ORNL-TM-1647.txt

 

 
      

v Y

OAK RIDGE NATIONAL LABORATO
operated by

Rfc g3cc

UNION CARBIDE CORPORATION
NUCLEAR DiVISION LT
for the

U.S. ATOMIC ENERGY COMMISSION
ORNL- TM~- 1647

o

COPY NO. - -+ '3,

DATE - October 13, 1966
GOV~
EXPERIMENTAL DYNAMIC ANALYSIS OF THE MOLTEN-SALT
REACTOR EXPERIMENT*

 

.

 

= oUING z
RELEASED FOR ANE: T. W. Kerlin and S. J. Ball

ABSTRACT

 

Dynamics tests were performed on the Molten-S5Salt Reactor Experiment
(MSRE) for the full range of operating power levels to determine the
power=to-reactivity frequency response. Three types of input disturbances
were used: the pseudo-random binary reactivity input, the pulse reactivity
input, and the step reactivity input.

The frequency response of the uncontrolled reactor system displayed
resonant behavior in which the frequency of oscillation and the damping
increased with increasing power level. Measured periods of natural
oscillation ranged from thirty minutes at 75 KW to two minutes at 7.5 MW.
Thege oscillations were lightly damped at low power, but strongly damped
at higher power.

The measured results generally were in good agreement with predictions.

The observed natural periods of oscillation and the shapes of the measured
frequency response agreed very well with predictions. The absolute amplitude
of the frequency response differed from predictions by a factor that was
approximately constant in any test (though different tests at the same power
level did not have the same bias). This bias difficulty is apparently partly
due to eguipment limitations (sta.ndard MSRE control rods were used) and
partly due to uncertainties in the parameters in the theoretical model.

The mein conclusion is that the system has no operational stability
problems and that the dynamic characteristics are essentially as predicted.

 

¥ Research sponsored by the U. S. Atomic Energy Commission under
contract with the Union Carbide Corporation.

For presentation at the Winter Meeting of the American Nuclear Society to be
held October 30-November 3%, 1966 in Pittsburgh, Pa.

NOTICE This document contains information of a preliminary nature
and was prepared primarily for internal use at the Oak Ridge Nationel
Laboratery. It is subject to revision or correction and therefore does
not represent a final report.

re LAV EE R BT R g o ceece .o Crape -

[ei8 DOUUMENT B0s T o
IVTRITOOEY S ey gy

MO INVENTCHZ U Bl Dinoeks]

10 THE A.E.C. A;;‘;;&j{g@flaaa THEREIN,

7 ] fo

  

—

  
 

 

 

LEGAL NOTICE

This report was prepared as an account of Government sponsored work, Neither the United States,

nor the Commission, nor any person acting on behalf of the Commission:

A. Mokes any warranty or representation, expressed or implied, with respect to the accuracy,
completeness, or usefulness of the information contained in this report, or that the use of
any information, apparatus, method, or process disclosed in this repert may not infringe
privately owned rights; or

B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of
any informotion, epparatus, method, or process disclosed in this report.

As used in the cbove, ‘‘parson acting on behalf of the Commission® includes any employee or

contractor of the Commission, or employse of such contractor, to the extent that such employee

or contractor of the Commission, or employee of such contracter prepares, disseminates, or
provides access to, any information pursuant to his employment or contract with the Commission,

or his employment with such contractor.

 

 
IT
TIT

v

Vi

VIiI

ViIT
IX

Appendix A.

Appendix B.

iii

CONTENTS

IntrodUCtiOn.eeseseseessossovrsosasansnons
Description of the MoRE . veeeeecrcssnves
Theoretical PredictionS.esecessssencenssne
A. Description of Mathematical Model...
B. ResulfS.ceerecscensrsstsossscsasesnas
Selection of Experimental MethodsS.seeess

A, Characteristics of the MSRE
RegulatingRod...l'......."‘..l..

B. Test Signals Used in the Experiments
1. Pseudo-random Binary Test..ev...
2. Pulse TestSiseeevrsarcscososncasns
3., Otep TestSeeeeeestesessssorenane
Experimental ProcedureSievseeeeescessess

A. TImplementation of Pseudo-random
Binary TestSeecsaececosonsrecscscns

B. Implementation of Pulse and
StepTests-.ll...ll....l...'l...‘.

Analysis ProcedUreSeceesrsecsssesssasssans
A. Direct Anelysis of PRES TestS.iervaess
B. Indirect Analysis of PRBS TestSeeese
C. Step Response Test Analysis.

®
-
»
L]
*
.
»
.

D. DPulse Response Test AnalysiS.cacesas
ResUltSieeceeessessasasercscanasssonasens
A. Transient RespOnseSesssscessscsssesns
B. Correlation FunctionS.eeessssesssens
C. TFrequency RespONSESeecsecssccssasseass
Interpretation of the Results..cevecevose
ReferenCesSieieenscescsesorenessnssscoscons
Potential Sources of Experimental
Error Due to Equipment Limitations..

The Direct Method for Cross~Power

Spectrlflfl Analysis.a-ooooc-OOOAA‘NOT‘CE

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-1

I. INTRODUCTION

A gseries of experiménts was performed on the Molten Salt Reactor
Experiment (MSRE) to determine the frequency response of the uncontrolled
reactor system. Tests were performed at eight different power levels
ranging from zero to full power. Three different types of input
disturbances were used to obtain the nuclear power to reacti#ity
frequency response: the pseudo-random binary reactivity input, the
pulse reactivity input, and the step reactivity input. Subsequent
sections of this report will give a description of the system, a
review of previously published theoretical predictions, a description

of the testing procedures, and the experimental results.
IT. DESCRIPTION OF THE MSRE

The MSRE is a graphite-moderated, circulating-fuel reactor. The
fuel is a mixture of the molten fluoride salts of uranium, lithium,
beryllium, and zirconium.l The basic flow diagram is shown in Fig.

1. The flows and temperatures shown are nominal values which were
calculated for operation at 10 MW, but heat transfer limitations at

the radiator currently restrict maximum power operation to about

Te5 MW, The molten fuel-bearing salt enters the core matrix at the
bottom and passes up through the core in channels machined out of
unclad, 2-inch graphite blocks. The heat generated in the fuel and
that transferred from the graphite raise the fuel temperature about
50°F. When the system operates at reduced power, the flow rate is

the same as at full power and the temperature rise through the core

is smaller. The heated fuel salt travels to the primary heat exchanger,
where it transfers heat to a non-fueled secondary salt before reentering
the core. The heated secondary salt travels to an air-cooled radiator
before returning to the primary heat exchanger. The design parameters
of major importance from the standpoint of dynamics are shown in Table

1. A detailed description of the MSRE appears in Ref. 1.
|| PRIMARY SALT
|| LIF - 70% 1025° F
Bef, ~ 23% .
“ 2rfy — 5% I LOOF -
Il They — 1% II
I UF, — 1% I SECONDARY SALT
| I LiF — 66%
I A ou - 34% |

Da

ORNL-LR-DWG 56870

         
   
   

      

 

 

 

REACTOR CELL |

300°F

AIR
167,000 cfm
100°F

 

SPARE FILL AND FLUSH COOLANT
FILL AND DRAIN TANK TANK DRAIN TANK
DRAIN TANK (68 cu ft) (68 cuft) (40 cu ft)
(68 cu ft)

Fig. 1 MSRE Flow Diagram
-3-

Table 1, MSRE Design Data

Nuclear

Flow

 

Heat

Temperature coefficient of reactivity
of the fuel, °F-1

Temperature coefficient of reactivity
of the graphite, °F-1

Neutron lifetime, sec.
Total delayed neutron fraction
Reactivity loss due to fuel

circulation, % 8K
K

Flow rate in the primary loop, gpm
Flow rate in the secondary loop, gpnm
Fuel transit time in the core, sec.

Fuel transit time in external primary
loop, sec.

Total secondary loop transit time, sec.
Transfer

Fuel salt heat capacity, MW sec/°F
Graphite heat capacity, MW sec/°F

Heat exchanger heat capacity, MH sec/°F

Bulk graphite ~ fuel salt heat transfer
coefficient, MW/°F

Fuel salt~heat exchanger metal heat
transfer coefficient, MW/°F

Heat exchanger metal, secondary
salt-heat transfer coefficient, MW/°F

Fraction of power generated in the fuel

b7 x 1077

2,6 x lO"'5
L0002k
.00666

~0,.,212

1200
830
8.5

16.7

24 .2

L.2
3.6

0.02
0.36

Q.17

0.93
=l

ITT. THEORETICAL PREDICTIONS

A. Description of Mathematical Model

Throughout the MSRE design effort, a wide variety of mathematical
models was used to predict the dynamic behavior. We will limit our
discussion here to the most up-to-date and detailed model reported,
referred to in Ref. 2 as the "complete'" model. The core fluid flow
and heat transfer eguations were represented by 18 fuel nodes and
9 graphite nodes. The nuclear power distribution and the nuclear
importances for each node were derived from a 2-group neutron diffusion
calculation. The flow rates and heat transfer coefficients for each
node were determined from calculations based on full-scale hydraulic
core mockup tests. The assumed flow mixing characteristics were
verified by transient tests on the mockup.

The neutron kinetic behavior was described by the usual space-

independent equations with six delayed-neutron groups, but with modi-
fications to include the dynamic effects of the circulation of precursors
around the primary loop. The thermal reactivity feedback was computed
by using a weighted nuclear importance for each of the 27 fuel and
graphite nodes. The xenon poisoning reactivity feedback included
iodine production and decay into xenon, xenon decay and burnup, and
xenon absorption into the graphite.

The transport of molten salt in the primary and secondary loop
piping was described by a plug flow model, where heat transfer to the
pipes was included. The primary heat exchanger and the salt-to-air

heat exchanger were each represented by a 50-node model.

B. Results of Theoretical Analysis

Several different methods of solution were used on the various MSRE
dynamics models, including analog and digital computer simulation
(time response), frequency response analysis, and pole configuration
analysis. The frequency response analyses can be directly compared
to the experimental results, since the latter are readily cast in
this form.

Fig. 2 shows the theoretical MSRE inherent frequency response
ORNL-DWG 65-9816

2

10,000

 

 

10
o0 2 5 001 2 5 o 2 5 t 2 S 10 2 5 100
FREQUENCY {rodians/sec)

90
80
70
PHASE OF E#%%
60 0
50
40
30

20

PHASE (deg)
o

NO FEEDBACK
-70 ———m ‘

 

-90
00001 2 5 0001 2 5 oo 2 5 ot 2 5 1 2 5 10

FREQUENCY (radians/sec)

Ffig. 2 MSRE Theoretical Frequency Response
-6

characteristics for normalized neutron level response to reactivity
perturbations at several power levels. It can be seen that the system
becomes more oscillatory at progressively lower frequencies as the
nominal power level decreases, though it is stable for all power levels
of interest. An explanation of the inherent stability characteristics

is given in Ref. 2.
IV. SELECTION OF EXPERIMENTAL METHODS

The selection of the experimental methods for the MSRE dynamics
tests was based on the information required and on the capabilities
of the available equipment. It may be seen from Fig. 2 that the most
significant part of the frequency response is in the range 0.0l to
0.1 radians per second, since the amplitude peaks are in this fregquency
range for the operating power levels of interest. This frequency range
corresponds to long periods of natural oscillation (10 min. to 1 min.).
This emphasis on low frequency results fortunately made it possible
to obtain the important part of the system frequency response using
the standard MSRE control rods to introduce the input reactivity
perturvations. In this section, we will examine the characteristics
of the MSRE regulating rod and the properties of the test signals used.

A, Characteristics of the MSRE Regulating Rod

The MSRE has three control rods, each with an active length of

59.4 inches. One rod is normally designated as the regulating rod and
is used for fine control. The other two rods are shim rods used for
coarse adjustments. The rods are actually flexible, stainless steel
hoses on which are strung gadolinium oxide poison cylinders. The rods
are mounted in thimbles which have two 30° offsetting bends so that
the rods can be centrally located even though there was no room for
the control rod drive assemblies above the central axis of the core.
The maximum rod speed is ~0.5 inches/second.

The three control rods are identical. Figures 3 and 4 show the
control rod and :drive assembly. The position indication for each rod
is obtained from two synchros geared t¢ the rod drive mechanism.

Synchro number 1 is used for coarse position indication and has a
ORNL-LR-DWG 67311

REVERSIBLE DRIVE MOTOR

  
  
  
  
  

COOLANT TO DRIVE
ASSEMBLY

COOLING
GAS INLET
COOLING GAS
SOLENOID ACTUATED CONTAINER
RELEASE

' POSITION INDICATOR
:.\ 7z SYNCHRO TRANSMITTER

‘l» FIXED DRIVE SUPPORT AND

 

 

 

 

   
  

|

|
UPPER LIMIT ——{ | 3in. CONTAINMENT TUBE
SWITCH i) —COOLANT TO
orwe uwT b ;'! POISON ELEMENTS

l i i % in.0.0.-3045.5.- FLEXIBLE
LOWER LT ,,.! '. HOSE CABLE
" I!"',"
5“"0 3 h"glth\
ORNE \ <IN
dfl‘ N\ Y 5 * SPRING LOADED ANTIBACKLASH
AN H HEAD AND IDLER GEAR
3 | w0
. % ‘. ‘\T‘

3in. x 2in, ECCENTRIC
REDUCER

16 In. RADIUS x 30° BEND

COOLANT EXHAUST

GUIDE BARS, ———.
4 AT 90°

 

BEADED POISON ELEMENTS =]

 

2in. CONTAINMENT THIMBLE —

 

 

 

 

Fig. 3 Control Rod Drive Assembly
ORNL-LR~-DWG 78808

 

SYNCHRO NO.2
60° PER INGH OF ROD MOTION ~o=

 

 

 

 

~s-TAGHOMETER

 

SERVO POTENTIOMETER
MOTOR SYNGHRO NO. !

 

 

 

 

5°PER INCH

 

 

 

BRAKE OF ROD MOTION-»

 

 

 

 

 

 

-=— GEAR REDUCER NO.2

 

 

ELECTRO-
MAGNETIC
GLUTCH

 

 

 

 

   

 

 

 

 

 

GEAR
REDUCER NO. {~e=

 

 

AIR IN

OVERRUNNING CLUTC

FLEXIBLE

TUBULAR ROD SUPPORT —a=

v=0.5 in/sec

{-TO-1 GEARS

INPUT SPROCKET

    
  

 

SPROCKET GHAIN

REACTOR VESSEL
CELL

 

 

 

 

 

Fig. 4 Diagram of Control Rod

   

 

POISON ELEMENTS

i He~TEMPERATURE ~ 41400°F

~= POSITION INDICATOR AIR
FLOW RESTRICTOR

 

\— /= AIR DISCHARGE - RADIAL

PORTS

Drive
~Q=

sensitivity of 5° per inch of rod motion. Synchro number 2 is used
for fine position indication and has a sensitivity of 60° per inch.
The signal from the coarse position synchro is transmitted to a
torque amplifier which drives a single-turn potentiometer feeding

a d-c signal to the MSRE on-line computer.

After the system had operated for some time, it became difficult
to obtain reproducible regulating rod position changes for a given
time of insert or withdraw. This wasg due to the wearing of this
rod and drive assembly caused by frequent use. For the dynamics
tests, one of the rods normally used as a shim rod was used as the
regulating rod. Since it is moved much less frequently than the
normal regulating rod, it had experienced less wear and had much
tighter response characteristics.

There are a number of factors which could adversely affect both
the accurate positioning of the rods and the indications of effective
rod position given by the instruments. BSome of the potential sources
of difficulty are listed in Appendix A.

These observations indicate that the MSRE control rods are hardly
ideally suited for dynamic testing. However, since no provision was
made for special control rods and since the main features of interest
occur at low frequency, the testing program was carried out using
the standard MSRE control rods. Fortunately, the rods performed
far better than expected by their designers, and the final results
were only slightly degraded by equipment problems.

B. Test Signals Used in the Experiments

Three different types of test signals which were used to obtain

the frequency response of the system are described in this section.

(&)

In this test, specially selected periodic series of positive

1) Pseudo-random Binary Test

and negative reactivity pulses called the pseudo-random binary
sequence (PRBS) were introduced. The PRBS has the advantage that its
frequency spectrum consists of a number of harmonics of approximately
equal size. This means that the frequency response may be evaluated

at a large number of frequencies in a single test. The spectrum of
=-10=-

the pseudo-random signal from one of the MSRE tests is shown in

Iig. 5. We note that the signal strength is concentrated in

discrete harmonic frequencies rather than distributed over a

continuous spectrum as in the case of non-periodic (e.g. pulse and
step) signals. This is helpful since it Improves the effective signal-
to-noise ratio at these frequencies.

A PRBS may be generated on-line at the test or may be pre-recorded
in some fashion and played back as a control signal. On-line
generation of the signal was used in these tests (see Section V).

A PRBS is characterized by the number of bits in the sequence and

the bit duration. A bit is defined as the minimum possible pulse
duration in the sequence. All pulses in a PRBS are minimum width or
integral multiples therecof. Numerous sequences may be generated, but
they are restricted to certain specific numbers of bits. In the MSRE,
PRBS tests were run with 19, 63, 127 and 511 bits. If the number

of bits is Z and the bit duration is At, then the PRBS has a period

Z At. The lowest harmonic radian frequency, w5 and the spacing of

the harmoniecs, Aw, is given by ® = Xy = 21t . The PRBS tests which
ZAE
were run and analyzed are shown in Table 2. In each test, the rod

motion was selected to give a reactivity change of 0.02% to 0.03%,
peak to peak. The maximum reactivity perturbation was determined on
the basis of keeping the resulting power level perturbations in the

linear range, i.e. 6N/NO maximum was kept below O.1.

2) Pulse Tests ™)

In theory, it is possible to excite a system with a single
pulse-type disturbance and obtain the frequency resonse by numerically
determining the ratio of the Fourier transform of the output to the
Fourier transform of the input. The frequency response can theoretically
be evaluated at all frequencies since the input has a continuous
frequency spectrum. In practice, the pulse test is often unsatisfactory
because the available signal strength is distributed over all
frequencies, resulting in a small amount of signal around the analysis

frequency and thus poor effective signal to noise ratio.
Input Power Spectrum

0 ORNL DWG. 66-11076

10

_'['[-

 

Frequency (radians/sec)

Fig. 5 Power Spectrum of the Input PRBS at 0.465 MW
-12-

Several tests using approximately square reactivity pulses of
between 0.01 and 0.02% were employed for the MSRE at zero power.
Tests were carried out both with circulating fuel and with stationary

fuel.

Table 2. Pseudo-Random Binary Sequence Tests

 

Power Bits in Bit Periodicity of Minimum
Levels (MW) Sequence Duration (sec) +the PRBS (sec) Frequency
(rad/sec)
0 19 6.58 125 .05
0 63 3.35 211 .03
075 511 3.35 1711 0037
L65 511 3.35 1712 0037
1.0 511 3.35 1711 0037
1.0 127 5.02 638 .0098
2.5 511 3.32 1699 .0037
2.5 127 4.97 631 .010
5.0 511 3.33 1701 .0037
540 127 4.97 631 010
6.7 511 3.32 1698 0037
7.5 511 3.33 1701 .0037
7.5 127 4.97 631 010

3) Step Tests!?)
If the output eventually levels off to some constant value

after a step disturbance, the frequency response may be obtained by
a modified Fourier analysis of the output and the input. As with the
pulse tests, the step input has a continuous spectrum, but is hampered
by a low effective signal-to-noise ratio at any analysis frequency.

Step tests were used at power levels where the temperature feedback
was adequate to cause the power to level off near the original power
after the reactivity change. These tests used a reactivity perturbation

of 0.01 to 0.02%. Step tests were done at 2.5, 5.0, 6.7, and 7.5 MW.
-13-

V. EXPERTMENTAL PROCEDURES

A. Implementation of Pseudo-random Binary Tests

The pseudo-random binary reactivity sequence required a very
precisely controlled series of regulating rod insertions and with-
drawals. Since the frequency range of interest was from about 0.002
to 1.0 radian/sec., the rod jogger was designed so that sequence with
a bit time of 3 to 5 seconds could be run for as long as 1 hour. The
rod jogger system, which required no special-purpose hardware,
consisted of a hybrid computer controller shown schematically in Fig.
6. The portable EAT-TR-10 analog computer was used to control the
bit time and the rod drive motor "on" times for the insert and withdraw
commands. The MSRE digital computer was programmed to control the
(3)
The number of bits in the sequence could be varied over a range between
3 and 33,554,431 bits.

The rod jogger system performed extremely well, as it was able to

sequencing of the pulse train by means of a shift-register algorithm.

position the rod with an indicated positioning accuracy of about

+0,01 in. (corresponding to +0.0005% 8k/k) out of 1/2 in. peak-to-peak
rod travel for over a 500 insert-withdraw operations. A typical PRBS
rod position signal is shown in Fig. 7 and a typical record of neutron
flux changes during a PRBS test is shown in Fig. 8.

The analog computer was also used to amplify and filter the rod-
position and power-level signals prior to digitizing. Throughout the
dynamic tests the MSRE computer was used in the fast scan mode to
digitize and store the data on magnetic tape. Each variable was
sampled at the rate of 4 per second.

B. Implementation of Pulse and Step Tests

The same set up as described in (A) was used for the pulse and

step tests with the PRBS generator omitted.
VI. ANALYSIS PROCEDURES
A. Direct Analysis of PRBS Tests

The "direct" method of analysis uses a digital computer simulation

of an analog filtering technique for obtaining cross-power spectral

(6)

density functions. A bplock diagram of the analyzer is shown 1n
 

INSERT

 

 

 

 

 

ROD-POSITION

:/ SIGNAL

 

 

‘-—-———-———-——-—-—f—-—-
]_I_‘ R%%-P&.VE WITHORAW
|
|
!

1l

ORNL-DWG 66~4T763

 

 

 

 

PORTABLE ANALOG MSRE DIGITAL
COMPUTER COMPUTER
________ - =~ — =
EAI-TR-{0O | BR-340

|
BIT-TIME | |
GENERATOR ||
|

 

PSEUDO-RANDOM
BINARY SEQUENCE

|
I
|
|
I
I
|
|
I
l COMPUTATION

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

REACTOR

 

 

o] AMPLIFYING —T—|" DIGITIZER AND
I AND || MAGNETIC TAPE
|[rOD ™™ FILTERING —1 RECORDER
POWER-LEVEL : |
SIGNAL
) N I -

 

 

 

 

 

 

 

Fig. 6 Block Diagram of MSRE Rod Jogging and
Data Logging System
Indicated Coutrol Pod Position

 

Low-Pass Filtered (Filter Transfer Function = T 1 -)

ORNL DWG. é8-11077

 

[Creuet EEL BUCInE B (Pl ERAOIORRAR dat £ RURIE £ OH BECLE A ;F HIEHL :rrrrrgy_mrr‘gnf iy wir ey ,{f Uu_r';,r_tz g:f'_'rfr';ag;f!
i LLfiLtL?‘i i 'm: Ifif\i:k.:\ i Uf Huu’fluu'fl G Rl HEeLL U RebiLE Gt B WL R uLh.fin MU G O LU GRAR BT G
ET j Jio 1; 20 2 b Y

KRapsed Pime (min.)

Fig. 7 Indicated Control Rod Position During a 511 Bit PRBES Test
NUCLEAR POWER, % OF 1.5 Me

-16=

ORNL-DWG 66~-4764

 

a2

 

 

I I I I I |

 

 

| I I I I I

 

22

o

4.47 8.34 12.5¢ 17.09 21.26 25.43 29.60
TIME ( 30-min FULL SCALE)

Fig. 8 Measured Flux Signal During a 511 Bit PRBS Test
-1~

Fig. 9. The narrow band-pass filter H(s) has the characteristics of

a quadratic lag and a transfer function

s

 

i(s) = =% 3
@, + 2¢ w, s + 5

where w, is the center freguency of the filter. Typically
the damping factor & is set equal to 0.05, giving a filter frequency
response as shown in Fig. 10, This analyzer is similar to that
described by Chang(7) and Van Deusen(8 ;3 the main difference is that
it includes extra cross-multiplication terms resulting in higher
accuracy. A mathematical description of the method is given in
Appendix B.
B. Indirect Analysis of PRBS Tests

The indirect analysis began with a calculation of the

autocorrelation function of the input and the cross-correlation function

of the input and the output

1 -\
¢ (D) =& j; I(t) I(t,+fT)fdt_
Coey a1
c (7)) =& fo I(t) oft + 7). at
where
Cll(m) = the autocorrelation function

012(1) the cross-correlation function

T = the lag time

T* = the integration time (a multiple of the signal

" I(t) = the input signal periodicity)
0(t) = the output signal
t = time
i)

ie—I /1

-18-

ORNL-DWG 68-10283

 

i{1) oft)

—_— G(s)

 

 

 

 

\SYSTEM TO BE ANALYZED

 

 

 

 

Hisky, Yo

!U

 

 

 

 

 

 

 

 

 

olt)

 

 

7 LI_'J \ - :: + -
L - [[ . ::]‘_’
f COPOWER

 

 

 

 

 

 

 

 

Fig. 9 Block

 

 

 

Hisky, %

 

 

 

 

 

H(sluo * NARROW BAND-PASS FILTER WITH CENTER FREQUENCY w,

_ COPOWER(s) + | QUAD POWER(s)

G(s) PSD; (s}

Diagram of Cross-Power Spectral Density Analyzer
PHASE (deg)

MAGNITUDE RATIO [H(s)]
N

10

o

o,

Q

-19-

ORNL-DWG 66-10635

S
52 + 2 Lwgs t wy?

H(s) =

PLOT SHOWN FOR { = 005, w, !

 

w/wg

 

 

 

 

 

 

 

\

 

\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 10

5 Of 2 5 10 2 5 10 2

w /wo

Frequency Response of Filter Used in CPSD Analyzer

100
w2

Subsequent Fourier analysis of C..(1) and 012(1) gave the input

ll(

power spectrum and the cross power .spectrum:

T
J; Cll(T) cos w T dt

10
|_’/"\
"
|
H

T T
¢32(mk) = T'J; 012(7) cos @y T At - % J; Cle(T) sin @ 7 d7

® (cuk) = input power spectrum at frequency, @,

cross power spectrum at frequency, wk

v
£
i

angular frequency of the kth harmonic (ak'= E%E)

>

T = periodicity of the test signal

Note that the input power spectrum is a real quantity since Cll(T) is
always an even function of 1.

The frequency response, G(jak), is given by
L JNTC (7) cos @ T dt
. T Jo 12
Re[G(J&k)] = T

1
= J; Cll(T) cos @ T dt

 

 

l -
Im[G(j&k)] = - T'J; Clg(T) Sin @ T dt
1
T

T
j; Cll(T) cos w T dt

The magnitude ratio, MR, and the phase, 6, are given by:

 

() = o (Releim) HF v (T, elin)])"
o )T (605w ) ]

e(wk) = tan fig-Tfizsa;Tj .

A Fortran computer code for the IBM-T7090 or IBM-360 called CABS(Q)

was prepared to carry out these computations.
-21-

C. Step Response Test Analysis

The step response tests were analyzed using a digital computer code
which implements Samulon‘s method.(5’lo)
D. Pulse Response Test Analysis

Although pulse response tests were attempted at power levels
of ~ 0, 75 KW, 465 KW, and 1 MW, the only successful runs were the
ones made at zero power. This was because the low frequency random
fluctuations in heat load at low (but non-zero) powers drastically
reduced the signal-to-noise ratio. At these powers, the radiator is
cooled primarily by natural convection and radiation, and is consequently
sensitive to atmospheric disturbances. At higher powers, where most
of the cooling was due to forced convection, these fluctuations were
not apparent.

The zero power tests required extremely accurate core temperature
control and rod positioning in order to avoid drift of the flux level.
Since a zero power reactor is an integrating system, a pulse reactivity
input results in a change in steady-state flux output, and Fourier
transforms are valid only if the response function eventually returns
to its initial value. We got around this problem by first numerically
filtering the output response data through a high-pass network HP(s)
with a transfer function

_50s
HP(s) = 55551

then performing the numerical Fourier transform, and finally compensating
the resulting frequency response for the high-pass filter characteristics.
The numerical Fourier transform calculations were done by a digital

(12) which is mathematically equivalent

computer code using a novel method
to the standard technique of summing the products of f(t) ¢ cos wt and

f(t) * sin wt, but which reduces the computing time by a factor of 3.
VII. RESUILTS

The experimental data were analyzed to give correlation functions,
input power spectra and cross power spectra, and frequency responses.

These results are given in this section along with the directly observable
00

transient response to system disturbances.
A. Transient Responses

At each power level, a transient was induced by inserting a
reactivity pulse or step, or by simply allowing the system to seek
equilibrium after the completion of some other test. These transient
responses are informative in themselves since they demonstrate the
damping and the natural frequency of oscillation of the system.

Figure 11 shows the observed transient response. At 0.075 MW,
it took over two hours for the flux to return to equilibrium.
B. Correlation Functions

The pseudo-random binary tests were analyzed by the direct method
and the indirect method. Autocorrelation functions of the input and
cross correlation functions of the input and output were obtained &s
intermediate results by the indirect method. The correlation functions
had the expected appearance in all tests until the 2-1/2 MW tests.

In that test, spikes appeared in the correlation functions which

have been present in all tests since then. The spikes always appeared
at points 432 sec. from each end of a period in the 511 bit tests

(~ 1700 sec. period), and 34 sec. from each end in the 127 bit tests

(~ 630 sec. period). Figures 12 through 19 show typical autocorrelation
functions and cross-correlation functions for tests before the 2-1/2 MW
test and after the 2-1/2 MW test.

The reason for the appearance of the spikes is not yet known. The
only significant difference noted in the input signal at 0.465 MW and
at 2-1/2 MW was a change in effective pulse duration for positive and
negative pulses. At 0.465 MW, the duration of a pulse above the mid-
point was the same as the duration of a pulse below the midpoint. At
2-1/2 MW, the duration of pulses below the midpoint was longer than
the duration of pulses above the midpoint. This was apparently due
to changes in the coasting characteristics of the rod. Calculations
were performed on pseudo-random binary sequences in which the pulse
duration was changed for positive and negative pulses to determine
whether this would cause spikes in the autocorrelation function.

These calculations showed spikes for some sequence lengths but not
for others. In particular, spikes were not found in these calculations

for a 511 bit sequence. The conflicting indications furnished by
POWER CHANGE (%)

-23-

ORNL-DWG 66-10284

0075 MW

 

 

 

 

0 7.5 MW

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0O 2 4 6 8 10 12 14 16 {18 20 22 24 26 28 30 32 34 36
TIME (min)

Fig. 11 MSRE Power Level Transients
Autocorrelation Function (Arbitrary Units)

ORNL DWG. 68-11078

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Correlation Time (sec)

Fig. 12 Input Autocorrelation Function for a 511 Bit
PRBS Test at 0.465 MW
Cross=Correlation Function {Arbitrary Units)

ORNL DWG. 66-11079

 

 

 

=

 

 

l,llllli?.

 

 

 

 

 

 

 

 

 

 

 

 

 

400 800 1200

Correlation Time (sec)

Fig. 13 Cross-Correlation Function for a 511 Bit
PRBS Test at 0.465 MW

1600
Autocorrelation Function (Arbitrary Units)

o
(‘lllll x =X =[x x x !=n x ] & F YTTrwrmwmymrTys

ORNL DWG. 66-11080

 

 

 

 

 

—98—

 

 

 

 

 

 

 

 

 

 

 

 

160 320 480 640

Correlation Tine (sec)

Fig. 14 Input Autocorrelation Function for a
127 Bit PRBS Test at 1.0 MW
Cross-Correlation Function (Arbitrary Units)

ORNL DWG. 4646-11081

 

T F I

*

L

 

 

lklllllllflflflllflll wlx ¥ 2 X

LeacertneorendT

 

 

 

 

 

 

 

 

 

 

 

EX N x M |x x =

 

 

160

Fig. 15 Cross-Correlation Function for a 127 Bit
PRBS Test at 1.0 MW

320

Correlation Time {sec)

640

-La_
Autocorrelation Function {Arbitrary Units)

ORNL DWG. 66-11082

 

 

 

 

 

 

 

 

 

W‘—wwfiw

 

 

 

 

 

 

 

 

 

400 800 1200
Correlation Time (sec)

Fig. 16 Input Autocorrelation Function for a 511 Bit
PRES Test at 2.5 MH
Crose-Carrelation Punction (Arbitrary Units)

ORNL DWG. 66-11083

 

 

 

X XN N X

 

ot XN X X

 

 

 

 

 

 

 

 

 

 

 

 

400 800 1200
Correlation Time (sec)

Fig. 17 Cross-Correlation Function for a 511 Bit
PRBS Test at 2.5 MW

"63"'
Autocorrelation Function (Arbitrary Units)

ORNL DWG. 65-11084

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 160 320 k8o

Correlation Time (sec)

Fig. 18 Input Autocorrelation Function for a 127 Bit
PRBS Test at 2.5 MW

640
Cross-Correlation Function (Arbitrary Unite)

ORNL DWG. 66-11085

 

 

 

 

 

» =
::nl“’" x> ®
»

qufl"'

 

:

 

 

 

 

 

 

 

 

 

 

© 160 320 180

Correlation Pime (sec)

Fig. 19 Cross-Correlation Function for a 127 Bit
PRBS Test at 2.5 MW

640

-Tg_
~30w

these calculations have not yet been resolved. However, we should
note that this unexpected feature of the correlation functions does
not invalidate the final result, the freguency response. It simply
means that the spectral properties of the input and output are slightly
different than anticipated, but that the ratio of cross power spectrum
to input power spectrum still gives a valid frequency response.
C. JFrequency Responses

The results of the frequency response analyses are shown in PFigures
20 through 28. The legend in each figure indicates the type of test
and the analysis procedure. The theoretical curves were taken from

(2)

corresponded to available calculations. The curves for the other cases

previously published results when the experimental power levels

were obtained using the same procedures as were used to obtain the

(12

the random noise in the neutron flux signal are also shown with the

published results. Results obtained by Roux and Fry ) by analyzing
zero-power results. These results were normalized to the theoretical
results at 9 rad/sec. These points were obtained by taking the square
root of the measured power spectral density (PSD) of the neutron flux
signal after subtracting the background PSD. This gives a result which
is proportional to the amplitude of the flux-to-reactivity frequency
response, assuming that the observed noise is the result of a white
noise reactivity input.

The natural period of oscillation of the system was determined
either by direct observation of a transient or by location of the peak
of the frequency response. These results, along with theoretical

predictions, are shown in Fig. 29.
VIII. INTERPRETATION OF THE RESULTS

The objectives of the dynamic testing program are twofold: 1)
provide information on the stability and operability of the system;
2) provide information for checking procedures for making
theoretical predictions so that future calculation on this and other
similar systems will be improved. The results are interpreted in

terms of these two objectives.
ORNL-DWG 66-10286

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

104
ZERO POWER
5 19 BIT PRBS DIRECT ANALYSIS @
19 BIT PRBS INDIRECT ANALYSIS &
63 BIT PRBS DIRECT ANALYSIS A
2 63 BIT PRBS INDIRECT ANALYSIS V¥
7 sec. PULSE, TEST NO. | v
103 7 sec. PULSE , TEST NO. 2 0
NOISE ANALYSIS o
5
m
M‘/ko ,
102
5
2
10'
0.001 0005001 002 005 Q! 02 ©0O5 10 20 50 100 200 500 1000
w, FREQUENCY (radians/sec)
20 ORNL—-DWG 65-8907A
ZERO POWER
1O | 19 BIT PRBS—DIRECT ANALYSIS A
0 | 19 BIT PRBS—INDIRECT ANALYSIS @
63 BIT PRBS—DIRECT ANALYSIS v 4
—10 163 BIT PRBS—INDIRECT ANALYSIS ¥ 11
-20 | 7 sec. PULSE, TEST NO. | 0 bn“ -
> 30 7 sec PULSE, TEST NO. 2 A Agfi,}'f ™
Q —
2 4 A A4‘3‘F;"‘ )
w —40 s X
2 4l a® TS Y vgig
< 18185 e Lil Lhvily
E -50 = Y Vv—ive
A/ )
_60 /
=70 / °
L
L4 0
—80 ’“’r )
‘_-__-—‘l/
_90 A
-100
0.001 0.01 04 1 10

w, FREQUENCY (radians/sec)

8N/N
' Q
Fig. 20 PFrequency Response of at Zero Power;
e
Iuel Circulating
. ORNL-DWG 66-10708

ZFRO POWER

T=See Pulse, Test No. 1
T=Sec Pulse, Test MNo. 2
3=1/2-Sec Pulse, Test Nu. 3
3-1'/2-Bec Pulse, Test No. 4

Ncise Analysis

 

Prequency o, red/sec

ORNL -DWG 65-8909A
20

PHASE (deg)

ZERO POWER

7sec PULSE , TEST NO. | e
7 sec PULSE , TEST Nu. 2 A
3.5 sec PULSE ,TESTNO. 3 &
3.5 sec PULSE ,TEST NO. 4 ©

 

0.001 0.01 0.1 1.0 10
w, FREQUENCY (radians /sec )
BN/NO
Fig. 21 Frequency Response of _B—K_TK_ at Zero Power;
Fuel Static
PHASE (deg)

-70
0.001

 

 

ORNL-DWG 66-476T7

POWER LEVEL 0.075 MW
51 BIT PRBS DIRECT ANALYSIS O
+ 511 BIT PRBS INDIRECT ANALYSIS O

THEORETICAL

0.001 0.002 0.005 0.0¢ 0.02 0.05 oA 0.2 0.5 1.0
FREQUENCY {radions /sec)

ORNL-DWG 66~4768

POWER LEVEL 0.075 MW
511 BIT PRBS DIRECT ANALYSIS @
511 BIT PRBS INDIRECT ANALYSIS O

THEORETICAL

o
0.002 0005 0.04 0.02 0.05 0. 0.2 0.5 {.0
FREQUENCY ({radians /sec)
BN /N

0
Fig. 22 Frequency Response of ; Power = 0.075 MW
6K7KO
 

-36_

ORNL-DWG 66~-4769

Ton
POWER LEVEL 0.465 MW

511 BIT PRBS DIRECT ANALYSIS O

5 511 BIT PRBS INDIRECT ANALYSIS O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2
/v, THEORETICAL
A
10°
5
2
102
0.00f 0.002 0.005 0.0f 0.02 0.05 0.4 0.2 0.5 {.0
FREQUENCY (radians /sec)
ORNL-OWG 66— 4770
60 T T T TTTTT T T T T
50 POWER LEVEL 0.465 MW 1]
511 BIT PRBS DIRECT ANALYSIS ®
40 511 BIT PRBS INDIRECT ANALYSIS O ||
30 i
20 .
10
= THEORETICAL
L 0
2
ué -10 Gl 1
a ~e0 5 - fi“ Attt - T
- g -V e . e e . - L R - L4711
=30 . k gi/ P; &
=40 o oo - 7 o
_—— o
‘&\r > ?%'.J_fl %lale
-50 o7 ol 1
o o°
~-60
-70
-80
0.00f 0.002 0.005 0.04 0.02 0.05 OA 0.2 0.5 1.0
FREQUENCY (radians/sec)
SN/N

o
Fig. 23 Frequency Response of ; Power = 0.465 MW
BK;KO
-37~-

4 ORNL-DWG 66-4771

POWER LEVEL 1.0 MW

127 BIT PRBS—~DIRECT ANALYSIS &
27 BIT PRBS—INDIRECT ANALYSIS O
511 BIT PRBS~DIRECT ANALYSIS ©
51 BIT PRBS—INDIRECT ANALYSIS O

10

 

 

 

Wyl 2 SAMPLING INTERVALS:
77 INDIRECT ANALYSIS —1 sec
3 DIRECT ANALYSIS-0.25 sec
10
THEORETICAL
5
2
10°

 

0.00¢ 0.002 0.005 0.0¢ 0.02 0.05 o} 0.2 0.5 1.0
FREQUENCY (radions /sec)

ORNL—-DWG 66-4772

POWER LEVEL 1.0 MW

5i1 BIT PRBS DIRECT ANALYSIS @
511 BIT PRBS INDIRECT ANALYSIS A
127 BIT PRBS DIRECT ANALYSIS A

PHASE (deg)

THEORETICAL

 

aAd

0
0.00¢ 0.002 0.005 0.0f 0.02 0.05 0.t 0.2 0.5 .0
FREQUENCY (radions/sec)
BN/N

o
Fig. 24 Frequency Response of ; Power = 1.0 MW
5K;KO
-38-

ORNL-DWG 86-10042

POWER LEVEL - 2.5 Mw

STEP TEST

127 BIT PRBS - DIRECT ANALYSIS
127 BIT PRBS - INDIRECT ANALYSIS
51t B8IT PRBS - DIRECT ANALYSIS
511 BIT PRBS - INDIRECT ANALYSIS

g > O

 

oo 0002 0005 001 002 005 Ol 02 05 10
FREQUENCY (radions/sec)

ORNL-DWG €6-10038

POWER LEVEL - 25 Mw

STEP TEST o
127 BIT PRBS - DIRECT ANALYSIS @
127 BIT PRBS - INDIRECT ANALYSIS A
511 BIT PRBS ~ DIRECT ANALYSIS &
511 BIT PRBS - INDIRECT ANALYSIS v

o

*
Y
A

PHASE (deg)

THEORET

®
A

O

- 48" APy g "

-50 VIrayg &

 

-70
0001 0002 0005 00! 002 005  Of 02 05 10
FREQUENCY (rodians/sec)

SN/N

o
Fig. 25 Frequency Response of ; Power = 2.5 MW
6K7KO
ORNL-DWG 66-10041

POWER LEVEL - 5.0 Mw

STEP TEST o
127 BIT PRBS -DIRECT ANALYSIS @
127 BIT PRBS - INDIRECT ANALYSIS &
51 BIT PRBS - DIRECT ANALYSIS &
511 BIT PRBS - INDIRECT ANALYSIS V

MWy
/i,

 

10
0.002 0.005 001 002 005 0! 02 03 1.0
FREQUENCY (rodions/sec)

ORNL-DWG 66-10035

POWER LEVEL - 50 Mw

STEP TEST o
127 BIT PRBS - DIRECT ANALYSIS e
127 BIT PRBS - INDIRECT ANALYSIS &
5i1 BIT PRBS - DIRECT ANALYSIS &
511 BIT PRBS - INDIRECT ANALYSIS ¢

A
O
v

PHASE (deg)

THEORE TICAL
»
0

a

 

0001 0002 0005 0Ot 002 005 o1 0.2 05 10
FREQUENCY {radians/sec)
&N/N

Fig. 26 Frequency Response of 2 ; Power = 5 M
SK;KO
40~

ORNL-OWG 66-10039

POWER LEVEL - 6.7 Mw

STEP TEST NO.1 e
STEP TEST NQ.2

511 BIT PRBS - INDIRECT ANALYSIS

v, 5
Bk

 

o?
Q002 0005 001 002 005 o1 02 05 10
FREQUENCY (radians/sec)

ORNL-DWG 66-10037

POWER LEVEL - 6.7 Mw

STEP TEST NO. 1 ®
STEP TEST NO.2 A
511 BIT PRBS - INDIRECT ANALYSIS v

THEORETICAL

PHASE (deg)

 

0.001 0002 0005 Q01 002 0.05 o1 02 05 10
FREQUENCY (radians/sec)
SN/N

o
Fig. 27 Frequency Response of ; Power = 6.7 MW
5K;KO
84/

PHASE (deg)

2

10

5

 

102
0002

-]~

 

ORNL-DWG 66-10040

POWER LEVEL - 7.5 Mw

STEP TEST o
127 BIT PRBS ~DIRECT ANALYSIS o
127 BIT PRBS - INDIRECT ANALYSIS A
511 BIT PRBS - DIRECT ANALYSIS &
5t BIT PRBS - INDIRECT ANALYSIS ¢

4
e,

v
v v

%
6’*@? i®

V‘O‘

0.005 oot 002 0.05 0l 02 05 10

FREQUENCY (rodions/sec)

ORNL-DWG 66-10036

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

80 T
-~ T '~ POWER LEVEL - 7.5 Mw
70 / o © ore o1d o % STEP TEST o[ T
60 b ol2 v A 127 BIT PRBS - DIRECT ANALYSIS el L
o s . A:'b 127 BIT PRBS- INDIRECT ANALYSIS A
30— | 0{: 511 BIT PRBS - DIRECT ANALYSIS Al
40 (=2 : 511 BIT PRBS - INDIRECT ANALYSIS V|-
30 4 %
20 n
v fo
10 a (—THEORET!CAL
0 AN
-10 ‘&l\ / Z
20 Qfif S~ TN L
-30 %99 7 —V__]
o Ly ?A<7VA ¢
-40 0 M-‘“
AL
- L
50 3
-60
-70
0001 0002 0005 001 002 005 04 02 05 10
FREQUENCY (radians/ec)
&N/N

0
Fig. 28 Frequency Response of ;
SK;KO

Power = 7.5 MW
PERIOD OF OSCILLATION (min)

ORNL-DWG 66-10285

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

° 1
® EXPERIMENTAL
&
\\\
2 \\
10 2
THEORETICAL
5 <
o
\\
\.
> N
N
\
1
002 005 01 02 05 10 20 50 100

POWER LEVEL (MW)

Fig. 29 MSRE Natural Periods of Oscillation
43~

The linear stability of the system is certainly adequate. The
frequency response shows a resonance which shifts to higher frequencies
and lower amplitudes as power increases. This means that the transient
response to a disturbance at low power will display a lightly damped,
low frequency return to equilibrium (period greater than ten minutes
for powers less than 500 KW). At higher power the system response
is much more strongly damped and much faster. For instance, a
disturbance at 7.5 MW causes a transient which is essentially completed
in 1-1/2 minutes. These observations are in good agreement with prior
predictions.

A detailed guantitative check of the theoretical predictions by
experimental tests is much more difficult than a comparison of more
general dynamic features such as stability, location of resonance
peaks and the changes expected in these performance measures with
power level. Early attempts to fit parameters in the theoretical
model to give agreement in the absolute amplitude of the frequency
response were abandoned bacause of uncertainties in the measured
amplitudes caused by equipment limitations. While all of the tests
at a given power level give results with the same shape, there is a
difference in the absolute magnitude ratios. ..Figure 27 clearly shows
this bias effect. PFurthermore, the portion of the frequency response
above 0.3 rad/sec should be the same for all power levels since
feedback effects are small in this frequency range and the zero power
frequency response should dominate. The experimental results for
various power levels show the same shape in this frequency region,
but different absolute ampiitudes. This further indicates a bias
problem. This bias problem is not surprising in view of the equipment
characteristics discussed in Section IV.

In spite of the blas difficulties, one feature of the theoretical
model i1s shown to be incorrect by experimental results. At high
power (greater than 5 MW) the theoretical magnitude ratio curve has
a dip at 0.2 rad/sec. This is due to the reappearance of a fuel
salt temperature slug in the core after traveling around the primary

loop. ©Since this dip was not observed experimentally, there must be
=Ll

more mixing and heat transfer in the primary locop than was included
in the theoretical model.

Because the predicted frequency response has the correct shape
and location on the frequency axis, we feel that the model used for
the MSRE dynamic analysis is a good representation of the system.
The only discrepancy observed in predicted and observed shapes is
the predicted dip at 0.2 rad/sec, just discussed. The apparent
bias in the measured amplitude unfortunately prohibits detailed

parameter fitting.
1.

lo.

11.

12,

13,

REFERENCES

R. C. Robertson, MSRE Design and Operations Report, Part 1:
Description of Reactor Design, USAEC Report ORNL-TM-728, Oak Ridge
National Laboratory, January 1965,

S. J. Ball and T. W. Kerlin, Stability Analysis of the Molten-Salt
Reactor Experiment, USAEC Report ORNL-TM-1070, Oak Ridge National
Laboratory, December 1965.

T. W. Kerlin, The Pseudo-Random Signal for Frequency Response
Testing, USAEC Report ORNL-TM-1662, Oask Ridge National Laboratory,
September 1966.

J. 0. Hougen and P. A. Walsh, Pulse Testing Method, Chem. Eng.
Prog., 57(3): 69-79 (March 1961).

H. A. Samulon, Spectrum Analysis of Transient Respounse Curves,
Proc. IRE, 39, 175-186 (1951).

. 8. J. Ball, Instrumentation and Controls Div. Annual Progr. Rept.

Sept. 1, 1965, USAEC Report ORNL-3875, pp. 126-7, Oak Ridge
National Laboratory.

5.5.L. Chang, Synthesis of Optimum Control Systems, Chap. 3,
McGraw-Hill, New York, 1961.

B. D. Van Deusen, Analysis of Vehicle Vibration, ISA Trans. 3,

T. W. Kerlin and J. L. Lucius, CABS — A Fortran Computer Program
for Calculating Correlation Functions, Power Spectra, and the
Frequency Response from Experimental Data, USAEC Report ORNL-TM-
166%, Oak Ridge National Laboratory, September 1966,

E. M. Grabbe, S. Ramo, and D, E. Wooldridge (Editors), Handbook of
Automation, Computation, and Control, Vol. 1, Chap. 22, J. Wiley
and Sons, New York, 1958.

S. J. Ball, A Digital Filtering Technique for Efficient Fourier
Transform Calculations, USAEC ORNL-TM report (in preparation).

D. P. Roux, and D. N. Fry, ORNL Instrumentation and Controls
Division, personal communication.

J. E. Gibson, Nonlinear Automatic Control, Chap. 1, McGraw-Hill,
New York, 1963.
1h.

15.

16,

l?' .

18.

19.

=46 -

G. C. Newton, L. A. Gould, and J. F. Kaiser, Analytical Design of
Linear Feedback Controls, pp. %6—381, J. Wiley and Sons, New York,

1957.

 

C. T. Morrow, Averaging Time and Data-Reducing Time for Random
Vibration Spectra, J. Acoust. Soc. Am., 30, 456 (1958).

T. J. Karras, Equivalent Noise Bandwidth Analysis from Transfer
Functions, NASA-TN-D2842, November 1965.

L. D. Enochson, Frequency Response Functions and Coherence Functions
for Multiple Input Linear Systems, NASA-CR-32, April 196k,

H. M. Paynter and J. Suez, Automatic Digital Setup and Scaling of
Analog Computers, Trans. ISA, 3, 55-64, January 196k.

S. J. Ball and R. K. Adams, MATEXP, A General Purpose Digital
Computer Program for Solving Nonlinear Ordinary Differential
Equations by the Matrix Exponential Method, USAEC ORNL report
(in preparation), Osk Ridge National Laboratory.
-L7-

APPENDIX A

Potential Sources of Experimental Error
Due to Equipment Limitations

As discussed in Section IV.A, there were a number of factors that
could have had adverse effects on the experimental results, and they
are listed below:

1) Friction in the bends in the thimbles. Rollers are mounted in
the bends of the thimble, but there is still considerable friction.
(Tests show that the rods fall with an acceleration of only about 0.4 g.)
This suggests that part of the motion at the rod drive might go into
boewing of the flexible hose rather than into motion of the bottom of the
poison section.

2) Bends in the hose. It was observed that an old MSRE control
rod used for out-of-pile testing did not hang straight when suspended
from the top. It had gradual bends that could be worked out by hand,
but which were not pulled out by the weight of the rod (6 .to 8 1b). If
such bends exist in the MSRE rod used for the tests, then the motion of
the bottom of the rod will not be the same as the motion at the top of
the rod if a bend in the hose is passing over a roller in the bend of the
thimble. |

3) Restricted twisting. The test rod showed a tendency to turn
when inserted into a mockup of the MSRE thimble. This twisting is pre-
vented in the reactor since the top of the rod is rigidly connected to
the chain drive (see. Fig. 3). If a tendency to twist is prevented, the
hose will bow and cause a difference in axial motion between upper and
lower sections.

4) Sprocket chain meshing. The action of the drive motor is trans-
mitted to the drive chain by a sprocket. This sprocket has a diameter
of 1.282 in. and the length of the links in the chain is 1/4 in. The
fact that the flat links cannot exactly follow the circular contour of
the sprocket means that some of the sprocket motion is taken up by a
lateral motion of the chain as well as the desired vertical motion.

5) Sticking of poison beads. The control rod thimble contains
1,8

vanes for centering the control rod. The vanes are held in place by
circular spacers located 4 in. apart. If the vanes became warped, they
could touch links in the poison chain in certain sections without touching
links in nearby sections. ©Since there is slack in the threading of the
poison cylinders on the central hose, this friction could hold up the
movement of certain poison elements.

6) Indicated poisition errors. It was necessary to use the coarse
synchro signal (5 deg of turn per inch of rod motion) for logging on the
MSRE computer and for subsequent data analysis. Since the l/2-in. rod
motion used in the tests corresponds to only 2.5 deg rotation of the
synchro, sizeable percentage error could be caused by only a few tenths
of a degree of deadband in the gears leading to the synchro. Periodic
calibrations of the logged rod position against the fine synchro, how-
ever, indicated that the error is less than +5% for a 1/2-in. rod travel.
The maximum deadband in the logger signal corresponded to about +0.008

in. of rod motion.
-49_

APPENDIX B

The Direct Method for Cross-Power Spectrum Analysis

Each of the terms used in the cross-power spectrum analysis is

computed in the following manner:

 

 

 

 

 

 

 

 

 

 

 

 

i(t) a(t)

e Hy (J0)

Input {
v ,
e 1 - £(d)

: a(t) b(t 10
msipty 0B L (a1

| -

o(t) ] () |

el () o

Output

 

 

 

W
where H, and H, are either H(jw) or 3% H(jw)} and f(¢IO) (depending on

which combination of H; and Hs are used) is related either to the real
(COPOWER) or the imaginary (QUAD POWER) part of the cross-power spectral
density (CPSD) ¢IO' Four combinations of the basic computation shown
above are used in the CPSD analysis as shown in Fig. 9.

To convert filtered time domain functions to frequency domain

functions, we make use of Parseval's theorem,'® which is

j'a;(t) b(t) dt = %E j'aé(jw) A(=jw) dw , (1)

=00 =

where B(jw) = Fourier transform of b(t), and A(—jw) = complex conjugate

of the Fourier transform of a(t).
_50_

Considering that only a finite integrating time T is available to

us:
T 1 o
J ale) () at ® == [ B(jw) Al—jw) du . (2)
0 ~®
Noting that
A(Jw) = H3(Ju) I(Jw) , (3)
B(Jjw) = Ha(jw) 0(jw) , (L)
where
I(jw) = Fourier transform of input, i(t),
0(jw) = Fourier transform of output, o(t).

From the definition of the complex conjugate, it can be shown that

A(=jw) = Hy(=jw) I(=jw) , (5)

H

B(=jw) = Ho(=jw) 0(=jw) . (6)

Hence (2) can be rewritten in terms of the Fourier transforms of the

~ input and output signals

T o
fo a(t) b(t) at % &= [ Ha(dw) Hy(=ju) I(=jw) 0(jw) dw . (7)

=00

Since the cross-power spectral density is defined as”
oy _ o iim 1., . .
9o (dw) = L, FlI(=jw) 0(iw)] (8)

it behooves us to operate on (7) in order to be able to incorporate

¢Io(jw). Taking the limit of both sides of (7) as T-» and dividing by

T, we get

lim1l T 1ore . oy fiim1 o, .

ron T J S08) 208) 86 = 5 | W) aloge) 320 5 1) (g} 0w
9

=

)

Substituting (8) into (9):
_5]__

. T o0
1 [ a0 500 0t = b [ el () S @ Qo

o0

Case 1

For the case where

Hy (Jw) = Ho(Jw) = H(jw)
and defining

T

TORIO E,%if % [ a(t) n(t) at ,
' 0

Eq. (10) becomes

a(t) b(t) = %E j'“fi(jw) H(-jw) 9;,(Jw) dw = %E m[H(jw)le ¢, (Jw) dv .
- (11)

=

Since the input and output signals are filtered identically, it
should be evident that this operation will yield information only about
the in-phase relationship, or the real part of ¢Io(jw). We can show
this by noting that since

 

a(t) b(t) = b(t) alt)

and
b(t) a(t) = %; m[H(jw)la ¢ p(iw) dw , (12)

the two integrals in Eqs. {11) and (12) must be equal; thus

L J TG0 2 (3or(a0) = brolau) Y aw = 0 . (1)

=00

If we assume that ¢IO(jw) and ¢OI(jw) do not change much over the
effective bandwidth of the filter, then ¢Io(jw) must equal ¢OI(jw). But

since

0o (Jw) = @7 (=dw) , (14)
_52-

this means that the imeaginary part of ¢Io(jw) mist be zero, or at least
no information about Im[¢IO(jw)] is present in the output. For case 1,

then

L %2 :
a(t) b(t) = T [H(Jw) | Re[d)IO(Jw)] dw . (15)
-0
If we assume that Re[¢IO(jw)] does not change much over the effective
bandwidth of H(jw),

o1 il e\ (2 .
a(t) b(t) & [fi-f [E(Jw) | dwjl l:Re[¢IO(Jw) ]] . (16)
- ;
For the present study, we used a filter with the following transfer
function:
Jw
H(jw) = . (17)
wi + jw 2§wo - wF

 

The filter "area" term can be evaluated using a table of integrals®®

 

%; !; IH(jm)IE dw = h%wo (rad/sec) . (18)
Thus
Re[d>IO(jw)] N b a(t) b(t) . (19)
Cage 2
For this case
Yo
Hy (Jw) = o H(jw) (20)
and
Ho(jw) = H(jw) .
Since

oy E=)
Hl( jw) = Ziw ’ (21)
=535~

Eq. (10) becomes

wy o H(Jw) H(=jw) 955(jw)

a(t) o(t) = 5= | : dw (22)

 

Since the input signal's filter has 90 deg more phase lag than the
output signal's filter, we should expect that this operation will yield
information only about the quadrature relationship, or the imaginary
part of ¢Io(jm). We can show this as follows:

In this case, revising the order of integration of the inputs makes

a difference, i.e., from (2)

T o
[ v(t) a(t) at = %; [ A(30) B(=jw) dw (23)
0

we can use (%), (6), (19), and (23} to get
0 0 . s
(t) a(t) —‘5% EIJ&%Z?I—QEQ-I(jw) O(=jw) dw . (24)

-0

 

Again, since a{t) b(t) = b(t) a(t), from (22) and (24) we can conclude

i

 

that
¢1o(dw) _ 9or(dw) (25)
—Jjw Jw
if we use the same argument as we did for case 1. Thus
9 ro(Jw) = — $5r(Jw) = = ¢p4(=dw) (26)

which is true only if the real part of ¢IO = 0, or at least if no

information about Re[¢IO(jw)] is present in the output. Hence, we can

substitute jIm[¢Io(jw)] in for ¢Io(jw) in Eq. (22). For case 2, then
W e |H(jw)]2

AT 00 = - 2 [

=00

= In[9;,(jw)] dw . (27)

If we assume that Im[¢10(jw)] does not change much over the
W
effective bandwidth of either H(jw) or 33 H(jw), then
-54_

alt) blt) & [— ;% m'lfifgfllf-dw} [Im[¢IO(jw)]} . (28)

-

For the particular filter used (Eq. 17):

W o0 IH(jw)IE

= do = grg (29)

Thus
Im[qSIO(,jw)]z - bw_ a(t) o(t) . (30)
Case 5
The case where
Yo
Hy(jw) = Ho(Jw) = 3o H( jw) (31)

can be developed similarly to case 1 as far as Eq. (16), since we could
redefine H(jw) as being equal to the expression in (31).

The integral to be evaluated is

 

1 4 E(Jw) H(=jw) WS o [H(Jw)|®
S J Hlw) mu-go) au - 2 [ SR Bl e - 0 ) o au ()
-0 - o0 eomo0 7 W
For the filter of Egq. (17), this integral is again equal %o héw s SO
o
Re[ézo(jw)] n Mgwo a(t) p(t) (%3)

assuming in this case, however, that Re[¢IO(jw)] does not change much
W

over the effective bandwidth of 3% H(jw).

Case L

The case where

iy (3) = H(3w) (54)
Ha(J0) = 2 B(30) (55)

can be developed similarly to case 2. Equation (10) becomes
...55..

W 0 . s
a(t) b(t) = 5% Elgeljgi_lfil ¢ (Jw) dw . (36)

-0
The expression on the right hand side is the hegative of that for
Eq. (22), case 2; hence for case 4, we get the expression corresponding
to Eq. (3) when the filter of Eq. (17) is used: |

Im[¢IO(jw)]z btw_ a(t) b(t) (37)

assuming again that Im[¢IO(jw)£ does not change much in the effective
bandwidths of either H(jw) or 3% H(jw).

The power spectral density (PSD) of the input function is required
for calculating the system transfer function G(jw). This is obtained
by squaring the outputs of both the in-phase and quadrature filters and

integrating the sum of the squares. In both cases, for the filter of
Eq. (17):

 

9. (Jw) & hlw_ a(t)? (38)

assuming that ¢II(jw) does not change much in the effective pass bands
of H(jw) and W, H(jw)/ jw.
The system transfer function G(jw) is then computed from

Re[¢..(Jw)] + § Im[d_ (jw)]
G(jw) = 10 ¢II(Jw) 10 ’ (39)

 

where each of the three terms on the right hand side of (39) are computed
using the sum of two estimates. (Note that all terms have the same gain
factor, hgwo.) The reason for the better accuracy of this method as
compared to using a single estimate of each term lies in the fact that
since the effective pass-bands of the in~phase and quadrature filters
are different, there is a bias in each of the quadrature, or imaginary
term, estimates. ©Since the two imaginary term estimates are of different
sign, this bias tends to be cancelled out.

Calculations of the percent standard deviations of both input and

output (PSD) estimates are made using (L0):15
-56 -

 

 

T
o
4 SD = 100 ¢ _ 100 , (40)
*xZ /BT
where

0 = standard deviation of mean square value,
x2 = mean square value,
B = equivalent noise bandwidth, rad/sec,
T = integration time, sec.

The equivalent noise bandwidthl® for the H(jw) filter used in this

study is
B = nlw_ » rad/sec. (41)
The coherence function Y& is also computed:

19

2
_ 1°10]
= W . (42)

72

The coherence function is useful for estimating expected errors in
transfer function calculations when the input and output signals are
random.r’ For periodic signals, however, such as the PRBS, the ex-
pressions for error estimates in the literature have been found to be
wildly pessimistic.

The calculation of the response of the digital filters is based on
Pzynterts matrix exponential method,18,1% and gives virtually exact

time-response solutions very efficiently.

-~
9

'
e
H O OCO—T O\ £\ o

22,

R NO o
A AN

6.

N oo
O 00—

\&\N\N\N\N\N\N
VT EW D HO

b AN AN
O C—J

==
= O

uo.

~
N

Li,

*

PESOLUNFOIHUEE D ROED L er

?Z'-f—il.-"jf«'i?@P?QiffigfiwhdtfimmgQOmt*me

. Adams
Adamson
Affel
Alexander
Apple
Baes
Baker
Baker
Ball
Beall
Rettis
. Blankenship
Blumberg
Bohlmann
Borkowskil
Bullock
Burger
Burke
antor
Carter
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Compere
Conlin
Cock
Corbin
Cristy
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Dandl
Danforth
Davis
Ditto
Donnelly
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Dunwoody
Engel
Fpler
Ergen
Ferguson
Fraas
Friedman
Fry
Frye, dJr.
Gabbard
Gallaher
. Grimes
Grindell

AREHEREERORR

=Ewse

AEEEHEE P VRN AP QUe N PP AR R

_57..

Internal Distribution

 

D7
58.

59.
60.
61.
62,
63.
64,
65,
66.
67.
68.
69.
70-109.
110.
111.
112,
113.
11k,
115.
116.
117.
118,
119.
120.
121.
122,
123.
12k,
125,
126.
127.
128,
129,
130.
131,
132,
133.
13k4.
135,
136.
137,
138.
139.
140
1h1,
kb2,

ORNL-TM-1647

Guymon
Hagen
Harley

. Harrill
Haubenreich
Herbert
Herndon
Holt

Holz
Hudson
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Kelly
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Knowles
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Lindauver
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Marable
Martin
Mathews
MacPherson
MacPherson
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McNeese
Miller
Miskell (Y-12)
Mixon
Moore

. O'Brien
atriarca
Payne
Perry
Piper
Prince

. Redford
Richardson

C. Robertson

F;F;P{F;5:flfugg5:?gpj?:?:Htficacnzgk4c4o:z:&qtamlfi HoQzE=2unm s o

P ="
_58-

143, H. C. Savage 163, W. C. Ulrich

ihk, A. W. Savolainen 164. B. H. Webster

145, D. Scott 165. A. M. Weinberg

146. H. E. Seagren 166. K. W. West

ih7. J. H. Shaffer 167. M. E. Whatley

ik8, M. J. Skinner 168. J. C. White

149, A. N. Smith 169. G. D. Whitman

150. P. G. Smith 170. R. E. Whitt

151. W. F. Spencer 171. H. D. Wills

152, I. Spiewak 172. J. V. Wilson

153, B. Squires 173%3. L. V. Wilson

154, R. C. Steffy i7hk. M. M. Yarosh

155, H. H. Stone 175. MSRP Director's Office,

156. R. S. Stone Rm. 219, 9204-1

157. A. Taboada 176-177. Central Research Library

158. J. R. Tallackson 178-179. 7Y-12 Document Reference Section
159, R. E. Thoma 180-18L, Laboratory Records Department
160. G. M. Tolson 185, ILaboratory Records Department,
161. D. B. Trauger LRD-RC

162, J. R. Trinko

External Distribution

 

186-200., Division of Technical Information Extension (DTIE)
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205. R. P. Gardner, Research Triangle Institute, Durham, N. C.
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207. S. H. Hanauer, University of Tennessee, Knoxville

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216. A. A. Wasserman, Westinghouse Astronuclear Laboratory,

P. O. Box 1086k, Pittsburgh, Pa.
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