IMPROVING SURFACE CURRENT INJECTION TECHNIQUES VIA ONE- AND TWO-DIMENSIONAL MODELS*
J. W. Williams and L. T. SimpsonMission Research Corporation
1400 San Mateo Blvd., S. E., Suite AAlbuquerque, New Mexico 87108
K. S. KunzLuTech, Inc.
First National Bank Building East, Suite 1619Albuquerque, New Mexico 87108
Abstract
A basic objective in the development of a surfacecurrent injection technique (SCIT) is to provide aninexpensive, transportable simulator which will allowelectromagnetic pulse (EMP) hardness checks of aircraftin the field. The development effort has been conductedas a combined theoretical and experimental study. De-tailed solutions for free field scattering problems areobtained from computer models such as the THREDE finitedifference code. It has been found that simplified one-and two-dimensional models can be used to provide in-sight and understanding less easily obtained from moredetailed numerical models. These models appear to beespecially useful in the study of basic symmetry andpolarity requirements which are likely to be employedin any direct injection scheme. In this paper, elec-trical and mechanical analogs are developed and appliedto EMP simulation by direct injection. Predictions ofthe simplified models are compared to results obtainedwith the THREDE finite difference code.
Introducti on
The concept of symmetry is often used to reduce thecomputational labor required in the solution of mathe-matical problems. Of perhaps greater importance, ex-aminations of symmetry can provide physical insight notreadily obtained from numerical solutions of complexequations. In the subatomic domain, dynamical symme-tries have yielded guidance in the absence of a morecomplete physical theory. Although the calculationalspeed and storage capacity of modern computers havealleviated some of the more stringent requirements forcomputational simplicity, the need for physical under-standing remains undiminished.
In this paper, we will discuss some of the symme-tries and simple physical models which were found toprovide practical guidance in the development of an EMPsimulation technique. The word "simple" is occasion-ally applied to some rather complex problems. We areusing "simple" to describe mathematically tractablemodels which do not yield detailed estimates of EMPresponses. These models are one- and two-dimensionalsimplifications of some real system. Both physicalpictures and mathematical solutions provided by thesemodels were found to be beneficial in the developmentprogram. The sections below summarize the specificdevelopment effort and discuss one- and two-dimensionalmodels which have been used in this program. Predic-tions of the simplified models are compared with esti-mates obtained from more detailed models.
Background
During the late 1970's, experimental and theoreticalefforts were directed toward development of a practicaltechnique for conducting EMP hardness checks of militaryaircraft in the field. From a pragmatic viewpoint, the
*Work supported by the Defense Nuclear Agency and per-formed under Naval Surface Weapons Center ContractN60921-77-C-0117.
equipment must be transportable and fairly simple toassure reliable operation under the environmentalstresses likely to be encountered during field tests.The quality of the technique is a primary concern,where quality refers to the capability to replicate adesired EMP environment. As is always the case, oneanticipates necessary compromises among fidelity, cost,and complexity.
One concept for a portable EMP hardness surveil-lance tool is the direct injection of charge and currenton the surface of a test object. This is historicallyreferred to as a surface current injection technique'.In the case of EMP hardne:s checks, as well as in moreelaborate EMP assessments conducted at fixed site EMPfacilities, responses of cables and subsystems insidethe enclosures are the primary concern. Analyticalsolutions for interior currents and voltages are dif-ficult to obtain due to the complexity of militarysystems. Hence, free field simulators are usuallyemployed to replicate as closely as possible the ex-terior surface responses expected to exist in an EMPenvironment. Since direct injection techniques areintended to complement fixed site simulators, one ob-jective in the SCIT development program is to repli-cate exterior responses obtained in a fixed site simu-lator. EMP evaluations of aircraft often include thecase in which an incident E-field is directed parallelto the fuselage with the aircraft centered relativeto the pulser. Hence, the development effort was ini-tially directed toward simulation of symmetric illumi-nation.
One of the simpler SCIT techniques one can envisionwould employ a high voltage generator to directly in-ject current and charge on the outer surface of a testobject at a single location. Early in the developmentprogram this approach was evaluated both experimentallyand with the finite difference code THREDEL. In termsof replicating the response of a symmetrically illumi-nated aircraft, results obtained with a single pulserand injection location were found to be of very lowquality. More elaborate schemes involving severalpulsers and injection locations yielded a modest degreeof success in simulating symmetric illumination. Figure1 illustrates a two-point injection scheme which was
Delta Synchronous DeltaModul e Tri gger Module
+ Polarity System - Polarity
Figure 1. Test configuration for two-point injection.
found to provide a noticeable improvement in qualitywhen compared to the results obtained with single pointinjection. Two-point injection was found to work bestwhen injection points of opposite polarity were locatednear the ends of the fuselage. Various values of theisolating resistance were investigated.. For values ofresistance large in comparison with the characteristicimpedance of the fuselage (treated as a cylinder over aperfect ground), the two-point injection scheme gavethe correct harmonic responses and yielded a reasonablyaccurate fit to the desired response at frequenciesnear the fundamental resonance of the aircraft fuselage.However, this scheme invariably overemphasized the highfrequency response. Improvements in the quality of thetechnique at higher frequencies were desirable, partic-ularly since the replication of the interior responseswas of ultimate interest in the program. At this pointsimplified models were investigated to provide a morefundamental understanding of physical processes occurr-ing during direct injection and to formulate generalapproaches toward improving the quality of the existingtechnique.
One-Dimensional Models
The physical systems of interest in this specificapplication is an aircraft. We can simplify the analy-sis by temporarily ignoring the wings and treating thefuselage as a conducting cylinder. Reducing the com-plexity still further, a one-dimensional analog couldbe selected as a balanced transmission line or a vibrat-ing string as illustrated in Figure 2. In the case of
0~~~~~-
ip(z)~~~~~~z
Figure 2a. Stringanalog.
-Q o Q
Figure 2b. Transmissionline analog.
a string with fixed end points immersed in a medium withelastic restoring force and a frictional force propor-tional to velocity, the equation of motion can be writ-ten as3:
22a2k =1 9 2k a + 112az2 c2 at2 C2 3t +F
where p(z,t) symbolized the displacement of the stringfrom equilibrium, and velocity of propagation c is thesquare root of tension divided by density. Here the2coefficient of friction is represented by k, while , 4denotes the restoring force. An analogous equation fora uniform, balanced transmission line is:
ar. a r + (QW + -?*) ->+.W;7- a 2a (2)
where r = (y) with v(z,t) denoting the line voltage andi(z,t) representing line current.
We denote the impedance per unit length as:49R+ SY
and admittance per unit length,& = (9+Ss,
(3)
(4)where s is the Laplace variable. The uniform transmis-sion line is characterized by propagation coefficienty - al + jS = IW9 and characteristic impedance Zo =
(1)
1/Yo = Vg/*. Equations 1 and 2 demonstrate the math-ematical equivalence of these simple mechanical andelectrical analogs. We now examine both one-dimensionalsystems in more detail beginning with the transmissionline model.
Solutions to the transmission line equations in thefrequency domain are obtained below using the Green'sfunction approach4. In order to emphasize the effectsof spatial symmetry upon mathematical solutions, thetransmission line will be positioned with origin atthe center of the line as shown in Figure 2b. If apoint voltage source is located at z = z', then Green'sfunction is given by:
IG(z,zo) = (Yo/A1)li + a2e2Y(z )1EeY(z z )
+ a1e UY(z+zz+L)U -Z) + (YO/AI)
[1 + a e-2Y(z+k)][e-y(z-z )
+ a2e (z+z )L)1 U(z-z,) (5)
where reflection coefficients al and U2 are defined interms of termination admittances Y1 at z = -L/2 = -Qand Y2 at z = Q as:
(6)=i (Yj - YO)/(Yj + YO) (i = 1,2)
Symbol A1 represents the quantity
l1 2(1 1a2e2)while U(x) is the usual step function
U(x) = 0 x < O
U(x) = 1 x > O
Similarly, the Green's function for a point currentsource at z = z' is given by:
vI(z,z') = (Z /A2)[1 + p2e ][e
+ pleY(z+z+L)]U(z -z) + (Z /A2)
1 + ple2Y(z+)] [e-Y(z-z-)+ p2eY(z+z L)]U(z_z )
where p, and P2 are reflection coefficients
andp= (Z i Z0)/(Zi + z0) (i = 1,2)
A2 = 2(1 - p1P2e 25L)
(7)
(8a)
(8b)
(9)
(10)
(11)
Since the analysis will eventually be applied to an
aircraft illuminated by an incident plane wave, wespecialize to the case of a transmission line which isopen at both ends. In the case of a point voltagesource, reflection coefficients al and G2 reduce to
a1= a2 = -1, and Equation 5 can be recast:
Iv(z,z9) = Y csch(yL) {sinh[y(k-z-)] sinh[y(k+z)]U(z8-z)
+ sinh[y(Q+zi)] sinh[y(t-z)]U(z-z)} . (12)
Similarly, for a line with open circuit terminations,reflection coefficients p, and P2 reduce to unity and
1846
in region 2 containing the sources, and
VG(z,z ) = Z csch(yL) {cosh[y(t-z)] cosh[y(k+z)3U(z"-Z)G ~~0
+ cosh[y(z+zl)] cosh[y(k-z)]U(z-z-)} (13)
Both the voltage and current Green's function are pro-portional to csch(yL) so that sharp resonances willoccur for a low loss line near singularities of csc(BL).The condition sin(3L) 0O defines resonant wavelengthsAn = 2L/n, n = 1,2...... . Thus, either type of exci-tation will in general excite even and odd harmonicson a transmission line with open circuit terminationat both ends.
Since symmetric illumination is of eventual interestin this application, we examine symmetric and antisym-metric excitations. For convenience, the line is sub-divided into three regions with sources confined toregion 2 as shown in Figure 3. Region 1(3) is to theleft(right) of current and voltage sources. General
.p I.
Region 1 Region 2 "Region 3(no (contains '(nosources)i sources) sources)
_p , t
- - z 0 z Q
Figure 3. Division of the transmission line intoregions.
solutions are given in terms of the Green's function as
IV(z) =f-(z)I V(z,z-)dz- (14a)
and
V(z) = -(1IS)f/(z ) dz IV(z,z)dz- (14b)
with analogous equations for the case of current sourceexcitation. For example, if the spatial location andpolarity of current sources is selected so that g(z-)= - z-z), then the line voltage is:
V(z) = -Zosech(y£) cosh[y(k+z)]z
x 0 .9 (z-)sinh(yz-)dz-
for region 1 to the left of the sources,
V(Z) = -Zosech(y£){cosh[Y(k+z)I],Zo0
xf . (z-)sinh(yz-)dz- + sinh(yz)
zX J6(zi) cosh[y(k+z-)]dz` (151-z0
V(z) = Z0sech(yz)>cosh[y(k-z)]I 6 (z )sinh(yz-)dz
Z>O~ ~ ~ oz>0 0inh(yz (
+ sinh(yz)f 9(z-) cosh[y(t-z-)1dz-1 (iSi
V(z) = Z sech(yQ) cosh[y(k-z)]0z
I(z-) sinh(yz')dz0
(15d)
in region 3. Line voltages and currents are propor-tional to sech(yZ) in this case. For a nearly losslessline, resonant frequencies will occur at the singular-ities of sec(SZ), or where Am = 2L/(2m + 1), m = 1,2...Thus, for the case .9(z-) = -o(-z'), only odd harmonicswill be excited. Similarly, Equations 12 and 14 can becombined to show that if 1/(z-) = -%i(-z'), it is alsotrue that only odd harmonics will be generated. Forthe cases .(zW) = 9(-z-) and t/(z") = -f(-z), linecurrents and voltages are proportional to csch(yQ), sothat only even harmonics are excited.
If we further restrict the source term to be of theform,
g(1W) = I06(z-z0) - I0(z+z0) I (16)
where symbol 6 represents the Dirac delta function,then Equation 15 gives;
V(z) = -I Z0 sech(y9) sinh(yz0) cosh[y(Q+z)]
V(z) = IoZ sech(yQ) cosh[y(k-z )] sinh(yz),2a 0
and
V(z) = I0Z0 sech(y£) sinh(yz0) cosh[y(i-z)],003
(17a)
(17b)
(17c)
where Vi(i = 1,2,3) denotes the line voltage in the ithregion.
Figure 4 summarizes the results for pairs of sym-metric and antisymmetric sources. Notice that theseone-dimensional solutions predict that source locationsalong the line can be selected so as to suppress orenhance individual harmonics in the responses. Forexample, in the case gI(z ) = -.(z), responses areproportional to sech(yk)sinh(yzo) or sech(yk)cosh[y(k-zo)]. Hence, position zo can be chosen so as to ex-tremize values of sinh(yzo) or cosh[y(k-zo)] at adesired frequency. Figure 5 illustrates the partialsuppression of the third and seventh harmonics on ahypothetical low loss line with current sources offsetslightly from zo = ±L/3. Suppression of the third har-monic in the line response could also be achieved bylocating point voltage sources near z = +L/6 with-V(z ) = V(-z )
Returning for a moment to the string analog, Figure6 illustrates the excitation of the third harmonicalong the string by a pair of point sources. In analogywith ideal current and voltage sources used in thetransmission line analysis, we can envision excitationby mechanical sources which either displace the stringnormal to the line which joins the supports at each endof the string, or sources which twist the string aboutsome symmetrically located positions. As indicated inthe figure, locations of the point sources can be se-lected to either enhance or suppress the third harmonic.It follows from this simple picture, that one can hypo-thetically add additional pairs of point sources so asto achieve complete control of the harmonic structure.
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Equation 9 can be rewritten:
(v)
9(Z ) = - #(-Z )
0 o
sinh(yz )or
cosh[y(Q- z )]
. sech(y),/( z') = /( -z )
-zo z
cosh(yzo )
(I) or
sinh[.tY(-z )]
Enhancement ofthe 3rd harmonicI
~I I~
Suppression ofthe 3rd harmonic
4)3
I
Figure 6. Excitation of the 3rd harmonic bysymmetrically located point source.
4a. Symmetric Excitation
(z ) = - V(-Z )
oo0z
cosh(yz )(V)~ - or
cosh[y(2-z )]
-zo z°
(V)sinh(yz )
or
sinh[y(k-z 0)]4b. Antisymmetric Excitation
Figure 4. Transmission line solutions for the caseof two point sources arranged symmetricallyor antisymmetrically.
+ I
_.~~I +-
z = -Q -2y/3 0 2Q/3 Q
Amplitude Isech(XL)sinh(2XL/3+6)I
0'4-0
04.)
Figure 5. Partial suppression of the third andseventh harmonics by sources slightlyoffset from z = +L/3.
Comparison of One-Dimensional PredictionsWith A Detailed Model
As an application of the predictions obtained fromthese one-dimensional models, excitation of a conduct-ing cylinder was examined for several types of excita-tion with the THREDE finite difference code. Figure 7shows the configuration tested for a cylinder 5 m inlength and having a 0.47 m diameter. For the completeband of voltage sources shown in Figure 7a, results arein agreement with predictions of the one-dimensionalmodels. Only the odd harmonics are excited by thissymmetric configuration and the third harmonic can besuppressed or enhanced by selection of the sourcelocations (see Figure 8). However, because the voltagesources form a complete band about the test object, azi-muthal variations in the response are precluded, thusreducing the response to effectively one-dimension.Figure 7b shows another type of symmetric excitation bypoint sources near opposite ends of the cylinder. Asindicated in Figure 9, the symmetry and polarity of thesources results in an odd order harmonic structure.However, suppression of the third harmonic was found tobe minimal for all of the injection locations which weretested including those which should have resulted in anoticeable reduction according to the one-dimensionalmodels.
Since the two-point injection scheme did not allowcontrol of the harmonic structure of surface responsesas predicted with the one-dimensional analogs, alterna-tive means of reducing the relative contribution of thehigher harmonics were investigated with the THREDEmodel. Recalling the test configuration illustratedin Figure 1, two potential methods are immediatelyclear. One method is to inject a waveform with rela-tively little energy content at the higher harmonicfrequencies. Another method is to vary values of theisolating resistance.
Figure 10 shows a comparison of the surface re-sponses obtained when the cylinder is excited with two-point injection and a pulser waveform which varies asAe-cttsin wot, where wo was selected to be the dominantresonant frequency for the conducting cylinder and acwas chosen to be the damping rate calculated for thecylinder in free space. The response of the cylinderwhen excited by a double exponential plane wave in freespace is also shown in Figure 10. For ease of compari-son the SCIT response has been multiplied by a factorof 17. Notice that the relative harmonic content ismuch improved when contrasted to the response shownin Figure 9 which was obtained by directly injecting adouble exponential waveform.
Results obtained by lowering the value of the iso-lating resistances are illustrated in Figure 11. Thismethod appears to be less promising than signal con-ditioning, since the relative reduction in amplitude
(v) -
1848
(V) , csch (yt)ITOA
AZ") = A-Z')
E E
)-4 - X
Figure 7a. Conductingcylinder excited by bandsof voltage sources.
io-64-'.,_
Cas:
14-'CcuS.-
S- -7
to
33
0 0-
4-'
'- 10
Figure 7b. Conductingcylinder above a groundplane excited by twopoint sources.
1UFrequency (MHz)
Figure 8. Suppression of the thirdbands of sources.
r0
LLJ
z10
S.-
4-'
14-0
a)-0
4-'
0.-
Frequency (MHz)
.4-
0
V)
a)
C.) I
(0
4-
a)
4-
E.1-0X
4E
10 100Frequency (MHz)
Figure 10. Comparison of surface response under SCITexcitation with damped sinusoid waveformand under illumination by double exponen-tial plane wave.
1 n- 55L100
harmonic with
iU
>)4-'
a)
4-
C
w-6S.
10
=3u
N
10o
a,
.C~0
4-'
QL
-\ / >\-
, \ SCIT with 2\\ Isolation Ri
Double ExpPlane Wave
VIK.
*i
io-8L
100
Figure 9. Typical results for two-point injection.
between the first and third harmonic was usually accom-panied by an undesired alteration in the shape of re-sponses near the fundamental resonance.
Thus, while the one-dimensional models are usefulin predicting basic symmetry and polarity requirementsfor the simulation, some predictions of the one-dimen-sional models are clearly at variance with those ob-tained from more detailed calculations. We will nowexamine a two-dimensional model which is intended tofurnish a more realistic assessment of the harmoniccontrol which can be expected for direct injectionschemes.
Scale Iactor = 14.5I ,1
Frequency (MHz) 100
)0 ohm!si stancenential
Figure 11. Comparison of surface response under SCITexcitation with isolation resistancelowered to 250 ohms and under illuminationby a double exponential plane wave.
Two-Dimensional Model
A two-dimensional model of the test object is re-quired when the excitation sources are point sources,as for direct current injection, or localized, as whenan E-field source illuminates only a small portion ofthe scatterer with appreciable fields. In this casenot only axial but circumferential variations are allowed.More complex modes can then exist than those posited bythe one-dimensional model. Adjustments in the attach-ment location to suppress higher order modes, calculatedusing the one-dimensional model, will only hold formodes that vary only axially. To understand how tocontrol all higher order modes requires a two-dimen-sional model.
1849
I 11 I
Bands L/3Inboard -
Bands L/6---Inboard
- o \
II~\11
/ \_L-
i r%3L
The simplest two-dimensional model treats the testobject as a cylinder in which the end caps are ignored.Point or localized sources are the only sources ofinterest since azimuthally invariant sources do notrequire a two-dimensional model. Only SCIT in theform of two point, opposite polarity current injec-tion will be considered here as this is a representa-tive system of significant interest. The SCIT pointsources are located along a line running longitudi-nally down the cylinder. Currents flowing into thecylinder at the injection points split circumferen-tially, running in opposite senses about the cylinder.If the cylinder is cut along the line between the in-jection points and unfolded to form a plate, the cir-cumferential and axial current behavior can be expressedin terms of the allowed modes which vary as
'inn=~Vmn (lPmn ^
+ ~~mn (18Imn V;mn ( ax ex ate) (18)
where x is the axial coordinate and g is the cylinderci rcumference.
If we require that current vanish at xexhibiting peaks of opposite polarity at g
== C, then
='mn -sin L sin C so thatmn L
I + I
= +L/2 w= 0 and
ihile
(19)
mn = mn)AXIAL "x + (Imn)CIR. g (20)
where
(Imn)AXIAL = L cos -sin C (21)
and(I ) niT sin
mTxcos
n (22)mn)CIR. C L C
as desired.
An arbitrary current can be expressed as
I = E mnImn (23)m n
Two of these modes are of most concern to us. Theyare I30 and Ill which for the cylinder we have beenstudying (L= 2.5 C) have frequencies within 10% of eachother. Locating the injection points to where they willsuppress I30 as predicted by the one-dimensional modelresults in only modest changes in the response as ob-tained by THREDE calculations (Figure 9). The reason isthat while I30 is being suppressed, Il is not. Clearlyall modes must be considered up to some frequency limitof interest if waveform fidelity is to be obtained.
For hardness surveillance, an upper frequency limitequal to the resonance frequency for I50, the fifth har-monic for a transmission line model, seems reasonable.Using the cylinder dimensions of L 2.5 C and statingfrequency as a multiple of the fundamental resonanceyields the results of Table 1. Only five modes must beconsidered. However, two of these modes involve circum-ferential as well as axial variations. By injecting topand bottom at each inboard location, selected to suppressthe axial variation only modes, these circumferentiallyvarying modes, Il and I31, will be pushed up in fre-quency to 5.1 fo and 5.8 fo respectively, just above thefrequency limit of interest. A higher frequency limitof interest would of course require more complex fixessuch as increasing the number of injection points orlocating the points circumferentially about the testobject.
Table 1. Modal Frequencies < 5 f
m 0 1
1 f0 2.7 fo3 3 fo 3.9 fo5 5f0 -
Conclusions
Simple one- and two-dimensional models offer insightand practical guidance in the development of a portablesimulation technique. One-dimensional models were foundto be useful in establishing general symmetry and polar-ity requirements for the simulation. However, the one-dimensional models are significant simplifications ofthe real system which lack sufficient detail to accountfor some effects associated with direct injection. Inparticular, circumferential variations in surface currentcould not be addressed with the one-dimensional model.Hence, two-dimensional models were developed to comple-ment the simpler one-dimensional analogs. Predictionsof simple models combined with frequent comparisonsagainst empirical data and calculations of more detailedanalytical models have been found to provide the bestresults.
For the specific application of interest, improve-ments in the quality of the SCIT simulation can be ob-tained if the test object is excited by bands of sourceswhich reduce the geometry to one-dimension. Point in-jection offers an advantage in simplicity, and may wellbe the more pragmatic approach, particularly if signalconditioning is employed to shape surface responses inthe frequency domain. Predictions of the simple modelsdiscussed above indicate that a two point injectionscheme is the least complex scheme which is likely toprovide a useable simulation of symmetric illumination.Control of the relative contribution of higher harmonicsin surface responses can be achieved by increasing thenumber of injection points beyond two. However, theadvantages of harmonic suppression or enhancement bythis method should be viewed in terms of the tradeoffbetween complexity and quality which is particularlyimportant for systems which must operate in the field.
References
1. Kunz, K. S., et al, "Surface Current InjectionTechniques: A Theoretical Investigation," IEEETrans. Nuc. Sci., Vol. NS-25, No. 6, December 1978.
2. Holland, Richard, "THREDE: A Free-Field EMP Cou-pling and Scattering Code," IEEE Trans. Nuc. Sci.,Vol. NS-24, No. 6, December 1977.
3. Morse, Philip M. and Herman Feshback, Methods ofTheoretical Physics, New York: McGraw-Hill, 1953,pp. 120-141 and pp. 1332-1349.
4. Weeks, W. L., Electromagnetic Theory for Engineer-ing Applications, New York: John Wiley and Sons,Inc., 1964, pp. 102-111.
