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'' TECHNICAL NOTE 1094

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

A Statistical Characterization of

Electroexplosive Devices

Relevant to Electromagnetic

Compatibility Assessment

Dennis S. Friday

John W. Adams
Tm he National Bureau of Standards' was established by an act of Congress on March 3, 1901. The

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.

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RESEARCH WR}RMATION

CENTER

A Statistical Characterization of

Electroexplosive Devices Relevant to

Electromagnetic Compatibility Assessment

Dennis S. Friday

John W. Adams

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

Sponsored by

U.S. Army Aviation Systems Command

St. Louis, Missouri 63120

T «•

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued May 1986
National Bureau of Standards Technical Note 1094

Natl. Bur. Stand. (U.S.), Tech Note 1094, 56 pages (May 1986)

CODEN:NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1986

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402
Contents

Page

Abstract i , . 1

1 . Background and Introduction 1

2. Sensitivity Testing - Prior Methods 3

3. The General Linear Model ^

k. The General EED Model and Its Symmetry 5

5. Parameter Estimation in the EED Model 7

6

.

Heat Flow Equations 9

6 .

1

Measurement Procedures 10

6. 2 Measured Data *. 12

5. 3 Instrumentation 14

7. The Statistical Experiment 14

7.1 Statistical and Thermodynamic Design of the Experiment 15

8. Experimental Results 18

8.

1

Statistical Analysis 21 8.2 Effects of Extremal Data 23

9. Implementation of the Firing Probability Plots 24

9.

1

Tolerance Intervals 24

9.2 Plot Implementation 25

9.3 Generating the Graphs 26

9.4 Interpreting the Graphs 27

1 0. Conclusions 28

1 1 . Acknowledgments 29

References 29

Supplemental Bibliography 30

Xll

A Statistical Characterization of Electroexplosive Devices

Relevant to Electromagnetic Compatibility Assessment

Dennis S. Friday and John W. Adams

National Bureau of Standards

Electromagnetic Fields Division

Boulder, Colorado 8O303

Electroexplosive devices (EEDs) are electrically fired explosive

initiators used in a wide variety of applications. The nature of

most of these applications requires that the devices function with

near certainty when required and remain inactive otherwise. Recent

concern with pulsed electromagnetic interference (EMI) and nuclear

electromagnetic pulse (EMP) made apparent the lack of methodology

for assessing FED vulnerability. A new and rigorous approach for

characterizing FED firing levels is developed in the context of

statistical linear models and is demonstrated in this paper. We

combine statistical theory and methodology with thermodynamic

modeling to determine the probability that an EED, of a particular

type, fires when excited by a pulse of a given width and amplitude.

The results can be applied to any type of FED for which the hot-wire

explosive binder does not melt below the firing temperature.

Included are methods for assessing model validity and for obtaining

probability plots, called "Firing Likelihood Plots". A method of

measuring the thermal time constant of an EED is given. This

parameter is necessary to evaluate the effect of a train of pulses.

These statistical methods are both more general and more efficient

than previous methods for EED assessment. The results provide

information which is crucial for evaluating the effects of currents

induced by impulsive electromagnetic fields of short duration

relative to the EEDs thermal time constant.

Key words: FED; FED response to pulsed currents; electroexplosive

device; electromagnetic compatibility; EMC; firing likelihood plots;

thermal time constant.

1 . Background and Introduction

A hot-wire electroexplosive device (EED) is an initiator that sets

off a small charge of primary explosive that surrounds a wire. Electrical

current flowing through this wire causes joule heating. When the primary

explosive reaches its critical temperature due to this heating, it explodes,

detonating a secondary explosive which serves as an actuator. A typical FED

is shown in figure 1. Commonly used parameters describe this process, but

additional parameters are needed for electromagnetic compatibility (EMC)

analysis

.

It is necessary to quantify both electrical and heat flow

characteristics of the EED. The heating power (p) is a function of current
(i),' and electrical resistance (R ). Parameters that must be measured are

e the critical temperature (0 ) of the explosive, the thermal resistance (R)

of the EED, the thermal capacity (C) of the EED and the firing energy (U).

U is the energy in a single pulse that will fire the EED when the pulse

width is so short that almost no heat energy has time to leak out during the

duration of the pulse. The thermal time constant (t) of the EED may be

calculated from the RC product. For single pulses, a family of curves may

be generated that relate pulse width and peak power of a rectangular pulse

to the likelihood of firing.

There are three distinct measurement procedures needed to obtain

these parameters and curves. Not all of these parameters can be measured on any one EED since each measurement destroys the EED. Subsequent

measurements must be made on other EEDs. For example, the firing current

and thermal resistance may be measured by using a slowly increasing current

ramp. The thermal capacity and energy to fire may be measured with another

measurement procedure which uses a very short pulse of current. These data

may then be used to calculate other parameters and to design the third

measurement procedure, the statistical experiment. All of the EEDs used in

the three measurement procedures should be randomly chosen from the same batch of EEDs.

The electromagnetic environment which may induce stray currents in

the wire of the EED is usually poorly known. The theory of how energy may

be transferred from this environment by unintended antennas (any electrical

conductor) is also poorly understood. These two very relevant topics are

not within the scope of this paper, but motivate the work reported. They

must be considered in any comprehensive EMC analysis.

There are several widely used standards or guidelines for evaluating

and handling EEDs in the presence of EMI [1,2,3,^]- The Bruceton up-down

procedure [5] has been used for years to measure the DC firing current of

EEDs. It will be described in the next section but in summary it provides

little information on extreme firing levels. In practice, since the

extremes of the firing current distribution are of interest, e.g., the

minimum all-fire current from an operational standpoint, and the maximum no- fire current from a safety standpoint, alternate procedures are used. In

one such procedure, fifty EEDs randomly selected from a lot are tested for

five minutes at an arbitrarily set no-fire current level. These same EEDs

are then fired at an arbitrarily set all-fire level. If any of the samples

fire at the no-fire level, or if any do not fire at the all-fire level, the

lot fails.

The reason that better, more direct measurement methods are not more widely used is due to a practical problem with the binder used to hold the

primary explosive around the wire of some of the older designs of EEDs. The

binder softens at a temperature below the critical temperature of the

primary explosive. This allows flow of the explosive-binder mixture, and

changes the thermal characteristics of the EED. Later designs do not have

this problem, and direct measurement of the DC firing current gives a better

measure of the average and standard deviation for a given sample size than

can be obtained with the Bruceton method.
An omission in existing analyses of the effect of EMI on EEDs is the

effect of impulsive EM fields, either from periodic pulses such as may be

generated by radar, or aperiodic pulses such as may be generated by

lightning, nuclear EMP, or arcing of DC machinery. Whether these transient

problems are of consequence is not clearly known. The methodology presented

later in this paper may help to answer these questions.

We propose a new way of characterizing the response of EEDs to

impulsive fields. The probability of firing is determined statistically as

a function of width and power of a rectangular input pulse. We present

general statistical procedures that can be used for characterizing EEDs and

describe the proper experimental methodology. The results of our experiments are presented to illustrate the methodology. These results are

not given for the purpose of characterizing the EED type used in this study.

The EEDs used, while all of the same type, were not suitably controlled

prior to testing. They were readily available, but the lot numbers were not

recorded and the storage time and conditions were not fully known. They

were used to develop and to demonstrate the method, and any variability from

these unknown factors, if any, was confounded with our random errors.

2. Sensitivity Testing - Prior Methods

Sensitivity testing is the name that has been used for the general

methodology associated with EED testing. The class of experiments is

characterized by a binary response, fire or no-fire in this case, and a continuous stimulus. The stimulus is adjusted to a predetermined set of

levels and the proportion of "fire" responses at each level is determined.

Many test specifications specify such procedures [2],

A procedure called the Bruceton method or the up-down method has

been used for such tests [5, 6, 7]. The stimulus in general may represent

very different attributes such as input voltage, height of drop,

temperature, etc. The experiment consists of selecting an equispaced

lattice of stimulus levels:... s_2, s_i , Sq, Sj, S2... centered at a nominal

"50 percent" firing level. Begin by applying the stimulus to a randomly

selected EED at level Sq • The remaining settings are determined by the

previous outcomes. If the first EED fires the second one is tested at level

s_i. If the first EED does not fire the second is tested at level Sj. Each

subsequent EED is tested according to this procedure; one level up if no response and one level down if a response. The advantage of this test is

that it concentrates the test levels near the mean and improves accuracy of

that estimate. Fewer EEDs are therefore required on the average for a given

accuracy. The disadvantages of the up-down method are that it obtains

relatively poor estimates of the dispersion, it requires one-at-a-t ime

sequential testing, and it deals with only one-dimensional stimuli.

Procedures for computing the estimates of the mean firing level and standard

deviation are given in the references.

Another possible approach to characterizing such devices is to adapt

methodology used in statistical bioassay known as quantal-response models.

In these methods the probability of response is expressed as a linear

function of the levels of the stimuli. Multiple stimuli are possible with

this approach. Considerable methodology is available on this topic because
of its many years of use and development in biological experiments. Two

models have evolved as the most commonly used. The Probit model is based on

a Gaussian probability distribution and the Logit model on a logistic

distribution. They are very similar in their results and assumptions, but

the Logit model is computationally simpler and appears to be the more

commonly used procedure. See [8, 9] for details.

Both procedures require information on the proportion of (in this

case) EEDs in independent trials that fired at preset levels of the stimuli.

To our knowledge Probit and Logit methods have not been used in EED

characterizations, but they are the natural choice for certain EED

experiments. They differ from the Bruceton method in several ways. They are not sequential in nature and the statistical design is predetermined

based on preliminary experimentation or a priori information. Stimulus

levels and the number of EEDs tested at each level are fixed. The model is

more specific than the Bruceton method and must be validated with data for

the results to be defensible. Good diagnostic tools are available for these

methods, however. All three methods are suited to experiments that are somewhat wasteful of information since exact firing levels are not observed.

EEDs not fired are wasted unless it can be determined that they have not

been affected by attempts at firing and can possibly be reused. We

originally used the Logit method for the EED characterization problem

addressed here but since it was possible with our measurement system and the

particular EEDs tested to obtain more information than is necessary for the

Logit model, we developed the more efficient procedure discussed herein.

The statistical basis for the methodology we propose will be developed over

the next three sections.

3. The General Linear Model

Some statistical concepts and notation are now introduced. The

general linear model is given by eq (1).

Y = X B + £ (1)

Where:

Y is an n X 1 vector of response variables.

X is an n X r matrix of regressor variables. Its elements are

fixed, not random variables. The dimension r (r<n) equals the

number of unknown parameters in the model. The structure of

this matrix fully specifies the form of the linear model.

3_ is a r X 1 vector of unknown parameters.

e is an n X 1 vector of random errors. Statistical assumptions

are zero mean (E(_e) = 0) and, in the most general case, that its

variances and covariances are specified by a positive definite

symmetric n x n matrix _^. The exact form of £ is often unknown.

i th

The elements e . of e are the errors in the i observation. When

1 —

the £. are uncorrelated , the off-diagonal elements of 2 will be zero. If

the e. also have constant variance a , then 2. = o 1 where I is the n x n identity matrix. However, this assumption is not made in this paper. When

it can also be assumed that the e. have a Gaussian distribution, then
uncorr elated c. are also statistically independent. With Gaussian errors

more can be said about the statistical properties of the model and its

parameter estimates than with arbitrary distributions.

The best (minimum variance) linear unbiased estimate

parameters B is then given by the expression

of the

= ( X I -1 -1

I -1 (2)

(where exponents ' and -1 are the transpose and inverse respectively).

This is the estimator for the general case with nondiagonal covariance

matrix 1. If e. are uncorrelated but variances are unequal, then ^ is a

diagonal matrix with components (o?

-1 It is usually written in terms of a weight matrix when a common variance is factored out. That is;

ioi)

^

1 , 1

__ 1 = - W = •(0^) ^ 0^ 0^ n

(3)

with weights w.; i = 1,2,...,n. Observations Y. with small w. have larger

1 1 1 variance than observations with large w.. The estimate, eq (2), then

reduces to

= ( X W

-1 X W (4)

This case is often called weighted least squares. An excellent reference

for linear models is the book by Graybill [10].

M. The General EED Model and Its Symmetry

In the classical linear model the response variable Y and the

columns of X, the regressor variables, represent distinct attributes. It is

sometimes possible to interchange attributes in a different linear model but

the parameter estimates and the error structure are different. The two

arrangements are distinct linear models. This restriction does not apply to

the type of EEDs considered here. We will demonstrate that a symmetry

exists between the two variables chosen for this study which is not

generally possible. Furthermore, it can be expressed within the linear

model given by eq (1).

The notation must first be generalized. The random vector Y will

always denote the response variable and the columns of X the regressor

variables. The symmetry arises because the attributes which represent X and

Y may be interchanged within the same linear model. We denote these two

possible arrangements as experiment 1 and experiment 2. The arrays X and Y are partitioned according to which experiment (or attribute arrangement) is

represented.
Let

Y =

(1)"

.(2)

and X = .(Dl

:2)

(5)

where superscripts

(2)

and (2) identify the experiment. Vectors Y (1) and

are of dimensions n x 1 and n x 1 respectively, where n + n = n

Elements of Y are written y.

(k)

; i = 1,2,...,n ; k = 1,2. Likewise the

K matrices X are n, x r with elements x. .; i = 1,2,...,n,; 1 — k 1 , J k

k = 1,2.

1.2, ,r

A general design matrix for this problem which admits an intercept

parameter and a parameter for experiment effects is the following (dimension

n X 3):

X =

+1

+1

-1

-1

.(1) 1 1,3

Hi ,3

1,3

"2.3.

(6)

A more restricted design might be to eliminate column 1 or column 2 of this

matrix, or even both columns 1 and 2. The final design is not dictated by

the analyst but by the data; the most parsimonious design for which a good

fit of the data is manifested in the diagnostic analysis.

While the above design matrix will be shown to be more general than

necessary for the particular EEDs which were tested for this study, we feel

it is important when applying this methodology to a new type of EED to

develop the model from first principles. Column 1 allows for a nonzero mean

( k

)

response when the value of the regressor x. . is set to zero. While

counterintuitive in a physical sense, one cannot rule out a priori that such

a shift might be exhibited by the data in the region where the experiment is

performed. The second column is perhaps more reasonable and is necessary to

ascertain whether there is any significant difference in mean value of the

two experiments. The question, "Does the phenomenon under study (the firing
level of the EED) depend on which attribute is fixed and which is random?"

is another way of expressing this. The symmetry can be illustrated by

considering two attributes u and v. Let indicator variable I(k) denote the

experiment. Kk) = +1 if k = 1 and I(k) = -1 if k = 2. I(k) changes with

the arrangement of u and v. Let 6o» Bi, 62 denote the elements of B_ and let

e have covariance matrix a^I_. If experiment 1 means u is the response and v is the regressor, and experiment 2 is the reverse, then (ignoring row

subscripts) either of the following arrangements is permitted in our linear

model.

_

1

_i _

1

_

1

u = So + 61 1(1) + B2 V + e V = So + Bi + 62 V + £ V (7a)

_ 1 _ 1

' _ 1 _ 1 V = 80 + Bi 1(2) + S2 u + e u = So - Bi + S2U + e u . (7b)

Variables u and v are present in both Y and X under the same model with the same parameters _B and 0^ . This is a situation which does not normally arise

in a linear model. The coefficients of e in eq (7) change the variance of

2 2 each row and correspond to the weights in eq (3) (i.e., w = v or u ).

In the case where the intercept is indeed the origin (So = 0) and

there are no effects due to the different experiments (Si = 0), the first

two columns of X are unnecessary. In this case eqs (7) reduce to

or

-1 -1 u = Sv+£V (8a,

v=Bu+eu. (8b;

These eqs (8) admit an interpretation of bilinearity when written in the

equivalent hyperbolic form

uv = S + e . (9)

While mathematically equivalent to eq (8), the form in eq (9) is not an adequate basis for a statistical analysis where u and v are being varied as separate factors, and the region of model validity must be determined. It

will be shown later that eq (9) enables simple computation of S once the

model has been proven valid for the given data from a particular type of

EED.

5. Parameter Estimation in the EED Model

We now consider the statistical estimation of the parameter vector B_ in the linear model given by eq (1) where the design matrix is given by eq

(5). First it is necessary to establish the statistical structure of the error vector e_. Each component of e_ corresponds to a different EED. It is

assumed that the random perturbation in firing level for each EED, about

their common firing level, is not affected by the errors of the other EEDs.
They were chosen randomly from the batch of EEDs and the time order of the

design levels and experiments was randomized. Therefore the elements of e are mutually uncorrelated . The variances of the e. are not assumed to be

constant, but to vary with the regressor variable as suggested by eqs (7).

.t r,

(k)

The e. may be represented as an independent random error e with variance a^

scaled by the value of the corresponding x. We may write

Therefore

^i = ^ ^.3^ '

, N r (k),2 2 v(e.) = { X. _} a

1 1,3

i,3

(10)

The covariance matrix of z_ is then:

(1),2

'1,3^

I- o' .(2) ,2

1,3 ^

n2,3

The least squares estimate of S is therefore given by

(11)

where

= ( X W X

,(i)r2

-1 X W Y ,

W =

(1) ,-2

'n^,3^

.(2), -2

1,3^

(12)

(13;

. (2) ,-2

.X _} n2,3

is consistent with eqs (2) and (3), and our assumptions. No assumptions

were made on the distribution of £ up to this point. We have simply derived

the least squares estimates. If a further assumption that the e. have a
the least squares estimates. If a further assumption that the e. have a

Gaussian distribution is possible, then _6 is the maximum likelihood

estimator of B- This estimator is unbiased and has the smallest variance of

all unbiased estimators. In this sense it is an optimal procedure. A practical advantage of this approach is that most statistical packages

contain software for computing these estimates and, equally important,

provide diagnostic information for assessing the quality of the fit of the

model to the data.

6. Heat Flow Equations

Some relevant thermodynamic concepts will now be introduced. These

concepts are important for a deeper understanding of the statistical model

and its limitations, for obtaining prior information on the region of model

validity, and for identifying thermodynamic parameters relevant for EED

characterizations. The methodology introduced in these sections is based on physical principles rather than statistical modeling. Only a small number

of EEDs could be allocated to these measurements and extensive model

validation was not possible. Parameter estimates are obtained directly from

the equations, not from statistical principles. These procedures are

adequate for the intended purposes. They follow basic heat flow equations

similar to those used in determining temperature distribution around a barretter wire [11]. In situations of temperatures less than red glow of

the wire, heat flow by radiation is minimal. Also, for small volumes, the

Grashoff number is low, indicating minimal heat flow due to convection.

Thus the simple models based on conductive heat flow should be adequate.

They provide approximate "estimates" of the thermodynamic parameters and a priori information on design boundaries for the statistical model. The

final statistical results demonstrate no inconsistencies with these initial

thermodynamic measurements. This theory is now developed.

The general heat flow equation that applies during joule heating is

0(t) = +p(t)R(1-e' ^^^)

; t>0 (14)

3. where

0(t) is temperature as a function of time (degrees Celsius),

is ambient temperature (degrees Celsius),

3. t is time after application of current (seconds),

2 p(t) is power due to joule heating, p(t) = i(t) R (watts), where

i(t) is current as a function of time, R is the electrical

e resistance of the EED wire (ohms). The current is often

constant over a given interval of time.

R is the thermal resistance of the heat-leak path out of the EED

(degrees Celsius per watt),

T is the thermal time constant (seconds), t is the product of R and

C, where C is the thermal capacity of the EED wire and

explosive-binder mixture (joules per degree Celsius).
Differentiating eq (14) gives the rate of rise of temperature

cl0(t)/dt = (p(t)/C)e"^''^. (15)

During cooling, the temperature decreases from initial temperature

(0 ) according to

e(t) = +(0 -0 ) e"^^^ . (16)

a o a

These equations have some approximations in them due to individual

variations of geometry and parameters among individual EEDs. There are also

nonl inear i ties due to parameter dependence on temperature. These effects

have been simulated for a similar structure, a barretter wire in air, with

an iterative mathematical procedure [11], and found to cause relatively

minor variations in absolute temperature.

The steady state temperature may be obtained from eq (14) by letting

time be much greater than the thermal time constant. For most EEDs, this

means time greater than 300 milliseconds. If R, p(t) and t are known, this

is a straight forward calculation. For most operational applications, the

temperature will exceed the critical temperature of the explosive well

before the steady state equilibrium conditions are reached. For some EED

designs, the bindei—explosive mixture may soften at a temperature below the

critical temperature of the explosive, resulting in a change of

characteristics, often indicated by dudding.

The temperature rise and fall due to a single impulse function of

current may be calculated using eqs (14) and (16). For times much less than

the thermal time constant of the EED, the t = limit of eq (15) may be used

as given by

d0(t)/dt = p(t)/C. (17)

This is a useful form to determine power and hence current needed to fire an EED within a specified time.

For the case where there is a train of pulses, either periodic or aperiodic, where the cooling time between pulses is of the same order as the

thermal time constant, a combination of eqs (14) and (16) may be used to

determine cumulative temperature rise, commonly known as stacking. This is

illustrated in figure 2.

If there is a single impulse with known amplitude and short duration

compared to the thermal time constant of an EED, the curves shown later in

this paper can be used to determine probability of firing.

6.1 Measurement Procedures

Three distinct measurement procedures are used to determine the

parameters of each EED. Since each measurement procedure destroys the EED,

only a subset of the relevant parameters can be measured on any one EED.

The first measurement procedure is as follows. Ambient temperature

is recorded. The critical temperature of the primary explosive is either

10
obtained from the manufacturer or measured with a special oven with five

solid walls and a sixth weak wall made of some easily replaceable material

that can serve as a pressure-release blast wall. The electrical resistance

of an EED is measured with special instrumentation that applies much less

current than the no-fire current. A slow current ramp is used to heat the

EED to detonation. The value of current at which the EED fires is recorded

as I . The average and standard deviation may be calculated for the

electrical resistance, the firing current, the firing power, and the thermal

resistance. The thermal resistance is obtained by

where

R = (0^-0 )/P„ , (18)

c a f

is the critical temperature of the primary explosive (degrees

Celsius)

is the ambient temperature (degrees Celsius),

P„ is the power (watts) applied at the time of firing, (i.e., P„ =

If the wire in the EED has a significant temperature coefficient of

resistance, a correction may be needed. This gives measured values of four

independent parameters, R , I , , and R. The fifth parameter, P , is

calculated from the first two.

The second measurement procedure also requires that critical

temperature of the primary explosive, ambient temperature and electrical

resistance be measured. A signal generator and amplifier are used to

generate a train of current pulses whose width is much less than the

expected time constant of the EED, and whose period is much greater than

this same time constant. The amplitude of the pulse can then be increased

slowly until the EED under test fires. The amplitude at which the EED fires

is recorded as I . Since the pulse width is much less than the thermal time

P constant of the EED, almost no heat energy has had time to leak out during

the duration of the pulse. Also, since the time between pulses is much

greater than the thermal time constant, the temperature of the wire nearly

returns to ambient temperature between each pulse. Therefore, the energy to

fire with a single rectangular pulse may be calculated as follows:

where

U^= P w (19)

U is energy (joules) to fire with a single impulse,

2

P is peak power (watts), P = I R , p e w is the pulse width (seconds).

The thermal capacity of the EED may be calculated as follows:

C = U„/(0 -0 ) (20)

f c a

n
where

C is the thermal capacity of the EED in joules per degree Celsius,

U is defined in eq (19),

and are critical and ambient temperatures respectively,

c a

The thermal time constant can be calculated as follows:

T = R C (21 )

where R is obtained from eq (18) and C is obtained from eq (20).

Four independent parameters are measured: R , dt , , and I . e a p

U and C are calculated from the above four, plus knowledge of . The third measurement procedure, the statistical experiment, is

discussed in section 7. Information obtained from the first two procedures

is used to determine the appropriate region for the statistical experiment.

6.2 Measured Data

A set of measurements were made on some squibs. A squib is an EED

of low explosive charge, usually without any secondary explosive. The

particular squibs used simulate a common form of EED, an electrically fired

commercial blasting cap. Due to the age and unsealed design of these

squibs, the data obtained are not considered typical of data that would be

obtained from normal EEDs. The purpose in making the measurements was to

allow evaluation and improvement in the procedures, test equipment, and

statistics needed to obtain the desired parameters and curves for EMC

analysis

.

Data from the first measurement procedure are summarized in the

table below. The average and standard deviation are calculated from 10

EEDs.

Table 5.1 Average and standard deviation of four parameters obtained

using first measurement procedure.

Parameter Average SD Units

Electrical 1 .42 0.123 ohms

Resistance

DC firing 0.39 0.015 amperes

Current

Thermal 1365.76 188.4 degrees Celsius

Resistance per watt

Power to 0.22 0.027 watts

Fire

12
Data from the second measurement procedure are summarized in Table

6.2, based on 1 1 EEDs.

Table 6.2 Average and standard deviation of three parameters

obtained using the second measurement procedure.

Units

ohms

millijoules

Parameter Average SD

Electrical 1 .45 0.09

resistance

Energy to 11.8 2.2

Fire

Thermal 39.1 7.4

Capacity

microjoules per

degree Celsius

The product of the average thermal resistance and the average thermal

capacity gives a calculated average value of thermal time constant, t, of

53.4 milliseconds.

The variation in the range of values for the thermal time constant

of individual EEDs must be estimated since this parameter cannot be measured

explicitly on any one EED. The destruction of each EED during either

measurement procedure precludes use of the same EED for subsequent

measurements

.

The lower and upper confidence limits of thermal resistance may be

calculated from the average and standard deviation of the measured values

for specified confidence levels. Since the two measurement procedures are

independent, a second set of confidence limits may be calculated for the

thermal capacity. Lower and upper 95 percent confidence limits are given in

the table below (10 degrees of freedom).

Table 6.3 Calculated values of confidence limits of thermal

parameters from measured values in previous tables.

Parameter

Thermal

resistance

Thermal

capacitance

Lower Average Upper

1230.9 1365.7 1500.5

33.8 39.1 44.4

Units

degrees Celsius

per watt

microjoules per

degree Celsius

An estimate of the thermal time constant may be obtained by taking

the product of the thermal resistance and the thermal capacitance. The

thermal time constant calculated from the two average values is 53.4

milliseconds.

13
6.3 Instrumentation

The instrumentation to make these measurements consists of four

pieces of equipment plus a restricted area for setting off the EEDs. See

figure 3 for a block diagram. The pulse generator, storage oscilloscope and

digital multimeter are commercially available items. The pulse amplifier is

a special design; it should be able to reach peak pulse currents of 20

amperes or more with a rise time of about 10 microseconds. It will require

some substantial ancillary power supplies. See figure 4 for one such

design. The multimeter must be able to measure resistance in the one ohm

range to three significant figures. Most good pulse generators and

oscilloscopes meet the rise time requirements. A dummy 1 ohm load (in place

of the EED) that can handle 10 amperes is needed to make adjustments on the

range scales of the pulse generator and oscilloscope.

The requirements for the area used for setting off the EEDs are

based mainly on safety and convenience. A 200 liter (55 gallon) drum with

about 30 cm of sand in the bottom serves as an absorber and director for

blast energy and shrapnel particles. Firing current is fed through a coaxial cable and a connector on the side of the drum. This allows final

activation of the circuit by connecting the cable to the outside of the

drum, and the person making the connection is never exposed to an armed EED.

EED resistance measurements are made through this same cable, subtracting

short circuit resistance of the cable from total resistance. This also

assures reliable electrical connection which prevents apparent dudding.

7. The Statistical Experiment

In this section we describe the actual physical experiment and its

statistical design. This is the direct application of the statistical

methodology developed in sections 3, ^ and 5 combined with the thermodynamic

concepts of section 6. The two attributes discussed in sections 4 and 5 are pulse width and pulse amplitude. The two experiments are defined according

to which of these attributes is fixed and which is measured.

Experiment 1. The pulse amplitude is fixed within a set

of predetermined design levels and the pulse width at

which each each EED fires is measured.

Experiment 2. The pulse width is fixed within a set of

predetermined design levels and the pulse amplitude at

which each EED fires is measured.

The physical interpretation of the symmetry discussed in eq (7) should now

be apparent. The pulse width, whether fixed or measured, is in units of

time (milliseconds).

The choice of units for pulse amplitude, however, is not as simple.

Two considerations arise here, namely model and error sources. Possible

choices for units are voltage, current or power. While the firing current

and voltage can be measured directly, the firing power must be computed as a function of either measured current (or voltage) and measured electrical

resistance.

14
Within the region where the instrumentation supplies a rectangular

pulse, the energy in a single pulse is simply the product of pulse width and

pulse power. If the pulse is short enough that there is minimal heat loss

during the pulse, then temperature rise is proportional to pulse energy.

Within these limitations we expect a simple pulse energy dominant model for

EED firing level. This reasoning suggests the symmetry discussed in section

The second consideration is related to minimizing the random error

in the experiment due to sources that can be eliminated by proper

statistical design. In this case the electrical resistance of the EED is

such a factor. If units of voltage or current were used for pulse

amplitude, the firing levels would implicitly depend on the electrical

resistance of each EED. This resistance, due to manufacturing variability,

material inhomogeneity and other reasons, fluctuates randomly among EEDs of

identical type and lot. Using power units for amplitude requires measuring

this resistance, but once it is known, a source of error is removed. While

there may be pragmatic reasons, depending on the applications of the

results, to include or exclude electrical resistance, our intent is to

explore the nature of the EED firing level in as error-free an experiment as possible.

The two considerations of model simplicity and error minimization

therefore lead to a common approach, power-width based measurements. Each

EED, after its electrical resistance is measured, will be excited by a train

of rectangular pulses of known width and amplitude. The amplitude, in units

of power, is determined for the particular EED under test, based on its

known electrical resistance. The relatively high accuracy of this

measurement justifies our assumption that electrical resistance is known.

The interpulse spacing is chosen so that the temperature returns to ambient

between pulses. We will return to this issue later. The pulse width or pulse amplitude, depending on the experiment, is increased by an infinitesimal amount for each succeeding pulse until the EED fires. The

width and amplitude of the pulse that fired each EED are the basic data for

the analysis.

The type of EED tested had the property that the binder material was not affected below the firing temperature. This is not true of all EEDs, as some binders soften at temperatures below the critical temperature of the

primary explosive. Such EEDs must be tested by a different procedure, as was stated earlier.

7.1 Statistical and Thermodynamic Design of the Experiment

The design of the experiment is done in two stages. The first stage

is based largely on physical considerations. Its objective is to determine

a range of pulse amplitudes and widths within which it is possible to

perform the experiment and be assured that crucial assumptions hold. The

second stage is the statistical design of the experiment within this

feasible region.

Limits of the instrumentation used in the experiment, thermodynamic

properties of the EED, and mathematical-statistical assumptions underlying

the analysis must be considered for the first stage of the design. A feasible region will then be established within which pulse width and pulse

15
amplitude may be set. The minimum interpulse spacing is also determined at

this time. This variable does not arise explicitly in the experiment but is

crucial for model validity. The details for determining the feasible

regions of pulse width, pulse amplitude and interpulse spacing are now

given.

There is a limitation on both the maximum pulse width and the

minimum pulse width. The minimum pulse width is limited by instrumentation

capabilities. The pulse amplifier rise time will distort the leading edge

of a pulse. For long pulse widths this rise time distortion is negligible

and the pulses may be assumed to be rectangular. For very short pulses the

imperfect leading edge will dominate its shape and the pulse can no longer

be considered rectangular. Pulse amplifier rise time limitations can be

determined from the amplifier design specifications and should also be

measured in the test system with a suitably fast oscilloscope. It was determined that pulses of about 0.1 ms minimum duration could be considered

rectangular. In the final statistical design the shortest pulse width

assigned for experimental purposes was 0.2 ms or twice the minimal pulse

width. The requirement of rectangular pulses enables use of the model

described in eq (1) and its special case eq (8). This is true because the

pulse can then be characterized by its width-power product, or equivalently,

its energy. The EED temperature is related to pulse energy in eq (1M), but

in order to simplify this relationship, a limit must be placed on the

maximum pulse width.

The maximum feasible width is not determined by instrument

limitations but by thermodynamic properties of the EED and the desire to

minimize the complexity of the model and the analysis. The rectangular

pulse requirement of the previous paragraph enables simple computation of

energy. Statistically this would enable the use of a simple model except

that the binary response (fire or no-fire) is in general not linear in

energy but in temperature. In order for the EED input energy and EED

bridgewire temperature to be approximately linearly related, there must be

minimal heat loss from the EED during the pulse. In this case (see eq 17)

the temperature at firing will be linearly related to the product of input

pulse width and pulse power, a highly desirable property. Using eq (17) we

can determine the percent error between the linear and exponential models

for various pulse widths. This error is 0.5 percent for x/lOO, 2.5 percent

for t/20, 5.1 percent for t/10, 8.2 percent for t/6, and 13 percent for t/^.

To enable as wide a range of pulse widths as possible t/20 was chosen as the

maximum preset pulse width for the experiment. The value of t = 53. ^ ms in

table 6.3 therefore limits pulse width to at most 2.7 ms in the statistical

design.

The maximum limits on pulse amplitude are due to the

instrumentation. The pulse amplifier exhibits some distortion at high

powers and begins to saturate above 50 watts. It was possible to generate

pulses over 50 watts, but for levels much greater, some distortion in the

rectangular shape was visible. Therefore the upper power limit is set at 50

watts. There is no lower limit on pulse amplitude per se, but it is limited

implicitly by the pulse width maximum. As pulse power decreases a longer

pulse width is required to fire an EED. Since, as discussed above, large

pulse widths allow heat leakage and therefore require more complex modeling

and experimentation. The pulse amplitude will, for this reason, have a lower limit to its range.

16
The minimal spacing between pulses must also be predetermined.

While not explicitly required for the statistical design, it is necessary to

space the pulses sufficiently far apart to allow all, or almost all, of the

heat from previous pulses to dissipate. The response of each pulse can therefore be related to its dimensions independently of previous pulses.

Equation (16), based on first order thermodynamic theory, describes how the

hot wire temperature returns to ambient after a pulse is turned off.

Approximately 95 percent of the heat energy from any pulse will be

dissipated in three time constants, 98 percent in four time constants, and

over 99 percent in five time constants. This is true for any ambient

temperature and any pulse width or power -- below the firing level. A minimum interpulse spacing of at least five time constants (267 ms) was

chosen. In the experiments described here an interpulse spacing of 500 ms

was used throughout.

The numerical values for these bounds are relevant only for the

particular type of EED and instrumentation used. The methods described here

may be used for obtaining the corresponding limits for any other EED for

which the binder material does not melt before firing. The experimental

limits could have been obtained by other methods, statistical for example.

The methods given, however, are simple to implement, have useful physical

interpretations, and appear to work reasonably well with minimal data (using

few EEDs). Note that the bounds which have been determined are used for the

preset values of pulse width or pulse amplitude only. If measured values

exceed these bounds, the effects will have to be evaluated statistically.

Given this preliminary information we chose to allocate the approximately

100 EEDs that remained in a 2x5 experiment. The two level factor is the

experiment type. The two experiments, described previously (section 7) are characterized by which variable is fixed and which is observed. The second

factor is the level at which the fixed variable is set. Five equally spaced

levels were chosen within the predetermined experimental region for each

experiment. The chosen levels are given in table 7.1.

Table 7.1 Factor levels chosen for pulse width and pulse power

Width: 0,2

Power: 2,0

Ideally the available EEDs should be allocated equally among all 10 design

points. This could not be done because a small proportion of the EEDs were

expected to be defective, i.e., duds. Remember that we used unsealed squibs

(EEDs with only bridge wire and primary explosive present) which are not as reliable as sealed EEDs. Since each EED is destroyed upon functioning, the

defective ones could be identified only when selected and tested.

After the levels of the experiment have been determined, it is then

desirable to randomize the order in which the experiment is performed. It was decided to randomize across the 10 levels of the two factors since to

randomly assign each EED would have been impractical. Setting the fixed

variable once and testing a set of EEDs insured better repeatability and

required considerably less time. While there is no reason to believe that

there is a time effect in this measurement system, the randomization is done

to insure against inadvertent time trends. Table A-36 from the book by

17

0,8 1 .4 2.0 2.6 ms

1 4.0 26,0 38.0 50.0 W
Natrella [5] was used and the ordering in table 7.2 was obtained. The

selection is done simply by assigning an integer to each level in table 7.1

and moving sequentially, in any direction, through Table A36 from a randomly

chosen starting point. The levels are then chosen without replacement as their integer code arises in the sequence.

Table 7.2 Randomized design sequence for measurement of EED design levels.

Test sequence Fixed variable Level

1 Width 0.8 ms

2 Width 2.0 ms

3 Power 38.0 W

H Width 0.2 ms

5 Power 2.0 W

6 Width 1 .

4

ms

7 Power 50 . W

8 Power 26.0 W

9 Width 2.6 ms

10 Power 14.0 W

8. Experimental Results

The experiment was conducted according to the chosen experimental

design. The resulting data are given in tables 8.1 and 8.2, along with some relevant transformations of the data. Table 8.1 contains the data from

experiment 1 (pulse power fixed) and table 8.2, the data from experiment 2 (pulse width fixed). All levels of experiment 1 have 7 or 8 observations.

This is reasonably balanced considering the random nature of the EED

reliability discussed previously. In experiment 2 all levels have 7 or 8

observations except one, the 0.2 ms width level, which contains 16

observations. This is due to an equipment failure that occurred in the

system which generates and measures the pulses. After the equipment was repaired and recalibrated, the level which was completed just prior to the

failure was repeated. When the new measurements exhibited no significant

change, the new data were combined with the old and the experiment

continued. Partly as a result of this failure, there were 46 EEDs tested in

experiment 2 as compared to 36 in experiment 1 .

Lineprinter plots of the data are given in figures 5 through 10.

Figure 5 shows the 36 data points for experiment 1 (power fixed) and figure

6 shows the 46 data points for experiment 2. In these plots the horizontal

axis is the fixed variable and the vertical axis the measured variable.

Figure 7 shows the combined data and figure 8 a log-log plot of the data.

The conjectured bilinear relationship and hyperbolic form are clearly

exhibited in the plots. We will demonstrate that the proposed model is also

supported by the statistical analysis.

Some additional plots are discussed before proceeding. The firing

energy (power-width product) for each EED tested is plotted in figure 9. They are plotted in the order in which they were tested. There are two

outlying data, EEDs which required larger firing energies than was typical

of the majority. The corresponding log energy for each EED is plotted in

figure 10. The choice of power for specifying pulse amplitude was based on
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20
knowledge of the electrical resistance of each EED. The values of the

individual EED resistances, which were measured with relatively high

accuracy, are plotted in figure 11. The ordering corresponds to the

previously discussed energy plots. The two EEDs which were outliers in

firing energy are also outliers in electrical resistance. In this case the

15th EED is a moderate outlier with lower resistance than all the others and

the 67th EED is a high resistance outlier. Both, however, required a higher

energy to fire than was typical of the other EEDs. Figure 12 is a plot of

firing energy vs resistance for all of the EEDs tested. The same two EEDs,

which fall in the upper left and right corners, are, once again, atypical.

Since the EEDs which required excessive firing energies also

exhibited nontypical resistances the possibility of screening EEDs by

measuring their electrical resistance is suggested. While this is possible,

it is not done here. The present study was not directed toward studying

outliers and there is insufficient information in these data to say anything

definite about them. The history of the EEDs used is unknown. Therefore

crucial knowledge about differences in age, manufacture, storage conditions,

etc. is unavailable. All of the data are therefore included in the main

analysis as it is possible that such behavior is typical. The effects on

the main analysis of adjusting for the outliers will be discussed, however.

8.1 Statistical Analysis

The linear model given in eq (1) is a general model which enables a wide choice of specific models. The exact model is specified by the choice

of the design matrix. It is good practice to fit several likely models to

the data and choose the best one. "Best" can be interpreted as the model

that provides a good fit determined by statistical methods, yet is

parsimonious (contains only necessary parameters).

Models were fit to the data beginning with the general design matrix

given by eq (6). Several iterations in the analysis determined that the

intercept parameter (column 1) and the experimental effect parameter (column

2) were unnecessary. The independent variable in column 3 is sufficient to

provide a good fit to the data. The firing level behavior of the EEDs can therefore be described by the relatively simple eqs (8) and (9).

The chosen design matrix has dimensions 82x1 (82 EEDs tested), and

its components are the inverse values of the fixed variable (width or power). The raw residual variances were nonhomogeneous; therefore weighted

least squares was necessary. The form of the nonhomogeneity is specified in

eq (8), and was suggested by statistical theory. The weights are the

inverse of the error variance under this model and proved effective at

stabilizing the residuals on the given data. The parameter vector 6_ reduces, in this case, to a scalar B.

The weighted least squares estimate of 6 is 11.60 and the standard

deviation of the estimate is 0.17 based on the data from 82 EEDs. The units

of B are watt-milliseconds. Therefore corresponding relationships

P = 1 1 .60 w"^ (22a) w = 11 .60 P""" (22b)

21
are the best predictors of P and w respectively within the range of

experimentation. The values must be in the correct units, milliseconds for

width (w) and watts for power (P). Equations (22) exhibit the symmetry of

eq (8).

Simplifying eq (2) for this model shows that the estimate of 6 is

simply the arithmetic mean of the firing energies. This fact suggests a simpler analysis. It is not recommended, however, except as an exploratory

tool or perhaps for testing a batch of EEDs for which this model has already

been proven correct and its valid experimental region determined. The full

linear model approach that we describe makes possible a thorough assessment

of the appropriateness of the model for the data. Foremost, it allows the

effects of the two factors, width and power, to be separated, whereas they

would be confounded in a simple energy model. Since the utility of the

method suggested here, and of the resulting graphics, is to determine the

likelihood of firing for a given pulse configuration (width and amplitude),

these factors must remain explicit in the experimentation and analysis.

Another advantage, since computation is done almost exclusively on

computers, is that good statistical software for linear models is readily

available. Such software usually provides excellent diagnostic tools for

use in the model validation. Good graphics and exploratory capabilities are also common.

Some diagnostic statistical plots which are of use in assessing the

fit of the model in eq (22) to the data (Tables 8.1 and 8.2) are displayed

in figures 13 through 16. The standardized residuals are plotted in figure

13- They should exhibit a random fluctuation with constant mean and

homogeneous variance. With the exception of the two outliers on the upper

margin and a cluster of low points in the right half, they appear

reasonable. Figure 1U is a Normal probability plot of the residuals which,

except for two outliers, is linear. This indicates the residuals are

Normally distributed, an important assumption for some of the diagnostic

tests and confidence intervals. The plot in figure 15 shows the

standardized residuals as a function of predicted values. This plot should

show homogeneity with respect to both location and variance of each group.

The plot exhibits this property with the exception of a slight drop in mean

on the right most group of points. Figure 16 is a plot of the standardized

residuals plotted against their adjacent (lag one) values. There should be no diagonal pattern, only a random cluster. Except for outliers this is

exhibited in the plot. Overall the plots indicate a good fit of the model

to the data. Except for the outliers there is only a slight suggestion that

the model is beginning to change at the extremes of the experimental region.

This will be discussed in more detail in the next section. We conclude with

the decomposition of the sum of the squares of the data given in table 8.3,

where df is degrees of freedom.

Table 8.3 Sum of squares decomposition for the fitted model in eq (22).

Total sum of squares: 1123^.64 (82 df)

Sum of squares due to model: 11041.05 (1 df)

Residual sum of squares: 193-59 (81 df)

22
The model therefore accounts for more than 98.25 percent of the variation in

the data.

8.2 Effects of Extremal Data

Various models and subsets of the data were considered in the

exploratory phase of this study. We will first evaluate the effects of the

two outlying data points which were discussed in the previous section.

These are the 15th observation in experiment 1 and the 31st observation in

experiment 2. Both points represented energies in excess of 17.4

millijoules while all the remaining data ranged between 8.8 and 13.7

millijoules with a mean of ] ] . ^5 and a standard deviation of 1.24. The

outliers are at least 4.8 standard deviations from the mean.

If the random variation has a Gaussian distribution, as it appears

to have, then this is an extremely unlikely event. If occasional outliers

are typical of EED behavior then our model must be extended to account for

the outliers. Further experimentation and analysis must be done to

determine if this is necessary. These points are therefore included in the

main analysis; it may be typical for this type of EED (squib). If these two

points are deleted, the point estimate of B is then 11.45 and its standard

deviation is 0.14. The classical estimates of mean and variance are known

to be sensitive to outliers. In this case, 2 out of 82 EEDs required

extreme firing energies and caused a shift in the mean of one standard

deviation. Considering the consequences of EED failures, the issue of

outliers requires a more complete investigation than was possible here. In

this case 98.8 percent of the variation is accounted for by the model and

the estimates are more typical of the majority of EEDs.

Another type of extreme is related to measurements falling on the

boundaries of the design region. In section 7.1 bounds were determined

using thermodynamic arguments for a region in which the model was likely to

be valid. The fixed factors in the experiments were constrained to remain

within these bounds. The measured factors, however, could not be

constrained a priori; the values were determined by the EED characteristics.

It can be seen from the data tables 8.1 and 8.2 or from plots of the data

that in each experiment responses in one of the levels exceeded these

limits. This occurred in level 1 of experiment 1 (EEDs 9 through 15) and

level 1 of experiment 2 (EEDs 53 through 68). In the former, the power was

fixed at nominal level of 2 watts and the measured pulse width ranges from

5.5 to 8.7 milliseconds which exceeded the chosen limit of 2.7 milliseconds.

In the latter, the width was fixed at 0.2 milliseconds and the measured

pulse power ranged from 44 to 87.9 watts whereas the chosen limit was 50

watts. Recall that heat loss during the pulse becomes significant for wide

pulses and that amplifier limitations cause pulse shape distortion at high

power levels.

All the data were retained in the main analysis but it was indicated

that some slight, but not significant, effects were noticeable. The two

levels discussed here, on the design boundary, contained the two outliers.

This is believed to be coincidental since the external behavior of those

EEDs is not believed to be related to the thermodynamic and instrumentation

limits.

23
Models were fit to the 59 data values which remain after deleting

these two levels. There was some improvement in the diagnostic plots due to

removal of the marginal points. The analysis, however, showed little change

from the one with all the data. The estimate of 6 is 11.61, almost

identical to the original, and the standard deviation of the estimate is

0.159. The total sum of squares is 7963.67 and 98.9 percent of this is

accounted for by the model.

Since the difference between the trimmed 59 point analysis and the

original 82 point analysis was insignificant, we retain the full analysis.

Note, however, that data in the extreme regions of the experiment are

exhibiting some slight changes. These are due to the thermodynamic and

instrumentation considerations discussed in section 7.1. In this sense there is reasonable agreement between the statistical results and the nonstatistical thermodynamic analysis. Extrapolation to higher power levels

and correspondingly shorter pulse durations can be done if a different pulse source that can generate these higher levels is available.

Physical theory suggests that these limits could be extended in all

but one direction. Pulse power may grow arbitrarily high and width

arbitrarily small and the relationship given by the model should still hold.

Statistically this is only a conjecture since we could not generate pulses

outside the given limits. The upper limit on pulse width, however, is a restriction of the model. Heat loss during the pulse would cause

nonl ineari ties which were not accounted for in the present model.

Extrapolation in this direction is not possible except as a bound. Clearly

more pulse power will be required for a given width due to heat energy loss.

Approximate firing likelihoods might also be obtained for non rectangular

pulses as long as the width is approximately correct (and less than t/6) and

total energy the same. The firing likelihood plots can therefore provide

approximations outside the established bounds based on physical

considerations.

9. Implementation of the Firing Probability Plots

A most useful tool for assessing EED behavior which is made possible

by this research is the firing likelihood plot (FLP). It graphically

summarizes relevant information from the analysis in a format that is useful

and easily interpreted. In this section we describe how to implement these

plots from the data analysis which was described in section 8. It is first

necessary to discuss two distinct topics related to the FLPs. This is done

in sections 9.1 and 9.2.

9.1 Tolerance Intervals

A commonly used statistical interval estimate is the confidence

interval. It is an interval which, given validity of the assumptions, has a known probability of containing the unknown parameter of interest. In this

case a 99 percent confidence interval could be derived for parameter B. It

would provide information on where the mean firing energy is for this type

of EED. An interval for the mean firing energy would be of little use

however. Concern in EED applications is for the probability that a given

type of EED fires (or does not fire) given a particular input pulse or train

of pulses.

2H
Tolerance intervals address this issue more directly. They are

intervals within which, with a given probability Y, a chosen proportion a of

the population will lie. In the case of EEDs, a T percent tolerance

interval estimates the range of firing energies, expressed in terms of pulse

width and pulse amplitude, which would fire 99 percent of the EEDs. The

coefficient Y is the probability that the resulting interval is correct. It

is analogous to the confidence coefficient in a confidence interval. There

are various types of tolerance intervals. One-sided intervals provide

either lower or upper bounds on the percentile of the distribution. While

these may also have applications to EED evaluation, we will develop only the

two-sided intervals. The one-sided intervals are simple adaptations of the

two-sided ones. Note that different tables are required depending on whether one-sided or two-sided intervals are being used. A good reference

for both tables and instructions for implementation of tolerance intervals

is Natrella [5]. They can also be implemented without distributional

assumptions.

The firing energy, the estimate, B, and the standard deviation, s,

are central in adapting tolerance intervals for firing likelihood plots. A

proportion P of the underlying population, Normal in this case, must be

chosen. Then a confidence coefficient Y, which is a probability that the

interval is correct, must be chosen. Table A6 of [5], a look-up table of

values of K for calculating two-sided tolerance limits, is then used to

obtain the specific value of K for the desired Y, P and df. Upper and

lower bounds for the tolerance interval are then given by the limits:

B + Ks (23a)

6 - Ks . (23b)

Each probability contour on the firing likelihood plot requires a pair of

these bounds. We must digress to discuss plot implementation before we can

use these intervals.

9.2 Plot Implementation

It is possible that a problem may arise in the algorithm for

generating plots. This will apply to most graphics software. The issue is

the hyperbolic nature of the function being plotted. Choosing equally

spaced points along either axis will generate plots which have a dense grid

on the chosen axis but are very sparse on the orthogonal axis. Even

logarithmic scaling will not alleviate this problem.

The approach taken was to derive a parametric equation, develop an equispaced index for the param.eter, and then generate the plotted points

from these values. The procedure is as follows:

2

The basic contours are described by the equation Pw = 9 where P is the

pulse power value and w is the pulse width value. We choose the following

pair of parametric equations:

25
w =

To determine bounds on t observe that:

1-t . min

w =9

max

1+t

P = 9

"^^". max

These equations are equivalent to the following:

log w

= 1 - max

min

log

log Pmax

t = - 1 max

log 9

Given the maximum w and P chosen for the plots, find t . and t . Then mm max

choose n, the number of points to be plotted.

Compute:

(t - t . )

' max min , . ^ ^ t.= t . + (i-1 ) ; 1 = 1,...,n

1 mm

n-1

and the corresponding pairs of points

1-t. w.= ^ (24a)

;i = 1 n

1 +t. p.= 9

^. (24b)

2

The chosen parameterization corresponds to pw = 9 , not 9. Therefore, if g is the estimate in the linear model, the proper relationship for plotting

"

1 /2 the mean firing level would be 9 = (6) . Details for relating this

parameterization to the plotting of other contours will be given in the

following section.

9.3 Generating the Graphs

A pair of contours for the firing likelihood plots are now simply

implemented. For a given P and "Y, obtain the limits given by eqs 23a and

1/2 ^ 1/2 23b. Then let = (6 + Ks) and = (B - Ks) , where 9 and Q are each

substituted for in eqs (24a) and (24b). Generate a set of points to be

26
plotted, as explained in section 9.2 for both and . These families of

points will generate the 100 P percent firing likelihood curves. This

process is repeated for each P for which a curve is desired. Examples are

given in figures 17 and l8 for the EEDs tested.

9.4 Interpreting the Graphs

Firing likelihood plots provide a convenient representation of an operating characteristic of a class of EEDs. One can graphically assess the

probability of firing for an EED from this class when it is subjected to a rectangular input pulse of a given width and amplitude. Once the experiment

described herein is performed on a representative sample of EED's and the

resulting statistical analysis is completed, then any relevant probability

level (P ) may be chosen for the plot contours. The contours define a set

of pulse dimensions which are almost certain (i.e., with probability Y;

sect. 9.1) to cause a proportion P of the EED's to fire. In most cases the

extreme values of P will be of interest, near zero and near one,

r corresponding to maximum no-fire and minimum all-fire levels, respectively.

In the example given the contours may be thought of as joint no-fire/allfire limits.

Alternative versions of firing likelihood plots are also possible,

for example, the one discussed above is based on a two-sided tolerance

interval and the two contours only have meaning when' considered as a pair.

An alternative might be to use one-sided tolerance intervals which have an individual interpretation. A P = 0.01 one-sided contour would provide a set of pulse widths and pulse amplitudes above which 99 percent of the EED's

will fire. Since the Gaussian distribution is symmetric, the same constant

is also used for a one-sided P = 0.99 contour. Natrella [5] also includes

tables and methods for one-sided tolerance intervals. Other alternatives

include plots based on distributional quantile estimates or confidence

intervals.

It is possible to relate the curves directly to electromagnetic

field intensity for a given physical configuration. Since pulse power is

2 proportional to E , electric field strength squared, the power axis of the

plots may also be expressed in terms of the peak field strength squared of

2 an impulsive field. A value of E that corresponds to a specific power

2 level is required for determining the position of the E scale. This may be

obtained by measuring steady state values of E and power for a particular

EED support structure. Usually the coupling of electromagnetic energy from

a field to the structure and hence to the EED is poorly known.

In a real world situation the pulses are not likely to be

rectangular, and even if they are, most antennas will cause ringing due to

phase distortion. Probabilities which are based on an ideal (rectangular)

test situation are therefore likely to be conservative because of the

decrease in actual energy coupled into the EED by an irregularly shaped

pulse whose maximum values of amplitude and duration are equal to those of a

27
rectangular pulse. Such a representation can extend the usefulness of the

firing likelihood plots to cover most irregular pulse shapes.

There is no physical reason that the left end of the firing

likelihood plots cannot be extended beyond the experimental region. The

limitation was imposed due to the inability of the test system to generate

rectangular pulses of higher power and lesser pulse width. These

limitations are related to maximum pulse amplifier power and/or oscilloscope

rise time specifications. Extension beyond this limit could not be

experimentally verified and statistical prediction was not considered. Such an extension of the plots should therefore be used with discretion as it is

possible that unanticipated factors may affect the model in this region.

Extension to the right of the experimental region is not possible under the

given model. Physical reasons previously stated limit extensions to the

right.

The interpretation assumes adequate cooling time between pulses. If

pulse repetition occurs in less than approximately 5 thermal time constants

then eqs (14) and (16) will have to be used to estimate the cumulative

heating effect. The consequent cumulative heating will cause the the EED to

fire at a lower pulse energy level. Since temperature increases

proportionally with power, it is possible to calculate a correction factor

for closely spaced periodic pulses. It is also possible to relate the

firing likelihood plots to aperiodic pulse trains but computations are more

complex and require consideration of specific aperiodic sequences.

10. Conclusions

A new method which integrates both statistical and engineering

concepts has been proposed for characterizing EEDs. The method provides a useful description of performance for a class of EEDs based on rigorous and

efficient statistical procedures. The methodology has been proven and

demonstrated in an actual experiment and is applicable to a wide class of

EEDs. The resulting firing likelihood plots provide information which is

relevant and not previously available.

While these contributions are unique, they suggest further

extensions of this work. These include models where the electrical

resistance is not measured, statistical methods for efficient estimation of

the thermodynamic parameters, extension of the present model outside of the

given limits, the link from this model to induced currents caused by

electromagnetic fields, and the extension to EED types which are affected by

previous heating (binder melts). We feel this work is only an important

first step toward new and rigorous methods of characterizing EED

performance. With the increasing use of EEDs, the increasingly complex

electromagnetic environment and the nature of many EED applications it

becomes a very important problem. The consequences of an inadvertent firing

or a failure to fire could be extremely costly. Optimal statistical

procedures for characterizing EEDs will help minimize the likelihood of such

events

.

28
1 1 . Acknowledgments

Support provided by the U.S. Army Aviation Systems Command, St.

Louis, Missouri, is greatly appreciated. This work, which has built on prior efforts, would not have been completed without this support.

Eric Vanzura designed and built the pulse amplifier and also helped

perform most of the measurements. Herbert Medley designed some of the

fixtures to hold the EEDs and Kim Dalton and Herbert Medley assisted with

some of the measurements. We gratefully acknowledge their valuable

contributions.

References

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32
LEAD wire:

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NBS-n4A IREV. 2-BCl U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS/TN-1094

2. Performing Organ. Report No. 3. Publ ication Date

May 1986

4. TITLE AND SUBTITLE

A Statistical Characterization of Electroexplosive Devices Relevant to

Electromagnetic Compatibility Assessment

5. AUTHOR(S)

Dennis S. Friday and John W. Adams

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State. ZIP)

U.S. Army Aviation Systems Command

St. Louis, Missouri 63120

10. SUPPLEMENTARY NOTES

I

j Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual sumnnary of most si gnificant information. If document includes a si gnifi cant bibliography or literature survey, mention it here) Electroexplosive devices (EEDs) are electrically fired explosive initiators used in

a wide variety of applications. The nature of most of these applications requires

that the devices function with near certainty when required and remain inactive

otherwise. Recent concern with pulsed electromagnetic interference (EMI) and nuclear

electromagnetic pulse (EMP) made apparent the lack of methodology for assessing EED

vulnerability. A new and rigorous approach for characterizing EED firing levels is

developed in the context of statistical linear models and is demonstrated in this

paper. We combine statistical theory and methodology with thermodynamic modeling to

determine the probability that an EED, of a particular type, fires when excited by a pulse of a given width and amplitude. The results can be applied to any type of EED

for which the hot-wire explosive binder does not melt below the firing temperature.

Included are methods for assessing model validity and for obtaining probability plots,

called "Firing Likelihood Plots". A method of measuring the thermal time constant of an EED is given. This parameter is necessary to evaluate the effect of a train of

pulses. These statistical methods are both more general and more efficient than

previous methods for EED assessment. The results provide information which is crucial

for evaluating the effects of currents induced by impulsive electromagnetic fields of

short duration relative to the EEDs thermal time constant.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

EED; EED response to pulsed currents; electroexplosive device; electromagnetic

compatibility; EMC; firing likelihood plots; thermal time constant

13. AVAILABILITY ^^ Unlimited

Q^ For Official Distribution. Do Not Release to NTIS ^^ Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402. Qn Order From National Technical In formation Service (NTIS), Springfield, VA. 22 I 6 I

14. NO. OF

PRINTED PAGES

56

15. Price

USCOMM-DC 6043-P80

;. U. S GOVERNMENT PRINTING OFFICE: 1986—671-865 45,212 REGION NO 8

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