IEEE Transactions on Nuclear Science, Vol. NS-29, No. 6, December 1982
PRELIMINARY CONSIDERATION OF THE MODIFICATION OF THE SGEMP RESPONSE OF ATRIAXIAL SATELLITE DUE TO ELECTRN CLOUDS*
John Dancz and Roger Stettner
Mission Research Corporation735 State Street, P.O. Drawer 719Santa Barbara, California 93102
ABSTRACT
In SGEMP experiments, extraneous electrons areemitted from the walls and damper of the vacuum tank.This paper examines the effect of these electronclouds on the electromagnetic modal response of satel-lites by considering a simple LC network model of amode. The results suggest that substantial frequencyand damping increases nay occur in high-fluenceexperiments.
1. INTRODUCTION
Changes in the electromagnetic response of satel-lites to SGEMP have been observed in photon experi-ments with the Skynet satellite and a structural modelof Skyneti,2 in the form of shifts in the character-istic frequency and damping time of internal struc-tural modes. These shifts are referenced to the freespace, electrical (no photons) excitation of the sat-ellite. An explanation for these shifts has been pro-posed by R. Stettner : photoelectrons pick-up energyfrom oscillating electric fields thereby changing themodal characteristics (produce a frequency shift) andremove energy from the system when these electronscollide with satellite structures (damp the mode). Inthe present work we extend these ideas of modal modi-fication due to an electron density and apply theresults to an interpretation of a high fluence SGEMPexperiment.
Comparison of calculations3 of the externalresponse of STARSAT in the Huron King experiment withdata suggest that the external modes may be veryrapidly damped at early times. A modal analysis4 ofthe Huron King data also suggests that the structuralmodes may be significantly shifted in frequency atlate times. The satellite-electron cloud interactonsinvestigated in this report are a possible mechanismfor the above suggestions.
Alteration of the free space structural responseof a system by a simulation may have imgortant simula-tion fidelity implications for the test . Seriousundertests at certain frequencies can result fromthese alterations. The conclusion of this report isthat under certain circumstances these alterations arepossible.
We are concerned with how an electron cloud inthe vicinity or within a satellite can alter thecharacteristics of the satellite's structural modes.These electrons may arise as a result of the directinteraction of X-ray photons with the satellite orwith the walls and damper of a simulator. Weinvestigate the nature of the satellite electron cloudinteraction by means of a simple model of a structuralmode, a capacitor and an inductor which is easilygeneralizable to a capacitor and a more generalimpedance. This simple model represents quiteaccurately the nature of many satellite modes. Theelectron cloud between the plates is introduced inseveral different ways in order to illustrate thediffering nature of the interactions that may result.
* Supported by the Defense Nuclear Agency underDNA001-81-C-0082.
Two separate interactions are reviewed withreference to external satellite modes, modal frequencyshifts and damping due to electron clouds. The keyparameter representing the electron cloud is the plas-ma frequency wp while the key parameter representingthe structural mode is the resonant frequency wo. Theintroduction of the plasma frequency here is not meantto imply the density oscillations of a neutral plasmabut rather a convenient parameter of the electron den-sity. The results of the calculations suggest thatwhen Wp is comparable or greater than w0 both modalfrequency shifts and strong damping are possible. Theelectron cloud in a typical high fluence SGEMP experi-ment is dense enough so that both modal damping andmodal frequency shifts are plausible.
Section 2 of this report discusses the equationsfor the interaction of the electron cloud and aparallel-plate capacitor and the resulting modal dis-tortion. Section 3 applies the results of Section 2to differing circumstances depending upon the sourceof electrons vis-a-vis the modeled capacitor and Sec-tion 4 discusses the application to a high fluenceSGEMP experiment. Section 5 presents a summary of theconcl usions.
2. BASIC EQUTIONS
In order to illustrate the physical processesoccurring when the quasi-static modes of a triaxialsatellite immersed in a free-electron cloud areexcited, a simple example of a large parallel platecapacitor will be considered. (Figure 1). Theassumption that the capacitor is large (that is, theplate area, A = QW, and the plate separation, d,satisfy d << X,W) allow the field quantities and thedynamical properties of the electrons to be treated asessentially a one-dimensional problem. In thefollowing paragraphs, we relate the current flowexterior to the plates to the voltge across the platesand charge flow within the plates.
From Maxwell's equations, the electric field, E,can be related to the electron density, n, by (MKSunits)
where co is the permittivity of free-space and e isthe electronic charge. This expression can beintegrated from the lower plate to some arbitraryposition inside the capacitor to yield
x
co E(x,t) = a(t) + e f dx' n(x' ,t) (2-2)0
where a(t) is the surface charge density on the lowerplate. This surface charge density is simply relatedto the charge on the capacitor, Q, by
Q(t) = a(t) A . (2-3)
The potential across these plates, which is meaningfulin the quasi-static limit, can then be defined,
d 1V(t) f dx E(x,t) = -
d x{a(t)d + e f dx f dx'n(x',t)} . (2-4)
0 0
In principle, the electron dynamics may be solved inorder to determine the electron density and hence thepotential; in practice., it is more useful to considerthe time rate-of-change of this expression and thenrelate the electron density to the electron current,J, by the continuity equation,
n(x,t) + -J(x,t) = 0 . (2-5)
and defining the average current flowing between thepl ates as,
I0(t) + eA fdx j(x,t),d o
(2-10)
a particularly simple expression results,
C a V(t) = I(t) - Io(t) (2-11)
which gives an explicit dynamical interpretation tothe current flowing between the plates. An under-standing of the behavior of Io(t) can only come fromthe dynamical properties of the electrons in thecapacitor.
Since the system is assumed connected to a con-ductor of inductance, L, the equation for the voltagedrop is also
av a2Ia - L
at At 2
or the equation for the system is
a2I I+ = I O/Cat2 C
(2-12)
(2-13)
This equation takes on a particularly simple formby introducing the frequency domain (t+w) and theresonant frequency of the system without the electrondensity associated with the capacitor,
Upon integration over x' by parts, this yields
a a dE-u V(t) = d - a(t) + e f dx [j(x=O,t) - j(x,t)]0
d= [a o(t) + e j(x=O,t)] d - e f dx j(x,t) . (2-6)
at o
The expression in square brackets has a simple inter-pretation in terms of the external current flowingonto the capacitor. Consider the surface charge den-sity which increases according to the external currentflowing onto the capacitor plate, I(t), and decreasesaccording to the current flowing between the capacitorplates, this implies,
(2-14)wo - (LC )- 1/ 2
then
(2-15)(1 -2 ) I(W) = IO()
wo
Modal characteristics are determined from the solutionof this equation in the absence of an electricdensity, Io = 0 and
W = ± Wo . (2-16)
The procedure to be utilized in determining thealteration of modal characteristics is to seek anexpressi on
a :(t) = I(t)/A - e j(x=O,t) (2-7)
and hence the time rate-of-change of capacitor voltageis related to the external current flowing onto thecapacitor less the spatially averaged (positivecharge) current flowing between the capacitor plates,
a V(t) = d I(t) - e d dx j(x,t) (2-8)at EOA Su o
which demonstrates a very important cancellation ofterms at the boundaries. Further note that by intro-ducing the capacitance of this parallel-plate geom-etry,
Io(W) = V(W)/Z(W) (2-17)
from the dynamical properties of the electrons andthen determine modal characteristics from equation2-11
(2-18)[ic + 1 V(w) = 1(w)Z( w)
In this form, the effect of the electron cloud is seen
as an impedance, Z(w), in parallel with the originalcapacitor, C.
For this loaded capacitor connected to aninductance, L,
(2-1) C =d
(2-9)
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V(w) = I(w) = - iwL I(w) ; (2-19)i WC + 1
Z( w)hence the resonant frequency is defined by solution of
(Ai2/ W= 1 + i ) (2-20)
amounts to neglecting the last term on the right handside of equation 2-2 in comparison to the first termon the right-hand side, in the determination of theelectron dynamics. This implies that acceleration dueto the electron density within the capacitor is smallcompared to the acceleration, due to the charge on thecapacitor,
note that as this impedance in parallel with the orig-inal capacitor becomes large
W + Wo . (2-21)
Further, the generalization to more sophisticated cir-cuit models of satellite response replaces
iwL + ± Zo
where ZO(w) is the effective impedance for the rest ofthe satellite (excluding the capacitor under consider-ation). Then the modal frequencies are determined bysolution of
1 + iuCZo(w) + ZO(W)/Z(w) = 0 (2-22)
3. APPLICATION
In this section we consider two examples of elec-tron cloud-capacitor interactions. These examples aremeant to be representative of some of the possibleinteractions in SGEMP simulations. The idea is that amodal oscillation has already been stimulated by thedirect interaction of the X-rays with the system. Anelectron cloud generated at the walls or damper of thesimulator drifts through the electric field region ofthe mode. In the example we are considering, theelectric field is normal to the direction of thedrift.
The first example in Section 3.1 deals with anelectron cloud between the capacitor plates where itis assumed (tenuous plasma assumption) that the totalcharge in the electron cloud is small compared to thecharge flowing through the inductor, from one plate ofthe capacitor to another. In addition, it is assumedthat negligible charge is lost at the capacitor plates(hence only the frequency of the LC current will beperturbed). This example shows that the frequency ofthe electron cloud-circuit system is larger than thefrequency (w0) of the circuit alone. The implicationis that when the total charge in the electron cloud islarge then the frequency can be considerablyincreased.
Section 3.2 considers the modal damping resultingfrom electrons which have a velocity parallel to theplates; again the charge in the electronic cloud istaken as small and the electrons are assumed not tostrike the plates of the capacitor. Apart from thefrequency modification term appearing in the exarnpleof Section 3.1 a damping tem also appears. Energy isessentially absorbed from the mode while the electronis between the plates. Since the plates have a finitelength, the energy is removed from the mode when theelectrons leave the region of the plates. Thisexample is analogous to a vacuum tank simulation inwhich electrons are emitted from a tank wall ordamper, absorb energy from a mode and then are lostfrom the tank due to impact with the wall or damper.
3.1 Motionless Electron Cloud
Here we assume that the electron cloud chargedensity is so small that the electron motion is deter-mined only by the charge density on the plates. This
w2d' << eaP e0n
(3-1)
where a is the charge density on the capacitor platesand wp the plasma frequency is defined by
2-2P2 e n
P cOm(3-2)
n is the average electron density, m is the mass of anelectron and d' is the extent of the electron cloud,where d > d'.
In addition, it is assumed that the electroncloud does not strike the capacitor plates. Thetendency for electrons to strike the plates dependsupon where the electrons are produced, over what dis-tance the electrons are accelerated by the electricfields of the capacitor and to what extent the elec-tron-electron interactions within the cloud can pushelectrons into the plates. Electrons which are accel-erated into the plates constitute a possible lossmechanism which will not be considered in this paper.In what follows, we consider the implications of theassumption that the electrons do not collide with theplates.
It should be noted that the electrons in thecloud are accelerated in a sinusoidal manner due tothe charge density on the capacitor. The chargedensity varies sinusoidally with the modal frequency,w. The oscillation of the electrons has an amplitude
eme0 w2
where a is the charge density on the capacitor. Therequirement that the electron not be accelerated intothe capacitor plates is then the statement that theaverage position of the electrons cannot be withind-d' of the capacitor plates
e a < d-d'meO W2 (3-3)
Equation 3-3 is assumed to hold for the problems underconsideration. Further, it is clear that the elec-trons produced or introduced withni d-d' of the capac-itor plates will be accelerated into the plates.
As already mentioned, in addition to the electronmotion produced by the capacitor fields, there is anadditional motion due to the Coulombic forces actingbetween the electrons in the electron cloud. An elec-tron near the edge of the cloud feels an accelerationof roughly
e2n d' 2 di= wmeo 2 P 2
We assume that the electron cloud is sufficiently ten-uous that over some, sufficiently long, period oftime, say roughly N modal periods (N/w), will notaccelerate the electron across the gap d-d' and col-lide with the capacitor plates. That is,
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2 21 (2 N dN ) N2 < d-d' . (3-4)2 P 2 w 4 wo
Note that this effect is important if the plasmafrequency is larger than or comparable with the modalfrequency, w.
We now calculate the frequency shift due to theelectron cloud. Assuming that the electric field,a/o =- E, has a time variation characteristic of themodal frequency
E = Eu exp[iwt], (3-5)the same time variation for the velocity of electrons,v, initially at rest is
e Euv =e- exp[iwtj . (3-6)m ilw
In order to predict the frequency shift due to theelectron cloud, the current flowing across thecapacitor, Io, defined by equation 2-10 must beevaluated. This integral can be determined byconsideration of the voltage drop over d'
I ° A= dx - Eod 0 miw
assuming E0 is constant in spaceequation 3-1 is valid)
, 2
o~= C P d VLw d
(3-7)
(i.e., the inequality
(3-8)
3.2 Electron Cloud With a Velocity Parallelto the Capacitor Plates
For the case where the electrons are moving in adirection perpendicular to the lines of electricfield, the nature of the interaction of the electroncloud with the capacitor changes. The situation isdepicted in Figure 1. Where the electrons have aninitial velocity Po parallel to the plates and it isagain assumed that the electrons do not strike theplate (wp/wu < 1). We redefine Iu to take accountof the dimension y
Jo =eA ffdxdy J (x,y,t)dQ 00 x
(3-12)
Where Jx is only the current in thex-direction, relative velocities within the cloud arenow important. We integrate the equation of motion asin Section 3.1 except that we now include a time towhen the initial velocity in the x-direction is zeroand y=0.
From Newton's third law,
d (x) = e Eo eiwtdt m
where E0 is again taken as o/e0, then
eEo iewt ei wt0timw
then,
w S(w) CViw d
(3-13)
(3-14)
(3-15)
Then from equation 2-17where
Z(W) - V/l = 1w2d'
iwC + P-Ciw d
(3-9)
which is equivalent to an inductance (Cw2 d'/d)-1 inparallel with the capacitor. P
From equation 2-20 we must also have
(3-10),k,= 1 + i WL I .p d C}w2 iw d
since LC = w-2 . Hence
w = (W P+ dwP d
S(w) E f dy(e -te )
Si nce
to = t - Y
Therefore
S(w) = e
(3-11)
In this form, it is clear that the frequency shift ispurely real (no lossy processes occur) and the shiftis toward higher frequencies. This interaction ismost important when the plasma frequency, wpapproaches the resonant modal frequency, wu, and isstronger if d' + d, i.e., the extent of the electroncloud is large.
The impedance of the electron-cloud capacitorinteraction which is taken as in parallel with thecapacitor is therefore (from equation 2-17)
2dW 2u0 xwZ( W) - A= C{1 - R- exp [-i n] ( (3-19)
and the modal frequency is the solution of (fromequation 2-20)
2 ~2 + d' 2u0 xw@ o= W y+- - exp[-i sin (2)} (3-20)
Note that in the limnit uo/2w is small (the length oftimne the electron spends in the capacitor is manymodal periods (i.e., slow electrons), the case of thetenuous plasma is recovered
W = Vw+wp d'/d (3-21)
If the transit time across the capacitor is short,u0/2w large, i.e., fast electrons
2 = 2 2 d' i wo P d 2uo
=w2 + w2 d' iw0_0 P d 2uo (3-22)
This is equivalent to a resistor in parallel with theoriginal capacitor, and has the explicit solution
w2= ± wo + i p d
4uo d(3-23)
In this case, the dainping again increases with theelectron density (plasma frequency) and the transittime of the electrons across the capacitor.
modal frequency and the extent of the electron cloudnearly fills the capacitor.
4. APPLICATION TO A HYPOTHETICAL HIGH FLUENCE SGEMPEXPERIMENT
In this section we apply the formulas arrived atin the last section to a hypothetical SGEMP experi-ment. The idea is to show that the effects described,namely damping and frequency increases, due toelectron clouds, could be important. The calculationsare not predictions since the particular circumstancesof any given experiment vary and large calculatedmodal shifts actually will violate the assumptions ofthe calculation, namely that w Iwo < 1. The intentof this section is to highlight potentially importantphysical phenomnena that should be investigatedfurther.
Assuming a fluence 1of 10-2 cal/cm2 a pulse widthof 10 ns, a yield of 10 elect/cal, and an averageelectron velocity of 5 cm/ns,
A typical satellite external mode has a period ofabout 60 ns, or
=o 108 (4-2)
Looking at equations 3-11 we can see that for d' =d, areasonable case, that the frequency shift could bequite substantial
Note that, in general, the sol ution of the trans-cendental equation 3-22 is not difficult numerically;and for small changes in the modal period
2
a (Dow0 (1 +_ -d 1w2 d
- 2uO exp [-i '0Q'] sin(L)})V/kw0 2u 0 2
(3-24)
which is a uniform expansion in powers of the transittime. Note that the magnitude of the imaginary partof this frequency is oscillating with transit time.It is further possible to expand this square-rootassuming w2/w2 d'/d is small, then
2-+ 1 _ d {1 --sin(-
2 w0 d wo0 2u
+ i -2. (1 -cos(-----))}.w0 2u 0
(3-25)
Note that the frequency shift (real part) is no longerclearly increasing in frequency as in the tenuousplasma limit but rather the shift varies in both mag-nitude and sign in a manner which would be hard topredict without a precise understanding of the parame-ters involved. It is clear from this last expression,equation 3-25, that the shift in modal frequencydepends upon w d' /w2d again being significant when theplasma frequency is greater than or comparable to the
Aw 1. 56w (4-3)
Equation 3-11 was derived under the circumstances thatup/wo < 1, however, and so equation 4-3 is not anexact result. Electron densities would also be verymuch reduced by space-charge limiting. A reduction indensity by a factor of 10 would still lead to a 30percent frequency shift, however.
We refer to equation 3-23 to obtain the modaldamping. The damping constant 6 is
26 = wp d'
4 o d
For d'd', with equation 4-1 and Q=lm
6 (2.5x108)2 = 1.6X1094x5x107
or a decay time of 3.2 ns. A typicalsatellite mode is 40 ns.
(4-4)
decay time for a
5. CONCLUSIONS
In this paper we have been concerned with theeffect electron clouds may have on the free spacemodal response of a satellite. Those extraneous elec-tron clouds which are generated only as a result ofthe simulation, and do not have a free space counter-part, can alter the modal response of a satellite.
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The alteration could lead to an incorrect interpreta-tion of an experiment or undue confidence in theresults of a system test.
We have shown that the square of the plasma fre-quency (proportional to the electron cloud density) isthe key parameter for estimating the frequency shiftand damping effects. Spatial extent of the cloud andthe average velocity of the cloud are also importantparameters. The equation, which demonstrate the fre-quency shiift and damping are 3-11 and 3-23 respec-tively. Although these equations were derived underrestrictive conditions they clearly demonstrate thequoted effects. Future work is necessary to bound theeffects in the high electron density limit.
6. REFERENCES
1. Fromme, D.A., et al, "Exploding Wire Photon Test-ing of the Skynet Satellite," IEEE Transactionson Nuclear Science, Vol. NS-25, No. 6, December1978, pg. 1349
2. van Lint, V.A.J., and D.A. Fromme, "Modal Char-acterization of the Skynet Response to Electricaland Photon Stimulation," IEEE Transactions onNuclear Science, Vol. NS-26, No. 6, December1979.
3. Dancz, J., J. Smyth, R. Stettner, V. van Lint,"SGEMP Studies - Final Report, Volume I - HuronKing Theoretical and Data Analysis," MRC R-642,May 1981.
4. Erler, J.W., V.A.J. van Lint, "Starsat Modes atHuron King," MRC/SD-R-94, December 1981.
5. Stettner, R., et al, "Satellite SGEMP SimulationFidelity Criteria," IEEE Transactions on NuclearScience, Vol. NS-26, No. 6, December 1981.
