## Value of the Signal for Transient in the Flux

If the flux is constant for t < о and differs from this constant value by 0(t) for t> o; i. e.,

 ф(0 = фо for t<0 ф(0 = ф0 + 0(t) for t>o, (3-47)

then the expected value of the signal from an a-c system is, from Equations (3-38) and (3-47),

/• со    <S(t> = АкФ0 I v2(x) dx •’o

At the input to the integrating circuit the expected value of voltage is, from Equations (3-39) and (3-47), ■ ■ ф (t-x)v2(x)dx

(3-49)

The application of Equation (3-48) to an actual case is usually quite difficult, leading to a large quantity of algebraic manipulation during which mistakes are very probable. An alternative method, that lends itself to some approximation and considerable simplification, is as follows.

Equation (3-49) is used to obtain the expected value of voltage at the input to the integrating circuit. This expression is replaced by a simpler one that should give essentially the same output; a knowledge of the properties of integrating circuits is useful here. This simpler expres­sion is then vised as the driving function in the differential Equation (3-2) of the integrating circuit to obtain the expected value of the signal, <S(t)> . Examples of this procedure are included in Subsection 3. 2. If an exact answer is desired, use either Equation (3-48), or Equations (3-49) and (3-2) with no approximations.

## CHARACTERISTICS OF THE ELECTRONIC SUBSYSTEM

5. 2. 1 Required Bandwidth

The required bandwidth of the subsystem is dependent upon the bandwidth of the information to be processed. The shape of the information pulse that is available from both the in-core and the out-of-core detectors has been described previously. The expression given was:

 І = 0 d ( 1 — — amperes, T T / where Po = charge of either sign/unit length of track in the gas 1 — oul | , cm / d = gas gap (cm), T = transit time of the electron or positive ion across the gas gap (seconds), and t = time after pulse initiation (seconds).

In the above expression, the time required to accelerate the ion pairs to their average drift velocity is assumed to be negligible. This information pulse is applied to the interconnecting cable for transmission to the remote amplifier input.

In the out-of-core subsystem, the interconnecting cable is modified RG-6A/U coaxial cable. Because of the almost ideal transmission characteristics of the RG-6A/U cable (attenuation/100 ft = 2. 9 dB at 100 Me), the information pulse is transmitted to the amplifier input essentially unchanged in shape. For the in-core subsystem, the interconnecting cable is 40 feet of stainless steel — quartz — stainless steel prototype in-core cable in addition to the RG-6A/U mentioned above. The transmission characteristics of the prototype in-core cable are far from ideal. * The attenuation

For an experimental comparison of these cables, see Section V.

per 100 feet of the prototype in-core cable is, typically, 6 dB/100 ft at 22 Me. This high frequency attenuation causes an increase in the rise time of the information pulse from on the order of 10 nsec to 0. 15 psec. The upper and lower frequency breakpoints of the electronic subsystem are made to conform to the requirements of this information pulse.

## Campbeller

Though the spurious signals arising from gamma and alpha pulses are not rejected by

-/

amplitude discrimination, they are effectively suppressed, since the response is proportional to

the square of the pulse amplitude. However, the MSV gamma signal does exceed the electronic

noise level, and is the predominant source of spurious signal in the system at high gamma levels.

Shown in Figure 4-4 is the gamma contribution to the MSV signal level at a measured gamma flux

level of 3 x 10° R/h and the projected level at 2.5 x 10 R/h. Also shown in this figure is the

-3 -1 -1

counting channel output at a counting sensitivity of 0. 67 x 10 sec nv.   Figure 4-4. MMSVM Neutron Flux Threshold

## Local Amplifier

The local amplifier serves three functions:

a. It provides a characteristic impedance termination for the signal cable from the remote amplifier,

b. It provides additional gain for driving the squaring circuit, and

c. It contains the attenuation networks necessary to divide the six decades of flux coverage into twelve ranges. Its gain is nominally 50, but it is adjustable over a ±30 percent range from its nominal value.

7. 1. 3 Inverter

The inverter is a unity gain amplifier whose function is to receive the signal from the local amplifier and provide two outputs to the mean square analog, one output 180 degrees out of phase with the other.

7.1. 4 Mean Square Analog

The mean square analog is a piece-wise linear circuit whose output current varies as the square of the input voltage. By passing the current through a 10, 000-ohm resistor, a voltage is obtained that is fed to a unity gain d-c amplifier which provides the readout signal and operates the trips. A capacitor in parallel with the IQ, 000-ohm resistor controls the averaging time constant of the system. .

## COUNTING STATISTICS

This section contains a discussion on some of the properties of a counting channel com­prising a detector, logarithmic count-rate meter, and period meter. The properties considered are:

a. Variance of the count-rate indication,

‘ b. Probability of false trip from the count-rate indication,

c. Variance of the period indication, and

d. Probability of false trip from the period indication.

The derivations are based on the assumption of constant average counting rate (infinite period), but the results should be valid for finite periods greater than the longest time-constant in the system.

2. 1 VARIANCE OF THE COUNT-RATE INDICATION

The logarithmic count-rate meter considered is of the multiple-diode pump type (Cooke^ Yarborough), of which one section is delineated in Figure 2-1, and in which R is the same for Figure 2-1. One Section of Multiple-Diode Pump Type of Log Count-Rate Meter

each section, C^/C ^ is the same for each section, and of any section is — Jq ^f °* preceding section. That is,

(Rcf)k = 10(RC()(k+D

and (RCT)k — 10(RCT)(k,

where к is the section number and starts at 1 for the section with the lowest frequency break­point. All sections of the multiple pump are driven from the output of a scale-of-two circuit.

Let the average counting rate, r, at the input of the scale-of-two be such that the output of the k— diode pump is one-half of its saturation value; i. e.,

 r 2

 (2-3)

 (RCt)i

Then individual pulses from the (k-1)-^, k—, and (k +-l)^ pumps are

 (2-4)

 ‘(к — 1)  (2-5) and (2-6)

 ‘(k+’l) respectively, where Tk ^ ^RCf]k, тк = (RCjk , and V is the voltage swing of the output of the scale-of-two. To obtain these expressions, note that the charge delivered to C, each time the scale-of-two output swings in the positive direction, is

VT

R (1 + 0. 5 rT)

and this charge leaks off through R with a timerconstant, r, hence,  » . VT—e-Vr.

t (1 + 0.5 ГТ) ‘

By combining Equation (2-7) with the relations,

. 0-5 r T(k_ 1} » 1 ,

0. 5 r Tk. = 1 ,

, . 0 5 rT(k+ 1) <<1 ’

Equations (2-4), (2-5), and (2-6) are obtained.  Now rewrite Equations (2-4), (2-5), and (2-6) as

An individual pulse from the log count-rate meter is a fraction, A, of the sum of these three pulses: AvfeV (2-13)

(Actually, the total pulse is a fraction of the sum of the outputs of all of the individual sections; however, the terms below (к — 1) and above (k+ 1) contribute an insignificant amount of energy to the total pulse and can be neglected. )

2

Now the variance-, a„ , of a voltage, S, that consists of a linear superposition of randomly •Э (11     arriving pulses of the form v (t) and an average arrival rate of 0. 5 r is given by’ ‘    (Equation (2-14) is true for pulses whose arrival satisfies Poisson’s distribution; pulses from a scale-of-two do not. However, it is shown in Appendix A that the variance obtained by the use of Equation (2-14) is in error by less than a factor of 2 and is on the large side.) Substitution of Equation (2-13) into Equation (2-14) yields the variance,   and this can be combined with Equation (2-3a) to obtain

The average voltage from one diode pump is given by

0. 5 r VTt 1+ 0. 5 r Tt ’  It should be remembered that the manner in which Equation (2-18) was obtained imposes the following restrictions:

a. It is valid when the counting rate is such that the k— pump is half saturated,

b. It is approximately valid for noninteger values of k,

c. It is too large by a factor less than /2 because of the slight regularity in the output of the scale-of-two, and

d. It is invalid for к = 1 or к = N (where N is the number of pumps in the circuit).

For к = N (high count rate, N— pump half saturated, all other pumps saturated), the   standard deviation is

The average output voltage is S    and the fractional standard deviation is

This expression is subject to restriction c. of the preceding paragraph.

## Frequency Response and Transient Pulse Characteristics of Two Types of In-Core Cable

The characteristics of the two cables are compared with each other, and with some repre­sentative types of standard coaxial cabies.

5. 2. 3.1 Nickel-Clad Copper — Quartz — Stainless Steel Cable

With a 21-rfoot length of this cable, a determination of the attenuation characteristics versus frequency was made (see Figure 5-1). It is shown in this illustration that the cable starts to attenuate at approximately 200 kc/sec, and exhibits a 3 dB attenuation at a frequency of approxi­mately 1. 4 Mc/sec. The attenuation then falls with a continually increasing negative slope, and has an attenuation factor of 57 dB at a frequency of 100 Mc/sec.

The transient pulse characteristics of this cable were investigated by inserting a pulse with a 1 msec width at the sending end of the cable, and determining the rise time of the pulse at the receiving bnd of the cable. Figure 5-2 shows these response characteristics. Note that the rise time of the pulse at the receiving end of this cable is 0. 312 psec, with a sending end pulse rise time of 14 nsec (uncorrected) or 7. 7 nsec (corrected). If the cable is considered a single time constant system, the upper 3 dB frequency is determined as

f = = Ml x 10+6 = L із Mc/SeC. .

7 r 031

## OUT-OF-CORE CAMPBELL SYSTEM

The range of the out-of-core Campbell system is < 1 x 10^ to 1 x 101^ nv (see Figure 8-21).

Gamma may limit the minimum measurable neutron flux in some applications. The detector can

be used to 300°C, and is limited because it is made with aluminum. A titanium detector of

similar design could be used; it would operate up to 1000°F (tested) and beyond (untested). The 19

specified life is 10 nvt. The same Campbell channel electronics is used as in the in-core system, with the exception of the polarizing voltage supply (the larger detector requires higher voltage).

■v I

 VT*

APPENDIX

## SPECIFIC RESULTS

To apply the above general results, it is necessary to obtain an expression for the pulse generated at the amplifier uulput by the detection of a single neutron, and this can be derived from the energy spectral density of the pulse or from a knowledge of the time constants of the circuit.

For example, consider a system whose linear portion (cable plus amplifier) has a single low-frequency break at шa single high-freauency break at and a mid-frequency

transfer impedance of Z^, and whose detector is a neutron-sensitive ion chamber. The energy

spectral density of a current pulse from the detector is flat and eaual to Qe / я between the angular frequencies 2 я/Tj and я/Те, where Qg is the product of the fraction of the chamber potential through which the electrons travel and the total charge of one sign released in the cham­ber by the detection of one neutron, T is the ion collection time, and is the electron collec­tion time. So if ^>2я/Т4 and и:ц<тт/Те (see Figure 3-2), the energy spectral density of v(t) is the product of Qe2/W and the square of the absolute value of the transfer function of the linear portion of the system:

Qe2zi22 “2/wL2

E(u>, v) = , (3-50)   and the pulse itself is given by

This expression for v(t) can be used in any of the preceding equations.

## Theoretical Bandwidth

The bandwidth of the electronic subsystem is a function of the circuit components, and is defined in the following manner: The lower half-power frequency is defined as the lower fre­quency at which the current gain of the subsystem is down to 70. 7 percent of the midband value; the upper half-power frequency is defined as the upper frequency at which the current gain of the subsystem is down to 70. 7 percent of the midband value. The bandwidth is defined as the difference of the upper half-power frequency and the lower half-power frequency. The lower half-power frequency is determined by the equation:

2* RL CC

where

f j = lower half-power frequency in cps Rl = load resistance of a stage in ohms, and

These components are the same components as those which provide the appropriate RC clipping. The minimum RC clipping time that can be used is determined by the characteristics of the leading edge of the information pulse. In general, in order to restrict the losses of pulse height in an RC clipping circuit to the order of 25 percent of the peak input pulse or less,, the clipping time must be at least ten times the rise time of the information pulse. As described earlier, the rise time of the slowest information pulse (the pulse from the in-core detector assembly) is approximately 0. 15 psec. It is therefore necessary that the clipping time be not less than 1. 5 psec to keep the loss in height to 25 percent. A clipping time of 2 psec was selected. This results in a calculated lower half-power frequency of approximately 21 kc/sec in the first stage of the remote amplifier and in the first stage of the discriminator.

The upper half-power frequency is determined by the high-frequency components or characteristics of the circuitry. These components and characteristics are, in general, the parallel interstage resistance, the shunt capacitance, and the frequency characteristics of the active devices used. On the basis of the parallel resistance and shunt capacitance, including the stray capacity, the upper half-power point will be in the region of 15 Mc/sec to 30 Mc/sec.

Considering a single stage with the type of transistor used in the amplifier stage of the

subsystem, the upper half-power frequency due to beta fall-off will be 7 Mc/sec minimum.

With five similar stages, the upper half-power point will be reduced by the additional breaks, and

(71

as described in Petit and McWhorter, ‘ ’ the new upper half-power frequency f’2 is given by

 f2 — 0. 833 f2 yflT where f2 = single-stage upper power half-frequency, and n = number of identical stages, and f2 “ (0.833) (7) = 2 5 Mc/sec 2. 29 6.2.3 Actual Measured Bandwidth An experimental determination of the frequency response characteristics of the amplifier is shown in Figure 6-1. This response curve indicates an experimental lower half-power fre­quency of 35 kc/sec and an experimental upper half-power frequency of 3. 1 Mc/sec.

## EXPERIMENTAL OUTPUT LEVELS —

4. 4.1 Measured Sensitivities

The following sensitivities are obtained directly from measured signal and flux levels. The units for d-c sensitivity are the usual. The counting sensitivity is given for three values of gamma flux background.

a. D-C Sensitivity D-C Sensitivity (Neutron) D-C Sensitivity (Gamma)

 Counter 1. 9 x 10 a = 2. 9 x 10"16 a/nv 9. 4 x 10" a ■ ■ = 3. 2 x 10"13 a/R/h No. 1 6. 6 x 109 nv 2. 9 x 106 R/h Counter 1. 9 x 10"6 a = 2. 9 x 10"16 a/nv 1. 2 x 10"6 a = 4. 1 x 10"13 a/R/h No. 2 6. 6 x 10‘9nv 2. 9 x 106 R/h

Counter 1- 00 x 104 cps = L52xl0’3 1.22 >: 106 cps = 0. 68 x 10’3

No — 1 6.6 x 106 nv 1.80 > 109 nv

Discriminator setting = 3.5 (arbitrary scale), 40 feet of 50-ohm in-core cable, 85 feet of standard 75-ohm cable (RG-6A/U) terminated in 75 ohms.

 -18

 -18*

 -18 

 •19*

 -18*

 -19**   c. Mean Square Outputs of Detector Assemblies  Out-Of — Core Chamber NA04

NOTE 1. The mean square output/cycle below breakpoint

2

due to ionic mobility in the < V > distribution curve of the ion produced signal is four-times that above the break­point. All sensitivities quoted are below this breakpoint.

NOTE 2. The fraction of the applied field through which the charge falls is about one-half if only electron motion is contributing to the signal; i. e., if the bandpass of the circuit lies above the breakpoint due to ion transit time in the power spectral density curve. It is unity if both electron and ion motion contribute to the signal; i. e., if the bandpass of the circuit lies below the breakpoint due to ion transit time in the power spectral density curve. . 