ORNL-CF-66-2-67.txt

'

 

OAK RIDGE NATIONAL LABORATORY

OPERATED BY

INTERNAL USE ONLY

 

UNION CARBIDE CORPORATION

NUCLlS!ON 0 R N L
rosvomes soxx CENTRAL FILES NUMBER

OAK RIDGE, TENNESSEE 37831

66-2-67

 

 

 

—
DATE: February 11, 1966 | copy NO. S

SUBJECT: Analysis of MSRE Zero-Power Flux Noise
Using Digital Techniques

TO: Distribution

FROM: T, W, Kerlin and S, J. Ball

ABSTRACT

The record of the flux noise obtained during the zero-power
operation of the MSRE with fuel circulating was analyzed by two differeht
digital computer techniques. The indirect method consisted of calcu-
lating the autocorrelation function of the flux noise and subsequent
Fourier analysis of this autocorrelation function to give fhe power
spectral density. The direct method used a digital simulation of a
band pass'filter to concentrate the signal in the desired frequency
range, The output of this filter was then squared and time-avefaged

to give the power spectral density,

Both methods were found to give comparable results at comparable
costs., The results were also found to give reasonable agreement with
previously published results obtained with analog methods; The value
of B/4 obtained by the digital ‘'method is 16.2 compared with 14.8

obtained in the earlier, analog study.

NOTICE

This document contains information of a preliminary nature and was prepared
primarily for internal use at the Oak Ridge National Laboratory. It is subject
to revision or correction and therefore does not represent a final report. The
information is not to be abstracted, reprinted or otherwise given public dis-
semination without the approval of the ORNL patent branch, Legal and Infor-
mation Control Department.
INTRODUCTION . . .
METHODS OF ANALYSIS

Indirect Method

Direct Method .
MSRE DATA . . . . .
'RESULTS . . . . .
Indirect Method

Direct Method .,

CONCLUSIONS . . . .
- Accuracy . . . .
" Cost . . ¢« ¢« o &

Flexibility . .

Speed . . . . .

APPENDIX . . . . .

CONTENTS

Page

N N 60 60 g oo

18
18
20
20
20
21
21
22

 
 
INTRODUCTION

.Digital téchniques were used to analyze the noise record obtained
during the zero-power run of the MSRE, These data were previously analyzed
by analog methods by Roux and Fry.l The purpose of the present analysis
was to supplement the analog results and to further test the digital
methods. One of the digital techniques used in this analysis had pre-

viously been used successfully in analysis of ORR noise data .2

METHODS OF ANALYSIS

Indiregt Methqg

 

The steps in the indirect method are:
1. Calculate the autocorrélation_function,,Cll(T), of the noise

record using the following expression:

1 Tfi
C,1(M == jw p(t) ot + 1) dt - (1)
m o
where
T, = maximum correlation time, and
¢ = the neutron flux signal,

2, Fourier analyze the autocorrelation function. Since it is an

even function with period’ZTm, we obtain:

Fk 'Cll(T) ='%— jflflm Cll(T) c«os%-TI dr . (2)
m o m
3. Apply necessary corrections., These include:
a. Spectral windows to compensate for the fact that the Fourier
analysis uses a finite integration time,
b, Filter corrections to remove the effect of a low-pass filter
used to eliminate aliasing.

c. Background corrections.

 

lD. N, Fry and D, P, Roux, 'Results of Neutron Fluctuation Measure-

ments Made During the MSRE Zero-Power Experiment,’' USAEC Report ORNL-CF-
65-10-18, October 29, 1965.

ZLetter from T. W, Kerlin to D, P. Roux, September 17, 1965,
Subject: Digital Calculation of the Power Spectral Density from Noise Data,
The corrected Fourier coefficient, Fk{fill(Ti}, at the frequency,'kfl/Tm

radians/sec, is the power spectral density (PSD) at that frequency.

; e

Direct Method

 

~In the direct method, the digitized noize signal is used as the input
or forcing function to the differential equations’ representing a narrow:
band pass filter, and the resulting output of the filter is squared and
integrated. The matrix exponential technique3 is used to solve for the
transient response of the filter, which has the characteristics of a

quadratic lag and a transfer function:

Jw

 

H(jw) = . ~ | (3)

WC + 25w jo— o

The center or resonant frequency of the filter iS~wb, and the band
width increases with increasing damping factor 3. The PSD may be com-
puted from

2 2
q volts

  

 

PSD = , | (9

@® 2
J/ [pGw]| aw\ rada/sec
0 ,
- | _ |
(where q is the mean square filter output) if it is assumed that the PSD

is constant within the band pass. For this filter

© N T . N
11 lH(JQD' d@w = ———— (radians/sec) .
4 5 w -
0 0
Provisions are also made in the code for correcting the PSD for any
‘low-pass filter that may have been used to prevent aliasing, and for cal-

culating the percent standard deviation of the PSD estimate.

MSRE DATA

The data previously used in the analog'analysis1 were digitized on

the Millisadic digitizer. The data included records taken for the

- | o L o = | o
S. J. Ball and R. K. Adams, 'MATEXP, A General Purpose Digital Com-
puter Program for Solving Nonlinear Ordinary Differential Equations by
the Matrix Exponential Method," USAEC ORNL' Report in preparation.
reactor critical and for the background noise observed when the reactor
was shutdown. The case considered was for the reactor primary salt cir-
culating with no bubbles, The noise record for the critical reactor was
passed through a low-pass filter consisting of a first order lag with a
time constant of 00,0047 sec, then digitized with a sampling interval of
0.00284 sec, The background noise was also filtered and digitized in
the same manner. Approximately 36,000 time points were used for both

cases,

| | RESULTS
Indirect Method

 

Figures 1 through 3 show the autocorrelation functions obtained in the
indirect analysis, All calculated points are plotted for the shorter
correlation times, but only every tenth point was included after the
curve had leveled out at longer correlation times., Figure 1 shows the
autocorrelation function for signal plus background. Figure 2 shows the
autocorrelation function for background only, The results shown in Fig,
3 were obtained by subtracting the background autocorrelation function
from the autocorrelation function for signal plus background. This can
be done if the signal and the background are uncorrelated., To show this,
take a signal composed of uncorrelated time functions x and y, and calcu-

late the autocorrelation function

011“) = % fOT [x(t) + y(t)][x(t + D + y(t + 7)] dt

1 T 1 T
= T*fo x(t) x(t + 1) dt + = fo y(t) y(t + 7 dt
1 [T 1
7 [0 x@ g+ at+z [T ye) x(t+ 1 at . (5)
T J, 0 |
Since x and y are uncorrelated, the last two integrals are zero and

T | T |
Cll("r) = -,-rl- fo x(t) x(t + 1) dt +,—},— fo y(t) y(t + ) dt . (€)

Thus, if x is the signal and y is the background, we see that we get the
autocorrelation function of the signal by subtracting the autocorrelation
function of the background from the autocorrelation function of the com-
posite signal. The improvement obtaineéd from the background correction

is quite apparent if one compares Fig. 1 with Fig,., 3.
66-2334

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Time Point (1 time point = 0,00284 sec)

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Figure 1. Autocorrelation Function — Signal + Background.
66-2335

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200 220 240 260 280 300 320 340 360 380 400 420 440

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Time Point (1 time point = 0.00284 sec)

Figure 2. Autocorrelation Function — Background Only.

 
ORNL DWG. 66-2336

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.........................................................
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..... - e R s e by e

   

o 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460

Time Point (1 ‘time point = 0,00284 sec)

Figure 3. Autocorrelation Function of Signal + Background minus Autocorrelation Function of
Background Only.

01
11

The autocorrelation function, Cll(T),'of the output of a constant

coefficient, linear system excited by white noise is given by:

n
— biT
= AE 7
€1 (M Z i s ()
i=1
where
. = a constant,
i , o
bi = a pole of the system transfer function, and
n =

order of the characteristic equation,

See the appendix for a derivation of Eq. (7). For a zero-power reactor,
the poles are at blg§0 and b, = —(A + B/4) for a one-délay—group model .
Thus the autocorreiation funétion of a zero-power reactor excited by
white noise is given by:

—(\h + B/L) T

Cll(T) = A_ + A2 e . (8)

1

If the output is low-pass filtered prior to autocorrelation with a first

order lag circuit, the overall transfer function is

LN . . 9
G Gw =G 0w 6.Gw , | | (9)
where
G (Gw = overall transfer function,
Go(jw) = reactor transfer function,
: . . . _ 1
G.(Jw) = filter transfer function = -— s and
Tf = filter time constant, sec.

In this case, the autocorrelation function is given by :

—T1/7T
Cll(T) —A 4+ A é—(x + B/L)T A e f

1" T2 3 ' (10)

For the MSRE experiment, 1/'rf = 213, Thus, a plot of 1og'C11(T)
vs T should show an initial slope of —213, .followed by a section of the

curve with a slope of —(A + R//£) before settling out to a constant value.
12

Figure 4, which shows the MSRE autocorrelation function for short =
correlation times, displays' the expected behavior. The equation that
fits this curve is

—213 —-16.3
f(T) = 4030 e 137 + 710 e 16.37 . (11)

This was obtained by assuming an exponent of —213 due to the filter
effect, then performing a least squares fit on the function: |

£(T) — 4030 & 213T .

Thus the result 6btained is

AN+ B/4=16,3 ,

Since A is approximately 0.1, we obtain

| 8/3216.2 .

- The value of B//4 obtained by this method (16.2) compares favorably -
to that obtained by Roux and Fry (14.8),1

Figure 5 shows the power spectral density of the signal prior to
the corrections(for the effects?of the filter and‘the background noise.
These results were obtained by Fourier analyzing the autocorrelation
function shown in Fig, 1. We note a sharp resonance at 376.1-radians/sec
(60 cps) due to line voltage pickup. This resonance could have intro-
duced an aliasing difficulty if the signal had been sampled at aSlower

rate and had been inadequately filtered.

Figure 6 shows the corrected power spectral density for the noise

4 | | , o
record. A Hanning window was used, the filter effect was removed, and
the background PSD was subtracted. The analog resultsl are also shown

for comparison.

 

1 -
- J. A, Thie, Reactor Noise, Rowan and Littlefield, Inc., New

York, 1963,

 
Autocorrelation Function (arbitrary units)

10,000

pt
-

o
Q
o

13

ORNL DWG. 66-2337

Data Processed as Follows:

(1) Background autocorrelation function
subtracted

(2) Filter autocorrelation function
subtracted and reactor autocorre-
lation function obtained by least
squares fit

- () Points for autocorrelation function of
reactor and filter

Points for autocorrelation function of
reactor only |

-16.37

Cll(T) = 710 e

Autocorrelation function of

~ . ~-213T
the low-pass filter = 4030 e

 

0.02 0.04 0.06 0.08 0.10 0.12
~ Correlation Time, T (sec)

Figure 4. MSRE Autocorrelation Function — Fuel Circulating.
PSD

   
   

1000 T =TT 1T T T wd ] e o e 2z Aaaa Y 1,0
Flux with Background sy 4 : - ORIL IW3. 66-2338 - .
No Filter Correction .
Hanning Window Used iRt AT T Ty Filter Weight
Digitizing Time Interval = 0.00284 sec dOTlp T LTI L LS :
500 Points in the Autocorrelation Function
. T . ;.,l..‘.. Tl . - o . N .

100 0.1

10 0.01

1

1

: 100 1000
Frequency (rad/sec)

Figure 5. Power Spectral Density of MSRE Flux Noise.

3
PSD

1000

100

10

 

4.

 

..... -

Background Subtracted

Filter Correction Made

Hanning Window Used ;

Digitizing Time Interval = 0.00284 sec b

......

& Analog Results (Normalized to PSD =
at 8.8 rad/sec)

......

10 100

® (rad/sec)

Figure. 6. Power Spectral Density of MSRE Flux Noise.

ORNL DWG. 66-233

9

  

¢l
16

. 2 . :
Earlier results indicated that the time required for indirect

analysis on the IBM 7090 is given by:

t=A Np £ (12)
where
t = IBM 7090 computing time (minutes),
A =2x 1076,
Np = number of points, and
Nt = number of values calculated for the autocorrelation function.

The calculations for the MSRE noise indicate that A is closer to

1.5 x 1075, For example, the IBM 7090 computing time required to obtain
the spectrum for Fig. 5 using 36,000 points to calculate 500 autocorre-
lation values was 26 minutes. The time required to numerically Fourier

analyze the autocorrelation function to obtain the PSD is negligible,

The number of values calculated for the autocorrelation function
was arbitrarily set at 500 in the calculations described above. It 1is
noted, however, in Fig. 3 that the autocorrelation function has leveled
off after about 100 time points. The points further out do not seem
to contain much information and possibly could be eliminated, cutting
the computation time by a factor of 5., This was done, and the results
are showh in Fig. 7. The use of this shorter maximum lag time results
in a smoothing of the autocorrelation function in a manner analogous to
the use of a broader filter in the direct method. The power spectral

density may be evaluated only for frequencies given by the relation:

kKT
w = — (radians/sec) ,
T |
m
where
k = any integer, and
T = the maximum correlation time (seconds).
17

)

ORNL DWG. 66-2340

»

   

v aal b o e dp A

 

F1ux WithHBdckground

No Filter Correction TG
Hanning Window Used

Digitizing Time Intervals = 0.00284 sec

100 Points in the Autocorrelation Function

......

100

.......

PSD

10

1 o o 10 100 | - 1000

Frequency (rad/sec)

Figure 7. Power Spectral Density of MSRE Flux Noise.

J
18

Thus, there are fewer points per unit frequency on the plot for the
shorter correlation time, However, it may be noted that the results

at appropriate frequencies agree,

Direct Method

 

The direct method was also used to analyze the same noise record
as was used in the indirect analysis. The resulting power spectral
density is shown in Fig. 8 with the background noise subtracted and
the filter correction applied. It is seen that the results are in
general agreement with the results obtained with the indirect method.
The drop-off of the points below 1.0 radian/sec is due to characteris-
tics of the wide-band ac amplifier that was used, and is not charac-
teristic of the PSD of the flux signal. This effect was also observed

in the earlier analog study.

The time required for the direct method is given approximately Dby

t =BN N 5 (13)
P w
where
t = IBM 7090 computing time (minutes),
B =7 x 1076,
N_ = number of points, and
~Nw = number of frequencies.

For instance, the results shown in Fig. 8 for 36,000 points and 26 fre-

quencies required about 7 1/2 minutes of IBM 7090 computing time,

CONCLUSIONS

Digital methods for processing reactor noise data have been success-
fully used in the anaiysis of noise records from the ORR and the MSRE,
These tests are considered to be an adequate verification of the two
different types of analysis employed. There seems to be no clear advan-
tage for either of the digital methods used. A complete power spectral
density calculation is probably a little faster by the direct method,
but the autocorrelation function obtained in the indirect method may be

worth the slight extra cost in many cases. A direct comparison of costs
PSD, Arbitrary Units

ORNL DWG. 66-2341

T T T T ';""T"""‘!“" T T LT T e e "’“’—

: b ' L :
cemeiema i e ‘177: 4o ogrmiardn g ."'::z. TR 7t

 

 

[ pirect Method _
® 1Indirect Method (Normalized at 9 rad/sec) AN

Chiednpe y o 1

1 0.01

0.1 1.0 10 | 100 .~ | 1000
. | | Frequency, rad/sec » '

Figure 8. MSRE Noise Analysis — Direct Method — Fuel Circulating, Band-Pass Filter ¢ = O.OS;V
AF = 0.00284 sec; Low-Pass Filter T.C. = 0.0047 sec; 36,000 Points. Corrected for background and
filter.

61
20

for a given spectrum calculation is not possible since the computing
time for the indirect method is proportional to the number of auto-
correlation points calculated, while for the direct method it is

proportional to the number of frequency points calculated.

It is worthwhile to compare the relative advantages of noise analysis
by analog methods and by digital methods. This comparison includes

considerations of accuracy, cost, flexibility, difficulty, and speed.
Accuracy

The accuracy of both the digital and analog methods depends on the
equipment accuracy and the length of the data record analyzed, but the
latter is usually the controlling factor. Thus the other considerations
which determine maximum record lengths that can be obtained reasonably

dictate ultimate accuracy.
Cost

The cost of an analysis includes the cost of equipment and the cost
of manpower, Both the analog method and the digital method require a
suitable detector and a high quality tape recorder; The digital method
requires a device for sampling the analog record, but since such a
machine is available at ORNL, no purchase of special equipment is
required. The remainder of the cost of digital analysis is digital
computer cost, At Oak Ridge costs, it is ecdnomical to run the computer
for an hour or more in order to replace one man-day of work, Analysis
of .data by the analog method requires special equipment, which is also
available at ORNL, and requires personal attention during analysis. It
appears that there is no clear cost advantage for either method at
ORNL., A decision by a potential noise analysf at some other installation
concerning the relative costs of the two methods would depend on the

equipment available at that installation.

Flexibility

 

The digital method seems to offer some advantages with regard to
flexibility. The frequencies to be analyzed may be selected to satisfy

the requirements of the system under consideration. Also, many frequencies

Y
)

oy

21

may be analyzed, and it is simple to go back and re-analyze the data
to clarify interesting, unexpected features of the power spectral
density curve, perhaps«by varying the band width over which the PSD is
averaged. The frequencies, and the band widths at these frequencies,
that may be analyzed by the analog method are determined by the avail-

ability of appropriate filters.

Cross correlation and cross power Spectral density (CPSD) analyses
can also be made on a "production' basis with digital techniques. |
Although analog techniques are available for CPSD analyses, practical

means are not available for production runs at ORNL,

Difficulty

 

The difficulty of carryingVout an‘anélYSis is not a major conr-

sideration unless you happen to be the analyst, Once the'data are

‘digitized, the digital method is virtually effortless. (The indirect

method code requires two input cards and the direct method code requires
one input card.) The analog method requires the attention of an atten-

dant for several hours.
Speed

It is sometimes advantageous to obtain noise analysis results
quickly. With the analog method, one could carry the taped noise to
the analyzer and immediately start grinding out results at a rate of
about 10 PSD points per hour with equipment available at ORNL, These
results are usually processed with a short digital computer code to
get the final spectra. With the digital method, one must feed the
taped data to the Millisadic digitizer, then feed the Millisadic's
cards through a digital program to correct digitizing blunders, and
then feed the good cards back to the computer for the PSD analysis

(the last two steps could be combined).

Thus the relative speeds of the two methods depends on the avail-
ability of the analog analyzer with attendant on one hand, and the

Millisadic on the other.

 
22

- APPENDIX
In this appendix, the followingaresult'will be proved:

n b t

\’ i ; | | -
Cll(T) =Z Ai e y - - (A.1)

i=1

where
Cil(T)‘= autocorrelation function of the output of a systém excited
by white noise,
i = a constant,
bi = a pole of the system transfer function (assumed to be
simple),»andv | |
n = order of‘thelcharacteristic equation,

This is simply proved if we use the fact that the spectral density

of the output, P_, is related to the spectral density of the input,

Pi’ by
P = |GGw|? P (A.2)
o i’
where |
: 2 : :
|G(Jw)| = G(Ew G(—jw ,
G(jw) = the system transfer function,

" If the input is white noise, Pi is a constant, K, and we obtain

P =kleGw ® . ©(A.3)

The transfer function of a lumped-parameter, constant-coefficient

linear system may be written:

(Jw-—,al)(Jw-— az) ces

 

Gw = (A.4)

(Gw — bl)(Ju>—-b2) .o
23

where

a zero of G(jub',

Il

2
b
1

a pole of G(jw .

The autocorrelation function of a 51gna1 is equal to the inverse Fourier

transform of its power spectral den31ty

c, (M =F1{p)=2 f GGw Gyw @ dw (A.5)

O - Q0

The term, G(jw) G(—jw) may be written in the form:

[Go=2a)Gu=—2a) ....][(ww- al)(—jm- a) ...]
G(jw G(—jw) = . (A.6)
[Gw = b)) Gw = D)) o] = b)) (jw—-Db,) ...]

 

The function, G(jw) G(—jw) ert has poles located as shown below:

 

 

The integral in Eq. (A.,5) may be evaluated as a contour integral using

the Cauchy residue theorem to give:
24

K ° ] jwt
o f 6w G—jw e dw

where

| | o o :th
Bi(bi) = the residue of the i

This result is_of the form,

and we may write

n .

= Kj ZBi(bi) e

n=1

pole.,

b. .t
i

(A7)

(A.8)
1.

3-12,.
13,
14,
15.

16.

17,
18,
19,
20,
21,
22,
23,
24,
25-34,
35.
36.
37.
38.
39.
40 ,
41 ,
42,
43,
44 ,
45,
46 ,
47 .
48-49 ,
°20=31,
92-54,
05,
06 ,
o7,

25

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1)

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