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Transform and filtration methods in fluctuation analysis
Annals of Nuclear Energy, Vol. 3, pp. 471 to 475. Pergamon Press, 1976. Printed in Northern Ireland
TRANSFORM
AND
FILTRATION METHODS ANALYSIS
IN F L U C T U A T I O N
R . F . SAXE Department of Nuclear Engineering, North Carolina State University, Raleigh, NC., 27607, U.S.A.
(Received 7 May 1975; and in revisedform 5 May 1976) Almtrnet--Digitized fluctuation signals from an ex-core ion-chamber of a PWR were analysed to produce a Power Spectral Density, (PSD), curve by two means: (1) by digital filter techniques and (2) by a Fast Fourier Transform program. For both these methods, the effects of the precision of the input data were investigated and it is shown that reasonably good PSD curves may be obtained using very poor input precision. 1. INTRODUCTION
2. DIGITAL FILTRATION TECHNIQUES
In the course of an investigation of system dynamic behavior by fluctuation analysis, it is normal to convert the fluctuating signal in the time-domain to a frequency response curve in the frequencydomain, usually by means of a Fast Fourier Transform, (FFT), operating on a repetitively sampled time-domain record. However, it is equally possible to achieve the same result by the use of a number of filters and this method, using analog filters, has been used in the past. Owing to the frequency ranges of interest in the nuclear field, it is difficult to perform such an analysis using analog methods, however, and recently little use has been made of filtration techniques. There is, however, no reason to use exclusively analog techniques for filtration and, with the relatively recent development of digital filtration methods, this method may be applied with some effect. This paper describes the use of digital filtration techniques to develop a frequency response of a nuclear power plant. With both methods of analysis, the question of required precision of input data also arises. If a high degree of precision is required in the input data, then not only is a higher accuracy analog-todigital convertor, (ADC), required, but also the manipulation of these data may become lengthier and more costly. There is therefore an incentive to reduce the required input data precision to the minimum consistent with the output requirements, especially for routine types of analysis such as malfunction diagnosis. Therefore, an analysis of the effects of input precision was performed and the results are shown below.
The process of digital filtration is one of convolution. If h(t) is the inpulse response of the required filter H([), then filtration of a time function g(z) is performed by integration of the product g(~') • h(t-'r) as a function of the displacement, ~-. The design of a digital filter includes the specification of H(f) such that the impulse response g(~-) dies out, essentially, or can be truncated within a required time interval, To. A n ideal filter has a sin (x)/x impulse response in the time domain and this does not decay to zero within a reasonable time; therefore, a departure from an ideal filter must be accepted. The filters used in this work were based on Chebyshev low-pass prototypes and a frequency transformation was used to arrive at the required band pass filter. The amplitude of the ripple within the pass-band was specified (1 db) as were the upper and lower cut-off frequencies and the order of the filter. The block diagram of the resultant filters is shown in Fig. 1 and the transfer function of each second-order section is of the form
H(z) = (1 - z-2)/1 + Bl( I)z -1 + B2(I)z-2. For the analysis to be described below, various filters were designed, as shown in Table 1. The frequency characteristics of these filters were calculated by a separate subroutine and a plot of some of these is shown in Fig. 2.
471
472
R . F . SAXE
Frequency analysis of a fluctuating signal
Fig. 1. Block diagram of digital filter.
Table 1. Lower cut-off frequency Filter number (Hz) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Upper cut-off frequency (Hz)
0.025 0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0
0.275 0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0
The fluctuating signal on which the frequency analysis was performed was that given by an excore power-range ion-chamber on the H. B. Robinson II P W R of Carolina Power and Light Company. The ion-chamber chosen was N0042A since the frequency analysis of signals from this ionchamber by an FFT program was immediately available and was typical of neutron-fluctuation signals for Core # 2 of this reactor. The fluctuating signal from N0042A had been digitized at 1/50 second intervals and stored on tape as a timeseries, each number in the series having fourdecade precision, see Saxe, Mayo and Saxe, (1976). To generate the frequency content of this timeseries, the time-series was convoluted with nineteen separate digital filters (see Table 1). The output time-series from a given filter was squared, summed, and normalized. The results from the nineteen filters could then be plotted as a nineteen point frequency curve. This procedure was repeated for ten blocks of data stored on the tape, each block consisting of 2048 data points forming a time-series. A plot of the running average for each filter output as a function of frequency and of number of data blocks analyzed is shown in Fig. 3. It will be seen that the running average for most filters FREQUENCY
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Fig. 3. Running average of filter-generated powerspectral-density as a function of the number of data blocks averaged.
Transform and filtration methods in fluctuation analysis
changes little after three or four blocks of data have been analyzed. In order to investigate the effect of the precision of the original data on the frequency analysis, the original datum was truncated to form a two-digit number and the filtering analysis was re-run. The plot of the PSD, averaged over ten blocks of input data is shown in Fig. 4; also shown in Fig. 4 is the similar curve using the full four-decade input precision for comparison. It will be seen that little change in the calculated frequency curve is apparent. A further reduction in input precision was made so that the entire recording range was defined by 20 integers, and the filtering analysis was performed. The resultant 10 block averaged PSD is shown in Fig. 5(a). Comparison with Fig. 4 shows that very little difference has occurred in the curve due to this reduction in input precision. A still further reduction in input precision so that the entire range is defined by 5 integers gives Fig. 5(b) while a further reduction so that the input is represented by a binary sequence (+1 and - 1 ) gives rise to Fig. 5(c) after analysis. It will be seen that deterioration of the resultant curve occurs for these last two cases, but that even so, the deterioration is not large. A similar investigation of the effects of input precision, using the F F T method of analysis with averaging over 10 blocks of information (20480 digitized points) gives the results shown in Figs. 6-10. It will be seen that the deterioration in PSDs, using the FFY analysis, closely parallels that using filtration techniques. A
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3. D I S C U S S I O N
The above time-domain domain curve method or by
results show that the analysis of a record to produce a frequencymay be performed either by the F F T the use of digital filters and that the
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Fig 11. Convergence of density estimates. results of using the two methods are closely comparable. The above results also show that, for the type of time-domain record obtained from a power-range ion-chamber on a typical PWR, a considerable relaxation of input data precision may be made without significantly affecting the resultant PSD curve. The PSD curves obtained by the F b T program as shown above exhibit considerably more frequency resolution than those obtained by the filtration
475
method as used here, but we suggest that this advantage is illusory in that it is highly unlikely that a real, physical phenomenon on the reactor could give rise to a resonance requiring such a narrow bandwidth for its resolution. Also, no attempt was made to use filters of a narrower pass-band for the above reason. A further difference between the two methods lies in the relative inflexibility of the analyzing bandwidth of the FFT program across the frequency scale and the complete flexibility of the filtration method to choose the bandwidth at any frequency. The digital filters, as used here, give more rapid convergence, as a function of the number of data blocks analysed, than does the analysis by FFT, as will be seen by a comparison of Fig. 3 above with Fig. 11. The frequency analysis results from a binary input resolution bring to mind the polarity correlation work of Pacilio and the various pseudorandom binary sequence experiments that have been performed and analysed. Acknowledgements--The author acknowledges the award of NSF Grant which made this work possible and also acknowledges Carolina Power and Light Company whose cooperation made possible the collection of the original ion-chamber data on which the above analyses were performed. REFERENCES
Pacilio, N. (1970) Reactor Noise Analysis in the Time Domain, AEC Critical Review Series. Saxe, R. F., C. W. Mayo and T. Saxe, Dynamic Analysis of a Pressurized Water Reactor, in this issue.
TRANSFORM
AND
FILTRATION METHODS ANALYSIS
IN F L U C T U A T I O N
R . F . SAXE Department of Nuclear Engineering, North Carolina State University, Raleigh, NC., 27607, U.S.A.
(Received 7 May 1975; and in revisedform 5 May 1976) Almtrnet--Digitized fluctuation signals from an ex-core ion-chamber of a PWR were analysed to produce a Power Spectral Density, (PSD), curve by two means: (1) by digital filter techniques and (2) by a Fast Fourier Transform program. For both these methods, the effects of the precision of the input data were investigated and it is shown that reasonably good PSD curves may be obtained using very poor input precision. 1. INTRODUCTION
2. DIGITAL FILTRATION TECHNIQUES
In the course of an investigation of system dynamic behavior by fluctuation analysis, it is normal to convert the fluctuating signal in the time-domain to a frequency response curve in the frequencydomain, usually by means of a Fast Fourier Transform, (FFT), operating on a repetitively sampled time-domain record. However, it is equally possible to achieve the same result by the use of a number of filters and this method, using analog filters, has been used in the past. Owing to the frequency ranges of interest in the nuclear field, it is difficult to perform such an analysis using analog methods, however, and recently little use has been made of filtration techniques. There is, however, no reason to use exclusively analog techniques for filtration and, with the relatively recent development of digital filtration methods, this method may be applied with some effect. This paper describes the use of digital filtration techniques to develop a frequency response of a nuclear power plant. With both methods of analysis, the question of required precision of input data also arises. If a high degree of precision is required in the input data, then not only is a higher accuracy analog-todigital convertor, (ADC), required, but also the manipulation of these data may become lengthier and more costly. There is therefore an incentive to reduce the required input data precision to the minimum consistent with the output requirements, especially for routine types of analysis such as malfunction diagnosis. Therefore, an analysis of the effects of input precision was performed and the results are shown below.
The process of digital filtration is one of convolution. If h(t) is the inpulse response of the required filter H([), then filtration of a time function g(z) is performed by integration of the product g(~') • h(t-'r) as a function of the displacement, ~-. The design of a digital filter includes the specification of H(f) such that the impulse response g(~-) dies out, essentially, or can be truncated within a required time interval, To. A n ideal filter has a sin (x)/x impulse response in the time domain and this does not decay to zero within a reasonable time; therefore, a departure from an ideal filter must be accepted. The filters used in this work were based on Chebyshev low-pass prototypes and a frequency transformation was used to arrive at the required band pass filter. The amplitude of the ripple within the pass-band was specified (1 db) as were the upper and lower cut-off frequencies and the order of the filter. The block diagram of the resultant filters is shown in Fig. 1 and the transfer function of each second-order section is of the form
H(z) = (1 - z-2)/1 + Bl( I)z -1 + B2(I)z-2. For the analysis to be described below, various filters were designed, as shown in Table 1. The frequency characteristics of these filters were calculated by a separate subroutine and a plot of some of these is shown in Fig. 2.
471
472
R . F . SAXE
Frequency analysis of a fluctuating signal
Fig. 1. Block diagram of digital filter.
Table 1. Lower cut-off frequency Filter number (Hz) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Upper cut-off frequency (Hz)
0.025 0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0
0.275 0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0
The fluctuating signal on which the frequency analysis was performed was that given by an excore power-range ion-chamber on the H. B. Robinson II P W R of Carolina Power and Light Company. The ion-chamber chosen was N0042A since the frequency analysis of signals from this ionchamber by an FFT program was immediately available and was typical of neutron-fluctuation signals for Core # 2 of this reactor. The fluctuating signal from N0042A had been digitized at 1/50 second intervals and stored on tape as a timeseries, each number in the series having fourdecade precision, see Saxe, Mayo and Saxe, (1976). To generate the frequency content of this timeseries, the time-series was convoluted with nineteen separate digital filters (see Table 1). The output time-series from a given filter was squared, summed, and normalized. The results from the nineteen filters could then be plotted as a nineteen point frequency curve. This procedure was repeated for ten blocks of data stored on the tape, each block consisting of 2048 data points forming a time-series. A plot of the running average for each filter output as a function of frequency and of number of data blocks analyzed is shown in Fig. 3. It will be seen that the running average for most filters FREQUENCY
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Fig. 2. Frequency characteristics of three tenth-order digital filters; arbitrary relative gains.
Fig. 3. Running average of filter-generated powerspectral-density as a function of the number of data blocks averaged.
Transform and filtration methods in fluctuation analysis
changes little after three or four blocks of data have been analyzed. In order to investigate the effect of the precision of the original data on the frequency analysis, the original datum was truncated to form a two-digit number and the filtering analysis was re-run. The plot of the PSD, averaged over ten blocks of input data is shown in Fig. 4; also shown in Fig. 4 is the similar curve using the full four-decade input precision for comparison. It will be seen that little change in the calculated frequency curve is apparent. A further reduction in input precision was made so that the entire recording range was defined by 20 integers, and the filtering analysis was performed. The resultant 10 block averaged PSD is shown in Fig. 5(a). Comparison with Fig. 4 shows that very little difference has occurred in the curve due to this reduction in input precision. A still further reduction in input precision so that the entire range is defined by 5 integers gives Fig. 5(b) while a further reduction so that the input is represented by a binary sequence (+1 and - 1 ) gives rise to Fig. 5(c) after analysis. It will be seen that deterioration of the resultant curve occurs for these last two cases, but that even so, the deterioration is not large. A similar investigation of the effects of input precision, using the F F T method of analysis with averaging over 10 blocks of information (20480 digitized points) gives the results shown in Figs. 6-10. It will be seen that the deterioration in PSDs, using the FFY analysis, closely parallels that using filtration techniques. A
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Fig. 4. Filter-generated power-spectral-density for 4decade input resolution and for 2-decade input resolution.
3. D I S C U S S I O N
The above time-domain domain curve method or by
results show that the analysis of a record to produce a frequencymay be performed either by the F F T the use of digital filters and that the
474
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SAXE LO-~_
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7.
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Fig. 9. FET-generated power-spectral-density with 5 integer input resolution.
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Fig. 10. FFT-generated power-spectral-density binary input resolution•
with
Transform and filtration methods in fluctuation analysis FREO,JEN CY - HZ
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Fig 11. Convergence of density estimates. results of using the two methods are closely comparable. The above results also show that, for the type of time-domain record obtained from a power-range ion-chamber on a typical PWR, a considerable relaxation of input data precision may be made without significantly affecting the resultant PSD curve. The PSD curves obtained by the F b T program as shown above exhibit considerably more frequency resolution than those obtained by the filtration
475
method as used here, but we suggest that this advantage is illusory in that it is highly unlikely that a real, physical phenomenon on the reactor could give rise to a resonance requiring such a narrow bandwidth for its resolution. Also, no attempt was made to use filters of a narrower pass-band for the above reason. A further difference between the two methods lies in the relative inflexibility of the analyzing bandwidth of the FFT program across the frequency scale and the complete flexibility of the filtration method to choose the bandwidth at any frequency. The digital filters, as used here, give more rapid convergence, as a function of the number of data blocks analysed, than does the analysis by FFT, as will be seen by a comparison of Fig. 3 above with Fig. 11. The frequency analysis results from a binary input resolution bring to mind the polarity correlation work of Pacilio and the various pseudorandom binary sequence experiments that have been performed and analysed. Acknowledgements--The author acknowledges the award of NSF Grant which made this work possible and also acknowledges Carolina Power and Light Company whose cooperation made possible the collection of the original ion-chamber data on which the above analyses were performed. REFERENCES
Pacilio, N. (1970) Reactor Noise Analysis in the Time Domain, AEC Critical Review Series. Saxe, R. F., C. W. Mayo and T. Saxe, Dynamic Analysis of a Pressurized Water Reactor, in this issue.