®(t) = Ф + 0 (t) ,

where Ф is the time-averaged value and <j>{t) has a power spectral density of G(w), then the time-averaged value of the signal from an a-c system, using Equation (3-51) for v(t) and Equation (3-46) for S(t), is. •

Since the first term on the right is the output due to the time-averaged flux, the second term represents the error due to the fluctuations of the flux. The fractional error is the ratio of these two terms: .

3,2.5 Value of the Signal for Transient in the Flux

In this section the expected value of the signal will be calculated for the case in which v(t) is given by Equation (3-51) and for three different transients in the flux.

The procedure that will be followed is:

a. Substitution of v(t) f Equation (3-51) J and the appropriate expression for <b(t) into Equation (3-49) to obtain < AV2 (t)> ;

b. Simplification of this expression for <AV2(t)>; .

2

c. Substitution of the simplified expression for <AV (t)>and the proper initial condition into the differential Equation (3-2) of the integrating circuit to obtain the expected value

. of the signal. <S(t)>; and

d. Simplification of <S(t) >. .

Case 1. The flux has one constant value for t<o and a second constant value for t>o; i. e

for t < o, and

®(t) = Ф0 + Д Ф for t > o.

So Equation (3-61) becomes

А(кДФ Qe Z12r ш

(шн + u’i).
yields the expected value of the signal
<S(t)>
Шн + WL
А кдф Qe2 Z122
WH + a, L
А (кДФОе Z12)^ 2t
-t/r
Since the correct value is
Шн + a, L
the error is E(t)
А к ДФ Qe2Z’i22 ‘-h2
-t/r e
WH+WL
	-t/т
wi>,H + a, L>
2t
and the fractional error is
Ф(0 = Фг	for t < o, and
ФМ = Ф0 + St for t >o,
then <Ht) = St , d(t-x) = St-Sx. and the expected value of Av (t) is

А к(Ф0 + St) Qe2 Z122
<AV2(t)> =	H
— AkSQe2Z122
WH-WL

-2ш1}

This expression approaches, with a time-constant of about l/w^, the simpler one

2

H

/kSQeZj2

Ш.

and, if t >l/wL, this simpler expression may be substituted into Equation (3-2) along with the initial condition

to obtain the expected value of the signal

* (i.. -,л’

Since the correct value is

and the fractional error is

Since we have assumed that т » l/w.^, this may be simplified to

Case 3. The flux is constant for t < о and increases exponentially (constant period, P) for t >o; i. e., ‘

then <(>(t) = Ф0(е1/,р-1), 0(t-x) = (e^’x^P-l), and the expected value of AV2(t) is

JH 2Р(ШН + Ші) (2P WHWL + РшН + PwL + t/P

шн + wL . (2Po>H + 1)(2Pwl + l)(PwH + Po>L + 1)

This is really the simplified expression for <AV^(t)>. The actual expression approaches this simpler one with a time constant of about 1/cuSo, if | P| l/o^ and т 3> 1/w^, this simpler Equation (3-74) may be substituted into the differential Equation (3-2) along with the initial condition

.. 2

<S(o)> =

to obtain the expected value of the a-c system signal

where

2P2(wh+wl)(2P2whcjl + Pwh+Pojl + 1)

(P + t) (2PwH+l) (2Pwl+1) (Ршн+Ршь+1)’

and

Since the correct value is

<S(t)>c = Get/P,

the error is.

. E(t) = G(H-l)(et/P-e’t/T ) + J (e2t/P-e’t//T ),’

and the fractional error is

235

In the out-of-core system, the chamber is one in which the U is coated on a base metal. The differential and integral spectra for out-of-core detector No. 6 are shown in Figures 6-4 and 6-5, respectively.

6. 4 THE EFFECT OF IN-CORE CABLE ON THE NEUTRON SPECTRA

As described previously, the in-core cable attenuates the magnitude of the signal available at the chamber. The theoretical attenuation varies as the inverse exponential of the attenuation factor and the length of cable, and is given by Equation (5-7):

a

resistivity of conductor material, in ohmmeters, frequency in cycles per second, relative dielectric constant of insulation, inner radius of outer conductor, in feet, outer radius of inner conductor, in feet, . length of cable, in feet, magnitude of voltage at sending end, and magnitude of voltage at receiving end.

05

I

As a result of this attenuation, there is a reduction in the pulse amplitudes and an increase

in the pulse rise time. The differential and integral pulse height spectra for the chamber with

in-core cable attached (referred to as "detector assembly No. 2") are shown in Figures 6-6 and

6-7, respectively. (It should be noted that these spectra were obtained with a 40-foot length of

cable. Since the attenuation is also a function of cable length, additional in-core cable length will

result in additional signal attenuation.) The peak on the differential pulse height distribution has

-14

moved toward the noise portion of the spectrum, and is positioned at approximately 13 x 10

■ -14

coulombs equivalent charge. Without the in-core cable, it was positioned at 16. 6 x 10 coulombs equivalent charge. .

-14

The electronic subsystem has a sensitivity of 6. 5 x 10 coulombs at a discriminator setting of 2. 6, and therefore is counting the major portion of the pulses in the peak distribution in both cases. Figures 6-8 and 6-9 are integral pulse height spectra for detector assemblies No. 2 and No. 1, respectively, with the complete subsystem in place. Figure 6-10 is a plot which relates equivalent charge input required to trigger to discriminator dial setting.

In the out-of-core subsystem, regular coaxial cables are used. These cables typically have 3 dB frequency attenuations at several hundreds of megacycles. The out-of-core information pulse is therefore essentially unchanged in its frequency content by the out-of-core interconnecting cable. Integral bias curves for out-of-core chambers No. 6 and No. 7 are shown in Figures 6-11 and 6-12, respectively. As before, Figure 6-10 provides the relationships of discriminator dial setting to equivalent input charge. .

Tests were conducted to determine what effects temperature might have on the in-core detectors and cables. Detector assemblies 1 and 3 were used for these tests.

4. 5.1 Temperature Effect On Breakdown

The first test was to determine what effect temperature might have on breakdown. For this test, both detector assemblies were placed in an oven and then slowly heated to 700°F. The breakdown was monitored on an oscilloscope connected to the output of the pulse amplifier. Fig­ure 4-5 shows the results of this test. There appears to be no significant change in breakdown voltage over the temperature range tested.

Previous tests indicated that breakdown voltage would decrease as the temperature was raised. However, in these previous tests only part of the detector assembly had been heated, the result being that the density of gas in the heated portion decreased as predicted by the ideal gas law (Equation 4-27):

where

number of gas molecules in whole cable, number of gas molecules in heated section of cable, temperature of heated section of cable, temperature of unheated section of cable, volume of heated section of cable, and volume of unheated section of cable.

A decrease in gas density can also be achieved by lowering the pressure while maintaining constant temperature, and under such conditions it is known that the breakdown voltage decreases Therefore, it seems reasonable that breakdown voltage should decrease when an in-core detector assembly is heated nonuniformly, thus making the two experimental results consistent with each other and with theoretical predictions.

> 600

О

«С о

Z 400

О

О

• < ш СИ

со 200

0 100 200 300 403 500 600 700

TEMPERATURE IN °F

The experimental results of system linearity for an in-core and out-of-core fission chamber are shown in Figures 7-1 and 7-2, respectively. In both cases, the system exhibits ±5 percent linearity over about 5. 5 decades. For the in-core detector, there is no noticeable roll-off at the maximum flux tested so it is very possible that its range can be extended. The range of the out-of-core detector can possibly be extended a little by operation at a higher voltage.

TABLE 7-1

VALUES OF SYSTEM PARAMETERS AND
RESULTING SYSTEM PROPERTIES

NOTES:

(1) Count rate is based on a calculated counting sensitivity of 1. 6 x 10”4 sec”7 nv”7.

(2) The pile-up factor is given by 2 k ф/ (u>H + u>L) and is a measure of the ratio of pulse width

to average interval between pulses. "

(3) The fractional standard deviation is given by Equation (3-55). .

(4) The false trip rate is given by Equation (3-110). Tabulations of values of this property for various operating conditions are shown in subsection 3.4.

(5) The fraction of the signal contributed to by reactor noise is given by Equation (3-100) or (3-108). These values are based on the following assumptions:

(a) Mass of U23® in detector is 0. 6 x 10”® g,

(b) Mass of U235 in reactor is 2 x 10® g,

(c) So e. is 3 x Ю’10,

(d) v(v-)/v’1 is 0. 795, .

(e) Kp is unity,

(f) v is 0. C5 x 10”3 sec, and

(g) /3 is 0. 65 x Ю”2.

(6) The flux transient error tabulated is the value of the second term in Equation (3-66) for

a +100 percent step in flux. . . ■

SECTION VIII

The false trip raLe is obtained from Equation (2-22), rewritten for the conditions at the output of the period meter. The average number of times per second that the signal crosses a voltage (with positive slope) corresponding to a period PT is given by

(2-34)

where P is the average period being presented at the time considered, f. is the frequency of the

D 2

upper half-power point of the power spectral density of the signal (Equation (2-32)), and Up is the variance (Equation (2-33)). .

5.3.1 Cable Terminated In Zn.

For alternating current, the transmission line equation^

when the line is terminated in its characteristic impedance. Power transmission as a function of frequency is, then,

(5-Ю)

where о is a function of frequency. The term о is given by the formidable expression

(5-11)

However, an examination of this equation for the case under consideration often leads to con­siderable simplification. For example, a stainless-steel quartz-fibre cable. used for Campbell operation in the frequency range 0. 3 to 0. 6 Me /sec has the following parameters:

An examination of the terms in Equation (5-11). using these values, results in

Г2 : ■ u2 (2 " — » — ‘ .,2 24, 2

a.’ c ■ g

(wre — u.’2 Г c) . /:> rg

so that for this case, Equation (5-11) may be written as

(5-13)

2

Hence, if the power spectral density of current from the detector is Gj(w) amp sec and the frequency response function of the amplifier fed by the cable is Y (ju>), then the power spectral density of voltage at the output of the cable is

2

volts sec, in the range of u> over which the approximation (12) is valid. The power spectral density of voltage at the output of the amplifier is

? — x }ТГш7с~ІЛуТ, ,2

G3(w) = Z^GjHe 1 I Y(jw) I (5-і!

o

volts sec, and the mean-square voltage at the amplifier output is

No range of validity of и was considered in Equations (5 = 15) and (Б-16) because it is assumed that! Y(J u.)!1 falls off rapidly, in this case, below 0. 3 Me/sec and above 0. 6 Me/sec.

= pulse rate,

voltage at amplifier output, and

at t = o, at amplifier

So the problem is to determine the pulse shape, v (t). at the amplifier output.

The conditions under which an improperly terminated cable will be used are: •

a. Relatively lossy cable with characteristic impedance of about 75 ohms, .

. b. Terminating resistance of 5000 ohms, and. .

c. Amplifier with bandpass of 8 kc/sec to 60 kc/sec.

Photographs were taken at the input and output of an amplifier having an input impedance of 5000 ohms and a bandpass of 10 kc/sec to 60 kc/sec. It was driven by either 200 feet of RG-59/U or 100 feet of. nickel-clad copper, quartz insulated. cable; the cable, in turn, was driven by 5 x 10 sec current pulses. The results of these tests are shown in Figures 5-7 through 5-10.

Figure 5-7* shows the amplifier input when driven by the low-loss RG-59/U cable. It

has two distinguishing features or components: (a) the original pulse and its reflections, and

(b) an exponentially decaying tail. The first component contains a considerable amount of energy,

but this energy is nearly uniformly distributed among frequencies from zero to 1/T, where T

is the width of one pulse, so very little energy lies in the narrow pass band of the amplifier. The

second. component decays with a time-constant of RC. where R is the input impedance of the

amplifier and C is the total capacitance of the cable: furthermore, if it is extrapolated back to

zero its amplitude is Q /С. where Q is the charge in the pulse. Hence, for this case, the pulse • ’ P — P • ‘

at the amplifier output is. essentially due to. an input pulse of the form

%e-t/RC.

C ’

♦See footnote at the bottom of Page 5^19.

5-16 . .

The amplifier output is shown in Figure 5-8. Note that the original pulse and its reflections are highly suppressed and that the’energy lies mainly in the slower components, as predicted in the above paragraph. ■

Figure 5-9* shows the amplifier input when driven by the high-loss nickel-clad copper, quartz insulated cable. The original pulse and its reflections contain very little energy, again distributed over a wide frequency band, so a negligible amount lies in the narrow pass band of the amplifier. The slow component decays with a time-constant of RC, and, extrapolated back to zero, has an amplitude of Qp/C. Hence, for this case, the assumption that the pulse at the amplifier output is essentially due to an input pulse of the form

S e — t/RC

C. .

is an even better approximation than for the case of low-loss cable drive. The amplifier output is shown in Figure 5-10. Note that it has practically no fine structure due to the original pulse and its reflections. . . ‘

Since the signal from a Campbell system is due to noise caused by the random nature of the de­tection process, it is reasonable to expect that reactor noise, could increase the size of the signal. This additional output is not an error, for it is shown on the following pages that it, too, is propor­tional to the reactor flux(except for some very low-frequency noises caused by mechanical disturban­ces; these very low-frequency noises are filtered out by the relatively high-frequency bandpass of the amplifier). The magnitude of the additional, flux-proportional signal, caused by reactor noise, is also very small due to low detection efficiency and the low frequency of the reactor noise coupled with the relatively high-frequency bandpass of the amplifier. The magnitude of additional signal caused by reactor noise is obtained in two ways.

The maximum sensitivity of the electronic subsystem is limited by amplifier and other noise

present at the input. For discriminator settings in which there is less than 0.1 cps due to noise,

-14

the electronic subsystem maximum sensitivity is approximately 5.2×10 coulomb equivalent

-14

charge at the input for the in-core sybsystem, and approximately 7×10 coulomb equivalent charge at the input for the out-of-core sybsystem with detector No. 6 (sensitivity ~ 0. 7 cps/nv).

0 4 8 12 16 20 24 28 32 36 40 44 48 EQUIVALENT INPUT CHARGE IN COULOMBS x 10“14

o> I СЛ

О 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

DISCRIMINATOR DIAL SETTING (TURNS)

Figure 6-10. Input Charge Required to Trigger Versus Discriminator Dial Setting

The second test was to determine what effect long-term temperature exposure might have on the neutron pulse height spectrum. For this test the detector and about 5 feet of the attached in-core cable were heated to about 700°F for approximately 300 hours. Curves of the chamber’s neutron pulse height spectrum were recorded at ambient and 700°F at the start and finish of the 300-hour test. These curves are shown in Figures 4-6, 4-7, 4-8, and 4-9. It can be noted that there is no visibly significant difference between the curves taken at the start of the tempera­ture run and those taken at the end.

This test brought to light another temperature effect. By comparing Figures 4-6, 4-8, and 4-10 it can be seen that the pulse height spectrum taken at 710°F and a polarizing potential of 250 volts (Figure 4-8) lies between the pulse height spectrum taken at 75°F and polarizing potentials of 300 (Figure 4-10) and 400 volts (Figure 4-6). This increase in pulse height with increased temperature is probably caused by either an increase in electron mobility or a decrease in recombination rate, or both.

Although the change appears to be of no great significance, the increase of pulse height from the chamber might well compensate for the increase in cable attenuation as the temperature is raised.

8.1 IN-CORE COUNTING SYSTEM

3 9

The range of the in-core counting system is from approximately 10 nv to 10 nv (see Figure 1-1 in Section I), Since the counter has a zero gamma sensitivity of approximately

О

1 x 10 counts/sec per nv. 100 counts/sec are available at a zero gamma startup,* which pro­vides excellent statistics and a three-decade upscale indication. Pulse overlap becomes large

fi ‘

above 10 counts/sec, thus limiting the range of the system at this level. The detector can be

retracted at this point to lower fluxes to continue power coverage, but since adequate overlap with the Campbell channel exists, this is not necessary. An integral bias curve for the system at zero gamma is shown in Figure 6-8 of Section VT.