IEEE Transactions on Nuclear Science, Vol. NS-27, No. 6, December 1980
ON ELECTROMAGNETIC ENVIRONMENTAL FIDELITY IN DAMPED CYLINDRICAL CAVITIES*
John Dancz and Roger Stettner
Mission Research Corporation735 State Street, P.O. Drawer 719
Santa Barbara, California 93102
ABSTRACTA study is made of the electromagnetic cavity mode
structure of a cylindrical tank containing a cylindricaldamper membrane. Maxwell's equations for the niost gen-eral mode of a cylindrical tank is considered andreduced to the problem of solving a single transcenden-tal equation written in terms of the Bessel function ofthe first and second kind. This equation is solvednumerically for some representative damped cavitiesyielding modal frequencies and damping rates.
INTRODUCTIONIt is generally accepted that electromagnetic
testing of spacecraft in a simulated orbital environmentis a desirable element in system hardening of space-craft. This, then, poses the question of what designconsiderations are required for proper simulation of therequisite orbital electromagnetic environment 1,2Typically, a large metallic test chamber is utilized toreproduce the vacuum conditions occurring in space; themetallic surfaces of such chambers (which cannot be toolarge due to financial limitations) always reflectelectromagnetic radiation which gives rise to cavitymodes in the chamber which interfere with the trueelectromagnetic response of the spacecraft. In orderto alleviate these effects, dampers have been suggestedto absorb the electromagnetic energy associated withthese electromagnetic cavity modes. The determinationof the optimal damper design has led to many studies ofan analytical,1,3',8 numerical ,''5'7 9 and experimentalnature4'6 which resulted in the development of largesheet dampers. Questions, however, still remain as tothe fidelity of such damped cavities in duplicating anorbital environment which have prompted us to undertakethe present study of electromagnetic environmentalfidelity in the simplified model of a cylindrical damperand cavity. Herein, we will consider the solution forthe cavity modes of a conducting cylinder (of radius Rand length L) with closed ends containing a thin axisym-metric cylindrical damper, of radius R0, with open ends.
This simple model permits an analytical analysis whichrelates the design parameters R, L, R0, Re (the sheet
damper resistance) to the damping rate of the cavitymodes. Such relationships can provide a general guideto the design of more sophisticated environments andserve as a useful comparison to other theoreticalstudies relating to dampers in spherical tanks1'5'6'7and elaborate upon other studies of cylindrical tanks.8'9
Herein, we examine the ability of sheet dampers toattenuate cylindrical cavity modes with the aim ofassuring that test objects in such a cavity will not beartificially driven by a cavity resonance. Other moresubjective questions arise, such as how well do thedampers meet the free space boundary condition or whatis the effect of putting a large object inside the tankon the damper effectiveness, which can not be answeredherein. Our present study is also limited to a parti-cular representative tank with a single damper geometrysuggested by previous studies.5'7-9 Further, we limitedthe damper to a single cylindrical surface instead ofincluding end-caps in order to achieve an analyticalexpression for the frequency within this model.
* Supported by the Defense Nuclear Agency undercontract DNA 001-79-C-0231.
MATHEMATICAL PRELIMINARIESIn this section, we derive the modal structure of
a conducting cylindrical cavity of length, L, andradius R, with a thin cylindrical membrane at a radius,RO. This geometry is depicted in Figure 1. The modes
of this cavity will be determined by the solution ofMaxwell's equations, for a medium of unit dielectricconstant and magnetization with time Fourier transformed(a/at) iw. The current density 3, and the chargedensity, p, is assumed to be non-zero only on the mem-brane which will be taken to be infinitely thin. Thatis, the current will be represented by,
I= at= R-I 6[r-ri]t (1)where Re is the sheet resistance. Due to the sym-metry of this cavity, Maxwell's equations are mosteasily discussed in cylindrical polar coordinates(r,O,z) in individual component form:
1 a a -iwBr ae -Ez az -O r
a E --aE-iEB=az -r ar -z iwB (2)
r (rE 1) - r a E = -iwB
1a~B ~- B = 1 (E + iw) E-r ae .z az e-0 C2 E:0 -
a Br a B = 12 ( a + i)Eaz -.r a'r -z c2 + i)0 (3)
1 a (rB ~ ra1 B 2 (" + iw)Err ~r ra0O-~r c2 E0 -
The boundary conditions appropriate to this cavityreflect the fact that the tangential components ofelectric field and the normal components of magneticfield must vanish on a perfectly conducting surface.These conditions result in a pair of boundary condi-tions for each field component. It should be stressedthat these boundary conditions are not independent ofone another.
To simplify these expressions, we will first sepa-rate out the e and z dependence of these components byexpanding in Fourier sine and the cosine series; it isthen apparent that the modes of the cavity are separableand can be represented by
where p,m are non-negative integers, 6 = O,ir/2, and
2 = 222/c- m It /L . Together, (p,m,6) are eigen-values which specify the mode of the cavity. The radialdependence of the field components is then determined:
-P E - m E = Bs z XL 0 r
mlf aL r --s =BE
1 a (sE ) + P E = Bs as s r z
R B +!'T Be = w12( wa ) E5 z AL 0 2Ac2B r
mflhr -a- 1 2r as z x2 i2 co2 E0
(6)
(7)
sas s- r = Ac 2 w) E
where s = Ar.
The radial dependence of these equations is nottransparent in the present form. It is more elucidatingto note,
[v2 + 2- j t = fP/ £0-
[2 +2 - wc2t ~ fxt/eoc2 (8)c c E:o ](8
In those regions of the cavity where the conductivityvanishes, the solution of these differential equationsare Bessel functions of integer order:
Er + Ee J4p4l(Ar), Yp+l (Xr)
Er - E - Jp_,(Xr), Yp_,(Xr)Ez- Jp (Ar), Yp(Ar)
Br+ B0 Jp_(Ar), Yp_V(Ar)
Br - Be Jp+1(Ar), Yp+1(Ar) (10)
Bz 3p(Ar), Yp(Ar)where, for r<R0, the coefficients of the Y's (Besselfunctions of the second kind) vanish because of therequirement that the fields be well behaved at theorigin, r = 0. For r>R0, both Bessel functions of thefirst and second kind are necessary to characterize thefields.
Further simplification is achieved by rewriting thefields using the following recursion relations which arevalid for both Bessel functions of the first and secondkind:"0
2[Cp l(z) + Cp,l(z)] = P C (z)
2 [Cp l(z) - Cp+l(z)] = Cp(z) (11)
where the prime (') indicates differentiation withrespect to the argument z. We therefore write, forr<R0,
Er= A1 P J (Ar) + A2 J(Ar)E= -A P J (Ar) - A J' (Ar)2AXr p 1p
(12)Ez = A3 Jp(Ar)Br = B1 ,P J (Ar) + B2 J(Ar)
Be =B2 r J(r) + B J'(Ar)B
2-B p(Ar
(13)
Bz -B3 Jp(r)which upon substitution into Equations 6 - 7 yield:
A1 = - B3
B2 AL 3
(14)A2 A-L A32
B = 22 AA c2
For r>R0, the structure of the fields are the same hence
we will define the coefficients in that region of spaceaccordingly. These results and this notation is sum-marized in Table 1. It should be noted that there arenow six undetermined coefficients: A3, B3, a3, b3, a3,
b . These, in turn, may be determined by six conditionsat r = R and four conditions at the membrane interfacederived from consideration of Equations 8 - the con-tinuity of EV, Ez and the jump discontinuity conditions
B (r=R) - B (r=R ) = li E (r=R )Z 0 Z 0 R cA 08 0e o
B -K1K2B3J(s)-K3A3J(s) -K1 K2(b3J (s)+bjY (s))-K3(a3%(s)+a3Y(s))
Bz -B3Jp(s) -b3ip(s)-bjYp(s )
Before proceeding with the solution of these equa-tions, it will be illustrative to discuss the specialcase of Re +00. Then,
a3 = A3
b = B3
a3 = b3 = 0
and the two boundary conditions at r = R yield,
(1 5)
1621
Ez(r=R) = A3Jp(AR) = 0
' B (r=R) = - AB3JJ(AR) = 09r z3
These equations may be satisfied only if
Jp (xR) = 0, B3 =0
or
(16)
(17)
J'(AR) = 0, A = 0p~~~~
That is, the solutions may be characterized by thelongitudinal (z) field component-being either electric(B3 = 0) or magnetic (A3 = 0). For the special case,
p = 0, the longitudinal electric field corresponds to an
azimuthal magnetic field and is sometimes referred to as
a "transverse magnetic (TM) mode"; similarly, the longi-tudinal magnetic field corresponds to an azimuthalelectric field and is referred to as a "transverse elec-tric (TE) mode." It is seen that for a general mode(pto), the significance of TE and TM modes becomesambiguous.
Returning now to the general case of finite con-
ductivity, the six equations defining the coefficientsare:
a3Jp(S) + a?Y (S) = 0
beJ%(S) + b3Y;(S) = 0
KIK2[A3-a3)Jp(S0)-ajYp(So + (B3-b3)J%(So)
bpY;(S0) = 0 (18)
(A3-a3)Jp(S)= a3Yp(S0) =0
(B3-b3)Jp(S) bYp(S )
RC K3 [K K A J (S) - BJ;(So)]R CE 3 L1 2 l3 p ~0I L3
K1K2[(B3-b3)jp(S )-b'Yp(S-)p + K3[(A3-a3)jp(S)
-a3Yp(So)] = - RCE K3A3ip(S0)
where S = AR, SO = AR0, and now K1 = p/ARo. The deter-
minant of the coefficients of the parameters A3, a3, a3,
B3' b3, b3, respectively are shown in Table 2, where
K4 = 1/RecE o.
Table 2. Matrix equation to be solved to determinefrequencies and decay times.
A3
&3
a3
This determinant, after much algebra, may be
reduced to
22) J (S)J'(S) - K2J (S )Jp( )S p -pTSp 4 Jp JP0~~~~~~~~~~~~(19)
i(-2 )K4(1+K2K2)K-2J (S )jp(S)pp + K2J (S)Jp(S )S
where
pp= J (S)Y (S) - Y (S)J (S )p po0 p po0
Sp= J'(S)Y'(So) - Y'(S)J'(So)and we have utilized the Wronskian,10
(20)
Jp SO)Yp(So) - J'(S)Yp(So) (21
Note that for p = 0 or m = 0, the constantK1K2 = 0 and Equation 19 may be factored:
[(7%) ~J (S) iK-K J (S )p
[(k0) Jp(S) iK2K4J;(S)Sp] = * (2;
The first factor in this expression agrees with theresults found in Reference 9 and the second factor isthe concommitant result for the TE modes.
1)
A numerical solution of Equation 19 has been per-formed for a 100 foot tank (in length) which is 55 feetin diameter with the sheet damper at four-fifths thedistance along the radius vector which has been pre-viously estimated as being optimal for a cylinder.9Results are presented in Figures 2 and 3 for a continu-ous spectrum of sheet resistivities calculated atintervals of 25 ohms per square. Results are also pre-sented in detail for three exemplary resistances of100, 200, and 300 ohms per square in Table 3. Theseresults were limited to the first three angular modes(p=0,1,2), the first four axial modes (m=0,1,2,3), andthe first four radial modes (roughly the first two TEand TM modes).
CONCLUSIONS
We shall now attempt to draw some general conclu-sions from these results and make comparisons toprevious work. In Figure 2, we observe the behavior ofmodal frequency with sheet resistance. It would beanticipated that the low resistance limit would corre-
spond to a cylindrical cylinder whose radius would begiven by the radius of the sheet damper and the highresistance limit would correspond to the cylinder with-out damper. This would imply frequencies which scaleas the ratio of these two radii. It is noteworthy thatthese curves do not follow this behavior for the fre-quencies shown above 275 megaHertz, but rather reversedirection. Such behavior has been previously noted in
spherical cavities.5 It is unclear whether thisbehavior is characteristic of high frequencies or will
again reverse at even higher frequencies than those
studied herein.
In Figure 3, the Q's of the resonances studied are
presented where
Q Real (2 Imag
It is especially significant that the Q's are fairlystrongly grouped together and are rather slow changing
Table 3. Modal angular frequencies in megaHertz for the 100's tank given for the foursymmetry. Imaginary parts (in megaHertz) are given in parenthesis.
Second ThirdLowest Mode Lowest Mode Lowest Mode
R= 100 Ohms oer Souare
functions of sheet resistance. Such behavior has alsobeen noted in numerical studies of spherical cavities. S7From a comparison of Figure 3 and Table 3, it can beinferred that the modes which display the highest mini-mum Q's correspond to the higher frequencies and hencereflect a trend toward decreasing damping effectivenesswith increasing frequency. From a perusal of Table 3,however, it is difficult to develop hard rules aboutdamping characteristics due to what may best be des-cribed as almost random behavior in damping. Evenbehavior which does stand out, such as the relativeindependence of our results upon the axial eigenvalue,m, will very probably go away with the introduction ofend-cap dampers. It would appear, though, that thebest damping for the lower frequencies would be achievedby the use of the 200 ohm resistance, while, at higherfrequencies, it is not clear if better results couldnot be achieved with a higher sheet resistance.
lowest modes of each
FourthLowest Mode
In conclusion, we would like to add that theseresults are generally applicable to the damping ofpure cavity modes. It has been found that radiallydampers operate much more efficiently for cavitieswith test objects present which have low frequency(essentially electrostatic) modes.11
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