J. A. Thie

5- 1 FUNDAMENTAL CONCEPTS

5- 1.1 Time and Frequency

The dynamic behavior of systems can be considered from either of two viewpoints as a function of time (time domain) or as a function of frequency (frequency domain) This dualism is quite natural, especially to mathematicians, because it stems from the well-known Fourier theorem a function of time can be represented by the sum (or integral) of sinusoidal functions of various frequencies. In this chapter both points of view are considered, although in certain specific applications we follow historically devel­oped conventions.

Table 6.1 lists the principal functions of time and frequency used in studying dynamic behavior. The func­tions are given as equivalent pairs, i. e., if one is known, the other can be obtained by computation. Thus we can measure functions in either the time or the frequency domain, whichever is the more convenient, and sub­sequently we can compute the Fourier transform function if it is preferred for purposes of interpretation.

5- 1.2 Transfer Functions

The concept of transfer functions was introduced in reactor plant analysis because of its proven utility in electrical engineering. As defined in Table 6.1, the transfer function is the ratio of output complex amplitude to input complex amplitude, this ratio is a complex number that depends on the amplitude ratio and the phase difference between two sinusoidal signals in a system of two or more dynamically related variables, all of which are oscillating at a given frequency. We may speak of a transfer function between any two variables. However, if the input driving function is oscillatory, it is conventionally used as one of the variables, in which case the transfer function is the output amplitude per unit input amplitude of sine-wave excitation. Several zero-power transfer functions are given in Fig. 6.1

CHAPTER CONTENTS

6-1 Fundamental Concepts………………………………………………………………………………… 140

6-1.1 Time and Frequency……………………………………………………………………… 140

6-1.2 Transfer Functions……………………………………………………………………….. 140

6-1.3 Impulse Response………………………………………………………………………… 141

6-1.4 Spectral Density…………………………………………………………………………… 143

6-1.5 Cross Spectral Density……………………………………………………………….. 144

6-16 Autocorrelation……………………………………………………………………………. 145

6-1.7 Cross Correlation………………………………………………………………………… 145

6-2 Reactor Applications………………………………………………………………………………….. 145

6-2.1 Neutron Kinetics…………………………………………………………………………. 145

6-2 2 Zero-Power Measurements…………………………………………………………… 146

6-2.3 Power-Reactor Feedback……………………………………………………………… 147

6-2.4 Power-Reactor Measurements …………………………………………………….. 148

6-3 Methods of Measurement……………………………………………………………………………. 149

6-3.1 Reactor Excitation………………………………………………………………………… 149

6-3.2 Noise Methods…………………………………………………………………………….. 151

6-3 3 Comparison of Methods………………………………………………………………. 153

6-4 Reactor Excitation Equipment…………………………………………………………………… 154

6-4.1 Excitation Signal…………………………………………………………………………. 154

6-4.2 Control Device……………………………………………………………… .. 155

6-5 Transfer-Function Analyzers………………………………………………………………………. 155

6-5.1 Usage……………………………………………………………………………………………… 155

6-5 2 Null-Balance Analyzer……………………………………………………………….. 156

6-5.3 Synchronous Transfer-Function Analyzer. . . 156

6-5.4 Cross Correlators…………………………………………………………………………. 157

6-5 5 Digital Techniques……………………………………………………………………….. 157

6 Frequency Analyzers………………………………………………………………………………….. 158

6-6.1 Usage……………………………………………………………………………………………… 158

6-6.2 Spectrum Analyzers…………………………………………………………………….. 159

6-6.3 Cross-Spectrum Analyzers………………………………………………………….. 160

6-6.4 Digital Spectrum Analysis………………………………………………………….. 160

6-7 Experimental Considerations ……………………………………………………………………. 161

6-7.1 Error Sources…………………………………………………………………………………. 161

6-7.2 Frequency Limits………………………………………………………………………… 161

6-7.3 Statistical Accuracies…………………………………………………………………… 162

6-7.4 Spectral-Analysis Data Planning……………………………………………….. 162

References…………………………………………………………………………………………………………. 163

Complete specification of a transfer function involves both an amplitude value and a phase difference given as a function of frequency. A complete specification of the dynamics of a system would involve all the transfer functions between all the pairs of variables given for the

entire band of frequencies of physical interest Often, however, the two most meaningful variables are related In reactor dynamics these variables might be the reactivity and the power of a reactor.

Table 6.2 contains a simple example of two variables, x and у (one input and one output), related by a differential equation having one time constant the complex transfer function is (1 + icorc)_1 and has an amplitude (1 + co2r2)~^ and a phase arctan(—corc) or real and imaginary parts of 1/(1 + w2r2)^and —сотс/(1 + согт) respectively.

Since almost all reactor dynamics analyses involve linear systems, linear systems are assumed in this chapter. In a linear system the transfer function at a given frequency is independent of the absolute magnitude used in its measure­ment. Usually a sufficiently large amplitude will cause nonlinear behavior in any system, but these cases are not treated with the techniques discussed in this chapter.

It should be mentioned, however, that the transfer — function concept may be applied to almost-hnear systems. Smets1 has presented a “describing function” approach to nuclear-reactor dynamic measurements

Describing function = (amplitude of fundamental Fourier component of output signal)/(amplitude of sinusoidal input signal) (6 1)

where the input signal is x(t) = a sin cot and the output signal is

y(t) = A! sin(cOit + фі) + A2 sin(co2t + 02) + .. (6.2)

The describing function is thus A!/a If Ai is not linear in a, then the describing function depends on the magnitude of the input amplitude a. If the system is linear, the describing function is synonymous with the transfer function.