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The envelopes of the power spectra of the PRBS, n sequence, and PRTS are shown in Fig. 3.11. All of the envelopes have [(sinx)/x]2 shape. The results shown in Fig. 3.11 are for signals with the same amplitude and the same value of Z. The envelopes for the n sequence and PRTS are higher than the
Normalized Harmonic Number (k/N) Fig. 3.11. Power spectrum envelopes for PRBS, PRTS, and лsequence signals. 
envelope for the PRBS because the even harmonics are zero in their spectra. This leaves more power to be distributed among the nonzero harmonics. The envelope for the PRTS signal is lower than for the n sequence because approximately one third of the inputs with a PRTS have a zero value, giving no contribution to the total signal power.
It is important to use the simplest available method to induce the desired controlrod motion for a test. Safety considerations also influence the selection of a procedure. In general, there are three possibilities.
(a) OpenLoop Rod Motion Control
In this case, the rod moves as long as the drive signal is supplied. Figure 7.1a shows the openloop control system. The motion may be continuous or stepwise. A binary pulse chain (or a finiterisetime approximation to it) can be introduced by actuating the insert and withdraw commands for a predetermined length of time. Also, it may be necessary to use different command durations for insert and withdraw signals because of gravitational effects on rod coasting characteristics.
(c)
Nuclear power has become a major factor in supplying the electric energy needs of the world. The scientists and engineers who brought nuclear power from laboratory discovery to commercial applications relied heavily on theorectical analysis of system performance. One reason for this is that the learnbyexperience approach involving numerous pilot plants is costly and potentially dangerous with nuclear reactors. An aspect of plant performance that has been analyzed extensively is the dynamic and safety behavior of the system. The results of these studies have had a strong influence on plant designs and on operating policies.
Now that a number of plants are going into operation, the opportunity to check the predictions is available. Some tests have been performed, but there has been too little emphasis on verification of theoretical reactor dynamics calculations by appropriate tests and only a small part of the available information has been extracted from most of the tests that have been performed. Procedures are available for performing dynamics tests on power reactors with very small cost and with insignificant interference to normal operation. Also, methods for interpreting the results to provide a great deal of information on system characteristics are available. The use of existing test procedures and interpretation methods along with the development of more advanced methods are in the best interest of safe and efficient power reactor operation.
The specific measurement emphasized in this monograph is the determination of the system frequency response using nonsinusoidal input perturbations. Nonsinusoidal perturbations are useful because normal power reactor hardware (such as control rods) can be used. Frequency response results are convenient because the interpretation is simple, correlation with theoretical results is straightforward, and the results can be used directly in control system design or modification.
A great deal of work has gone into the development of procedures to design, implement, analyze and interpret such tests. The newcomer is likely to be overwhelmed if he tries to gather and digest the reports and papers on this subject. This monograph is intended to collect and cull the required information.
The analysis required for these tests involves numerical Fourier transformations using analog or digital equipment. Fourier analysis is basically a very simple process, but users have shown great ingenuity in finding ways to do it incorrectly. Much of the material in this book is intended to steer the analyst around the multitude of traps.
The contributions of my colleagues and students are gratefully acknowledged. In particular, S. J. Ball of Oak Ridge National Laboratory and J. C. Robinson of the University of Tennessee are thanked. Also the cooperation of Duke Power Company, Babcock and Wilcox Company, and Carolina Power and Light Company in the application of the procedures described in this monograph is gratefully acknowledged. Wes Kerlin and Randy Pasqua are thanked for suffering through the preparation of the illustrations.
An algorithm called the fast Fourier transform (FFT) has been developed for computing the DFT on digital computers. For a signal that has been sampled to give N evenly spaced samples, the Nyquist frequency c% is given by
coN = л/At = nN/T (4.6.1)
This corresponds to harmonic number к = N/2. Thus, if the signal energy at frequencies above the Nyquist frequency is negligible, then the Fourier transform at these N/2 frequencies gives a complete spectral representation of the signal. The complete Fourier analysis at N/2 frequencies to give all N Fourier coefficients (real and imaginary at each frequency) requires N2 multiplications when Eq. (4.3.5) is implemented directly.
The FFT algorithm is an efficient method that can greatly reduce the required number of arithmetic operations if the total number of data points can be restricted to some integer power of two (N = 2"). It has been found that the FFT algorithm requires 2N log2 N arithmetic operations to evaluate the N Fourier coefficients. The ratio of the number of arithmetic operations for the FFT to the number for a conventional DFT is
(2 log2 N)/N (4.6.2)
This represents a substantial savings for larger values of N. For example, a data record containing 32,768 samples would take 1260 times as long to Fourier analyze using the conventional method as compared to the FFT. This makes it economically feasible to analyze signals containing many more data points than was possible prior to implementation of the FFT.
The principle underlying the FFT may be explained quite simply using a development of Cochran et al. (7). We begin by dividing the set of N samples Xp into two sets of N/2 samples each. The first set, Yp, is composed of all evennumbered samples and the second set, Zp, is composed of all odd — numbered samples. Thus
Yp = X2p, Zp = X2p+l, 0 < p < (N/2) — 1 (4.6.3)
The desired Fourier transform is
Ak = Y Xp exp( — 2njkp/N) (4.6.4)
p = о
It is possible to develop a formula for Ak in terms of the Fourier transform of Yp and Zp. This leads to the economy of the FFT algorithm. The Fourier
t See the literature (79).
transforms of Yp and Zp are
(N12) 1
Bk = X Ypexp(4njkp/N)
P = 0 (N12> 1
Ck = X Zpexp(4njkp/N)
P = o
The Fourier transform of the total signal may be written in terms of the odd — and evennumbered points as follows:
Inspection of Eqs. (4.6.5) and (4.6.6) reveals the following:
Bk = Bk+(N/2) (4.6.9)
Ck = Ck + iNi2) (4.6.10)
Thus we obtain
Equations (4.6.8) and (4.6.11) provide the formulas required to construct Ak from Bk and Ck.
Since the evaluations of Bk and Ck involved N/2 samples, conventional Fourier analysis to obtain these quantities requires 2 x (N/2)2 = N2 multiplications. Therefore, the analysis based on splitting the data record halves the computing time. Of course the halving can be repeated over and over as long as the remaining record is divisible by two. This is why the complete FFT algorithm is based on 2" data points for integer values of n.
Efficient computer programs have been developed for FFT calculations. One important feature of many of these programs is the capability of “in place” computation. This means that the computer is required to store only the N data points, because after they are used, the values at any stage in the calculation are no longer needed and their storage locations may be allotted to new intermediate values. Also, the details of the logic required to perform the FFT calculation are well documented in the literature (8).
It should be noted that other forms of the FFT are available and that FFT algorithms exist for N = 3", N = 4", and so on. However, the method outlined here illustrates the principles involved, and the algorithm for N = 2" is by far the most common.
A number of BWRs have been subjected to frequency response tests. All of these measurements used the oscillator method.
The BORAX reactor and the experimental boilingwater reactor (EBWR) were early boilingwater reactors that were operated in the 1950s and 1960s to evaluate the feasibility of the BWR concept. One of the major points of concern was the stability of this type of reactor. The EBWR experience has already been mentioned in Section 6.1.
As the BWR concept was developed into a commercial power reactor by the General Electric Company, continued emphasis was placed on plant stability and control. In the theoretical phase of this work a major effort has been devoted to techniques for computing the frequency response of the system. A great deal of theoretical work has been done, and a number of frequency response measurements have been made on test loops and on reactors to check the theoretical predictions. The standard method for theoretical dynamic analysis of large BWRs involves a frequency response analysis using a computer code called FABLE (43. 44).
The first large commercial BWR power plant in the United States was Dresden 1 (a 613MW dualcycle plant). The control rods in Dresden 1 and all subsequently designed BWRs to date are the bottom entry, hydraulically driven type of rod described in Section 7.3. In the Dresden 1 tests, a standard rod was slightly modified so that it could introduce sinusoidal reactivity perturbations with frequencies up to 0.27 Hz (25). Oscillator tests were performed at eleven power levels up to full power, and satisfactory stability performance was observed at all power levels.
Further control rod oscillator tests have been made at Big Rock Point, Garigliano, and Gundremmingen (42). The Gundremmingen test program also included measurement of the response of the system pressure to oscillations in the steam control valve position (45). The tests at Garigliano (26) (a 506MW dualcycle plant) provide a good case study. These oscillator tests were performed using a special purpose hydraulic oscillator on the central control rod. In a discussion on the Garigliano plant, Mr. F. Santasilia (of Ente Nazionale per l’Energia Elettrica in Rome) indicated that it would have been much easier and less expensive to introduce trapezoidal reactivity perturbations than sinusoidal perturbations. He stated that the effort
required to achieve the sinusoidal input was justified because the results are easily correlated with theoretical frequency response measurements. This suggests that measurement techniques based on binary signals that have been developed since the Garigliano tests would have been valuable for that application.
The frequency response of a BWR depends on the operating conditions (power, flow, void content) and on the location of the detector. These features are accounted for in the FABLE model, permitting a direct comparison between theory and experiment. Typical results from Garigliano are shown in Fig. 8.1.
In all of the newer BWRs that are being planned or built, step response tests are to be used to measure decay ratios (see Section 6.1) for certain process variables. These tests will suffice for checking stability margins and for rough comparisons with theoretical models. However, a detailed comparison with the FABLE theoretical calculations would require a complete frequency response measurement.
In a frequency response measurement that uses a sinusoidal test signal, the input is В sin cot and the output is M sin (cot + ф). The amplitude of the frequency response is M/B, and the phase is ф. If two sinusoidal signals with different frequencies were used simultaneously, the input SI would be
SI = B, sin со^ + B2 sin co2t
Because of the principle of superposition for linear systems, the output would be
SO = M1 sin(aj! Г + Фі) + M2 sin(a>2t + ф2)
or
80 = S! IG(jcoі) sin^t + фк) + B2G(jco2) sin(a>2t + ф2)
The same idea may be used with general, nonsinusoidal, periodic signals. The input may be expressed (exactly) as a Fourier series
00
8l(t) = (aj2) + £ (ak cos cokt + bk sin cokt) (2.9.1)
k= і
where <ok = 2knjT and T is the period. The output is
00
80(t) = (aJ2)G(0) cos ф0 + £ akG(jcok) cos(cokt + фк)
k= 1
+ bkG(jcok) sin(o)kt + фк) (2.9.2)
The complex form of these expansions is
00
<5J(t) = X ck exp()wkt) (2.9.3)
к = — oo
00
50(t) = £ CkG(jcok) exp(jcokt) (2.9.4)
к = — oo
If the input and output signals are Fourier analyzed at a harmonic frequency, the results are
j»nT
(1 /nT) 80(t) exp(~jcokt) dt = Ck G(jcok
J 0
where n is the number of periods of data analyzed. The frequency response is obtained as follows:
C(. , (1 /nT) j"r 50(f) exp( — jcokt) dt }C°k (1/nT) jjr dl(t) exp( —j(okt) dt 
(2.9.7) 
The results are obtained from the following expressions: 

G(M) = (R + jS)/( V + jW) 
(2.9.8) 
where 

лпГ R = (1/nT) SO(t) cos cokt dt J 0 
(2.9.9) 
лпГ S = —(1/nT) SO(t) sin cokt dt J 0 
(2.9.10) 
/* nT V = (1/nT) dl(t) cos cokt dt J 0 
(2.9.11) 
W= (1/nT) SI(t) sin coktdt Jo 
(2.9.12) 
These lead to the following results: 

Re{G(jcut)} = (RF+ SW)/(V2 + W2) 
(2.9.13) 
Im{G(M)} = (SF — RW)/(V2 + W2) 
(2.9.14) 
G(M) = [(R2 + S2)/(V2 + IF2)]1/2 
(2.9.15) 
^{G(M)} = arctan[(SF — RW)/(RV + S1F)] 
(2.9.16) 
The frequency response also may be obtained from the power spectra. If the input signal is given by 

CO dl{t) = X Ck exp{jcokt) к = — со 
(2.9.17) 
then the output is 

oo dO(t) = X °k exp(jwtt) k = — со 
(2.9.18) 
where 
Dk = CkG(j(ak)
The crosspower spectrum PI0 and the power spectrum of the input P,, are given by
PI0 = DkC. k = G(jcok)CkC_k (2.9.20)
P„ = CkC. k (2.9.21)
Then the frequency response is simply
G(M) = P, o/P„ (2922)
One way to obtain these power spectra is by Fourier analysis of the crosscorrelation function and the autocorrelation function of the input (see Eqs. (2.7.3) and (2.7.4)). Thus a prc :edure that may be used to give the frequency response is
G(jw) = F{C12}/F{Cn} (2.9.23)
The analyst would compute the correlation functions and then Fourier analyze them. As will be seen later, this requires greater computational effort than direct Fourier analysis of the input and output signals. But the correlation functions are sometimes worth the effort because they can be interpreted directly to provide information on system dynamics (see Section 3.6).
Dynamics tests may be used to determine the degree of subcriticality in a subcritical reactor. The method for performing and interpreting this type of measurement may be deduced by considering the kinetics equations for a subcritical reactor with a source (spaceindependent model):
6
dn/dt = [(p — P)/]n + £ А, с, + S (5.4.1)
;= і
dCJdt = Pi/An — A, C; (5.4.2)
where S is the source strength. A subcritical reactor can be perturbed by modulating the source or by modulating the reactivity. The pertinent transfer functions are
We note that these two transfer functions differ only by a gain factor. The shapes of the frequency response functions are identical.
Figure 5.7 shows the frequency responses for various levels of subcriticality. The highfrequency break is due to a pole with a value (P — p0)/A. A frequency response measurement using a source modulation or a reactivity modulation around the fixed subcritical reactivity can be used to determine (P — p0)/A by identifying the break frequency. If p and A are adequately known, p0 can
then be determined. We also note that the phase shift in the range 0.001 to 1 rad/sec is quite sensitive to the subcritical reactivity for subcriticalities of less than a few dollars. This suggests that lowfrequency measurements can be used to determine the subcritical reactivity by relating the phase shift to subcriticality.
The frequency response has been measured in research reactors using the sourcemodulation technique. This requires the use of a pulsed neutron
source. It is not usually feasible to use this technique in power reactors, and other techniques for measuring subcritical reactivity have been developed. These are the randomnoise technique (35), the sourcemultiplication technique (5), and the inversekinetics technique (5).
Even though two of the signals considered previously in this chapter are binary and have signal power in a number of harmonic frequencies, we will reserve the name multifrequency binary sequence for signals that can be generated with a spectrum whose shape is (within certain restrictions) specified by the user. This differs from the other signals whose harmonic amplitudes are predetermined once the number of bits is selected.
A continuouslevel, periodic, multifrequency signal may be developed by simply adding sinusoids with selected harmonic frequencies:
N
I(t) = Y, at s’n (3.5.1)
i= 1
where N is the number of sinusoids, щ the selected amplitude of the ith sinusoid, and щ a harmonic frequency. A signal of this type is shown in Fig. 3.12. The complicated waveform even for this simple example would be difficult to implement with usual system hardware.
Fig. 3.12. A continuous multifrequency signal and its binary approximation. 
Jensen (62) proposed a binary multifrequency signal that is obtained simply by setting the input equal to +A when I(t) from Eq. (3.5.1) is positive and — A when I(t) is negative. That is,
I'(t) = A{sgn[I(t)]} (3.5.2)
where I'(t) is a binary signal with values 4A and — A. A binary signal obtained this way is also shown in Fig. 3.12. This approach gives a signal that largely overcomes the hardware problem and still retains a large fraction of
t See the literature (49, 50, 5254, 6265).
the signal power in the selected frequencies (usually 4060%). However, because of the quantization, the distribution of the energy among the selected harmonics may be far from the desired distribution.
Later work (52,65) provides MFBS signals that achieve both a concentration of signal energy in selected harmonics and a good matching of the distribution to the desired distribution. This is accomplished by a computer optimization that modifies the polarities of the bits in a binary pulse chain until the difference between the desired spectrum and the obtained spectrum is minimized. Since antisymmetry is achieved simply by making the last half of a period the negative of the first half, this property is easily obtained.
A computer code (52) has been developed for generating MFBS signals and has been used to prepare a number of different signals. Experience shows that the procedure gives sequences that concentrate 7080% of the total
TABLE 3.4 Selected MFBS Signals”
“ Only the first, half of each signal is presented. The second half is obtained by inverting the first half. bThe notation J+ or J— means the signal is positive or negative over J minimumwidth intervals. 
signal power in the selected harmonics. The user specifies the number of bits in the sequence, the harmonics at which power is to be concentrated, and the magnitude desired at each of these harmonics. Of course, there is no need to specify harmonic amplitudes that are greater than the maximum possible amplitude given by Eq. (2.12.7). Several MFBS signals and their spectra are shown in Tables 3.4 and 3.5.
TABLE 3.5 Spectral Characteristics of Selected MFBS Signals Fraction of signal power in indicated harmonic

The MFBS must be generated on a computer using the optimization procedure. The sequence is then loaded into an input device for feeding into the system being tested. The input device might be an online computer, a papertape reader, or an electronic storage device such as a ringtail counter.
In this case, a rodposition demand signal is fed into a controlrod position controller. The controller moves the rod until the difference between the desired position and the measured position is zero. Figure 7.1b shows the closedloop rod control system. This is a very convenient method for dynamic testing, but it is not the way control rods are normally positioned in nuclear reactors. Usually, it would be necessary to install a specialpurpose controller for dynamic testing. This is generally a simple technical problem, but may require considerable effort to satisfy operational and licensing requirements.
1.1. The Need for Frequency Response Testing
Frequency response testing should play an important role in evaluating the performance and safety of modern power reactors. The main motivations for testing are:
1. To assess the system stability margin and to detect trends in the stability margin caused by changes in operating conditions.
2. To check mathematical models and coefficients used in theoretical studies.
3. To provide information needed to optimize controller parameters.
4. To provide information that can be used to tune theoretical models so that they can be used to predict the plant response with assured confidence.
A number of frequency response measurements have been made on research reactors and some of the early central station power reactors. Most of these tests have used the oscillator method, which employs a sinusoidal reactivity input. This procedure requires special (and expensive) hardware, modification of the system design to accommodate this hardware, and interference with normal system operation during the lengthy tests. In most of the new commercial power reactors, economic and operational considerations have dictated that the expense and the inconvenience are too great to justify oscillator test programs. The alternatives are to forfeit the information or to develop new procedures without the disadvantages of the
і
oscillator method. These new methods have been developed and are the subject of this book.
The emphasis in this book is on optimum testing procedures for measurements in power reactors. The criteria for suitability of tests in power reactors are:
1. The tests should make maximum use of standard system hardware and instrumentation so as to minimize costs.
2. The tests must be virtually incapable of causing a scram or a component malfunction.
3. The test must cause insignificant interference with normal power generation. Dynamic testing must be done while the system continues to satisfy load demands.
4. The duration of the test must be as short as possible. The presence of test personnel in the control room, the use of system equipment (such as the online computer), the possible suspension of other tests, and the possible departure from normal control policies can be tolerated only briefly.
5. The results must be suitable for quick and easy interpretation. The tests should be planned and set up by specialists, but the plant engineer should be able to understand the significance of the results and their implications in connection with plant operation.