When this system was designed, it was desired that the controlling frequency breaks should be those of the remote amplifier. For this reason, the frequency breaks of the local amplifier, inverter, and mean square analog were designed to be approximately a decade below and above the extremes set by the remote amplifier, or about 800 cps to 6 Me.

7.2.1 Selection of System Parameters,

The critical system parameters (viz., input impedance Z, lower and upper half-power points u>L and Mg, and averaging time-constant r) must be so chosen that the system performs satisfactorily with respect to the following properties:

a. Signal to noise ratio, .

b. Dynamic range of the electronics subsystem,

c. Standard deviation of the signal (steadiness of the reactor power indication),

d. Probability of false trip or level alarm,

e. Extraneous signal due to reactor noise, and

f. Transient response.

The choice of these parameters requires the use of Equations (3-51), (3-52), (3-55), (3-66), (3-100) or (3-108), and (3-110), together with cognizance of the following facts:

a. For the high range, the input impedance of the system, Z, is fixed to match the cable impedance. For the low range, increasing Z increases the value of the signal and improves the signal-to-noise ratio; but it also decreases for is determined by Z and the cable capacitance.

The half-power points, and co^, must be large enough to ensure satisfactory values of standard deviation of the signal, false trip rate, response to reactor noise, and error due to flux transients; at the same time they must be small enough to ensure sufficient pulse pile-up and hence a realizable dynamic range of the electronics subsystem.

c. The averaging time-constant, т, must be large enough to ensure satisfactory values of standard deviation of the signal, false trip rate, and error due to flux transients; at the same time it must be small enough to ensure sufficiently rapid response to flux transients.

The values of these system parameters that were chosen, and the resulting values of the system properties at various flux levels, are listed in Table 7-1.

For a signal composed of the linear superposition of randomly-arriving pulses, as the product of count rate and pulse width becomes large with respect to unity, the amplitude distri­bution of the signal approaches a normal distribution; also, the average number of times per

(2-22)

where fg is the frequency of the upper half-power point of the power spectral density of the signal.

For the system considered in this section, the pulse width is of the order of r^, . and the counting rate is of the order of 1/T^; so the product is т^/Т^, and this product is much greater than unity. Also, a single pulse is described approximately by.

(2-23)

so the upper cutoff frequency is, approximately,

The false trip rate is obtained by combining Equation (2-22) with Equation (2-24): .

rT. MO e-(sT-s)2/(2«s2) (2_.

. . rk.

Values of S and Og, for use in this equation, can be obtained by plotting S and <jg’ versus k, using Equations (2-15), (2-17), and (2-19), and then transforming the linear к scale of the plot to a logarithmic count-rate scale. Note that for this purpose the values of A and V do not have to be known, for they divide out in the exponent.

With a 42-foot length of this cable, a determination of the attenuation characteristics versus frequency was also made (see Figure 5-3). It is shown in this illustration that the cable starts to attenuate at approximately 5 Mc/sec, and exhibits a 3 dB attenuation at approximately 20 Mc/sec. The attenuation then falls with a continually increasing negative slope, and has an attenuation factor of 16 dB at a frequency of 100 Mc/sec.

. The transient pulse characteristics for this cable are shown in Figure 5-4. Note that the rise time of the pulse at the receiving end of this cable is 25 nsec (uncorrected), or 22 nsec. (corrected), for the same sending end pulse rise time of 7. 7 nsec (corrected). If this cable is considered as a single time constant system, the upper 3 dB frequency is calculated to be:

0, 35 = 0. 35

7 t 22 x 10"9

сл I 05

ги INPUT PULSE SWEEP SPEED: 0.2Msec/cm	INPUT PUI. SE SWEEP SPEED: 20nsec/cm

0.01 0.1 1.0 10.0 100 FREQUENCY IN MEGACYCLES/SEC

Figure 5-3. Gain Versus Frequency — Prototype Cable

,r INPUT PULSE

1 ; SWEEP SPEED: 0.2/zsec/cm

m INPUT PULSE

SWEEP SPEED: 20nsec./cm

The purpose of this appendix is to determine the effect of the scale-of-two on the variance of the count-rate indication. The method used consists of determining the distributed portion of the power spectral density of this indication and comparing it with the power spectral density of the indication that would be obtained without a scale-of-two. Since the variance is the integral of the power spectral density, a comparison of the variances can be obtained in this manner.

In this derivation it is assumed that the average counting rate is constant. The power. spectral density of a signal, S(t), is given by

. G(o;) = f ф(5) cos w 6d6 , (A-l)

ir J о

where ф(6) is the correlation function of S(t) and is defined as

ф(6) = ave [S (t) S(t+ б)] . (A-2)

Now the signal, S(t), can be considered as consisting of two parts: a d-c component generated by those diode pumps that are saturated, and a second part composed of a linear superposition of pulses from the unsaturated diode pumps. We will neglect the first part since it only contributes an impulse function at f = o, representing a d-c component of S(t).

The second part can be written as

co

S2(t)

where the kth pulse starts at tk and where v(t) describes a pulse that starts at t = o. This equation can be written in a form suitable for statistical analysis in the following manner. Divide the time axis into small equal intervals of length At; then the onset of a pulse during the nth interval will produce a signal at time t of v (t — n At), and the total signal due to all of the

A-l

previous pulses will be. T

S2^ = 2v^’nAt^’ (A-4)

n

where the summation includes only those intervals during which a pulse starts. If we define a random variable 77 , which equals one if a pulse starts during the 11th time«interval and equals zero if no pulse starts during the nin time interval, then the total signal can be written as

where the summation now includes all the time intervals that precede t.

Likewise,

. (t + 6 )/At.

S2(t + 6) = ^ ^ 77m v (t + б — m At) , ,r (A-6)

m = -00 ‘ • ‘

and hence

For a stationary random process (constant average counting rate in this case), the time average of a function is equal to the ensemble average, so we can write.

Ф(6) = .<S2(t). S2(t + 6)> , • (A-8)

where the brackets <> indicate expected value at time t, or ensemble average. Hence,

t/At (t + 6)/At IK

d{6) = y<

. n = -00 m = — .rl

since the average of a sum of random variables is equal to the sum of the averages.

A-2

It is convenient at this point to make a change of variables.

oO 00

We now must evaluate < n/ f V / t я >> anc* we use the fact that the

( —-a] ( —+ — — b] ‘

At J V At At j

expected value of a random variable is the sum of the products of each value the variable can as­sume and the probability of the variable assuming that value. Since 77 or any product of 77’s can assume only the values one or zero, and the zeros will not contribute to the expected value, we need consider only the value unity and the probability of assuming this value.

The first case to consider is — — a = — + — — b; i. e., both time intervals are one

and the same. A pulse will start in the ( — — a V*1 interval if a neutron is detected during this

. — v* /

■ interval and if the scale-of-two output is in its less positive state during the preceding interval. Hence, .

At At

for a = b——— . The first factor on the right-hand side of Equation (A-12) is the probability

At

that the scale-of-two output is in its less positive state, and the second factor is the probability of detecting a neutron during the/ — — aV*1 interval; к is the counting efficiency, Ф is the flux,

W )

and hence кФ. is the counting rate at the input of the scale-of-two.

GE АР — 4900 Г

If — — a precedes — + . —— b (і. e., if a > b — б / At), then it is necessary that a

• At. . 1’"’ At At ‘ ‘ ■■ ‘ ■

pulse start in the first of these two intervals, an odd number of neutron detections occur between them, and a neutron detection occur in the second of these two intervals. Hence,

for a>b — — . The first factor on the right-hand side of Equation (A-13) is the probability

. At ‘ ■ • .

that a pulse starts in the first of these two intervals, the second factor is the probability of an odd number of neutron detections occurring between them, and the third factor is the probability of detecting a neutron in the second of these two intervals. This can be written as

Likewise, if — + — — b precedes — — a (i. e., if a < b — б /At), then At At At.

— 2кФ

Substitution of Equations (A-12). (A-14), and (A-15) into (A-11) yields

0° ‘ • .

Ф(6) = 0, 5 к Ф At v (a At) v (a At t 6) 4

a = о.

-2кф/ь-а — — — lAt ^ ^ 0. 5 (кФ At)^ * ~ e———— ——— v (a At) v (b At) .

which becomes, as At — o,

This last Equation (A-20) can be rearranged to

ф(6) = 0. 5кФ ] v(x) v(x + 6) dx

J Л

(A-21)

The second term on the right-hand side of Equation (A-21) contributes to the power spectral density an impulse function at f = o. (It represents the d-c component of the pulses that make up S(t)). We will neglect this term and write the following as the correlation function of the a-c component of the signal:

(A-22)

The first term on the right-hand side of Equation (A-22) isv ‘ the cor relation, function of the a-c component of a signal composed of a linear superposition of pulses of the form v(t), arriving, randomly at an average rate of 0. 5кФ ; the second term, therefore, is the contribution to the spectrum resulting from the regularizing action of the scale-of-two.

To illustrate the magnitude of this effect, let

. v(t) = vQe’a;ot. (A-23)

and calculate: (a) the power spectral density of the signal obtained with the scale-of-two present, and (b) the power spectral density of the signal obtained without the scale-of-two present and with half the average counting rate. The first calculation is performed by combining Equations (A-l), (A-23), and (A-22); the second by combining Equations (A-l), (A-23), and the first term of (A-22). The results are: ‘ .

with the scale-of-two, present,

without the scale-of-two.

A comparison of these two equations shows: (a) Ga(u-‘) is one-half G^^’) between the frequencies zero and /2 к Ф, and (b) Ga(w) equals G^(<^’) between the frequencies 2кФ and infinity. Hence, the integral of G^u,’) is less than twice the integral of G (<*:)• and the variance obtained by neglecting the scale-of-two is too large by less than a factor of 2.

BY AN AMOUNT (1/P)E

Rate of Exceeding (1/P)T+ (1/P)E

0.050 nepers sec 0.060 nepers sec 0.100 nepers sec 0.125 nepers sec 0.150 nepers sec 0.175 nepers sec 0. 200 nepers sec

[2]

01) (a’L+ °’) (a’ H+ “l)

But the first term on the right, above, is the value of the signal without reactor noise, so the second term is the contribution made by the reactor noise. Hence, the reactor-noise contribution expressed as a fraction of the "correct” signal is the ratio of the second to the first term: i. e.,

[3] 40 feet, of 50-ohm cable, 125 feet of 75-ohm (RG-6) cable terminated in 5000-ohms,

• electronic system. frequency breakpoint at 5 and 9 kc, 100 volts applied.

[4] 40 feet of 75-ohm cable, 85 feet of 185-ohm. (RG-114) cable terminated in 5000 ohms, electronic — system. frequency breakpoints at 8 and 18 kc, 100 volts applied.-

[5] 85 feet ofr 185-ohm cable (RG-114) cable terminated in 5000 ohms, electronic system ■frequency breakpoints at 8 and 60 kc, 400 volts applied. ■

[6]The rise time of the measuring device was 11. 7 nsec.

[7]The sample chamber is chamber No. 17, which was later made a part of detector assembly No. 2.

[8]Also called a multi-range mean square voltage monitor (MMSVM).

Substitution of Equation (3-51) into Equation (3-14) yields the following expression for the expected value of the signal:

If both neutrons and gamma photons are being detected, this expression becomes

Note that the total signal is the sum of one part due to neutrons and one due to photons.

A measure of the gamma-discrimination ability of a system is the ratio of the flux of gammas to the flux of neutrons when both are producing the same output. This ratio can be obtained by equating the two terms in the right-hand side of Equation (3-53):

Ф

Ф

POWER SPECTRUM OF CURRENT FROM ION CHAMBER

SQUARE OF ABSOLUTE VALUE OF TRANSFER FUNCTION OF LINEAR PART OF SYSTEM

1010

102

ANGULAR FREQ, (radians/sec.)

GO • ‘

I

-J. Figure 3-2. Frequency Considerations

The basic requirements for subsystem sensitivity are derived from the system requirements on measurable neutron flux, detector locations possible, and types of detectors being used. The system requirements are covered in other sections, along with the detector locations possible.

6. 3. 1 In-Core System

235

In the in-core system, the counting chamber is a fission chamber in which the U is diffused into a base metal. The differential and integral spectra for neutrons from a Pu-Be source for this chamber are shown in Figures 6-2 and 6-3, respectively. [7] Note that the dis­tribution of pulse heights of the chamber shows a peak at small relative energies. The equivalent input charge is defined as that amount of charge which, when introduced into the input of the subsystem or device, will produce a given output level. The charge is introduced by applying a negative step voltage of V volts to the input of the device through a General Radio 10 pf standard air capacitor. The value of the charge is equal to the value of the step multiplied by 10-1*, the value of the capacitance.

0.001 0.01 fj 0.1 1.0 f2 10 100 FREQUENCY IN MEGACYCLES PER SECOND

o> і СЛ

а> о>
DISCRIMINATOR SETTING: 1.7 BASELINE: 0.00 GAIN: 1.00
6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 EQUIVALENT INPUT CHARGE IN COULOMBS x 10"14	36 38 40 42
Figure 6-2, Differential Bias Curve — Chamber No. 17

I -a

The integral pulse height curve for the sample chamber shows that with a system sensi — -14

tivity of 2 x 10 coulombs equivalent charge, the total number of counts per unit time is 28 counts/sec in a neutron flux of 1. 45 x 10^ nv.

To compare the measured output of a detector in the Campbell mode with the predicted value, experimental values were applied in the following expression:

where

к = detector counting efficiency (with discriminator setting of zero),

Zjg = mid-bandpass transfer impedance,

Q2 = [the product of charge of either sign per pulse and fraction of applied

field through which it falls | 2 average,

Ы1Н = uPPer break frequency of bandpass of system, and = lower break frequency of bandpass of system.

An in-core counter (No. 1 assembly) was chosen for comparison because (a) the load impedance seen by the chamber can be expressed simply, being 5000 ohms; (b) a pulse height spectrum for determination of Q could be obtained from the detector in its actual operating condition (the Camp­bell detectors, having low internal gas pressure, do not provide pulses of adequate amplitude). For this counter, the ion collection time (0. 56 x 10’^ sec) and the bandpass of the system ( 8 to 10 kHz) result in an equivalent fraction of applied field through which the charge falls, of 3/4.

The experimentally determined values are:

2

Q is determined in the following manner. On a differential pulse height spectrum, the charge in a given pulse, q^, is related to the indicated pulse height, Pj, by a constant C.

(4-25)

where N is total number of pulses in the counting interval, and n is the number of pulse in the interval of average height Pj. .

The constant, C, can be evaluated from d-c sensitivity of the chamber.

where is the d-c sensitivity of the chamber. Hence,

The maximum noise level (input open) referred to the input of the remote amplifier is З p V rms, and the minimum (input shorted) is 1. 6 pV rms. This means that to ensure a signal — to-noise ratio of 10 to 1 mean square, the input signal should be about 10 p V rms. In experiments with 2000 picofarads of detector and cable capacity attached to the input, the noise level is about 2. 2 p V rms.

7.3 RANGE

8 14

The MMSVM was designed to monitor flux over the six-decade range of 10 to 10 nv (in-core detector). Assuming a linear chamber, the electronics are capable of ±5 percent linearity operating over 6. 5 decades.

The variance of the period indication (the voltage proportional to growth rate) can be obtained by using Equation (2-14) with the expression for an individual pulse at the period-meter output,, or by integrating the power spectral density of the period indication. The latter method is used here because the power spectral density is also needed to calculate the false trip probability.

For a signal composed of the linear superposition of randomly-arriving pulses, the distributed portion of the power spectral density is given^ by

• so

G (W) = — Г!/■• (б) COS w 6 d б.’ (2-26)

77 Jo

where

‘ <jC. .

Ф(б) = 0. 5 г I v(t) v(t + 6)dt. (2-27)

‘ ‘ о,

In this last expression. 0. 5 r is the average arrival rate of the pulses and v(t) is the shape of one pulse that starts at t = o. But v(t) at the output of the log count-raie meter is given by Equation (2-13). Substitution of Equation (2-13) into Equations (2-27) and (2-26) gives the power spectral density there as

The transfer function of the rest of the system (period meter) is

where Rj, Rg, Cj. and Cg are identified in the schematic (Figure 2-2) of the period meter.

(2-30)

2 -1 in (volts) /(radian sec ) .

To normalize this expression, note that at a period of 1 second per neper the flux changes by a factor of 10 in 2. 303 seconds. This corresponds to a change of unity in к or. from Equation (2-17). a change of AV volts at the output of the log count-rale meter. Hence, at 1-second period.

dS = AV dt 2.303

At the input to the period meter, the rate of change of voltage is

AV H — — "A *

2. 303 _

and this produces a voltage at the output of the period meter of

— 1 “ — I

The power spectral density at the output of the period meter, in (nepers sec ) / (radian sec ), is obtained by dividing the right-hand side of Equation (2-30) by the square of Equation (2-31). The result is

(Equation (2-3a) was also used in obtaining this expression.)

The variance of the period indication is obtained by integrating Equation (2-32) over all frequencies; i. e.., ‘

For comparison, these cables show the following attenuation characteristics*:

RG-6A/U: Down approximately 3 dB for 100 feet at a frequency of 100 Mc/sec.

RG-114A/U: Down approximately 3 dB for 100 feet at a frequency of 100 Mc/sec.

5.2.4 The Effect of Temperature on Pulse-Height Attenuation and D-C Resistance

A variety of cables were tested for pulse-height attenuation at several temperatures. A description of the cables is given in Table 5-2, and the results of the tests are shown in Figure 5-5.

The main contributor to variation of attenuation with temperature is variation of the resistivity, д of the conductor. A plot of measured resistance of several materials, normalized to 20 mils diameter by 100 feet long, versus temperature is given in Figure 5-6.

Note that the theoretical data do not include calculations for the cables using nickel or nickel clad copper as conductors, because the attenuation equations are not valid for cables having conductors with magnetic properties.

Note, also, that the above procedure is valid only for cables in which the attenuation at f = 1/(2T) is not much greater than that at low frequencies. If these two attenuations are significantly different, the shape of the pulse changes in traveling along the cable and the method of calculating output pulse height would be more involved than the approximate method outlined above. The ratio of high-frequency to low-frequency attenuations is given by

where p is in ohm-meters and a, b, and x are in feet.

♦Catalog W2, American Phenolic Corporation.

TABLE 5-2

CABLE DESCRIPTION

СЛ. —————————- .

1 ‘

~ NOTE: Quartz fiber insulated cables have braided guards and shields of 4-mil wire.

0 TOO 200 300 400 500 600 700 800 TEMPERATURE (°F) ‘

Figure 5-6. D-C Resistance Variation With Temperature — Normalized to 20 Mils Diameter