A DIRECT METHOD FOR THE
MEASUREMENT OF CONTROL ROD WORTH
IN A NUCLEAR REACTOR

Thesis for the Degree of M. S.
MICHIGAN STATE UNIVERSITY
WILLIAM ARTHUR EMSLIE
1972

4"-
............................ ; LIBRARY

~ IIIIIIIIIIIIIIIIIIIIII/IIIIIIIIIIIIIIIIIIII 'Michiganstam

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University

 

 

 

 

 

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ABSTRACT

A DIRECT METHOD FOR THE MEASUREMENT
OF CONTROL ROD WORTH IN A.NUCLEAR REACTOR

By

William Arthur Emslie

It is often necessary to perform control rod calibration
in a nuclear reactor. Such calibration requires the measurement
of the reactivity produced when discrete sections of a control rod
are withdrawn from a critical reactor pile. From these data a
total reactivity worth curve can be complied for the length of the
rod. The subject of this thesis is the development of a means by
which reactivity worth can be measured.

A solution to this measurement problem has been developed
using the current output of a compensated ionization chamber
(adjacent to the reactor) as a signal source. When a section of
control rod is withdrawn from a critical reactor, the current from
the ion chamber is

c/e

. = p
18(t) Ioe

is

where t is time, 9p is the reactor stable period and 10

the initial ion chamber current.
The stable period is proportional to the amount of reactivity

produced since

William Arthur Emslie

9.."

x loo:
'0 loo

9 =
P ex

90 is the mean neutron lifetime, kex is the excess multiplication
factor and p is the reactivity. This approximate relationship
between 9p and p makes it possible to obtain the reactivity

from the output current of the ion chamber since

t
/O pt/OO

e p22 I e

is (t) = I0 0

Actual measurement of p is accomplished with a circuit
composed of a logarithmic amplifier followed by a differentiator
and filter. The transfer function of this circuit is

('Rlczs) 1

out(8) = V1(S) (R1C18+l)(R2023+1) (Rfo

V 3+1)

where V1(s) is the frequency domain expression for V1(t) (the

output voltage of the logarithmic amplifier) and is equal to

 

_ -A log10(Io/Ik) A logloe

V1(s) — - -————7f—_
8 9 s
P

In the time domain, the logarithm of the ion chamber current

produces a linear ramp voltage v1(t) where

i(t) Ie P I
v1(t) A log10 I A log10 I A log10 I O logloe
k k k p

This ramp voltage is then differentiated and filtered to
obtain a d-c voltage pr0portiona1 to the inverse of p. The re-

sulting voltage (V is

2)

William Arthur Emslie

I
_ .d. = - d_. - _Q _ 4.5
V2 ' ‘K dt “1“” K dt ( A 1°g10 1k 9 1°g1oe)
= _ a. £19.
9 logloe._ 90 logloe

V2 may be calibrated on a d-c voltmeter to indicate the
corresponding reactivity.

A circuit to synthesize the transfer function was designed
and constructed. Results of tests made on circuit performance show
the total accuracy in the measurement of p to be within‘i 0.5

percent of the meter scale.

A DIRECT METHOD FOR THE MEASUREMENT
OF CONTROL ROD WORTH IN A NUCLEAR REACTOR

By

William Arthur Emslie

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Electrical Engineering and Systems Science

1972

TO MY PARENTS

ii

 

ACKNOWLEDGMENTS

The author wishes to express his sincere thanks to his
major professor, Dr. G.L. Park, for his guidance and support during
the course of the research. Sincere appreciation is also expressed
to the other members of the guidance committee, Dr. B. Wilkinson
for his advice on nuclear reactor theory and Dr. L. Giacoletto for
his guidance on the design of the reactivity measuring circuit.

Special thanks is extended to Mr. E. Brockbank who made

possible extensive testing and calibration of the reactivity meter.

iii

TABLE OF CONTENTS

Page
LIST OF TABLES .................................. v
LIST OF FIGURES ............... ...... .... ........ vi

LISTOF SMOLS OOOOOOOOOOOOOOOOOOOOOOOO000...... Vii-i

Chapter
I. INRODUCTION OO0....0..OOOOOOOOOOOOOIOOOOOOOOO0.0. 1
II. REACTOR TRANSIENT BEHAVIOR 5
2.1 Reactivity, Reactor Period and Theoretical
RelationShi-ps 0.0.0....COOOOOOOCOOOOOOOOCOOO 5
2.2 Relationship Between Reactivity and the
Reactor Stable Period ...................... 7
2.3 Effects of Prompt and Delayed Neutrons on
Reactor Transient Behavior ................. 7
2.4 The Compensated Ion Chamber ................ 11
III. DESIGN OF THE REACTIVITY MEASURING DEVICE ....... 15
3.1 Formulation of the Problem ................. 15
3.2 Possible Methods of Solution ............... 16
3.3 The Circuit and Theory of Operation ........ 20
3.4 Calibration 00......OOOOOOOCOOOOCOCOOOOOO... 23
Iv. CONSTRUCTION OF THE REACTIVITY METER 27
4.1 Circuit Tests, Performance and
Specifications ............................. 27
V. CONCLUSIONS ..................................... 30

BIBLIOGRAPHY ............................................... 32
APPENDIX A DESIGN OF THE LOGARITHMIC AMPLIFIER ........... 33
APPENDIX B DETERMINATION OF THE RAMP FREQUENCY COMPONENTS 39
APPENDIX C DESIGN OF THE DIFFERENTIATOR .................. 42
APPENDIX D DESIGN OF THE FILTER .......................... 47

APPENDIX E SPECIFICATIONS OF THE COMPENSATED ION CHAMBER . 49

iv

Table

4.1

A.1

LIST OF TABLES

Specifications of the reactivity meter ..........

Electrical specifications of the Burr-Brown
modular logarithmic amplifier model 4116 (current
input only) 000......0.0.0.0...OOOOOOOOOOOOOOOOOO

Specifications of the Westinghouse type WL-8501
compensated ion chamber .........................

Page

29

34

49

Figure

1.1

2.1

2.2

2.3

2.4

3.1

3.3

3.4

3.5

3.6

4.1

A.l

LIST OF FIGURES

Page
Calibration curve for the shim rod in the
Michigan State University TRIGA Mark I
duel-ear reactor ......OOOOOOOOOOOOOOOOOOOOOOOOOO. 3
Relationship between reactivity and reactor
stable period for the Michigan State University
TRIGA Mark I nuclear reactor (Courtesy of Gulf
GeneralAtan1C) 00.0.00.........OOOOOOOOOOOOOOOOO 8
Combined effect of prompt and delayed neutrons
in a supercritical reactor ...................... 10
Simplified diagram of a compensated ion chamber . 12

Graph of ion chamber output current as a function
of reactor output power level for the Michigan
State University TRIGA.Mark I nuclear reactor ... 14

Block diagram of a circuit which provides a d-c
voltage prOportional to the inverse of the
reactor Stable periOd 0............OOOOOOOOOOOOO. 16

Circuit for obtaining a d-c voltage proportional

to e O......OOOOOOOOCQOOOOOOOO00.000.000.000... 17

Circuit for obtaining a discrete voltage
proportional to the slope of is(t) ............. 18

Circuit diagram of the reactivity measuring
instrument .....OOOOOOOIOOOOO......OOOOOOOOOOOOO. 21

Apparatus used to calibrate the reactivity meter 24
Graph showing the relationship between reactivity

and reactor stable period for the Michigan State
University TRIGA Mark I nuclear reactor

(Courtesy of Gulf General Atomic) ............... 26
Photographs of the reactivity meter ............. 28

Burr-Brown modular logarithmic amplifier model
4116 with external connections .................. 35

vi

Figure

A.2

D.1

Design curves for the Burr-Brown model 4116
log amplifier, (a) Reference current vs. R ,
(b) Voltage constant vs. Rx .............¥ .....

Transfer function plot for an ideal
differentiator, (a) Imaginary part, (b) Real

part .............OOOOOOOOOOOO.....OOOOOOOOOOOOO.

Transfer function plot for a compensated
differentiator, (a) Imaginary part, (b) Real

part 00......OO.I.......OOOOOOOOOOO0.00.00...0..O

Equivalent circuit of a compensated
differentiator ....OOOOOOOOOOOOOO......OOOOOOOOOO

Output filter for the reactivity meter ..........

vii

Page

42

43

44

47

LIST OF SYMBOLS

Voltage constant for log amplifier
Open 100p voltage gain

Capacitance

Initial ion chamber output current
Reference current for log amplifier
Ion chamber output current

Reactor effective multiplication factor
Reactor excess multiplication factor
Neutron

Reactor power level

Initial reactor power level
Resistance

Time

Voltage

Impedance

Beta particle

Fraction of delayed fission neutrons in the ith group
Gamma photon

Mean neutron lifetime

Reactor power level doubling time
Reactor stable period

Minimum ramp on time

viii

Decay constant of ith group
Excess reactivity (reactivity)
Reactor neutron flux

Initial reactor neutron flux

Radian frequency

ix

CHAPTER I

INTRODUCTION

The necessity to perform calibration of the control rods
in a nuclear reactor requires a means by which to measure the
reactiVity produced by withdrawal of discrete sections of a con-
trol rod from a critical reactor pile.

Use of an ionization chamber adjacent to the reactor pro-
vides an exponential current (is(t)) from which the reactivity
may be derived. When a section of control rod is withdrawn from
a critical reactor is(t) becomes

t/e

13(t) = Ioe (1.1)

where I0 is the initial ion chamber current, t is time and O
is the reactor stable period.

The reactivity may be obtained from equation (1.1) since

9
e 22-41 (1.2)
P P

2
where is the mean neutron lifetime.

90

 

The reactor stable period used in this expression is the time
required for the ion chamber current (is(t)) to increase by a
factor of e.

This approximation is explained and justified in Section 2.1.

The relationship between p and i8(t) expressed in
Equations (1.1) and (1.2) makes it possible to obtain p from
is(t). This may be accomplished by means of a circuit which first
takes the logarithm of is(t) and then differentiates and filters
the result to yield a d-c voltage pr0portiona1 to p.

An example of a control rod calibration curve is illustrated
in Figure 1.1. The curve is compiled by withdrawing the rod a
short distance and measuring the corresponding reactivity produced.
This procedure is repeated until the entire length of the rod has
been withdrawn.

Chapter II of this study details the theory underlying
reactivity and the ionization chamber. The effects of prompt and
delayed neutrons are also discussed.

In Chapter III the reactivity measurement problem is
formulated with a set of possible solutions. A particular solution
method from these alternatives is then presented. Analysis of the
design, mathematical properties, and calibration of the circuit follows.

I

Further design analysis is presented in the Appendices. I
‘ I

|

 

I
3 i
The units of reactivity as expressed in Figure 1.1 are in dollars 5

and cents where 1 dollar = 100 cents and is defined as

k
1 dollar = £3; 2 P- (1.3)

Bi 81
where k.ex is the reactor excess multiplication factor and Bi is
a normalizing constant equal to the total fraction of delayed neutrons
and varies with different reactors. For the Michigan State University
TRIGA'Mark l reactor Bi: 0.0073 and therefore one dollar's worth
of reactivity is that amount required to produce an excess
multiplication factor of 0.0073.

p, Reactivity (dollars)

4.00

3.00

2.00

1.00

 

L l l i l l

100 200 300 400 500 600 700

Rod Position (arbitrary units)

Figure 1.1 Calibration curve for the shim rod in the Michigan
State University TRIGA Mark I nuclear reactor

800

Chapter IV describes a prototype instrument which was built
from the design in the previous chapter. Test results and circuit
specifications are also given.

Conclusions are made in Chapter V.

CHAPTER II

REACTOR TRANSIENT BEHAVIOR

2.1 Reactivity, Reactor Period and Theoretical Equations4

The excess reactivity (hereafter referred to as reactivity)
in a nuclear reactor is defined as the ratio of excess to effective

multiplication factors and is given by
ex
= 2.1
p k ()

In the above formula the effective multiplication factor,

keff is equal to unity for a critical reactor. The excess multi-

plication factor is
k = k - l (2.2)

and is equal to zero for a critical reactor.

Substitution of Equation (2.2) into Equation (2.1) yields

k - l
ff
9 = _ek__ (2.3)
eff

Since the values of Re used in this analysis are to be no

ff

 

4 The content of this section, except for that material noted
otherwise, is taken from.M.M. El-Wakil, Nuclear Power Engineering
(New York; MCGraw-Hill Book Co., 1962), pp. 128-130.

greater than 1.035, Equation (2.3) may be approximated as
p ask - l = k (2.4)

Using the above approximation for reactivity it is possible
to relate p to the instantaneous neutron flux within the reactor.
The equation describing the flux in a supercritical reactor

is
¢=€be p (2.5)

The reactor stable period (9p) may be written as

9
e = ESL (2.6)
p ex
and substitution of Equation (2.4) for kex relates p to 9p
where
9
a 2—0' (1.2)
P P

This approximate expression for 9p may be substituted
into Equation (2.5) to form a relationship between reactor flux

and reactivity such that

pt/O0

0:: @0e (2.7)

 

Reference to the rod calibration curve in Figure 1.1 shows the
maximum reactivity to be approximately four dollars. This value
corresponds to an effective multiplication factor of 1.0292 as
defined in the Nuclear Reactor Operations and Traininnganuel,
Reactor Theory (Michigan State University, 1969), pp. 4-5.

2.2 Relationship Between Reactivity and the Reactor Stable Period

As Equation (1.2) illustrates, the approximate relation-
ship between the reactor stable period and the reactivity is weighted
by the mean neutron lifetime. This equation is characterized by a
nonlinear property of 90 which is not constant but rather varies

slightly with p. Examination of the exact expression for reactivity

reveals the source of this nonlinearity. p may be written exactly as

9 B.

epkeff i 1 + xiep

where 31 is the fraction of delayed fission neutrons in the ith
group and I1 is the respective decay constant.

For large 9p, Equation (2.8) may be approximated as7

1
p 3'3“ E
p i

(2.9)

VIID
...:
I
(DCD
«a lo

This approximation, however, is only valid for 9p > 100 and as
6p decreases in value both the first term on the right in Equa-
tion (2.8) and the l in the denominator of the summation no longer
remain negligible. The net effect is a nonlinear relationship
between 9p and p as illustrated in Figure 2.1.

2.3 Effects of Prompt and Delayed Neutrons on Reactor Transient
Behavior

The fission process in the Michigan State University TRIGA

Mark I nuclear reactor is described by the following nuclear

 

Samuel Glasstone, Nuclear Reactor Engineering (New York:
D. Van Nostrand Co. Inc., 1963), pp. 244-245.

Ibid., pp. 244-245.

p, Reactivity (cents)

100

50

40

30

20

10

I—-

     

I I I I I

 

 

I l l I I I I II J L 1 J. I I IIII
10 20 30 40 50 100 200 300 400 500 1000

9p, Period (seconds)

Figure 2.1 Relationship between reactivity and reactor stable
period for the Michigan State University TRIGA Mark I
nuclear reactor (Courtesy of Gulf General Atomic)

II

i if"; 32".-

'- 1.411 , _
. TwI'Tbtnl '

:—

'»]fl%‘- f’ ‘

fifir

 

"I-" -Jl . . .

 

 

equations:

U235 +- n1 a U236 + energy (2.10)

92 O 92
or

235 1 , 1
92U +|0n a 2 prS + energy +'y + con (2.11)

Equation (2.10) occurs about twenty percent of the time. The
fission products (fips) in Equation (2.11) have a variety of atomic
weights but will generally weigh about half as much as the U235.
Heat is generated by the energy given off in both reactions and the
energy is about 200 Mev per fission. On the average between two and
three (2 s c s 3) neutrons are given off per reaction in Equation
(2.11). This equation is critical to reactor operation since the
neturons produced must sustain the fission chain reaction.

Neutrons emitted in.Equation (2.11) are of two types:
Prompt neutrons and delayed neutrons. Prompt neutrons make up
about 99.27 percent of all neutrons emitted in the TRIGA reactor.
The total mean lifetime of these prompt neutrons is less than one
millisecond.9

Delayed neutrons are the result of the decay of certain

fission products in Equation (2.11). An example of a delayed

neutron emission is given by the nuclear equation

531137 B 54X137 n 54x136 (2.12)
22 sec e instantaneous e

 

Training Manuel, Reactor Theory, p. l.

9 Ibid., p. 4.

10

where 531137 is one of the fission products. The 22 second delay

in the neutron emission gives it a half life of appearance of 22
seconds.1

Even though the fraction of delayed neutrons is small, their
long apparent lifetimes make the average lifetime of all neutrons
emitted much longer than the lifetime of prompt neutrons alone.

This effect is vital since a reactor period caused by prompt neutrons
alone would make control of the fission process impossible.

The effect of prompt and delayed neutrons on the change in
neutron flux can be described as follows: In a critical reactor
withdrawal of a control rod some discrete distance will produce a
transient period due to prompt neutrons and a stable period due to
the combined effect of prompt and delayed neutrons. The effect of
the transient period will result in a "bulge" in the neutron flux.

This effect is shown graphically in Figure 2.2.11

p positive

 

p negative

 

 

V

time (seconds)

Figure 2.2 Combined effect of prompt and delayed neutrons in
a supercritical reactor

 

10 El-Wakil, Nuclear Power Engineering, p. 131.

11 Ibid., pp. 133-134.

11

Since p is proportional to the slope of the curves in
Figure 2.1 it is necessary to let the transient period die out

prior to measurement of p.

2.4 The Compensated Ion Chamber

It has been stated in Equation (2.7) that the reactivity
produced when a discrete portion of control rod is withdrawn from
a critical reactor pile is exponentially proportional to the change
in neutron flux. In order to measure p it is therefore necessary
to monitor this flux. This is accomplished in the TRIGA Mark I
reactor by means of the output current of a compensated ion chamber
located near the reactor core.

The output of the ion chamber is a very low current which
is directly preportional to the neutron flux in Equation (2.5)

such that

t/e

= P
is(t) Ioe (1.1)

where 9p may be substituted by the approximate expression

0
a 21—0- (1.2)
P P

as previously stated.

Figure 2.3 is a simplified diagram of the compensated ion
chamber. The device is actually composed of two chambers where
R8 is the equivalent output resistance. One chamber is lined

with boron enriched with B10

and is biased at + 580 volts. From
this sub-chamber a current proportional to neutron flux and gamma

flux is obtained. The other sub-chamber has a variable bias from

12

0 to -37 volts and is sensitive to gamma flux only. If the gamma
sensitive current is subtracted from the gamma, neutron current
of the boron coated sub-chamber, a current proportional to neutron

flux alone is obtained.12

 

Boron I
Coating

I
I
I
I
I
I
W .4; #1.
I
I
I
l
I

 

 

Equivalent Output
Resistance

-37svcs0v --

 

Figure 2.3 Simplified diagram of a compensated ion chamber

At low neutron flux levels (Q < 2.5 X lOZn/cmZ/sec), the
current contribution from the gamma sensitive chamber is nearly
equal to that of the gamma, neutron sensitive chamber. It is
therefore important to set the compensating chamber voltage (Vc)
to the proper value in order to achieve a linear response due to
neutron flux alone. This leaves room for error in the linearity
of is(t) if Vc is not properly adjusted. Fortunately, as the

neutron flux level increases the current contribution to is(t)

 

Training Manuel, Instrumentation, p. 5.

13

due to gamma flux alone becomes negligible and i8(t) approaches
a linear dependance on neutron flux alone.

Experimental results reveal that improper adjustment of
the compensating voltage at low reactor flux levels (correSponding
to reactor output power levels below ten watts) causes the ontput
current of the ion chamber (is(t)) to become nonlinear. Figure
2.4 shows the maximum nonlinear effects at low power levels when
the compensating voltage is adjusted to zero volts.

Since a linear relationship between ion chamber current
(i8(t)) and neutron flux (also reactor output power level) is
necessary, it is best to rely only upon output currents which
correspond to reactor power levels greater than ten watts.

Specifications of the ion chamber are given in Appendix E.
Since the output resistance is greater than 1013 ohms, the chamber

may be considered as a current source.

Ion Chamber Output Current (nA)

1000

500
400

300

200

100

50
4O

30

20

10

14

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

// vC = 0 v
/r
//
//
1 2 3 4 5 10 20 30 40 50 100 200 300 400 500

Reactor Power Level (watts)

Figure 2.4 Graph of ion chamber output current as a function of
output power level for the Michigan State University
TRIGA Mark I nuclear reactor

1000

CHAPTER III

DESIGN OF THE REACTIVITY MEASURING DEVICE

3.1 Formulation of the Problem

Having presented the equations defining reactivity and its
relationship to the output current of the ion chamber it now re-
mains to develOp a means by which p can be measured.

Equation (1.1) expresses the relationship between 9p

and IS where

t/ep
is(t) = Ioe (1.1)
The objective is to process this current so as to produce a voltage
proportional to the reactor stable period. If this can be accomplished
effectively then 6P may be related to p as in.Equation (1.2)

since

9
.,._Q
9p _ p (1.2)

Because this relationship is slightly nonlinear as the
graph in Figure 2.2 illustrates, the value of p corresponding

to a given 9p may be taken from this graph and used to calibrate

the reactivity meter.

15

16

3.2 Possible Methods of Solution13

Given the desired output p and the input is(t) several
methods for determining p will now be considered.

One method makes use of the circuit in Figure 3.1 where a
d-c voltage pr0portional to l/ep is obtained from is(t) using

a logarithmic amplifier followed by a differentiator and filter.

 

 

 

v1(t) Differen- V2 Low-Pass vout

A

. Meter

Logarithmic

o—>——I 9
Amplifier tiator Filter

4b

 

 

 

 

 

 

 

 

 

 

 

Figure 3.1 Block diagram of a circuit which provides a d-c
voltage proportional to the inverse of the
reactor stable period

In a circuit of this type the logarithm of the ion chamber
current is first taken such that

t/e

p At
— - . — - — - -_ .
v1(t) A log1018(t) A loglOIOe — A loglOI logloe (3 1)

The voltage v1(t) is a linear ramp function whose slope
is inversely proportional to the negative inverse of 9p. Dif-

ferentiation of this ramp voltage yields a d-c voltage V2 where

= — - _ - F = — 3 .
V i V1(t) - d ( A IOgIOIO O logloe) logloe ( 2)

 

3
1 The first two methods presented were suggested by Dr. L. Giacoletto

of Michigan State University. The third method was developed by
the author.

17

The result is a d-c voltage inversely pr0portional to 9p. This

voltage is then passed through a low-pass filter to eliminate any

low-frequency noise components passed by the differentiator.
Finally, due to the slight nonlinear relationship between

9p and p the voltage Vo (Figure 3.1) is calibrated on a

ut
meter via Figure 3.6 to indicate exact values of reactivity.
A second method for obtaining a d-c voltage proportional

to 9p involves the use of an integrator and a quarter-square

multiplier. The circuit is pictured in Figure 3.2.

 

 

 

 

 

I -

130) +

”if

i

 

 

V Quarter-

 

N

 

 

 

0

Square 3

Multiplier . Meter

Figure 3.2 Circuit for obtaining a d-c voltage prOportional

to
9?

 

 

 

 

 

In this circuit the input current (iS(t)) is first con-

verted to a voltage

t/e

v1(t) = is(t)Rv = Ione p (3.3)

using a buffer amplifier whose input resistance is much greater

than R .
v

This voltage is then integrated to obtain v2(t) where

t -I R 9 t/e

v2(t) = -I;%£VI(t)dt = JFK—2 (e P - 1) (3.4)

18

The quarter-square multiplier is next used to divide out

the exponential term in v2(t) such that

 

V = v2(t) = -IORVO (et/Cp - 1).____l___..,.:EE (3.5)
out v1(t) RC t/e “ RC
p
I R e
0 v

This resultant d-c voltage is preportional to e and
can therefore be calibrated on a meter using the graph of Figure
3.6 to represent the corresponding values of reactivity.

A third method for finding 9p involves the integration
of a voltage proportional to the change of the input current.

Figure 3.3 is a simplified diagram of the circuit.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

v (t)
c
T c
V
V _ I
U HI Reed R
- -—o I»— .
v (t) A V35 ”ind“ co 3”“
i (t) 1 ‘ T’ Comparator “II? + A Vout
s + 0— r0
V
R L Meter
v
Figure 3.3 Circuit for obtaining a discrete voltage propor-
tional to the lepe of is(t)
The input current is first converted to a voltage v1(t)
such that
t/ep
v1(t) = 18(t)Rv = IORve (3.3)

This voltage is then fed into a window comparator whose "G0" out-
put (Figure 3.3) acts as a current sink for VL s v1(t) S‘Vu and
as a large resistance whenever v1(t) <‘VL or v1(t) > Vu' The

”G0" condition will allow the current iC(t) to flow in the coil

19

of the reed switch and close its contacts. When this occurs the

d-c voltage V is switched to the input of the integrator and

I
is integrated for the period of time that v1(t) remains within

the bounds of Vu and VL. This period of time is pr0portional

to 9p and is also proportional to the final integrator output

voltage. The relationship between Vout and 9p can be found

by starting with the output voltage integral
9
-1 t

out =RC' VIdt (3.6)

The total integration time (at) is that period of time that

VL s v1(t) s'Vu and may be calculated by substituting Vu and

VL into the equation for v1(t) where

ablep
v1(eL) = V‘L = vaoe (3.7)
and
eu/e
v1(9u) - Vu = RvIOe (3.8)

at and en are the times for which v1(t) is equal to VL and
Vu respectively.
Equations (3.7) and (3.8) are then solved for the time

increment 9t where at = eu - QC and

Vu
_ _ ZL__)

Using Equation (3.9) for the value of 9t in the integral

 

of Equation (3.6), the voltage V may be written as

out

Vu

_.:l “.12..
vout — RC 9p(£n R:IO% nR v10) (3°10)

 

20

which is a d-c voltage proportional to the stable reactor period.
As in the first two methods this voltage may be calibrated
on a meter using the nonlinear graphical relationship of Figure 3.6
to indicate the corresponding value of reactivity.
The above method for measurement of reactivity differs
from the preceeding two in that the output voltage is not a con-
tinuous representation of 9p but rather a discrete value obtained
after integrating a d-c voltage over a discrete time period at.
After consideration of the three alternatives the method
chosen to measure p was the first one described using a
logarithmic amplifier followed by a differentiator and filter

(Figure 3.1).

3.3 The Circuit and Theory of Operation

With the reactivity measurement solution method chosen,
the circuit of Figure 3.4 was designed to produce a d-c voltage
proportional to 9p.

The transfer function of the circuit is

(-chls) 1

V = V (3)
out 1 (R1C18+1)(R2C28+l) (Rfos+l)

(3.11)

where V1(s) is the frequency domain expression for v1(t) given

88

-A loglo(IO/Ik) A logloe

s 0 82
P

 

V1(8) = (3.12)

and A is the logarithmic amplifier voltage constant determined

approximately by

 

OOHINN HovoE xmamo

1: App ppmuuo p-p

a: o
a
THAMHH<>N1|.
OH

,- Eo
Ifiwm

ucosspumcH wcwusmmoe muw>wuomou ocu mo Emuwmwv

App “sweep p-p

 

 

N

Emm

>
w qum mm

8x 5

"magnum umBom

 

 

 

 

 

uwsouwo ¢.m muswflm

 

 

 

upausflae< mos

oaaq mm

 

> ma +

CZ m.HH

 

 

 

22

Ry + 5000 +R3 (26 X 10-3)

)
R3 .434

A a ( Volts (3.13)

where R3 is a temperature dependant resistor within the logarithmic

amplifier module.14

IR is the log amplifier reference current determined

primarily by Rx.15
The concept by which the Circuit works is as follows:
Using the ion chamber current as the input, the logarithm of is(t)

is first taken. The output voltage (v1(t)) is a linear ramp

function since

i (t) I t/e
= - _§___ = - .9. P
v1(t) A log10 I A log10(I e )
k k
I0 At
= -A log10 IEI- Eg'logloe (3.14)

v1(t) is then differentiated to produce a d-c voltage V where16

2

 

14 Modular Logarithmic Amplifier Model 4116 (Burr-Brown Research
Corp., 1971), p.3.

15 A more complete discription of the logarithmic amplifier and

its parameters is given in Appendix A.

16 Since the input current (is(t)) is an exponentially increasing

function and since the circuit (Figure 3.4) used to process this
function has a limit to the absolute magnitude of the input, iS(t)

can only remain "on" for a discrete period of time et' Therefore,
in order to be absolutely correct when defining is(t) the gate

function notation should be used where

Get(t - et/Z) = 1 0 s t s at

Get(t-et/2) 0 t<0,t>et

8 is the time of the input pulse.
t Using this notation i8(t) becomes

23

I

 

V2 R201 dt 1(t) R201 dt< A 10g10 Ik 9 logloe)
R C A
2 l
= 3. 5
9p logloe ( l )

This d-c voltage is then filtered to eliminate any low-
frequency noise enhanced by the differentiator. The resultant

voltage V0 is inversely proportional to the reactor stable

ut

period and approximately proportional to the reactivity such that

R A R A
V = 2C1 log e a: p —2-C—1- 10
out 0p 10

(3.16)
90

 

g1oe

As previously discussed (Section 2.2) the relationship
between 9p and p is nonlinear. Calibration of the meter may
be accomplished by producing known reactor periods and.marking
the meter deflection with the corresponding values of p via the
graph in Figure 3.6.

A comprehensive analysis of the circuit design is given in

Appendices A, B, C and D.

3.4 Calibration

The output voltage in the circuit of Figure 3.4 is a d-c

voltage proportional to the inverse of 9p and approximately

 

t/ep
is“) = I0e E9t(t-et/2):|

If the gate function notation is carried with iS(t) two finite
strength impulses will appear when v1(t) is differentiated in Equa-

tion (3.15) due to differentiation of the gate function.
Although these finite strength impulses are of importance (partic-
ularly since they show up on the output when iS(t) is switched on

and off) they shall not be carried in the analysis. It should be under-
stood, however, that is(t) is on for a finite period of time. This

period of time (at) is discussed further in Appendix B.

24

proportional to p where

R C A R C A
— _.2._1._. _2_.1___
Vout 9p logloc a p 90 logloe (3.16)

As previously mentioned the relationship between 9p and
p is nonlinear. Figure 3.6 illustrates this relationship for the
Michigan State University TRIGA Mark I reactor.

The range of reactor stable periods for which the meter is
to be calibrated is 10 sec s 6p s 40 sec. This range corresponds
to a range of reactivity between 40 and 19 cents respectively.
(Figure 3.6)

Calibration of the reactivity meter is accomplished with

the apparatus in Figure 3.5.

 

 

 

 

 

 

 

 

 

Reactor Power ............... --J
Level Indicator Reactivity Meter

 

 

 

 

Ion Chamber

Figure 3.5 Apparatus used to calibrate the reactivity meter

The reactor is first made critical with the ion chamber

current equal to I A control rod is then pulled out a discrete

0.
distance and the ion chamber current becomes

t/O

- = p
18(t) Ioe (1.1)

25

The corresponding d-c voltage on the reactivity meter will

be

 

log (3.16)

10‘3

This voltage is then marked on the meter.

In order to mark the reactivity corresponding to Vout
the graph of Figure 3.6 is used. As this graph indicates the
reactor stable period must be known. This value is determined by
using the reactor power level indicator and a timer to find the

power level doubling time ed. The period can be calculated from

the doubling time since

 

9p = CH 2 (3.17)

This process is repeated until the range of p is

adequately covered.

p, Reactivity (cents)

100

50
4O

30

20

10

26

 

1 2 3 4 5 10 20 30 40 50 100
Op, Period (seconds)

Figure 3.6 Graph showing the relationship between reactivity and
reactor stable period for the Michigan State University
TRIGA Mark I nuclear reactor (Courtesy of Gulf General
Atomic)

CHAPTER IV

CONSTRUCTION OF THE REACTIVITY METER

4.1 Circuit Tests, Performance and Specifications

The circuit of Figure 3.4 was constructed and tested.
Figure 4.1 shows several photographs of the instrument.

Testing of the circuit was accomplished by connecting
the meter input to the ion chamber via a coaxial cable. The re-
actor was then made critical and a control rod withdrawn to pro-
duce a reactor period within the desired range. The range of
the meter was checked for consistancy and variation due to noise.
These tests revealed the meter deflection to be approximately 77
percent for a 10 second period and approximately 24 percent for a
40 second period thus adequately covering the desired range of
9p. The maximum variation in meter reading due to noise was less
than i 0.4 percent of full scale.

The minimum reactor power level for which the circuit was
designed to operate is ten watts. This corresponds to an ion
chamber current, I0 = 3nA. Values of input current below this

level will put the logarithmic amplifier in saturation and there-

fore cause V0 to register inaccurately on the meter.

ut
The upper power level was chosen to be 200 watts resulting

in an ion chamber current of 60 nA. This value is not an absolute

upper limit and the meter will continue to read accurately up to

27

. shaman
mama mufi>auommu onu mo anonymouonm H q
u

28

 

 

29

a reactor power level of 1000 watts (is(t) = 280 na) at which time
the negative temperature coefficient inherent in the reactor will
cause the meter to lose accuracy.

The final specifications of the reactivity meter are listed

in Table 4.1.

Table 4.1 Specifications of the reactivity meter

 

t/e

Input: Exponentially increasing current...is(t) = I e p

0

Op min ........................................ 10 seconds
ep max ........................................ 40 seconds
IS min .................................. 3 X 10.9 Amperes
S max ................................. 60 X 10.9 Amperes

outeut 00.0.00.........OOOOOOOOOOOOO reactiVity (p) in cents
p 0.0... ..... .............OOOOOOOOIIOOOOOO0.... 19 cents

p .....OOOOOOOOOOOOOOOO.......OOOOOOOOOOOOOOOO.40 Cents

Accuracy (after calibration) ...... ........... :_0.5 percent

 

 

 

Since the values of p on the meter are calibrated from
a graph, the only error will be that variation caused by noise on

the output voltage of the filter (Figure 3.4).

CHAPTER V

CONCLUSIONS

The control rod calibration curve in Figure 1.1 is of the
type for which the reactivity meter was designed to produced.
Original design Specifications called for the measurement of periods
between 10 and 40 seconds correSponding to reactivities between
40 and 19 cents. As the graph indicates, these values cover only
a small portion of the total rod calibration curve.

In order to cover the entire range of the calibration
curve the measured values of rod reactivity must be added in incre-
ments within the range of the meter. For example: Reference to
the graph of Figure 1.1 shows the reactivity corresponding to a
rod position of 300 to be approximately 60 cents. If at this
position the reactor were made critical and the rod pulled out to
a position of 350, a reactivity of about 40 cents would be recorded
on the meter. In order to get the total rod worth at this new
position this value of reactivity would have to be added to the
previous total reactivity of 60 cents and would therefore yield
the new total reactivity of approximately 1 dollar.

It should also be noted that since the total rod worth is
determined by the sum of incremental reactivities the error in
measurement of each incremental rod worth becomes very important

and the total error increases with each sum.

30

31

The overall results of the tests made on the instrument
reveal that the reactivity meter works well for the range of input

currents and reactor periods for which is was designed.

BIBLIOGRAPHY

BIBLIOGRAPHY

El-Wakil, MQM. Nuclear Power Engineering. New York: McGraw-
Hill Book Co., 1962.

Glasstone, Samuel. Nuclear Reactor Engineering. New York:
D. Van Nostrand Co. Inc., 1963.

Jackson, Albert 8. Analog Computation. New York: McGraw-Hill
Book Co., 1960.

Modular Logarithmic Amplifier Model 4116. Burr-Brown Research
Corp., 1971.

Nuclear Reactor Operations and Training Manuel. Michigan State
University, 1969.

RCA Linear Integrated Circuits. RCA, 1967.

Selby, Samuel M. CRC Standard Mathematical Tables. Cleveland,
Ohio: The Chemical Rubber Co., 1965.

Westinghouse WL-8105. New York: Westinghouse Electric Co., 1965.

32

APPEND ICE S

APPENDIX A

DESIGN OF THE LOGARITHMIC AMPLIFIER

APPENDIX A

DESIGN OF THE LOGARITHMIC AMPLIFIER17

The logarithmic amplifier used in constructing the circuit
of Figure 3.4 is a Burr-Brown model 4116 modular logarithmic
amplifier. Its purpose in the circuit is to take the logarithm
of the input current is(t) where

Us
180:) = Ioe P (1.1)

The resultant output is a linear ramp function given by

1 (t) I Us
3 - —S_——— = - £ p
v1(t) A log10 I A log10 I e
k k
3.0 At
= -A log10 1k - a; logloe (3-14)

There are two design parameters for this amplifier. These
are the voltage scale factor A and the reference current Ik-
Figure A.l shows the logarithmic amplifier with exterior connections.

The electrical specifications are given in Table A.l.

 

17 The design equations, graphs, figures and specifications in this
section were taken from the pamphlet Modular Logarithmic Amplifier
Model 4116 (BurréBrown Research Corp., 1971), pp. 1-5.

33

34

Table A.l Electrical specifications of the Burr-Brown modular
logarithmic amplifier model 4116 (current input only)

 

 

Accuracy

Accuracy, percent of full scale with current
source input for 0.4nA 5 IS s 400 uA ......... i_l percent

Input

Current Source Input ...... ................ 400 pA to 400 uA
Absolute Maximum Current ........................... i_10 mA

Reference Current Range (1k)............ 400 nA s I s 40 uA

k
Voltage Scale Factor Range (A) ............ 2/3 V s A s 10 V

911.22%

current ......OOOOOOOCOOO ...... .... ..... ......OOOOOOCisnlA
VOltage ....0.0.0.0000.........OOOOOOOOOOOO00.0.00...ilov

Output Imedance atA=5 0...........OOOOOOOOOOOOOO 10 Ohm

Stability

Scale Factor Drift (AA/0C) ................... i_0.0005 A/OC

Reference Current Drift (AIk/OC)
for 0.4 pA s IR 5 1 pA ..................... i 0.003 Ik/OC

Input Offset Current Drift (AIS/0C)...10 pA at 25°C, 1 pA/°C

Input Noise - Current Input .... l pA rms, 10 Hz. to 10k Hz.

Power Supply
RatedVOItage ............OOOOOOOOOO......OOOOOOOO.:15 Vdc

Supply Drain Gluiescent) ........................... i_22 mA

 

 

35

 

 

 

 

R
x
1r——d’\/\———T+ 15 V
3 -I- R
Current Input OLIJLIX BB - 4116 5 y
Voltage Input Oégéfifi]. Log Amplifier
(optional) 4 ‘3 V
2 l
f Y -

 

 

 

’ I

+ lS‘VA H-c offset 5- 15 V

adjust

 

Figure A.l Burr-Brown modular logarithmic amplifier model
4116 with external connections

A design that takes advantage of the full range of out-
put voltage is best. The values of Rx and Ry required to
accomplish this may be obtained by first solving Equation (3.14)

for the full i_10 volt range where

 

I . .
v1 max = -A loglo 1““ = 10 Volts (11.1)
k
I a
1 min = -A 10g10 I—“‘--’3 = -10 Volts (A.2)
k

These equations may be added to solve for Ik with the

result

I =(I I )35 (A.3)

k max min

The corresponding voltage scale factor (A) may next be
found by substitution of Equation (A.3) into Equation (A.l) for

Ik' This provides an expression for A where

36

10
A = 5 (A.4)

log10 Imax

Imin

 

In Chapter IV it was established that the value of 1min
would be 3nA and Imax would be 60nA. Substitution of these

values into the expressions for IR and A yields

Ik = 12.96 nA (A.5)

A = 15.75 Volts (A.6)

At this point a problem is encountered. Figure A.2 shows

the graphical relationships between I A and their design re-

k’
sistors. Reference to these graphs and to the specifications in
Table A.l indicate that the values of IR and A in Equations
(A.5) and (A.6) are outside the design limits.

A compromise must be made to reduce the output voltage
range in order to put IR and A within their respective tolerances.
For the final design a value of Rx = 11.3 m was selected. This

corresponds to a reference current of .56pA.

Using this value for I the best value for A can be

 

 

 

k
calculated by letting the output be 10 volts for IS min' Sub-
stitution into Equation (A.l) yields
A = '10 = '10 = 4.5 (A.7)
I . -9
min 3X10
1°g10 I 10g10 -6
k .56X10

Reference to the graph in Figure A.2b indicates a value of
28 RH for Ry will make A equal to 4.5 volts.
The actual output range of the logarithmic amplifier may

now be calculated where

37

 

 

 

 

 

 

 

 

 

 

100 pA
10 uA ~\\\\S§\\
..f‘ 1 “A N
100 nA
10 nA
10 k 100 k l M 10 M 100 M
R ohms
y ( )
(a)

 

10 V //

.v //

< /

 

 

 

 

 

 

 

 

 

 

4 V
2 V
0
O 20 k 40 k 60 k 80 k
Rx (ohms)
(b)

Figure A.2 Design curves for the Burr-Brown model 4116
log amplifier, (a) Reference current vs. R ,
(b) Voltage constant vs. Rx y

38

1 max = 10 Volts (A.8)
I -9
60x10
. = -A log __m_____ax = -4.5 log ——--_
l min 10 IR 10 .56X10 6
= 4.36 Volts (A.9)

The design parameters of the logarithmic amplifier may

be summarized where

A = 4.5 Volts
R = 28. k Ohms
y
Ik = .56 “A
Rx = 11.3 M Ohms
Input Range (is(t)) 3nA to 60nA

Output Range (v1(t)) 10V to 4.36V

APPENDIX B

DETERMINATION OF THE RAMP FREQUENCY COMPONENTS

APPENDIX B

DETERMINATION OF THE RAMP FREQUENCY COMPONENTS

The ideal output of the logarithmic amplifier in the circuit

of Figure 3.4 is a linear ramp function of the form

I
=- 41-45 3
v1(t) A loglo I 9p logloe ( .14)

k

Differentiation of this function will yield a d-c voltage

 

where
d d 39, At
v2 = 'R2C1 E? V1(t) = 'R2C1 3? (TA 1°810 IR "'5’ 1°gloe)
p
R C A
_ 2 1
— 9p logloe (3.15)

In order to obtain the true differential as in Equation
(3.15) it is necessary to differentiate all frequencies of the
ramp function v1(t). This therefore requires the knowledge of
the range of ramp frequency components.

The highest frequency components occur when the ramp func-
tion is on for the shortest period of time. This corresponds to
a reactor stable period of ten seconds.

The "on" time for this ramp may be found by substituting

the limits of is(t) into Equation (1.1) and solving for 9t where

t p min (1.1)

39

40

or
0 /10
60X109=3x109et
6t = lOLn 2022 30 seconds
The Fourier series half-range cosine expansion for
f(x) = x as a recurrent triangular wave is1
_ _t it ms _1_. in}; 1. 51X.
f(x) x — 2 - “2 (cos k. + 9 cos k +25 cos k
+...+ -——l—-§ cos LgELlZEE) O < x < k
(Zn-1) k
n = 1,2,... (B.1)

Substituting the ramp function of Equation (3.14) for

f(x) the Fourier series expansion for v1(t) is

 

 

I I O A log e
At _Q_ t 10
v (t) = -A log "Q'-——'1og e = -A log - -
1 10 Ik 9p 10 10 IR 2 9p min
49 A
'-—--- log e cos 1JJET-I-l-cos §fl£+.l_ cos-ZEE
2 10 e 9 0 25 9
TI' 6 t t t
p min
_1_ 7nt _1_ 9nt 1 llnt
+ 49 cos-—6--+ 81 cos -E—-+ I2I cos 9
t t t
+...+ --l-- cos 2n-l fit 0 < t < e = 30 sec.
(Zn-l)2 9t t
n=l,2,... (3.2)

The radian frequency components for the minimum duration
ramp and the respective amplitudes of each (Pi) can be determined

from Equation (B.2). The values are

 

l
8 Samuel M. Selby, CRC Standard Mathematical Tables (Cleveland,

Ohio: The Chemical Rubber Co., 1960), p. 147.

41

 

 

 

 

 

 

49 A lo e
wl = g—‘= 0.1048 rad/sec, P1 = 2t glO = 7.9
t n 9p min
49 A log e
w3 3 in = 0.314 rad/sec, P3 = g 10 = 0.878
at 9n 0 .
p min
48 A log e
ms = 2E'= 0.524 rad/sec, P5 = t2 10 = 0.316
9t 2511' e
p min
49 A log e
07 = h = 0.734 rad/sec, P7 = ‘2 1° = 0.161
6t 4911 e .
p min
49 A log e
019 = g”- - 0.943 rad/sec, P9 = t2 1° = 0.0975
t 81w 6 .
p min
49 A log e
wll = llfl = 1.15 rad/sec, P11 = t 2 10 = 0.0653
6t 12111 0 ,
p min

Since the amplitudes of higher frequency components are

less than 1/144 of the amplitude of wl it is reasonable to omit

them from the analysis. The value of wll is therefore the highest

radian frequency to be differentiated.

APPENDIX C

DESIGN OF THE DIFFERENTIATOR

APPENDIX C

DESIGN OF THE DIFFERENTIATOR

Figure C.1 is a magnitude plot for the real and imaginary

components of an ideal differentiator whose transfer function is

(to)

V
out _ .

V. (w) JKw (C°1)
1n

 

 

 

 

 

A A
V (00) V (w)
Im OUt Re OUt
V. (w; . V (m;
In in
w T: w I
(a) (b)

Figure 0.1 Transfer function plot for an ideal differentiator,
(a) Imaginary part, (b) Real part

A differentiator circuit with the type of transfer function
in Equation (C.l) has the disadvantage of enhancing noise.

If the range of frequencies to be differentiated is limited,
it is possible to reduce unwanted noise by using a compensated

differentiator whose transfer function is

 

19 Albert S. Jackson, Analog Computation (New York: McGraw-Hill
Book Co., 1960), p. 147.

 

42

43

 

 

 

 

 

 

 

Vout(w) = .ngg (C 2)
Vin(w) (ij + 1) (JKw + 1)
where
2
Re M = 2K2") (C 3)
vin(w) (1 - K2012)2 + 4K2w2
and
voutm) inil - K2031
1‘“ W a 2 2 2 2 2 (CA)
in- (1-Kw)+4Ku)
These real and imaginary components are plotted in Figure
C.2
A A
V
Im out(w) Re vout(w)
Vin(w) I J I 44’ Vin(w)
I
w
I I I l I;
m
(a) (b)

Figure C.2 Transfer function plot for a compensated
differentiator, (a) Imaginary part, (b) Real part

In Appendix B it was determined that the maximum radian
frequency to be differentiated in the linear ramp function was
wll = 1.15 rad/sec. It is therefore desirable to make 1/K greater
than 1.15 rad/sec.

2
Figure C.3 is a circuit diagram of the differentiator.

 

2
0 RCA Linear Integrated Circuits (RCA, 1967), pp. 75-78.

 

 

 

 

 

 

 

 

Ti?

Figure C.3 Equivalent circuit of a compensated differentiator

The transfer function may be derived as follows:

 

 

 

 

Define:
R2
z£=R2HC2 =ij0 +1 (C's)
2 2
z =R +—L- (C.6)
r 1 ijl
Ve’ Vi and V01 may be written as
Z + Z Z V Z Z
V = V ( f Oi)// i + 01( r// i) (C.7)
° + +
e In zr + (zf + ZOi)//zi zf in zrl/zi
V, = 0 (C.8)
1
V01 = -AO(01)(Ve - V1) (0.9)
Substitution of Ve and Vi into the expression for
V01 yields
-V A (0.))(Z +2 .)2,
V in 0 f 01 1 (C.10)

= +
01 (zf+in)zi zr(zf+in+zi)+A0((p)zizr

If R a m the exact expression for the transfer function

L

can be written as

45

Vout _ z011211 " A0(w)zizf

Vin A0(w)Zin+ (Zf+201)Zi+Zr(Zf-I-Z

 

(C.11)
01 + Zi)

This expression may be simplified if the following assump-
tions are made:

Assume:

21 >> zfl/zr (C.12)
Z01 << Zf (C.13)
Zf << Ao(w) (C.14)
Zr << A0011) (Col-5)

2
Equation (0.9) may then be written 1

V t -Zf
ou = ___
V Zr (C.16)

 

When the expressions for Zr and 2f are substituted into

this equation the differentiator transfer function becomes

V -RC
_0‘2£= jgzrlw ((3.17)

vin (leclp +-1)(jR2C2m +.1)

If RIC1 = RZCZ then the magnitude plot in Figure C.2

will correspond to Equation (C.17) with l/K = 1/R101.
For a differentiator error of one percent maximum it is

necessary to establish a corner frequency about ten times the

maximum frequency to be differentiated. In other words

= 1 =16... =10w (0.18)
max 11

 

2
1 Ibid., pp. 75-78.

46

22
where

wll = 1.15 rad/sec and was determined in Appendix B.
A trade off of accuracy vs. noise is encountered if Equa-
tion (C.18) is used to design the differentiator. The prupose of

the low-pass filter in Figure 3.4 is to attenuate the noise passed

by the differentiator. It was discovered that for l/RIC1 = 10 wll
= 11.5 rad/sec the filter could not effectively reduce the noise

without causing anexcessive delay in the d-c response due to its

long time constant. A decision was made to limit the filter time
constant to 0.6 second. This meant the corner frequency of the

differentiator had to be reduced until the noise level was tolerable.

The experimental resultant corner frequency was l/R C = 3.8 rad/sec

l l
or about three times as great as the highest frequency to be dif-

ferentiated.
The magnitude of the error caused by inexact differentiation
of the maximum ramp frequency will be small since the amplitude of

this frequency component is only 1/121 of the major ramp frequency.

 

22 Jackson, Analog Computation, p. 147.

APPENDIX D

DESIGN OF THE FILTER

APPENDIX D

DESIGN OF THE FILTER

Figure D.l is a schematic of the output filter in Figure

3.4.

The filter is a simple RC low-pass filter with a unity

gain output buffer stage to prevent loading.

 

 

 

 

 

_v Iv—
- ’ 2
1n Rf “ uA 741 -———o Vout
Ioo'kni
Cf 6 uF

adjust
- 15 V

I d-c offset

Figure D.1 Output filter for the reactivity meter

The transfer function for the filter is

 

 

V
out(w) = 1 (D 1)
Vin(m) JRfow + 1
Since the output voltage of the differentiator (V2) is

a d-c voltage with superimposed low-frequency noise it is desirable

to make the filter time constant long enough to reduce this noise

47

48

while at the same time have a reasonable reSponse time to the d-c
voltage.

Final tests of the reactivity meter indicated that a filter
time constant of Rfo = 0.6 seconds reduced the output noise level
on the meter to less than :_0.4 percent full scale. This value

for Rfo also gave a d-c reSponse time of about 3 seconds.

I"

 

APPENDIX E

SPECIFICATIONS OF THE COMPENSATED ION CHAMBER

APPENDIX E

SPECIFICATIONS OF THE COMPENSATED ION CHAMBER

The TRIGA Mark I nuclear reactor at Michigan State University

contains a Westinghouse type WL-8501 compensated ion chamber. Table

2
E.1 tabulates the specifications of this chamber. 3

Table E.1 Specifications of the Westinghouse type WL-8501
Compensated ion chamber

 

MaximumTVoltage Between Electrodes (dc)...............1000 V
Operating Voltage for Boron Chamber (dc).............. 580 V

Operating Voltage for Gamma Chamber (dc).. ...... 0 to -37 V

, 13
Output Resistance, Signal to Case, Minimum ........10 Ohms

Output Capacitance, Signal to Case, Approximately.....130 pF

1
0 n/cmZ/sec

~14

Thermal Neutron Flux Range.....2.5X102 to 1.5X10
Thermal Neutron Sensitivity...................l.5X10 A/nV

Gamma Sensitivity, Compensated..............2.5><10-13 A/R/hr

 

 

 

Since the output resistance of the ion chamber is greater

than 1013 Ohms the chamber may be treated as a current source.

 

Westinghouse WL-8501 (New York: Westinghouse Electric Co.,
1965), pp. 1-20

49

‘IIIIIIIIIIII