Frequency response tests that use periodic test signals typically involve the use of multiple cycles to achieve adequate accuracy. In principle, the Fourier analysis methods described earlier in this chapter could be used to analyze the whole data record from start to finish. That is,

F(jwk) = WnT) Г x(t)exp(-jwkt)dt (4.7.1)

Vo

where n is the number of cycles in the data record.

Three factors require that this procedure be modified. These are:

1. Some analysis methods cannot handle an unlimited number of data points because of computer memory requirements. The FFT has this feature. The analysis must be broken into blocks, each with a record portion containing a number of samples equal to some integer power of two. The results from each block would then be used to construct the frequency response results.

2. Some analysis methods experience numerical problems when the num­ber of points in a record becomes large. For example, the algorithm of Section 4.4 has been found to have difficulty for records with more than about five thousand samples. A way to overcome this problem is to break the record into blocks and reinitialize the analysis at the start of each block. As above, the results from the block analyses are used to construct the frequency response results.

3. Fourier analysis becomes more sensitive to timing and periodicity errors as the length of the data record increases. Timing errors include imperfect specification of analysis frequencies and slight differences in the duration of a period from one cycle to another. Periodicity errors are due to differences in the wave form from one period to another that are usually encountered in test situations. One way to view the cause for these problems is by considering the effect of the length of the data record on the effective filtering in Fourier analysis (see Section 2.11). As the record length increases the filtering becomes sharper, and slight errors in placing the peak of the filter on the correct signal frequency can result in large errors.

Because of these problems, block averaging methods are generally used in data analysis. These will be described below along with a discussion of their ability to discriminate against background noise. The Fourier estimate for block і is

(A; + eii) + j(Bki + є2і) (4.7.2)

where Akj is the real part of the Fourier coefficient for block i, Bki the

imaginary part of the Fourier coefficient for block i, eu the error in Aki, and £2l the error in Bki. The three types of average used most often are:

1. Average the real and imaginary parts for the Fourier coefficients calculated for each block for each signal:

M

Ak = (1 /М) £ (Aki + eu)

i = 1 M

Bk = (1 /М) £ (Bki + e2i)

i = 1

The final frequency response is obtained by complex division of the average for the output signal by the average for the input signal. The errors due to background noise will be reduced in this averaging method because the noise will generally have random phase shifts, thereby giving cancellations that reduce the error contribution. For this method, the error asymptotically approaches zero as the number of blocks in the average approaches infinity.

However, it may be impossible to exploit this asymptotic approach to the correct result because this method is equivalent to Fourier analysis of the whole data record and the problems described above that occur because the effective filtering is too sharp may occur. As a consequence, this averaging method can be used to advantage only in tests where the timing and periodi­city are precisely controlled. In many practical situations, this cannot be done and the averaging methods described below must be used even though we will find that they do not converge to the true result as the length of the data record increases.

2. Compute an estimate of the frequency response for each block:

Gki = DJCki (4.7.5)

where Gki is the frequency response estimate, D the Fourier transform of the output, C the Fourier transform of the input, к the harmonic number, and і the block number. The final estimate of the frequency response is obtained by

и

Gk = (1/M) X Gki

Each estimate of Dki and Cki will have errors due to noise

The complex division gives

DkiCti + Єі. Є2. + £1 A* + Dki£ 2i

Скіпкі + Є2 iC*i + Є*іАі + Є2іЄ2і

We observe that the third and fourth terms in the numerator and the second and third terms in the denominator are uncorrelated and decrease as the number of blocks increases. However, the last term in the denominator is a positive real number and cannot be reduced by averaging. Also, if the error in the input is correlated with the error in the output, then the second term in the numerator does not have an average of zero.

The conclusion here is that this type of averaging does not provide some of the error cancellation obtained in the first averaging procedure, and the result does not converge to the correct value as the number of blocks approaches infinity. However, this method is suitable for test situations with imperfect timing and periodicity. Since this is usually encountered, this method will often be preferred over the first method.

3. The frequency repsonse in a third averaging procedure is obtained by

_ (l/MlXfl, DklC*

k (i/M)£"=1cwC£

If the estimates contain errors, then

(1/M)E"=, №ki + еи)(СЪ + S*2i) k (УМ)^=1(Скі + s2i)(C*ki + s*2i)

or

r (l/M)Xf=1 DkiC*ki + ЄіА*. + + ЧҐ2І (47Ш

* (1/Af) Xf=1 CkiCti + е2іСІ + е? А,- + e2is*2i 1 " ]

As in the second averaging method, this method does not give an estimate that converges to the correct result as a number of blocks approaches infinity. Of course, if one has a separate measurement of

M

(1 /М) X Є2іЄ2і

i= 1

then this term can be subtracted from the denominator term.

Thus, we conclude that a trade-off exists. Method 1 is preferred for tests with precisely controlled timing and periodicity because the error approaches zero as the record length approaches infinity. However, because this ideal situation rarely exists, methods 2 and 3 must be used to avoid problems that

arise because of imperfect timing or periodicity. The price paid for this is that the analyst must accept a nonzero error regardless of the length of the data record.

Frequency response measurements have been used extensively in almost all of the fast reactors that have been operated so far. This included EBR-I, EBR-II, Fermi, SEFOR, LAMPRE, and Dounreay. Tests are also planned for the fast flux test facility (FFTF) in the United States and for the prototype fast reactor (PFR) in Britain.

In commercial liquid-metal-cooled breeder reactors, suitable dynamic behavior will be achieved by designing to achieve a satisfactory balance between stabilizing and destabilizing effects while maintaining good economic performance. Dynamics tests are useful to determine whether stability margins have decreased due to changes in core characteristics because of aging and whether the models and parameters used in dynamics and safety studies are accurate. These needs are well recognized and account for the wide use of frequency response measurements in fast reactors.

Almost all of the frequency response measurements in fast reactors have used the oscillator method. The sole exception is the measurement at Fermi (69), which used single reactivity pulses for frequency response determination as a supplement to the oscillator tests. The oscillator rods used in fast reactors have been of the rotating type and the linear (in-out) type. Both have been notoriously unreliable. A great deal of work has gone into the design of oscillator mechanisms that give pure sine waves (no harmonics) and permit accurate determination of the time-varying reactivity at all frequencies of interest. These systems can be installed in new fast reactors, but the cost will be several hundred thousand dollars. This cost can be eliminated if newer testing methods that utilize standard rods can be used.

The suitability of standard control rods for frequency response measure­ments in fast reactors is largely determined by the ability of the rods to move fast enough to give the highest frequencies of interest. Since there is little uncertainty in the zero-power kinetics of the system, the frequency range of interest is determined by the frequency range over which feedback effects are significant. In all operating and planned fast reactors, the highest frequency of practical interest for observing feedback effects is about 10 rad/ sec. Standard control rods will probably suffice for measurements up to this frequency.

Experiences with EBR-1 (45) provide an interesting example of the use of frequency response results. EBR-I was a small (1.2 MW) reactor that began operation in 1951 with the main purposes of demonstrating that a fast reactor could have a breeding ratio greater than unity and that liquid-metal- cooled fast reactors are feasible for producing electric power. Two dynamic effects were apparent in the first core (Mark 1) that caused concern. The first was evidence that the system had a prompt positive reactivity coefficient and a (larger) delayed negative reactivity coefficient. The second was a dependence of system stability on coolant flow rate.

A number of frequency response and transient response tests were run on the EBR-1. During a transient response test on the Mark 11 core, the power rise was excessive and part of the fuel melted.

Further study suggested that bowing of the fuel because of temperature gradients was responsible for the prompt positive coefficient. To confirm this, rigid cores with little bowing were constructed. Frequency response tests showed that this reduced the prompt positive coefficient. The delayed negative coefficient was found to be due to thermal expansion of core structural members. In the interpretation of the results, the feedback frequency response was determined using Eq. (5.2.1). These results clearly showed that rigidizing the core increased the stability of the system.

The analysis of the EBR-1 system indicated that the stability problems were due to the design peculiarities of that system, and that other systems could readily be designed without those features. However, the possibility of other peculiarities suggests that experimental confirmation of stability margins during core life will continue to be useful.

Interesting tests were performed at Fermi in which the frequency response was measured using the pulse technique (69). These tests were performed to show the suitability of a standard control rod for frequency response measure­ments. Measurements were made over the range from 0.0001 to 0.15 Hz and were found to agree favorably with results from oscillator measurements.

So far, no frequency response tests on fast reactors have used periodic, binary input signals. However, they will be used in the FFTF and in the PFR.

The coherence function may be constructed from experimental data to check the influence of uncorrelated system noise on test results. The coherence function involves averaging the power spectra from different segments (or blocks) of the data record. Each block data must contain an integral number of periods for periodic signals. The coherence function is defined as (4)

(2.10.1)

where

and N is the number of blocks of the data record, Fn + еи the Fourier trans­form of [input signal plus input noise] for segment i, F0i + e0i the Fourier transform of [output signal plus output noise] for segment i, and * denotes complex conjugate.

From these definitions, we obtain

P, o = (1/ЛГ) І FnF*0i + FIi(*0i + (iiF*0i + e, i(*oi

І= 1

p„ = №) £ F„F,*; + F, tft + ‘„FT, + Cjtf,

І= 1

Poo = (1/W) Y FoiFoi + Роі€оі + €oiFoi + eo;eoi

i= 1

Each of these terms is recognized as a power-spectrum estimate. As the number of blocks in the average increases, the contributions of the cross­power spectra of uncorrelated signals diminish toward zero. Thus:

P’I0 = lim PI0 = lim (1/ЛГ) £ FnF*0i

N-oo N-oo (tTj

P’„ = lim P„ = lim (1/ЛГ) Y FnF*i + (n(*i

N-*oo N -»oo

_ N

Poo = lim P00 = lim (1 /N) Y FoiFoi + tofoi

N~* oo N -* oo j _ j

Then the coherence function based on these limiting values is

„_________ ИГ-, ___________

II,"., F, n + <n«SI Ig., Vl‘ + <<„<11

If the process is stationary (Fu and F0i have the same values for any block of the data record), then the coherence function goes to its maximum value of unity as the noise contamination goes to zero. The departure from unity of (y2)’ is taken as a measure of the influence of noise on the test results (assum­ing a stationary signal).

Several words of caution are appropriate in connection with the use of coherence functions. First, the interpretation presented above depends on the disappearance of uncorrelated terms as a result of a long data analysis, but actual tests use finite data records. Furthermore, it can be shown that the coherence function has a value of unity for a record of zero length and decreases to its limiting value as the quantity of data analyzed increases. This indicates that errors due to using insufficient data records will cause the estimated coherence function to be too high (nonconservative). The second
point is that the interpretation of the coherence function depends on the existence of exact Fourier coefficients. If the data analysis is imperfect (i. e., if incorrect harmonic frequencies are used in the analysis because of inaccu­rate timing), then the coherence function may be meaningless.

This chapter deals with techniques for extracting useful information from test results. In general, results from dynamics tests have been put to these uses:

1. Assessment of system stability reserve and changes in the stability reserve with changing system characteristics.

2. System identification. This broad category contains three subtopics: (a) construction of empirical models, (b) verification of theoretical models, and (c) parameter estimation.

The experimenter should select input signals that provide frequency response results over the range of interest with the desired frequency resolu­tion. In order to obtain an adequate signal-to-noise ratio in the shortest possible time, he must minimize the waste of available signal energy in frequencies other than those desired.

The MFBS signal has the greatest potential for minimizing signal-energy waste, since the energy can be concentrated in harmonics of interest. The spectra for the other signals (PRBS, PRTS, and n sequence) are completely predetermined once the number of bits is chosen. Their harmonics are evenly spaced on a linear scale, but frequency response results are usually inter­preted using Bode plots, which employ a logarithmic frequency scale. This means that the harmonics for the PRBS, PRTS, and n sequence fall closer together on a Bode plot as the frequency increases. This is shown in Fig. 3.13, which shows a PRBS spectrum on a logarithmic frequency scale. The bunch­ing of the points at higher frequencies is apparent. In most applications, much of the energy in these closely spaced harmonics would be used up in identifying frequency response points that are only slightly different from one another.

It is informative to compare the testing time required to obtain the same harmonic amplitude at desired frequencies in two separate tests instead of a single wide-band test. For simplicity, let us assume that all nonzero harmonics within the bandwidth in a signal have the same amplitude. Then the energy in each harmonic is

Ek = E/L

where Ek is the energy in harmonic к, E is the total energy, and L is the
number of harmonics in the bandwidth. Since the total energy is proportional to the test duration Tt, we obtain

Ek = C(TJL)

where C is a constant. This shows that a reduction in the number of fre­quencies in the bandwidth can be accompanied by a proportional reduction in the testing time without a loss of signal energy in the harmonics of interest. Thus an MFBS test with most of the energy concentrated in the desired harmonics can be completed in a much shorter time than a PRBS test that covers the same frequency range.

The same considerations are important even if the MFBS is not used. The experimenter usually has to decide whether to use a single test to cover the whole range of interest or to use several tests with frequencies that cover different portions of the range of interest. In this case, there is a trade-off between convenience and efficiency. For example, a PRBS that has a band­width of D decades will have 10й harmonics in the bandwidth. The harmonics will be distributed as follows:

Number of harmonics 9 90 900

We observe that PRBS signals that span more than about two decades concentrate an excessive fraction of the signal energy in the higher fre­quencies.

As an example, let us consider the design of a PRBS test to cover three decades of frequencies. The 2047-bit PRBS has a bandwidth that extends to the 900th (2047 x 0.44) harmonic. Thus this signal is the appropriate PRBS to cover approximately three decades. However, the test must be run long enough to build up enough signal energy in each of 900 harmonics. As an alternate selection, consider two 127-bit sequences. The first would have the same period as the 2047-bit sequence. This would cover the range shown in Fig. 3.14. Then a second test would use a 127-bit sequence with a period equal to one eighteenth of the first 127-bit sequence. Figure 3.14 shows that this test extends out to the highest frequency of interest. Also, we note that there is an overlap of the results from the two tests. This is important to check whether the tests were both run at the same conditions. The two-test procedure provides results at 110 frequencies (127 x 0.44 x 2) compared to 900 for the single-test procedure. If this resolution is suitable, then the two-test procedure would require 110/900 as much time to give approxi­mately the same energy in the useful harmonics.

A possible difficulty is that the low-frequency 127-bit signal will have longer runs with the signal at a fixed value. This can be a problem if it is necessary to insure that the output stays within prescribed limits. The bits in the 127-bit sequence must be 2047/127 as long as the bits in the 2047-bit sequence in order to obtain the same period. The longest runs are 7 bits long for the 127-bit PRBS and 11 bits long for the 2047-bit PRBS. Thus the longest run for the 127-bit PRBS will be longer than the 2047-bit PRBS by a factor (2047 x 7)/(127 x 11) = 10.2.

In general, the length of the longest run in a PRBS with a given period decreases as the number of bits increases. The period is (2" — 1) At, where n is an integer and At the bit duration. The duration of the longest run is n At. Then the ratio of the duration of the longest run to the period is n/(2" — 1). Table 3.6 shows this ratio for various PRBS signals. This demonstrates that problems caused by long runs can be eased by using longer sequences.

This method is convenient when a rod motion controller is used to change the power to satisfy a change in a power-demand set point. The binary
pulse chain is fed into the control system as a power-demand set point. The controller changes reactivity to attempt to satisfy this demand. Figure 7.1c shows the power-demand rod control system. In this case, the output would have a power spectrum that approximates the power spectrum of the binary pulse chain (approximation depends on control-system perform­ance). The input power spectrum would approximate the reciprocal of the square of the reactor gain. This is completely acceptable for frequency response measurement purposes. Of course, if the rod-position controller uses some other signal rather than reactor power, then the method will also work if that demand signal is used.

The value of frequency response testing in nuclear systems has been appreciated for a number of years. Tests have been run to check stability margins and theoretical dynamics models. All of the early tests used sinu­soidal reactivity perturbations to excite the system. This approach was a direct implementation of the basic definition of the frequency response, but the equipment was expensive and not very durable (particularly in the hostile environment in the more advanced reactors). Nevertheless, a number of excellent tests were performed on reactor systems.

In the early 1960s, alternate testing procedures involving periodic binary (two-level) input signals were first used on reactors for frequency response measurements and for impulse response measurements (see Section 3.1). The work of Balcomb et al. (1) was the key contribution in the development of these procedures. The pseudo-random binary sequence was used in most of the measurements during this period. Tests using the pseudo-random binary sequence have two features that make them superior to oscillator tests for power reactor measurements:

1. The two-level inputs can be introduced by standard hardware, such as control rods, in many reactors.

2. The signal contains many harmonics, permitting the determination of the frequency response at a number of frequencies in a single test.

After the introduction and use of the pseudo-random binary sequence, other binary and ternary (three-level) signals with advantages over the pseudo-random binary sequence were developed. The needs for the improve­ments achievable with the newer signals and the manner in which the improvements were made are discussed in Chapter 3.

Frequency response measurements may also be made using nonperiodic inputs such as reactivity pulses or steps. These also allow the determination of the frequency response at a number of frequencies in a single test and have simple hardware requirements. The problem with this type of signal is that it may be difficult to achieve a high enough signal-to-noise ratio to achieve good accuracy.

Information on system dynamics can also be obtained by analyzing the inherent statistical fluctuations (noise) in the system output. If the frequency dependence of the statistical driving function is known, the shape of the amplitude of the system frequency response can be determined. If the fre­quency dependence of the driving function is not known, less quantitative information can be obtained, but the results can still be used for a diagnostic to indicate changing conditions. Noise analysis is very well documented (2-4) and will not be included in this book.

Other developments besides the improvements in testing procedures have occurred that further increase the practicality of frequency response testing. The first development has to do with data analysis. Particularly significant is the fast Fourier transform technique, which allows digital computer analysis of test data for a small fraction of the cost and time previously required. The data analysis problem has also benefited from the availability of new digital computers, particularly the small minicomputers that can be taken to the test site and can provide at least a first look at the results in seconds or minutes. The other major development has to do with data inter­pretation. A new technology called system identification has evolved to aid in extracting useful information from test results. Examples of the informa­tion that may be obtained are specific system coefficients such as temperature coefficients of reactivity or heat-transfer coefficients. This technology is still growing rapidly, but already it has been applied successfully and profitably on several nuclear reactor tests.

Fourier analysis of a periodic signal is valid at all harmonic frequencies. The harmonics in a signal with duration T are at angular frequencies 2л/T’, 4л/Т’, 6л/Т’,…. The data record may be a single period of data, or it may be an integral multiple of a period. If the data record of length T’ contains exactly n periods, then only every nth harmonic may be nonzero. This is demonstrated in Fig. 4.11 for a З-bit PRBS signal. For one period

of data, every harmonic may be nonzero. For two periods of data, every other harmonic may be nonzero. It should be emphasized that the term harmonic means the harmonic based on the total data record, not the length of a period. However, all of the harmonics obtained in a single period are also contained in a record that contains an integral number of periods.

A nonideal situation that is sometimes encountered in practical measure­ments is that the total data record does not contain an integral number of periods. This sometimes occurs in the use of the fast Fourier transform algorithm, which requires that the data contain 2" data points for integral values of n (see Section 4.6). The analyst usually picks the period T to give the desired lowest frequency and frequency spacing. The sampling time At is selected to give the desired highest frequency and to prevent aliasing. These
two quantities fix the number of points per period N:

N = T/At (4.8.1)

Then the total number of samples in M periods is MN. If the FFT algorithm is to be used and if M is to be an integer, then

MT/At = 2" (4.8.2)

or

M = (At/T)(2n) (4.8.3)

The only way the number of periods can be an integer is for the number of samples per period to be an integral power of 2. That is,

T/At = 2m (4.8.4)

Since the desired value of T is set by the test objectives and the value of At is a compromise between test objectives and the capability of the digitizing equipment, it may sometimes be inconvenient to satisfy Eq. (4.8.4). This leads us to consider the use of the FFT algorithm on a data record that contains a nonintegral number of periods.

The harmonics in the FFT analysis must be based on the length of the data record used in the analysis:

o)k = 2kn/T’ (4.8.5)

where T is the length of the data record. Since T is not necessarily equal to an integral number of periods,

T = nT + 8 (4.8.6)

where n is the number of periods, T the period, and <5 the portion of a period. The relationship between the Fourier coefficients obtained by analysis of a nonintegral number of periods, and those that would be obtained from analysis of an integral number of periods is easily derived.

where the interval — (nT 4- 8)/2 to (nT + S)/2 has been chosen for conve­nience. The signal x(t) may be represented as

Using Eq. (4.8.7) in Eq. (4.8.8) gives

sin(mn — к + md/T)n

(mn — к + md/T)n

Thus the Fourier coefficient calculated by analysis of a noninteger number of periods is related to all of the Fourier coefficients of the original signal, and the contribution of each is given by the previously encountered (sin x)/x filter (see Section 2.11). The peak of this filter occurs at m — k/(n + S/T). The displacement of the nearest harmonic from the peak is m8/T, and the first null in the filter is at m = (1 + k)/(n + 8/T). The results are shown in Fig. 4.12.

A phenomenon called the picket-fence effect appears when the Fourier analysis is based on a nonintegral number of periods. This is caused by the shift in the displacement between analysis harmonics and signal harmonics as the analysis frequency changes. For example, let us assume that the first harmonic analysis frequency is 10 % higher, than the first signal harmonic. Then the second analysis harmonic would be 20 % higher than the second signal harmonic. Figure 4.13 shows this effect. For this example, the syn­chronization between the signal harmonics and the analysis harmonics would become worse until the fifth harmonic, then would improve until they coincided at the eleventh harmonic.

Drift is also sometimes a problem. For example, the process may experience some unmeasurable input in addition to the test signal, and this additional input could cause the response to drift. One might be inclined to assume that Fourier analysis would automatically eliminate the drift effect because it is a low-frequency phenomenon and usually outside the frequency range of interest. Unfortunately, this is not true. A truly periodic signal begins and ends each period at the same value. Because of drift, the last point will not be

Fig. 4.14. Effect of drift.

the same as the initial point as shown in Fig. 4.14. However, Fourier analysis assumes that they are equal, and this would require a jump in the signal as shown in Fig. 4.14. This jump introduces high-frequency errors. High-pass filtering can be used to alleviate this problem. Also, if the drift has a simple form, such as a ramp, the drift effect can be subtracted, point by point, from the signal before Fourier analysis.

Essentially all of the published information on frequency response testing in gas-cooled reactors comes from Britain. A great deal of excellent work has been done in Britain on testing procedures and interpretation of results as well as on actual tests.

Measurements were made on the Dragon high-temperature gas-cooled reactor at full power (200 MW). Pseudo-random binary reactivity perturba­tions were introduced using a standard control rod. The frequency range covered was approximately 0.01-0.6 rad/sec.

In the Dragon experiments, the results from the PRBS tests were also used to obtain the step response of the system. As described in Section 4.10, the cross-correlation function between input and output is approximately equal to the impulse response if the autocorrelation function of the input is approximately a delta function. Also, the step response is the integral of the impulse response. The step response for the Dragon reactor was obtained by integrating the input-output cross-correlation function obtained in a PRBS test.

It is informative to investigate the filtering that is implicit in the Fourier analysis of a signal. Let us consider a general periodic signal I(t) with the following (exact) Fourier series representation :

I(t)= £ Qexp(M’) (2.11.1)

к = — oo

Now let us Fourier transform that signal using n periods of data:

1 ЛлГ/2

I(co,) = F{I(t)} = — I(t) exp( —jcoit) dt

n* J — nT/2 oo £ /*пГ/2

= z exp[;(cut — 0J,)t] dt

fc= — 00 ПІ J —nT/2

(2.11.2)

The function (sinx)/x appears often in the frequency-domain analysis of signals. It is usually called the sampling function or sine function. It is shown in Fig. 2.20. (The sampling function and its square are tabulated in Appendix A.) We observe that the function is zero at all integer multiples of n except at

zero. This is a way to view the orthogonality property discussed earlier. Using orthogonality properties to evaluate Fourier coefficients is equivalent to analyzing so that all of the harmonics fall on zeros of the (sin x)/x filter except for the analysis harmonic.

This approach also illustrates how noise enters the results and what happens if the analysis is done at nonharmonic frequencies. The noise contribution will be that which is passed by the (sin x)/x filter. The effect of analysis at a nonharmonic frequency will be to mix the contributions from all the harmonics in a way that depends on the difference between the harmonic frequency and the analysis frequency (see Fig. 2.21).

The analysis also shows the effect of the number of periods of data analyzed. Figure 2.22 compares results for one and two periods of data. Clearly, using more data results in a sharper filtering effect.