48 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. 1, FEBRUARY 1988

Scaling Laws for Coupling Quantities on EMP MICHAEL P. BERNARDIN

Abstract-A set of scaling laws relating three potential electromagnetic coupling quantities to three electromagnetic pulse (EMP) characteristics is derived. The electromagnetic coupling quantities attempt to characterize in a very rough sense the ability of an electromagnetic wave to induce an electronic upset in an electric circuit. The coupling quantities used in this study are chosen based on voltage transients, power transients, and energy-induced overload as mechanisms for electronic upset. The inde- pendent quantities in the scaling laws are chosen to be the pulse rise time, peak field, and total energy density. Against these scaling laws, experi- mental data may be tested as a means of further enhancing the understanding of electronic upset from circuit exposure to electromag- netic waves.

Key Words-EMP, coupling, scaling laws. Index Code-Mle.

I. INTRODUCTION A. Background

PROBLEM under active research is the study of the A coupling of electromagentic energy to electronic systems. Typically, the problem is described by a plane-wave electro- magnetic signal incident on a cannister containing electronic circuits. The electromagnetic wave may permeate the cannis- ter by coupling to antennas and/or cables, or through a variety of cannister-apertures, including seams, windows, and cracks. The transfer function describing the coupling depends, in general, on the frequency content, amplitude, and wave- polarization of the wave. Once inside the cannister, the electromagnetic wave may couple energy into electronic components. Wave-energy coupling to circuit board compo- nents is a very complex phenomenon, depending on such factors as circuit lead lengths (for wave-circuit resonances), cannister spatial dimensions (for cavity resonances), and the relative spacing of circuit boards to cavity walls (for whole- circuit resonances). Because of the large number of variables in any given problem, attempts to predict an electronic upset or failure of a system from exposure to electromagetic radiation are extremely difficult.

In light of this difficulty, one way to proceed toward developing a predictive capability for electronic upset (the approach taken here) is to neglect the severe complexities that exist for each of the problem subelements and to attempt to prescribe an all-encompassing or global coupling quantity. I

An immediate question that arises with this approach is just

Manuscript received March 17, 1987; revised September 22, 1987. The author is with Los Alamos National Laboratory, Los Alamos, NM

IEEE Log Number 8719086. ' The approach of replacing a series of complicated problems with a single,

rather simplistic one has proven to be fruitful in the field of magnetic fusion, where plasma confinement times for new experimental devices have been traditionally based on semi-empirical scaling laws, as opposed to detailed theory.

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which global coupling quantity out of an exhaustive set of choices is most relevant for this type of prediction?

In order to progress toward discerning a most relevant coupling quantity, a more fundamental question needs to be addressed. Suppose that it is possible to experimentally subject electronic systems of a particular target set to electromagentic pulses (EMP's) where the rise time of the pulse, the peak field, and the total energy density in the pulse are varied as independent quantities. In such a systematic experiment, one might be able to empirically determine the relative importance and scalings of the three experimentally controllable parame- ters on electronic system upset or burnout. In order to interpret the data in terms of a most relevant coupling quantity for the particular target set being tested, one must address the question, how do candidate coupling quantities depend on the pulse rise time, peak field value, and total energy density?

B. Relationship of Coupling Quantities to Pulse Characteristics

It is the purpose of this paper to derive relationships between a few candidate coupling quantities and the three aforementioned characteristics of the incident EMP. These relationships are a mechanism to provide a connection between engineers performing the experimental vulnerability tests of electronic hardware to electromagnetic pulses, and the theorists who are attempting to provide a predictive capability of electronic upset levels based on electromagnetic coupling studies. The connection to be constructed is to provide a relationship between parameters that the engineer may control and electromagnetic coupling quantities of interest to the theorist.

To select potentially relevant coupling quantities for study, it is necessary to consider potential mechanisms that may cause electronic upset or kill. Clearly, rapid fluctuations or large transients in either voltages or power may disrupt the normal functioning of a circuit. The voltage between two circuit nodes is equivalent to the integral of the electric field along the connecting path:

v= ib Ec . 2 a

In this equation, is the electric field in the circuit at the point of interest. A surge in voltage in the circuit is then related to the local electric field by simply

V-Ec.

To relate the local electric field in the circuit to the imposed external electric field (of the EMP) requires detailed knowl-

0018-9375/88/0200-0048$01 .OO 0 1988 IEEE

BERNARDIN: SCALING LAWS FOR COUPLING QUANTITIES ON EMP 49

edge of all the aforementioned complexities that we wish to avoid. Thus, we make the assumption that the fields coupled to the circuit are proportional to the externally imposed fields. In this way, a voltage surge is to be characterized by the coupling quantity (E),,,, the maximum value of the time derivative of the EMP electric field. We make a similar assumption about a power surge in an electronic circuit. Since the power imposed by an external EMP waveform is proportional to the square of the electric field (through the Poynting vector relation), we characterize power surge by the coupling quantity (E,!?)max. Thus, (,!?>,,, and (E,!?),ax will be chosen as two coupling quantities for this investigation. Another potential mechanism for electronic upset or kill may consist of a more gradual hardware degradation from a prolonged energy overload. Here, it is likely that the most pertinent spectral components of the pulse have wavelengths that make relevant points-of-entry dimensions resonant. Electronic systems under experimental testing for vulnerability levels at two U.S. Department of Energy (DOE) national laboratories include computers, ra- dios, circuits in cannisters, and missiles. For these objects with spatial dimensions on the order of meters, the most important frequencies lie between roughly 100 MHz, and 1 GHz. Thus, the quantity E * , designating the energy density in the 100-MHz-to-1-GHz frequency regime, will be chosen as the third coupling quantity.

It should be pointed out here that we are not proving that these are the only or necessarily the most relevant coupling quantities. Rather, the intent is to state that these are three potential coupling quantities with some heuristic justification that are to be related to three EMP characteristics. Once the scaling laws have been developed, it will be up to the engineers conducting experimental tests to identify which, if any, of the coupling quantities apply to their test objects. Then, in order for any theoretical vulnerability models or numerical codes to be able to establish robustness in predicting vulnerability levels for these tests objects, it will be necessary for these models or codes to be able to recover the results of the scaling law relating the identified coupling quantity to the EMP characteristics. This is the essence of the connection we are attempting to construct between the theoretical and experimental RF-vulnerability communities.

The relationships we wish to establish between the coupling quantities and the three EMP characteristics are to take a power-law form. Hence, if Q designates a coupling quantity, then the mathematical relationship to be defined is of the form

Q - ~ ; E ; E T (1)

where a, 0, and y are the parameters to be determined. In (I) , t,, E,,, and cT denote the rise time, peak field, and total energy density of the pulse, respectively.

At first glance, it might appear that the three quantities chosen to characterize the EMP may not be independent quantities. We choose to defer detailed discussion of this point until the waveform used in this study is presented; suffice it to say, for the moment, that a function of three independent quantities, e.g., f = f (a, b, c), can be formally inverted such that the dependent quantity and an independent quantity interchange roles, e.g., c = c (a, b, f).

An outline for the remainder of this paper is as follows. In the next section, the waveform used for this study will be defined, the relevant EMP characteristics will be calculated, and parameter studies to tabulate coupling quantity values as a function of EMP parameter values will be described. In the following section, scaling laws will be developed from the parameter-study data. In the final section, results shall be discussed and some conclusions will be presented.

11. COUPLING QUANTITY CALCULATIONS In order to calculate relationships between the coupling

quantities and the EMP characteristics, it is necessary to choose an electric field waveform. The waveform chosen should closely approximate actual EMP waveforms, yet not be so complicated as to make calculations extremely difficult.

The waveform chosen for this study is the ratio of exponentials :

(2)

In (2), the quantities A , a, and b are like the amplitude, rise rate, and decay rate of the pulse. This waveform is advanta- geous from the viewpoint of closely approximating actual EMP waveforms. Further, it does not suffer from the infinite initial rise rate inherent in the double-exponential waveform. A deleterious characteristic of the chosen waveform is that it is noncausal; that is, it has a finite value at t = 0. However, this aspect is not a serious detriment from a practical standpoint since the value of the electric field at t = 0 can be chosen to be as small as one wishes.

We now derive expressions or describe methods for obtaining the three EMP characteristics we wish to vary from the waveform given in (2).

This rise time and the peak field are easily calculated. The rise time used in this study is defined to be the time interval during the rise of the pulse bounded by the electric field values of 10 and 90 percent of the pulse's maximum value. This time can be calculated numerically by setting the left-hand side of (2) to O.lEp and O.9Ep, respectively, calculating the associ- ated times, and computing the time differential. The peak field can be calculated analytically from (2). The result is

e a ( [ - @ E ( t ) = A

1 + & + b K - @ )

E,,=A (~>"""+"' a . (3) 1 +-

b Calculating the energy density in the pulse is somewhat

more involved. Since we also need to calculate the energy density in the 100-MHz-to- 1 -GHz frequency range, we shall develop both calculations in tandem.

Let E(t) be the electric field in the radiated waveform; then the radiated power per unit area as given by the Poynting vector is

(4)

The total radiated energy per unit area is obtained by

50 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. I , FEBRUARY 1988

integrating (4) over time; designating this quantity by E , we have

parameter study are roughly 0.6-2.6 ns, 29-100 kV/m, and 1- 20 pJ/cm2. The parameter values associated with a nuclear explosion have been reported to be on the order of a few nanoseconds, 30 kV/m (saturated value), and a few micro- joules per square centimeter, respectively [ 11. If one assumes these values, then the ranges of data selected encompass these

coupling quantities. In this way, the introduction of noise through the statistical data reductions should be minimized.

E = - i I , ! ? ( W ) ~ ~ dw (6) For illustrative purposes, an overlay-plot of the waveforms used in the cT variations is displayed in Fig. 1.

1 m

POC - m ( 5 ) E = - s IE(t)I2 dt.

This can be converted to an integration Over frequency data but extend in directions so as to enhance the values of the space by using Parseval's theorem. The resulting expression is

1 -

27rpOc - a

111. SCALING LAWS

function of the three EMP characteristics are formulated. The scaling of each individual coupling quantity on each individual EMP characteristic is developed independently from the data

where ,!?(a) is the Fourier transform of E(t). From this expression, it is apparent that the quantity I,??(o)l 2/2ap0c

interval. represents the energy radiated per unit area per unit frequency In this section, scaling laws for the quantities as a

The Fourier transform of the electric field given in (2) is

7r csc (S .) . (7) ,!?(a) =Ae-iw'o - a+b

The square of the magnitude of this quantity is straightforward to calculate and is given by

where

A = sin2 (2) cosh2 (""> a + b

+ cos* (E) a+b s i h 2 (2) .

It is now quite easy to calculate the energy density in any frequency interval. Defining the limits to be o0 and 01, we have

A 2 n w l d o poc(a + b)2 SUO d E * = (9)

To calculate the total energy density, w0 and wl are set to zero and infinity, respectively.

In this study, the three coupling quantities are calculated numerically. The quantity E* is calculated from (9), while the quantities (l?),,,,, and (El?),,,ax are easily calculated from (2).

Scaling laws relating the coupling quantities to the three EMP characteristics are most efficiently effectuated if the coupling quantities can be computed by varying one EMP characteristic at a time, a process complicated by the fact that the controllable parameters are a, b, and A , and that these quantities are related to the three EMP characteristics through highly nonlinear equations. However, with a sufficient amount of patience, single-parameter-variation calculations can be achieved.

In our study, we have calculated 12 problems in which t, and ET are held fixed (eT variations), ten problems in which t, and E T are held fixed (Ep variations), and five problems in which Ep and eT are held fixed (t, variations). The selected ranges of rise time, peak field, and energy density used in this

contained in Tables 1-111. Once the scaling law for each EMP characteristic is formulated, the scaling laws will be combined into a single law for each coupling constant, each as a function of the three EMP characteristics. The final scaling laws will be tested with a few generic waveforms as an exam for the overall accuracy of the models.

A. Scalings on From the first four problems in Table I, it is evident that for

wide variations in c T , the coupling quantities (E),,,ax and (E&,,ax are relatively constant. It is also evident from these data that there is a weak dependence of E* on E T . A plot of E* as a function of f T appears in Fig. 2. The data in Table I show that for values of the quantity cT/Ep less than approximately 5 / 40 [pJ/cm2]/[kV/m], E* scales with E T as

(10) E * - E -0.14

while for values of E T / E ~ in excess of 5/40, E* is nearly independent of ET.

The reason for this type of scaling is that for low values of €T/Ep, both the portion of the waveform before the peak and the portion of the waveform after the peak contribute to the energy density in the high-frequency regime. As the slope of the latter portion of the waveform is decreased (corresponding to an increase in eT) , the contribution of this portion of the waveform to the high-frequency regime decreases. Eventu- ally, as the slope of the latter portion of the waveform continues to be decreased, a point is reached at which the contribution of the latter portion of the waveform to E* is negligible (i.e., E* has reached an asymptotic value).

The scalings of the coupling quantities on the three EMP characteristics developed here are confirmed by the data in Table I corresponding to problems 5-12.

B. Scalings on Ep It is immediately apparent from a quick scan of the data in

Table I1 that each of the coupling quantities depends, in a significant way, on the peak value of the waveform electric field.

These scaling relationships are developed using a least squares f i t to the data.

BERNARDIN: SCALING LAWS FOR COUPLING QUANTITIES ON EMP

E ( T ) VS T

IO -

.O . 2 . 4 . e . 8 1.0 1.2 1.4 1 .6 1 . 8 2 . 0

eo

50

40

W

20

10

0

51

E(T) VS T

40

SO

25

w 15

IO

5

0

.o LLXd .1 .4 .e .e 1.0 1 . 2 1 . 4 1.6 1.8 2 . 0 ~~

T (SH)

(C)

Fig. 1. (a)-(c) Electric field profiles for c y variations.

One might predict from a dimensional analysis and from a study of the shape of the waveforms that the scalings of the coupling quantities on Ep might be the following:

(l?)max-Ep (EE),,,ax-Ei €*-E;.

Indeed, the scaling laws developed through the plausibility arguments are fairly close to those obtained through statistical analysis of the data. Least-squares-fit analysis results in the following relationships:

from a dimensional analysis and from a study of the waveform shapes, one might predict that the quantities and (El?),,,ax scale inversely with the rise time. The scaling of E* on t, is difficult to predict since decreasing t, might increase the portion of the frequency spectrum between, say, 500 MHz and 1 GHz at the expense of the portion of the frequency spectrum located below 500 MHz.

Detailed analysis of the data in Table I11 gives the following three scaling laws:

C. Scalings on t, It is apparent from the data in Table I11 that there is also a

significant dependence of the coupling quantities on t,. Again,

D. Comprehensive Scaling Laws Thus far we have derived relationships between the three

coupling quantities and the three EMP characteristics by

5 2 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 30, NO. I , FEBRUARY 1988

TABLE Ill TABLE I 6~ VARIATIONS t , VARIATIONS

Problem NO.

~

1 2 3 4 5 6 I 8 9

10 11 12

o(8h- I ) __

50.0 50.0 46.0 43.0 27.1 26.6 24.0 22.0 50.0 50.0 46.0 43.0

b(sh-') __

0.25 0.50 1.50 3.00 0.25 0.50 1.50 3.00 0.25 0.50 1.50 3.00

Problem No. (k)maz [AI __

1 513 2 523 3 514 4 518 5 289 6 286 I 285 8 291 9 770

10 185 11 171 12 117

A(k"/m)

41.28 42.28 46.02 50.90 42.10 43.85 50.03 57.73 61.91 63.42 69.02 76.35

0.84 40.0

1.50 40.0

0.84 60.0

3.29 3.40 3.46 3.62 1.87 1.89 1.98 2.11 7.40 7.65 1.79 8.14

0.15 0.16 0.18 0.20 0.11 0.11 0.14 0.17 0.34 0.36 0.40 0 45

8.95 4.65 1.76 1.02 9.24 4.92 2.00 1.23

20.13 10 45 3.96 2 29

Problem No. a(sh- ' ) b ( s h - ' ) A(kV/m) t.(na)

1 65.0 1.42 44.35 0.62 2 55.0 1.45 45.09 0.70 3 46.0 1.50 46.02 0.84 4 20.0 1.88 53.60 1.12 5 12.0 2.38 62 64 2.62

~~~~~

1 706 4.69 2 605 4.05 3 514 3.46 4 250 1.78 5 169 1.26

40.0 1.16

W I I I I 1 I I I I

1 2 3 4 5 6 7 8 9 sT (pJ/cm*)

Energy density in the high-frequency range (100 MHz-1 GH?) as a Fig. 2. function of the total energy density in the pulse.

TABLE I1 Ep VARIATIONS

E * (s) 0.19 0.18 0.18 0.13 0.10

the following scaling relations:

Problem No. a(sh- ' ) b(sh-') A(kV/m) t,(ns) E, (kV/m) ET (5)

1 51.0 0.25 29.15 0.84 28.9 4.65 2 50.0 0.50 42.28 40.0

_ _ ~ ~ ~ _ _ _ _ _

3 46.0 1.32 10.00 61.6 ET pJ/cm2

I (13) 8 E T 5 pJ/cm2

4 44.0 2.40 96.25 78.5 -0.14E2.2t-0.44 for -52 [ =] 9 E2.2t -0.44 for->- [-I .

5 6

€ T p r 9 Ep 40 40.0 4.70 139.70 - 99.8 48.0 0.80 33.10 0.84 30.4 1.76 46.0 1.50 46.02 40.0 41.0 3.80 16.40 57.2 36.5 9.00 123.80 - 15.3

10 31.0 24.0 194.40 - 98.0 P r ' Ep 40 kV/m

Problem No.

1 2 3 4 5 6 I 8 9

10

377 523 184 1012 1290 391 514 131 1010 1431

1.74 3.40 8.10 13.70 23.15 1.96 3.46 7.42 14.28 28.19

0.08 0.16 0.41 0.75 1.36 0.09 0.18 0.43 0.81 1.22

considering variations in one EMP characteristic at a time. Now the individual scalings will be combined to yield a single, comprehensive scaling law for each coupling quantity on the EMP characteristics. After formulating the comprehensive scaling laws, the predictive accuracy of the models will be checked by testing the models with some generic pulses and comparing the predicted results to detailed calculations.

Using (1) as the desired format of the scaling laws, combining the results of Sections 111-A, 111-B, and 111-C gives

Note that the strongest dependence of the coupling quantities on the characteristics tr, Ep, and eT are respectively by (E),,,a, and (E E),,,, (E E),,, and E *.

To test the accuracy of these scaling laws, three generic pulses will be used to calculate the coupling quantities exactly, and the results will be compared to predictions. Since the relationships in (13) are scaling laws, a baseline waveform is needed to calculate the constants of proportionality. The waveform chosen for this baseline is that given in problem 3 of Table I.

The generic waveforms chosen for this comparison are those given in Table IV. Note that there exists a factor-of-2- type variation in the rise time, peak field, and total energy density between the three pulses. The latter two sets of data in Table IV contain the exact calculations and the predictions for the coupling quantities. Also contained in this table is the relative error between the predictions and the exact calcula- tions.

It is clear from these data that over these ranges of rise times, peak field values, and total energy density, the scaling

BERNARDIN: SCALING LAWS FOR COUPLING QUANTITIES O N EMP 53

TABLE IV GENERIC CHECKS

Problem No a ( s k - ’ ) b ( s k - ’ ) A(kV/m) f , ( n s ) E , (kV/m) cT (si)

~~~ __

1 65.0 2.00 70.00 0.60 61.2 3.05 2 30.0 0.75 50.00 1.30 44.6 4.22 3 50.0 0.75 34.00 0.82 31.5 1.98

Exact Calculations

I 1106 11.39 0.46 2 366 2.73 0.16 3 419 2.16 0.10

PredictionsfError

Problem __ No. ~ o r ~ o r 2 r s

1 1101 0.4% 11.36 0.3% 0.49 6.5% 2 370 1% 2.78 2% 0.17 6% 3 414 1% 2.20 2% 0.105 6%

laws given in (1 3) quite accurately represent the way in which the three coupling quantities under consideration depend on the three EMP characteristics. The models for (E),,,ax and

are particularly accurate. The model for E* is somewhat less accurate due to its asymptotic-like dependence on f T for values of ET/E,, exceeding 5/40 (pJ/cm*)l(kV/m).

IV . CONCLUSIONS

Using a ratio of exponentials as an EMP waveform, scaling laws have been developed to relate three potential coupling quantities to three EMP characteristics. The three coupling quantities, and E * , were chosen for study based on a consideration of potential electronic upset mecha- nisms. Each relationship between a coupling quantity and an EMP characteristic was developed independently. After deri- vation of individual scalings, the relationships were combined to yield a comprehensive scaling law for each coupling quantity on the three EMP characteristics. The final scaling laws where then checked for accuracy by comparing their predictions with detailed calculations for generic pulses. Agreement between the models and exact calculations was found to be quite good.

It should be noted, however, that the scaling laws derived here have been shown to hold only over the parameter ranges used in the study. We expect that the scaling laws for the coupling quantities (E),,,ax and (EE)max apply over wider parameter ranges. The scaling law for E* though may be sensitive to the parameter range of interest because of the complicated way in which it depends on the EMP characteris- tics (see (9)).

With the scaling laws derived here in this paper, the experimental RF-vulnerability community is challenged to identify a most relevant EMP coupling quantity from this set, or to help identify other potentially applicable EMP coupling quantities for their test objects of interest. In order to meet this challenge, experimenters must be able to conduct studies subjecting electronic hardware to plane-wave electromagnetic pulses, where the pulse rise time, peak field, and total energy density form a complete set of controllable parameters. If these studies can indeed be carried out for parameter values in the ranges we have defined, then the data can be treated for the predicted dependencies on tr, E,,, and cT as given in (13). One should be able to distinguish between the applicability of the scaling laws for and (El?),,,ax to vulnerability test results by their respective dependencies on the peak EMP field (first- power versus second-power dependency). These two scaling laws should in turn be distinguishable from the scaling law for e* by the different dependencies on the rise time (first power versus roughly the 1/2 power). This empirical determination of a most relevant coupling quantity then supplies an observa- tional fact that must be predicted by any theoretical model on the coupling of electromagnetic energy to electronic circuits hypothesized by the theoretical RF-vulnerability community. If a model cannot be identified to fit the data, one may be able to conclude that for the target set under testing, electronic upset may not depend on just a single mechanism, such as a voltage surge, a power surge, or an energy-induced overload in the circuits. This would also be an important fundamental result in making progress in the understanding of electronic upset.

REFERENCES [ I ] C. L. Longmire, “On the electromagnetic pulse produced by nuclear

explosions,” IEEE Trans. Antennas Propagat., vol. AP-26, no. 3, pp. 3-13, 1978.