## Derivation in Time Domain

Since the voltage at the output of the amplifier is given by

 V(t) t/At = 2 ^n v ft ■ 11 At) • ‘ (3-3) n = -°°

then

 v2(t) Jl t/At • ’ ■ У] . Vm Vn V(t — mAt) V (t — IlAt) . (3-81) m = — ^ n =-<*■• since the expectation of the sum of a number of random variables is equal to the sum of their expectations.   If the average fission rate in the reactor is F and the counting efficiency of the detector is e counts per fission, then the probability of detecting a neutron during the n^ time interval is F є At. and the probability of detecting no neutron during the n*^ time interval is 1 — Fe At. So the expected value of ^ nm^2 is

The expected value of v?>n has two terms: one due to accidental pairs, and one due to coupled

pairs. An accidental pair is caused by a count registering in the m*’*1 time interval due to a

neutron in a family from one primary source neutron, and a count registering in the n*1*1 time

interval due to a neutron in an entirely separate family from another independent primary neutron.

A coupled pair is caused by counts in the mth and ntfl time intervals due to neutrons that can be

(2)    traced back to some common ancestor; i. e.. one fission (refer to de Hoffman’ ‘ for a discussion of accidental and coupled pairs). So the expected value of rm r is

probability of registering a count in the time interval, conditional probability that if a count is registered in the iL

min time interval an accidental count will be registered in the n*’*1 time interval, and.

conditional probability that if a count is registered in the m*11 time interval a coupled count will be registered in the nth time interval. de Hoffmann^ that  P (count in m) (count in n) + = (F€ At)2 + Ffc2 vlv-1) c-a (tg — 4) (At)2

. 2o Tf2

where

v = number of neutrons emitted per fission,

a = Rossi alpha = v (1 — Kp)/~f •

tg-tj = time between mm and nl time intervals.

Kp = promp’t neutron multiplication factor, and Tf = mean life of neutron for fission;

and Orndoff^ gives.    P (count in m) ^Рд (count in n) + Pf, (count in n) J

where

t = time between m**1 and n^ time intervals.

: ‘ r0 = mean life of neutron including absorption and leakage, and

о = Rossi alpha = (1 — Kp)/~0   But since

these two expressions are equivalent within a factor of Kp (which is essentially unity in a critical reactor). We will use Equation (3-87) and write, by combining Equations (3-85) with (3-8.7),

<VmVt>= F £ Д t [ F £ At + Aee’a (mAt ‘ n At) At] ,

for m > n

= F £ At [ F£ At + A£e-° (nAt" mAt)At] for m <n,  where

So, substitution of (3-84) and (3-89) into (3-83) yields t/At

<V2(t)> = ^ F £ At і/2 (t — m At)

. m = — 00 . t/At m

+ 2 2 [<F£At>2 + F

m = — °° n = -°°

x v (t — m At) v (t — n At)

t/At t/At _ , n

JT £ [(F£At)2 + F£2 Ae~a (n At — m At) (At)2 J x v (t.- m At) v (t — n At) . Note that we have dropped the requirement m *,n from the second and third summations; the argument for the validity of this step is given earlier in this section. Now let

 ■* 01 *. ш: ОДІ

 m,

 At

 At a = о

 (F t At)2 + Ft2 Ae"° (aAt ‘ bAt) (At)2

 * E E a = о b = о

 x v (a At), v (b At)

 00 oo r ■ . + ^ (Ft At)2 + Ft2 Ae _

 x v (a At) v (b At) , which becomes, as At — o,

 (3-92)

 = T Ft v2 (x) dxj r

 I (Ft)S (x) V (y) dy

 dx dx dx.

 (3-93)

 + Fe2AQe2Z122
 < V2(t)> c (WH+. ") (U’L+ a

Because of Equation (3-11), the second term in Equation (3-93) is equal to zero; hence 2

the expected value of V (t) is

 + F£A I e‘°’xv(x)

 + Fe2A I eaxv(x)      <r(t)> = Fc v (x) dx ‘o

 2 FeQe2zi22 = S—-

 “h*  substitution of (3-51) into.(3-94) yields

The product A a is  v{v-) Kp2

9-2 _ 2

2 " "О

A < V2(t)> = ev ^ ~ ^ KP2__________  <V2(t)>C і~2r02 (*H + «) (a;L+ a)

And finally, since

a ~ /3/т0 ,         where /3 is the delayed-neutron fraction, then

Equation (3-101) can be rearranged into the canonical form:  ■ ‘ У ‘ ‘ <!I(w)|2> = q2<; 11

7ГТ,

‘ ‘■ . ‘ ‘

(Equation (3-102) has been somewhat simplified by using the assumption that /3 » т’ X ) .

‘ f. ‘ ‘ ■ ■

But the efficiency c , in terms of counts per neutrons lost, is related to the efficiency e in-terms of counts per fission by «!• . .h.

So Equation (3-102) becomes     у

 n Q2 t Kp n - = ————- b- ■ ■ 1,7 ov

 і * JL  Since the low-frequency, break point of the system is at a frequency much greater than X, we will disregard that part of the spectrum below w = Л and write

which is equivalent to Equation (3-104) for u; > Л.

GEAP-4900.

 (3-106) and the ratio of reactor-noise contribution to the mean square voltage at the amplifier output to the "correct" value is given by (3-107)

 or

 A c

 6 „ (и — 1) Кр (Зг108)  Comparison of Equations (3-100) and (3-108) shows them to be equal within a fgctor of Kp.

## RANGE OF ELECTRONIC SUBSYSTEM

• — 1 fi ‘

The electronic subsystem has a range of 10 to 10 counts/sec true random input. Because of the random nature of the input signal, the pulse width of the incoming signal should be minimized in order to minimize counting loss due to random coincidences of pulses. With the in-core system, the pulse width — limited by rise time and collection time — is approximately 0. 3 psec. Thus, at a

О •

true random input of 1 x 10 cps there will be a counting loss of 23 percent of true count rate.

This is equivalent to 1. 63 percent of full-scale analog output current or voltage, and is well within the allowed tolerance for a seven-decade logarithmic instrument. An experimental determination of the counting loss for the in-core system is shown in Figure 6-13. The counting, loss of the ° out-of-core system is shown in Figures 6-14 and 6-15. These curves compare well with the theoretical loss calculated. Improvements in the speed of the subsystem are possible, and can be made where upper half-power frequency of the information pulse will allow it. .

## INTRODUCTION

The objective of Project Agreement 22 was to determine the feasibility of covering the

2 13

complete reactor neutron flux range from 1 x 10 to 5 x 10 nv in-core by using in-core or out-of-core ion chambers.

This report describes the analytical predictions and the experimental results of a nuclear instrumentation development program undertaken by General Electric to determine the feasibility of this approach. The program has been concerned with the development of a fujl-range reactor control system incorporating either in-core or out-of-core ion chambers, and the purpose of the program has been satisfied by the development of a full-range instrumentation system consisting of two electronics subsystems. Counting techniques are utilized for source and lower intermediate range neutron fluxes, and Campbell — or mean-square voltage — techniques are utilized for inter­mediate and power range fluxes.

Historically, power reactors have used out-of-core neutron detectors for source and inter­mediate range coverage during a plant startup. In present power reactors, an in-core neutron

4′

source producing about 3×10 nv is attenuated by the water annulus surrounding the core so that the incident flux at the detector is approximately 0. 5 to 3. 0 nv. Larger power plants, especially those with internal steam separation, provide for much larger water annuli and neutron source

_9

attenuations of perhaps 10 have been projected. Additionally, there is considerable interest in the in-core flux as opposed to the out-of-core flux, and consequently, it has become necessary to develop suitable in-core detectors for the source and intermediate range fluxes.

The development program has investigated the problems associated with the in-core counting detector, the in-core Campbelling detector, the in-core transmission lines, as well as those unique problems of a two-channel, full-range instrumentation system. In addition, the effort has been expanded to ensure a compatible system for either in-core or out-of-core applications. In this manner, flexibility of the developed hardware for a wide range of applications has been assured.

## IN-CORE CABLE DATA

Manufacturer: New England Electric Wire Corporation

Configuration: Triax; stainless steel center conductor, quartz insulation, stainless steel

guard, quartz insulation, double braid copper shield.

Preparation at Manufacturer: Starch and oil impregnated, heat cleaned.

NEPS Preparation: Baked out in vacuum at 700°F, back-filled with Argon at 200 psig.

## Performance in the Presence of Gamma Flux’

Tests have been performed in the GETR to determine the effect of gamma flux on the counter’s sensitivity. A bundle of three detectors was lowered into the Z trail cable with the reactor at full power. The detectors in the bundle were Counter No. 1, Campbellbr No. 3, and an unlined 1/4-inch — diameter ion chamber. The reactor was then scrammed and the outputs of the three detectors re­corded as a function of time.

The gamma flux at any time was determined by the reading of the unlined ion chamber. The neutron flux was determined by the reading of the Campbell channel using a neutron sensitivity of

О • • •

1. 89.x 10 nv/div, after correcting the reading for gamma contribution using a gamma sensitivity of 3.23 x 10 R hr /div. The counting rate was determined by the reading of Counter No. 1 after correcting for counting losses by using the data of Figure 6-13. ■

The results are shown in the following graphs. Figure 8-1 shows the gamma flux, Figure 8-2 shows the total reading of the Campbell (MSV) channel and those parts of this reading that are caused by gammas and by neutrons, Figure 8-3 shows the observed and corrected counting rates, and Figure 8-4 shows the calculated counting sensitivity. There is’ considerable spread in the count­ing sensitivity before 19 minutes, when the correction for counting loss is large, but after that time the correction is smaller and the spread in the counting sensitivity is quite small.

-3 -1 -1

The average counting sensitivity, after 18 minutes from scram, is 0. 67 x 10 sec nv in an

7 -1

average gamma flux of 2.2 x 10 R/h. This is to be compared with a counting sensitivity of 1. 52 x 10’^ sec”^ nvat gamma flux of about 10^ R/h

————————- • 5 ■

♦Assuming sources provide 10 nv to the detector at startup. ,

 4 x 107 TIME AFTER SCRAM (min.)

 TIME AFTER SCRAM (min.)

## SPECIFIC RESULTS: PERIOD INDICATION

Standard deviations and trip rates were calculated for the period indication (voltage pro­portional to growth rate) for a system consisting of eight diode pumps satisfying Equations (2-1) and (2-2), and for which Tg = 2 seconds, T2 = 80 seconds, and R^Cj = RgCg = 2 seconds.

The standard deviations at various average counting rates are given In Table 2-6.

STANDARD DEVIATION, ap, OF PERIOD INDICATION VERSUS COUNTING RATE Count Rate (sec-1)

0.5 5

5×10і 5xl02 5ХІ03 5X104

-2

Then, using the largest стр listed in Table 2-6 (4. 8×10 nepers/sec), the average rate at which the period Indication exceeds the true growth rate (1/P)T by an amount (1/P)E is shown in Table 2-7.

TABLE 2-7

RATE AT WHICH PERIOD INDICATION EXCEEDS TRUE GROWTH RATE, (1/P)T

SECTION III

## Test Conditions ‘

The test facility consists of dry thimbles in an operating reactor core. The total length of the thimbles is about 30 feet. Half of this length is outside the pressure vessel; of the remaining half, the last 5 feet is actually in the core. The cable temperature external to the pressure vessel is about 80°F. Inside the pressure vessel but not in the core, the cable temperature is about 550°F. In the core, the cable temperature due to gamma heating, as well as ambient temperature, is about 650°F. ■ .

The profile of both the gamma flux level and the neutron flux level is roughly a cosine curve

with a zero intercept 1 foot beyond the end of the thimbles and 5 feet along the thimble. The peak

12 12 neutron flux was about 5×10 nv during the first portion of the test and 7. 5 x 10 nv during the

latter portion of the test. •

5.4. 3 Test Data ‘

•Figure 5^11 is a plot of the d-c leakage versus applied potential across the cable insulation for various levels of nvt. The d-c leakage of the quartz did not increase with increasing nvt. This was illustrated by measurements-of d-c leakage at 100 volts applied potential for various nvt values (see Figure 5-12). On the other hand, the leakage currents through the S-glass insulation generally increased with increasing nvt.

## PROBABILITY OF FALSE TRIP FROM MEAN SQUARE VOLTAGE

If noise having a Gaussian amplitude distribution is passed through a nonlinear device (3uch as a squaring circuit), and if the resulting. voltage or current is passed through a relatively narrow bandpass filter, the statistical properties of the output voltage or current approximate those of a random noise voltage or currentThese conditions are satisfied by a Campbell system operating at high reactor fluxes, for then the average pulse spacing is much less than the pulse width, ensuring a Gaussian amplitude distribution at the squaring-circuit input; also, since t, the time constant of the averaging circuit after squaring, satisfies the relation (3-109)

T   the filter after squaring has a relatively narrow bandpass. Hence we can use the expression for the frequency with which a random-noise voltage crosses a given level. This rate of crossing is given^ approximately by

where

rT = rate, per second, at which signal crosses the value ST with positive slope,

. f2 = 1/(2 7T t ) .

<S> = expected value of signal

= Ak<*Qe 2g122 “H2

2 + WL

ST = trip level setting, and

dg2 = variance of signal,

, (Ak»)2 (q6Z12)4 ^h4

4T (“h* “l)3

Neglected in this analysis are the effects of internally generated noise within the subsystem or system components, and the susceptibility of these components to externally generated noise. These effects tend to increase the probability of false trips and, therefore, the results of this analysis will relate to a more stable subsystem or system than can generally be achieved in practice.

The Campbell subsystem covers its range with a linear scale in PERCENT POWER and a 10-position range switch. If the following operating conditions apply:

 Scale Reading (Percent Power) Equivalent Percent of Full Scale System Function Required 125 . 100 (For reference only) 120 96 Scram trip 115 92 Alarm indication 100 80 Maximum operating point

dnd if the level is allowed to reach 67 percent of the full-scale value, the false trip probability, r™,, will be as tabulated in Table 3-1 for the various ranges (the equivalent neutron flux is based

A. IQ ‘ О

on an assumed subsystem sensitivity of 8. 1 x 10 volts /nv). If the level is allowed to reach only 63 percent of the full-scale value, the false trip probability will be as tabulated in Table 3-2 for the various ranges. From the value of rT (Events/year) in Tables 3-1 and 3-2. it can be seen that the probability of false trip for either operating option is very small. A determination of the variation of r^, as a function of operating level for the most sensitive range and a trip setting of 96 percent of full scale has been made as follows:

 Operating Upper Trip Scale Point (Max) Point Reading rT (nv equiv.) (nv equiv.) (125 = full scale) (Events/year) 6.67X108 . lxlO9 80 . 2.89×10"136 8.33X108 lxlO9 100 2.89xl0"lba 8.75X108 lxlO9 105 2. 89X10"4- 6 9.17X108 lxlO9 110 2. 89xl0+3‘ 74 9. 58X108 О ^4 X гЦ 115 ‘ 2. 89xl0+5- 90

Thus, the "one false scram per year" yardstick occurs when the MMSVM is operated at an approximate scale reading of 108 on its most sensitive range. ‘    TABLE 3-2 OPTION 2 ANALYSIS

(<S>___ = 63 PERCENT OF FULL SCALE)

——— Шал — _

 Operating Range (nv eq’uiv.) Upper Trip Point (nv equiv. ) ,x-—— max . % ^S^max 10"10v2 (sT — ) Trip 10”10v2 as 10”12v2 (sT —  ## LINEARITY OF THE ELECTRONIC SUBSYSTEM

The electronic subsystem output does not deviate from the true random analog output current or voltage by more than a span of 10 percent of the equivalent linear full-scale output current or voltage over the top six decades, nor by more than 20 percent of the equivalent linear full-scale output current or voltage over the bottom decade, under the following conditions:

Temperature — 5 to 50°C, .

Humidity — 20 to 90 percent R. H.,

Line — 19 Vdc to 29. 5 Vdc.

6.8 TIME CONSTANTS — ANALOG CIRCUIT

The time constants of the analog outputs (level and period) of the electronic subsystem are established on the basis of a compromise between speed of response to changing input rates and fluctuations at the analog outputs. Fluctuations at the analog outputs are of special interest in the case where these outputs are coupled into reactor safety systems.

## SECTION I GENERAL

1. 1 SUMMARY OF PERFORMANCE ‘

Two prototype full-range instrumentation systems have been developed, each capable of covering about 10 decades of reactor power. ■ • ‘

• і ■■ ■ ‘ ■

235

The in-core system contains two detectors. The first of these is a U — lined fission

3 9 3

counter that covers the range 10 nv to 1.5 x 10 nv; at 10 nv the counting rate is 1. 5 cps, and 9 7

at 1. 5 x 10 nv (and 2.5×10 R/h) the counting loss due to the electronic subsystem is 23 percent. 235

The second is a U — lined fission chamber, operating in the MSV mode, that covers the range

О ‘ 1 О Q

2.2 x 10 nv to 5 x 10 nv; at 2.2 x 10 nv the neutron-produced signal equals that produced by 2.5 x 10 R/h of gammas, and at 5 x 10 J nv the signal is still proportional to flux (indicating that even higher fluxes could be measured).

235

The out-of-core system contains one detector, a U — lined fission counter-chamber. In

• C .

the counting mode, this chamber covers the range 1.4 nv to 1.4 x 10° nv; at 1. 4 nv, the counting

fi

rate is 1. 0 cps, and at 1. 4 x 10 nv the counting loss due to. the electronics subsystem is 23 percent.

3 10

In the MSV mode, this chamber covers the range 10 nv to 10 • nv; the lower limit may be some — ■

4

what greater than 10 nv, depending on the gamma field in the vicinity of the detector, and the higher limit may be greater than 10^ nv for the signal is still proportional to flux at that level.