from the half-spate fields due to the slot magnetic currents, 1~.The elemental integrations are performed use Gauss-Legendre quadrature.

For a swept-frequency response calculation, the elemental assemblies for

[GJ4i.i,[GJW2.],and [GAJ3C]are done once; the global matrix at each frequen-cy is reconstructed by updating the multipliers: and finall] the right-handside terms which are explicitly a function of frequency, are re-computed. Thissaves considerable computational time. Thus, given a presumed slot current

3.3 Hybrid Solution

The hybrid moment llletllod/fillite-elel~le~lt method solution to find the nlag-netic current along the cavity-backed slot is now presented. Equations (26)and (37) together are a set of algebraic equations to solve for the unknown

slot magnetic currents and cavity difference fields. ‘These can be solved eithersimultaneously or in a two-step process. We have chosen to utilize the latterto avoid the problems of having to deal with a sparse matrix having a block

dense portion.In the two-step process, the cavity is excited at each frequency by the

half-space fields NslO~times; that is, we compute ~FJs and ~Hs due to eachN1OJI slot. basis function and determine AfiZ. While the values from theFE\ I solution are available we compute a column of the [Cc] matrix by

testing the fields resulting from each J’IONI basis excitation. Solving theFEJI problem NslOt times yields the complete :V,Lotx :V,{Otmatrix and theMOLI problem solution can proceed. As implemented here, after the solution

is completed, the [Cc] matrix is stored on disk. It is then read and the MOMproblem is solved via (26). At this juncture, we must also save the FEN1 and

half-space fields (AE and lYHs), due to excitation by each MOM basis, at therne~<urement point, w-ithin the cavity. If the fields everywhere in the interior

region are desired, it is necessary to store the half-space and difference fieldsat every node within the finite-element mesh for each excitation. In thisstudy, we are interested in comparison to laboratory measurements performed

at a single probe location; thus. only the fields at the measurement point arerequired.

Once the vector of magnetic current amplitudes, {A}, is known, the mag-

netic current distribution along the slot is determined. The fields scatteredby the cavity-backed aperture can then be computed by substituting {A}into (2:3) and then using (9). The total cavity fields. in local coorclinat~s atthe measurement point, are then computed as:

(38)

13

.\-,Lo:

fit+ = E(.A7,Ht:s + Lh’7t,,)7L=1

lv,,o~

(39)

(40)

Note that a t ransformat ion of these fields mqst be done to find the fields in

the global Cartesian coordinate system. The electric field can be calculated

from the computed values of ~Hs -and AH. Given the finite element withinthe mesh and the position within that subdomain, we numerically take thecurl of the magnetic field to find the electric field.

It should be ,noted that the computation of {Cc] in (26) is dependent

on the radius parameter po. It wouid seem that ‘the- antenna radius wouldgreatly affect the energy stored in the vicinity of the slot; however, we wish to

determine AIYZ (p.,., -~ rather than the half-space field in the non-local region.In order to cancel the pOdependence of the local transmission line terms it\vas necessary to etaluate the half-space field in the + region at p = a, the

equivalent antenna radius. The question then becomes “what is the depen-dence of the z-component of the difference field near the conducting wall’?”We know that for a perfectly conducting wall the total tangential magnetic

field doubles. From resonant cavity theory we know that the field in thisregion is relatively slowly varying. -Rigorously satisfying the constraints in(1), we know that pO should be greater than the slot cross-sectional dimen-

sion. However, since the half-space term evaluated at the equivalent radius

accounts for most of the slot physics and the difference field does not varyradically in the region near the slot, we will evaluate the difference term onthe slot wall, z = -td/2. The foilowing results will show this to be a goodchoice for pO.

- k ya=o?.

4 Results ‘

A large number of experiments were performed to verify that the numericalslot-cavity model would produce accurate results. For the sake of brevity:we present some se~ect cases. Full results can be found in [30], All measure-

ments were done in che anechoic chamber located in che Electromagnet icsLaboratory at Xeu- SIexico State University. The measurements presented

14

I 1 I 1 I

VI I .0016” ..50” I 2.003” \ 2.180” I1 I t

\ VII \ .0095” .~o:! I2.008”I2.045”

Table 2: Measured aluminum slot dimensions

i Cavity I Width Height Depth (dC)

/1 I 3.4” 1.7” 1 1.995”~ 3.4 1.7’ 2.637”3 3,4:, >: 3.295”

4 ~,~), :,;,> 4,2(3.5”

Table 3: Ca~-ity dimensions

were clone with elect ric field probes fabricated and calibrated in the Elect ro-magnetics Laboratory. The details of the measurement setup can be foundin [30]. Four different cavities and seven different slots were constructed sothat the effects of slot width and depth as well as cavity size could be exam-ined. All the slots and cavities ~~ere constructed from aluminum. Tables 2and 3 below give the dimensions of each of the seven slots and four cavities

used. ‘The slots u-ere mounted to the front of the cavity test fixture as

shown in Figure .5.’For all cases, the incident electric field is polarized per-pendicular to the slot. Due to the limited size of the cavity and to preclude

undesired loading of the cavity only one probe was used. A measured value

of a = 2.6 x 10T for the aluminum conductivity was used for all the computermodels. In all cases an effective slot length, m. determined by comparison tohalf-space measurements, was used to compensate for the loading producedby imperfect short circuits at the slot ends.

The first set of results are for cavity 1. The first cavity resonant frequency(TEIOl) is well above the slot resonance for cavity 1. The computer modelused a 28 element finite element mesh with 21 mol~~ent-met hod basis func-tions for the slot. Figures 6 - 7 show the coupling for cavity 1, slots I, H,III, and VII respectively. The reader should note that the follo[ving charac-

teristics in each of the plots: 1) cavity resonance frequency 2) slot resonance

frequency: 3) cavity resonance Q, 4) slot resonance 0, and 5) peak coupling.In each of the cases, the computer model correctly tracks the ex~erimental

results. As expected the Q decreases with increasing slot width. The peakcoupling also increases with increasing slot width. Increasing the slot depth

clecreases the coupling due to wall losses. It is interesting to note that the

slot greatly affects the coupling at the resonant frecluency of the cavity evenwhen the slot and cavity frequencies are widely spaced.

The second set of resuits are for cavity 2. The slot and cavity resonant

frequencies are nearly coincident for this case. The computer model used

a 35 element finite element mesh with 21 moment -nlethocl basis functionsfor the slot. Figures 8 and 9 show the coupling for cavity 2, slots I and 11

respectively. In both cases, the computer model is calculating the complexinteraction of the slot and cavity. The proximity in frequency of the tworesonances reduces the Q of the system as a whole.

The third set of results are for cavity 4. The slot resonant frequency liesbetween the first t~~wcavity resonant frequencies. The computer model useda 63 element, finite element mesh with 21 moment-method basis functionsfor the slot. Figures 10-12 show the coupling for cavity 4, slots 1, 2, and7 respectively Again, the computer model is able to predict the couplingaccurately.

\l~e now- consider the effects of loading the cavity with dielectrics and

conductors. The first combination is for cavity 1 partially filled with sty-rofoam with slot VII as given in Fia~re 13. The calculated and measured

]:’ curves shown in Figure 14 display excellent ae~eement. The measured cavity7.+

resonance decreases- from 3.43-5 to 3.415 GHz. Ignoring losses, a volumet--% ~rlc mixting model results in c,cff = 1.018. This results in a cavity resonant

frequency decrease of 29.5 MHz: from 3.4465 to 3.418 GHz. The numericalmodel with 42 elements and 21 slot moment method basis functions predicts

a decrease of 31.5 LIHz.The seconcl combination is for cavity 1 partially filled with styrofoam with

slot WI as given in Figure- 1.5. The calculated and measured curves shownin Figure 16 display excellent ag-cement. The measured cavity resonance

decre~es from 3.43.5 to 3.417 GHz. Ignoring losses, a volumetric mixing/.< “r::’ model results in C,,ff = 1.017. This results in a cavity resonant frequency

~ decrease of 28.5 \lHz, from 3.4465 to 3.418 GHz. Tl~e numerical model with

42 elements and 21 slot moment method basis functions predicts a decreaseof 27.5 \IHz.

For each of the last m-o resuits, we \vere able to use perturbation theoryto predict the cavic:- resonant frequency shift since the slot was narrow and

has a large depth so that the cavity fields are not perturbed significantly.‘The third combination is cavity 1 [rith slot VII having a small teflon

block placeci near the front wall (Figure 17). The published value used in9 1 – jO.0003. The frequency response curvesthe computer model was E, = . .

calculated by the computer model again agree well with the measured results

(Figure 1S). The measured peak shifted from 3.435 GHz in the air-filledcavity to 3.345 G Hz when the dielectric slab was inserted, a downward shift

of 90 MHz. Perturbation theory can be used to preciict the resonant frequency

shift assuming that the dielectric occlusion in the cavity is small. The shiftis given as

(41)

where A]$’ is the additional enerO~ stored in the cavity with the dielectric in

place: and W is the energy stored without the dielectric. The shift computedwith perturbation theory is 35.8 MHz. The computer model (84 elements and

21 slot moment method basis functions) predicted a resonant frequency of~ 3 3.408 GHz which is close agreement with the shift predicted by perturbation

theory. The larger measured resonant frequency shift may be indicative ofan error in the measured location of the dielectric slab.

The final result is for ca~~ity 2 with slot VII having a metal block set onthe lower surface of the cavity (Figure 19). The block was not held in placewith screws or bolts. It was set in the cavity so that a small gap between the

block and rear wall of the cavity was created. The measured and calculated

curves are shown in Fieg.ue 20. The computer model used 100 elements and21 slot moment method basis functions. Two main features of the response

are observed. First, there is a dramatic decrease in the cavity resonantfrequency. The cavity resonant frequency shifts downward nearly 800 MHzto 2 GHz when the metal block is introduced. A second resonant peak of theloaded cavity is observed ‘near 3.5 GHz. A second cavity resonance is seennear 3..5 GHz, The coupling in the vicinity of the slot resonant frequency.approximately 2.S GHz, is perturbed by the presence of the conducting block,

Although not as close as pre}-ious examples, the agreement between thecomputed and ex~erimental results is still adequate for practical situations.JIost importantly the h~-brid \IOkI/FE}I model predicts the general vicini-

ty of the cwo cavity peaks. These are the points in the frequency spectrum

17

where maximum collpling occurs. Thus in terins of decerrnining system sus-

ceptibility-, the numerical moclel performs well. Furthermore, che computedcur~’e clisplays trencls similar to those measured in the laboratory near theslot peak.

Reasons for the u-eaker relationship between modeled and measured data

are difficult to determine in this last case. Imprecise knowledge of the dimen-sions of the cavity, irregularities in the dimensions of the metal block, and

uncertainties in its location may have all contributed to the discrepancies.

5 Conclusions

The primary focus of this research was the formulation ancl numerical so-lution of slot/cavity problems incorporating realistic apertures. This was

accomplished by modification of the antenna/local transmission-line modelof Warne and Chen, which represents the physical attributes of an actualseam by incorporating slot depth and losses, and appiying it to arbitrarily-shaped cavity-backed apertures. A hybrid MOM./FELI technique was appliedto solve the two-region problem utilizing standard piecewise-sinusoidal basisfunctions for the moment method implementation and mixed-order, covari-

ant projection finite elements. The hybrid model ~~as used successfully to

compute the electromagnet ic coupling into a variety of slot/cavity geometricconfigurations with varying dielectric inhomogeneities.

A series of numerical and laboratory esperirnents for slot/cavity con-

figurations with right-rectangular geometries was performed, with excellentagreement between the experimental data and numerical computations. In-

consistencies were mainly attributed to loss mechanisms not represented in

the hy-brid MOhf/FEkJi slot/cavity model, caused by uncertainties in the me-

chanical configuration. However. the model successfully indicated the relativeposition of slot and cavity resonances, demonstrating the interaction betweenthe slot and the highly reactive cavity load, and the resulting enhancementin coupling. ‘The ca~ity fields increased w-ith increasing slot width and/ordecreased slot depth. Frequency shifts measured ~vhen the right-rectangularcavities were partially loaded with dielectric materials were predicted -withthe hybrid 11011/FELI model, and found to be in good agreement with theresults of a perturbational analysis.

Theinterior

hybrici teclmique developed here utilizes an FEJI solutioncavity problem and allows for arbitrarily-shaped geometries.

18

of theAn ir-

,

regular cavit~’ was assembled and coupling measurements performed. The

hybrid lIOM/FEl I model predicted the nominal frequencies at v.hich highcoupling ievels occurred. Although wall losses were included, the hybridhIOM/FENI model somewhat overestimated the peak coupling values, thus

providing a conservative estimate of the maximum field ~vithin the cavity.That is. the measured fields in the cavity l~ere smaller. the internal elec-tromagnetic environment was less severe than that predicted by the hybridMOM/FELI modei.

The mixed-order. covariant-projection finite elements described by Crow-ley allow curvilinear geometries to be accurately modeled with a minimum

number of elements. The use of the slot model of JVarne and Chen avoids

the need for using m fine cliscretization in the slot region further adding to

the efficiency of the method.An additional advantage of the hybrid numerical moclel used here is that

in the course of computin: the coupling, the fields everywhere within thecavit~ can be determined. For a given confiewration. this ensures that themaximum field and its location are determined. This is often difficult toaccomplish in a measurement program. Furthermore: the numerical analysisallows determination of the ener~~ distribution within the system.

To enhai-ice the capabilities of the slot/cavity formulation presented here,

realism should be added to the exterior problem and terminated wires as wellas complex loads should be included within the interior cavity region.

6

This

Acknowledgements

work was supported by Sandia National Laboratories under contracts#7.~.3706, #12-~ 6(J4, and #AA-.5698, the National Science Foundation under

~~ant #ECS-91-5-i3.504, and the Air Force Phillips Laboratory under prime

contract F29601-92-C-O1O9 through the Kaman Sciences Corporation sub-contract KD-SC-2216-1O. SandMis @nwlti~rogr~rnlaboratory

operatedby Saridk Corporation, aLockheed ‘Martin Compky. for theUoited States Depmmnerit of LiWQY

References undercontract D“E-AW4-94AL8~@10.

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[19].Jin, J.JI. and tJ.L. Volakis: “Electromagnetic Scattering by and Trans-mission Through a Three-dimensional Slot in a Thick Conducting Plane,”IEEE Tmnsactions on Antennas and Propagation, Volume AP-39, No. 4,April 1991, pp .543-.5.50.

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f20] Jin, .J.\I. and ,J.L. Volakis, “.-l Finite Element-130 undary Integral For-mulation for Scattering by Three Dimensional Ca.vicy-Backed Apertures,;:IEEE Transactions on Antennas and Propagation, Volume AP-39, Yo. 1,

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23

List of Figures

12

34

567

89101112

1314151617181920

Bolced seam . . . . . . . . . . . . . . . . . . . . . . . . . ...25Narrow slot aperture with depth . . . . . . . . . . . . . . . 25

Local cross-section of slot . . . . . . . . . . . . . . . . . . . . 26Narrow- slot aperture with depth . . . . . . . . . . . . . . . . 27

Cavity fixture and probe location . . . . . . . . . . . . . ‘. . . 28Coupling versus frequency for cavity l/alunlinum slot I . . . . 29

Coupling versus frequency for cavity I/aluminum slot VII . . 29

Coupling versus frequency for cavity 2/aluminum slot I . . . . 30

Coupling versus frequency for cavity 2/aluminum slot II . . . 30Coupling versus frequency for cavity 4/aluminum slot I . . . 31Coupling versus frequency for cavity 4/aluminum slot II . . 31

Coupling versus frequency for cavity 4/aluminum slot VII

StJTofoam on bottom . . . . . . . . . . . . . . . . . . .Cavity l/aluminum slot VII (styrofoam on bottom) . . .Styrofoam on front wad! . . . . . . . . . . . . . . . . . . .Cavity l/aluminum slot VII (styrofoam on front wall) . .‘Teflon block near front wall . . . . . . . . . . . . . .Cavity l/aluminum slot VII (teflon block near front wall)Metal block on bottom wall . . . . . . . . . . . . . . .

Cavity 2/aluminum slot VII (metal block on front wall) .

. . 32

. . 33. . . 33. . . 34. . . 34. . . 35. . . 35

. . 36. . . 36

a

/

Figure 1: Bolted seam

1-

Y

1—————4Yslot

z

Figure 2: Narrow slot aperture with depth

x.

cross-section of slot

SLOT

Figure4: Narrow slot aperture withdepth

i

Cavity fixture and probe location

.

Gvim l/JMmlimml slot I

I— Measured — Calculzkd I

Figure 6: Coupling versus frequency for cavity l/aluminum slot I

Cavity l/Ahnminum Slot %’11

Frequmcy (GHz!

— }leasurd ----- Cdcdiad ~

Figure 7: Coupling versus frequency for ca~ity 1j aiuminum slot VII

wo

z?1

(x,.

!,

.

cavity4/MmlinuIu Siot I32] I I I I i I I i I

t23 : I I I I I

. I I ; I 1

.- -“

-= %&.= l...-%?“%....-----

-: -3d I -----’%. /’””t--LLk--- 1 w“+..

‘t--% ! I ,:, 1 1

/

i 1 I

I-so‘ I I 1 I

i

I I I I I

i-7C ! I I i I I I I I 1

‘2.2 2.4 2.6 2.9 3 3.2 3.4 3.6 3.8 4Ikequency (GHZ)

l— Mtm.w-ed — C“cukikd I

Figure 10: Coupling versus frequency for cavity 4/aluminum slot I

Frequency (GHz)

Figure 11: Coupling-versus frequency forcavicy 4/aluminum slot II

31

,.,

Cavity 4/Ahmlirtmn Slot W1.

Freqmlcy (GH2)

— MeMJred — Caku12teci

Figure 12: Coupling versus frequency forcavity 4/aluminum slot VII

32

.. , I

.

I

1.995”7<

Figure 13: Styrofoam on bottom

Cavity l/Ahmninmn Sot W(S3*ofoanl 1.1” on Bottom)

3~ ‘

20j

10 t<

0,. j’ ;,

-10- %.,

-20--%..

-30

-40,.<,:s

-w ,2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

— Me.mired “—””” calculated

Figure 14: cavity I/aluminum ~iot VII (~ty~~foam on bo~t~m)

33

.,$

.

1.995”7<

Figure 15: Styrofoam on front wall

.‘ Mminuin slotWI

(PI&%oam on Front Wall)30

20i

E@., I t I 1 1 1 t I I

.G~“-

2.2 2.4 2;5 2:8 3 3:2 3.4 3:6 3.8 4FqIIency {.GIW

— was-wed — Cdcuiatcd

Figure 16: Cavity l/aluminum slot VII (sty-ofoam on front wall)

~<

.,,,

Figure 17: Teflon block near front. wall

cavityl/Mmiimm slotVIITEFLON BLOCK

~(-J-

2c- ‘

10 8.

== 0.-.~

-10,;”

~--z -2(J‘J

-~Q

-m-

=0$‘“ 2.2 2.4 3.8 4

Frequency (GHz)

— Masure(i — CaicuiateclI

Figure 18: Cavity l/aluminum slot VII(teflon lJlocknear front wall)

3.5

. * .>

/ .515”

Y

Figure 19: Metal block on bottom wall

cavity 2/Mwnirnml slotWI(MetalBlock on Bottom Wail)

l— >Ica’wrc!d “-–- Calculated \

Figure 20: Cavity 2/aluminum slot VII (metal block on front wail)

36