Noise techniques, the class of reactor dynamics ex­periments in which no external excitation signal is used, are among the categories listed in Table 6 6, namely

1. Vanance-to-mean method (sometimes called the Feynman method)

2 Spectral analysis or the time-domain equivalent, autocorrelation.

3. Cross spectral analysis or the time-domain equivalent, cross correlation.

Other noise-analysis techniques (such as various kinds of probability analysis of individual pulses) are not sufficiently related to transfer functions to warrant discussion here, but they have been treated in Refs. 113, 113a, and 113b along with the three techniques cited above. In power reactors only the last two methods are used, whereas in zero-power reactors all three methods can be used.

Whether reactor dynamics are studied by introducing external excitation or by relying on the reactor’s intrinsic self-induced noise, the data-acquisition and data-processing hardware are almost the same. Noise methods, of course, process no signals from excitation equipment Table 6.11, having much in common with Table 6.10, shows the types of equipment used in the various noise-analysis experiments described briefly here. Most of the equipment is used for spectral and cross spectral analysis, and only that involving the gate scaler or ion-chamber current integrator is used for the variance-to-mean method

In the variance-to-mean method, the dynamic constants in the neutron kinetic equations can be determined by using a digital computer to give _

1 The variance of neutron-detector counts [c2 — (c) 2 ] taken many times over a time interval or “gate,” r.

2. The average count, c, during г

Results for various gate times30 can be shown to conform to the following equation

7

59eE Ъ Go(7j) V

J=1

Figure 6 8 shows an example of this relation. 0 001 0 01 0 1 1 10 GATE TIME sec

Fig. 6.8—Results of determinations of the variance-to-mean ratio for many counter gate times on the Ford reactor.1 13

The constants of Eq 6 21 appear in the zero-power transfer function (6 22) and are given in Table 6.12. By fitting Eq. 6.21 to data, as in Fig 6.8, you can evaluate the constants, especially 7i = (0 ~P)/1

Related to this method are others, such as the Mogil’ner method,33 based on the probability of no counts in a time interval, and a count interval-distribution method of Babala 32 These and many similar techniques of time-domain
analysis of neutron pulses in zero-power reactors have been extensively reviewed in Refs 113a and 113b

The constant є in Eq 6.21 is important in noise measurements of zero-power reactors It is the detector efficiency and is defined as

number of counts/sec (6 23)

number of reactor fissions/sec ‘

Evidently є is the probability of detecting an individual fission. Counters are located in or quite near the reactor core to obtain the values above about 10~s which are needed for a successful experiment. For very large reactors only a zone near the detector contributes neutrons and determines an effective efficiency.

Spectral analyses of ion chambers or fluctuations of other variables are usually accomplished experimentally in one of two ways (1) by passing the signal through a narrow band-pass filter tuned sequentially to the various desired frequencies or (2) by obtaining the autocorrelation function of the signal and then performing a Fourier analysis at the various desired frequencies. (Direct Fourier analysis of the signal is rarely done.) Table 6.11 indicates the various combinations of equipment that can be used to accomplish one or the other of these approaches

In spectral analysis of the signal from an ion chamber in a zero-power reactor, the shape of the transfer function, G0, is obtained directly from the measured P(f) of the detection of particles by using the relation11 3

— = 1 + 0.795 elG0(f)l2 (6 24)

eFo

where F0 is the number of reactor fissions per second Again є must exceed about lCf5 for successful experiments For the ion-chamber noise in a power reactor, the spectrum Pn(f) of n(t) in Fig 6 4 is measured, its fluctua­tions being induced by an internal noise source, kln(t) Evidently,

Pn(f)= lkln(f)l2 lG(f)l2 (6 25)

where G(f) is given by Eq 6 20 Thus ion chamber noise analysis in a power reactor gives information about both the transfer function and the input reactivity noise

Figure 6 9 shows typical results for power operation and how the spectrum differs from the zero-power spectrum Figure 6 10 indicates that large pressurized-water reactors of similar structure have similar noise spectrums

Fig. 6.9—Spectral density measurements of ion-chamber noise in the Hanford Test Reactor at powers of 1 watt (x), 5 watts (•), 500 watts (л), 5 kW (□), and 100 kW (o) 80

In the more informative power reactor experiments, two signals are observed simultaneously and correlated to obtain a transfer function between them One way to do this follows from Eqs 6.19 and 6 20 The terms x(t) and y(t) are any two system variables whose fluctuations are related After their cross correlation function has been measured, their cross spectrum can then be determined An accurate transfer function can be obtained from Eq. 6 20 if x(t) and y(t) depend primarily on the same noise-source excitation, і e, if they have a high coherence, Eq 6 8.

As indicated in Table 6 11, a cross-spectrum analyzer can be used directly, it is not necessary to determine the cross-correlation function first Equation 6.20 is still used to obtain a transfer function As in cross correlation, only two variables at a time are treated in multivariable systems In Table 6.6 a number of cross-correlation and cross­spectrum experiments in zero-power reactors are noted This approach has been used to measure a quantity proportional to just the second term of Eq 6 24 since the cross spectral density of detection events in two ion chambers is70,72

Pxy(f) = 0.7956162F0 lG0(f)l2 (6 26)

where є] and e2 are their efficiencies Although accuracy analysis70 indicates that G0(f) may theoretically be de­termined to the same precision from Eq 6 24 or 6 26 (assuming the same total detection volume, location, and data-collection time for the single detector and the pair of detectors), experimentalists have indicated preference for the method using two detectors

Fig. 6.10—Spectral-density measurements in three large pressurized-water reactors62 o, Yankee, 385 Mw(th) ‘, Indian Point, 290 MW(th) о N. S. Savannah, 59 MW(th). All show the usual reduction in frequency content at higher frequencies caused by intrinsic values of system time con­stants