Issues in the Design of Amplitude–Phase Analyzers

Yu. K. Rybin

(Presented by the scientific seminar of the Department of Radio Engineering)

Spectral analysis of periodic oscillations by Fourier-transform algorithms (1, 2, 3):

\[ A_n=\frac{2}{T}\int_0^T U(t)\cos n\omega t\,dt, \tag{1} \]

\[ B_n=\frac{2}{T}\int_0^T U(t)\sin n\omega t\,dt, \tag{2} \]

where \(U(t)\) is the input signal, or

\[ C_n=\frac{2}{T}\int_0^T U(t)\cos(n\omega t-\varphi_n)\,dt, \tag{3} \]

\[ C_n=\sqrt{A_n^2+B_n^2}, \qquad \varphi_n=\operatorname{arctg}\frac{B_n}{A_n}, \tag{4} \]

has at least two significant advantages: the first of these consists in the possibility of unlimited improvement of the resolving power of the analysis, and the second in the possibility of obtaining the phase spectrum. Therefore, in recent times, interest has sharply increased in questions of hardware implementation of Fourier-transform algorithms. However, the possibilities of hardware analysis are to a considerable extent limited by how ideally this algorithm is implemented, i.e., by the accuracy of the operation of multiplying the input signal by orthogonal harmonic voltages and by the error of integration.

Below, various variants of hardware implementation of algorithms (1–4) are analyzed according to the criterion of ensuring the minimum measurement error.

If the interval over which the input signal is specified is relatively large and if no special requirements are imposed on the analyzer with respect to analysis time, then the integration operation can be replaced by filtering the result of multiplication with the simplest \(RC\)-filter. Taking this into account, Fig. 1 presents the block diagram of the analyzer. In blocks 1, 2 the input signal is amplified, normalized, and fed to the multipli-

multipliers 3, 3′, is then filtered by low-frequency filters 4, 4′ and, in the form of constant voltages proportional to the real and imaginary components of the measured harmonic, is fed to blocks 5 and 6, in which operations are performed to compute the modulus and argument \(C_n\) and \(\varphi_n\). Block 7 generates orthogonal voltages with the frequency of the measured harmonic, synchronized with the first harmonic of the input reference voltage. According to this structural scheme, analyzers have been implemented,

Fig. 1. 1 — input device; 2 — amplifier; 3, 3′ — multipliers; 4, 4′ — low-frequency filters; 5 — functional converter \(C_n=\sqrt{A_n^2+B_n^2}\); 6 — functional converter \(\varphi_n=\operatorname{arc tg}\dfrac{B_n}{A_n}\); 7 — generator of reference orthogonal voltages; 8 — generator of the fundamental harmonic

described in [1, 2, 3]. In order to simplify the multipliers, synchronous or phase-sensitive detectors operating in switching mode are often used as the latter. In that case algorithms (1, 2) are replaced by the approximate ones

\[ A'_n=\frac{2}{T}\int_0^T U(t)\cdot \frac{4}{\pi}\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\cos(2n-1)\omega t}{2n-1}\,dt, \tag{1′} \]

\[ B'_n=\frac{2}{T}\int_0^T U(t)\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\sin(2n-1)\omega t}{2n-1}\,dt. \tag{2′} \]

This leads to the fact that the coefficients \(A'_n\) and \(B'_n\), and consequently \(C'_n\) and \(\varphi_n\), depend on the content of higher harmonics and their phase relationships. The maximum possible error in determining \(A_n\), \(B_n\), and \(C_n\) can reach [3]

\[ \delta_n=\frac{C'_n-C_n}{C_n} = \frac{\displaystyle\sum_{i=1}^{\infty}\frac{C_{(2i+1)n}}{2i+1}}{C_n}, \tag{5} \]

where

\(C'_n\) is the amplitude measured with allowance for the influence of higher harmonics,
\(C_n\) is the amplitude of the desired harmonic.

In this case there is no phase error, since if \(A_n=0\), then \(\varphi=\pm 90^\circ\), while if \(B_n=0\), then \(\varphi=0^\circ\). The maximum phase error is equal to [4]

\[ \Delta\varphi_{\max}=\pm\arcsin \frac{\displaystyle\sum_{i=1}^{\infty}\frac{C_{(2i+1)n}}{2i+1}}{C_n}. \]

These errors in measuring spectra, whose components decrease inversely proportional to the harmonic number, can under unfavorable conditions reach, respectively,

\[ \delta_n \simeq 23.3\%, \qquad \Delta\varphi_{\max} \simeq 13^\circ 30'. \]

In [4] it is proposed, in order to reduce the errors \(\delta\) and \(\Delta\varphi\), that the analyzer be made according to the principle of parallel analysis and that the results of measuring the first harmonics be corrected using the results of measuring the higher harmonics. Such a correction method can also be extended to an analyzer constructed according to the principle of sequential analysis. The essence of the correction consists in measuring \(k\) harmonics and further processing the measurement results in accordance with (6 and 7):

\[ A_n = A'_n + \frac{1}{3}A'_{3n} - \frac{1}{5}A'_{5n} + \ldots \pm \frac{A'_{(2i-1)n}}{2i-1} \quad \begin{gathered} \text{for}\\ (2i-1) \ne (2j-1)^2 \end{gathered} \tag{6} \]

\[ B_n = B'_n - \frac{1}{3}B'_{3n} - \frac{1}{5}B'_{5n} - \ldots - \frac{B'_{(2i-1)n}}{2i-1} \quad \begin{gathered} \text{for}\\ (2i-1) \ne (2j-1)^2,\ \text{where } j=2,3,4\ldots \end{gathered} \tag{7} \]

Thus, to correct the result of measuring the first harmonic it is necessary to measure the third, fifth, etc. harmonics and to sum the measurement results of these harmonics, with a certain weight and sign, with the result of measuring the first harmonic.

However, such a path leads to a complication of the analyzer circuit (in parallel analysis) or to an increase in the analysis time (in sequential analysis), and, most importantly, it does not make it possible to correct the measurement results of all harmonics.

Fig. 2a. \(1'\), \(1''\), \(1'''\)—switch multipliers; \(2'\), \(2''\), \(2'''\)—weighting-coefficient blocks; 3—summer.

Fig. 2b.

A considerably smaller error from the influence of neighboring harmonics can be ensured if, as the multiplier, a synchronous detector is used, implemented on several switching elements, the voltages applied to which arrive with definite phases, while the output signals from them are summed at the detector output with a definite weight, thereby making it possible in the general case to eliminate the influence of \(n\)-odd harmonics. For example, for the synchronous detector shown in Fig. 2a, b, the multiplying function is written in the form

\[ U(t)=\frac{4}{\pi}\sum_{n=1}^{\infty}\cdot \]

\[ \cdot \frac{ \dfrac{1}{3}+\dfrac{1}{\sqrt{3}}\cos(2n-1)\dfrac{\pi}{6} +\dfrac{1}{3}\cos(2n-1)\dfrac{\pi}{3} }{ 2n-1 } \sin(2n-1)\omega t. \tag{8} \]

In this function, the coefficients for even harmonics, and also for \(\sin 3\omega t\), \(\sin 5\omega t\), \(\sin 7\omega t\), and \(\sin 9\omega t\), are equal to zero. Consequently, when an input signal with a sufficiently broad spectrum is applied to a synchronous detector, its output signal is determined practically only by the harmonic being measured, thereby achieving high selectivity of the multiplier with respect both to the even and to certain odd harmonics of the signal being measured. Obviously, on this principle one can construct a multiplier that is insensitive to any preassigned harmonics; in this case the multiplier and the circuit for obtaining the reference voltages become somewhat more complicated, which, however, is quite permissible when measuring signals with a rapidly decreasing spectrum. For measuring the harmonic components of a signal whose spectrum decreases slowly, however, the multiplier becomes unacceptably complex.

In the situation that has arisen, a way out can be found only by using multipliers with a multiplying function close to a harmonic one, or by shifting the spectrum in order to eliminate the influence of harmonics. Leaving aside the second method (we shall only note that it has independent significance), let us consider multipliers with a harmonic multiplying function. The simplest such multiplier can be implemented on the basis of a dc differential amplifier. It is shown in [5] that the selectivity with respect to the harmonics of the input signal, both even and odd, can be sufficiently high and is determined by the magnitude of the reference voltage. Thus, even for \(U_{op} \le 50\) mV the rectification coefficient of the third harmonic is less than 6%, and with the introduction of negative feedback with respect to the reference voltage [6] it can be reduced still further, to the depth of the feedback. Thus, the use of [6] as a synchronous-detector multiplier makes it possible practically completely to eliminate the influence of harmonics on the measurement error. High selectivity in such a multiplier is possible only in the case of a purely sinusoidal reference voltage; therefore block 7 for forming the reference voltages must generate orthogonal sinusoidal voltages of multiple frequencies with a low harmonic content.

A survey of existing frequency-multiplication methods makes it possible to identify two methods suitable in the present case, which make it possible to obtain a multiplication coefficient switchable over wide limits while maintaining a sufficiently sinusoidal form of the output voltages.

1. Preliminary distortion of the reference voltage of the fundamental harmonic, followed by extraction of the required harmonic by a selective filter.

2. Use of a phase-locked loop system (PLL).

The first method is less preferable because of the difficulties arising in the creation of selective amplifiers with high equivalent \(Q\) and a stable gain in the low-frequency range, and also because of the phase errors that arise in this case. Therefore the only solution truly suitable in the present case is the use of phase locking. In such a system, not only frequency multiplication but also the generation of two orthogonal voltages is solved very simply if a two-phase generator is used as the tunable element. By switching the tuning frequency of the two-phase generator discretely, one can obtain oscillations with a frequency that is a multiple of the frequency of the fundamental harmonic,

and by jointly retuning the generators it is possible to vary their frequency smoothly. As the two-phase generator, any two-phase generator is used, for example, a generator based on an oscillatory-circuit model.

It follows from (4) that determining the amplitude and phase reduces to carrying out operations that are sufficiently simple mathematically and that are very easily performed manually with the aid of a slide rule. The hardware implementation of these algorithms, however, involves considerable difficulties; this applies especially to implementation of the function \(\operatorname{arctg}\). This is due primarily to the fact that the domain of definition of the function \(\operatorname{arctg} x\) extends from \(-\infty\) to \(+\infty\); therefore, in the neighborhood of the angles \(\pm \dfrac{\pi}{2}\), the error in determining the phase may be very large. In [4], in order to reduce the error in phase measurement, it is proposed to restrict the domain of definition of the function \(\operatorname{arctg} x\) to the interval from 0 to 1. This made it possible to simplify the implementation of the functional converter and to reduce the measurement error, but at the cost of a substantial complication of the analyzer as a whole. But this approach, too, is scarcely acceptable, since, in general, the performance of any functional transformations is usually associated only with an increase in error. From this point of view, when determining amplitude and phase it is expedient to dispense entirely with the functional transformations (4). For this it is necessary to use, for example, the algorithm

Fig. 3. 1, 2, 3, \(3'\), 4, \(4'\), 7, 8 are the same as in the circuit of Fig. 1; 5 — quadrature phase splitter; 6 — phase shifter; 9 — automatic-control system.

\[ C_n = \operatorname{MAX}\,\frac{2}{T}\int_0^T U(t)\cos(n\omega t-\varphi_n)\,dt. \tag{9} \]

Obviously, in order to implement this algorithm it is necessary, by varying the phase of the reference voltage, to obtain the maximum value of \(C_n\). In this case the phase of the harmonic can be determined by means of a calibrated phase shifter. An analyzer based on algorithm (9) was proposed in [7].

The methodological error in determining the phase in such an analyzer depends on the static error of the extremal system:

\[ \Delta\varphi = \operatorname{arctg}\sqrt{\frac{\delta}{1-\delta}}, \]

where \(\Delta\varphi\) is the error in determining the phase,
\(\delta\) is the static error of the extremal-control system,

and may be rather large. Thus, already at \(\delta = 1\%\), \(\Delta \varphi\) is \(5^\circ 40'\). To reduce the systematic error in phase measurement, it is advisable to implement the analyzer according to the structural scheme shown in Fig. 3. A feature of this scheme is that control of the phase shifter is carried out from the output of the multiplier of the quadrature channel. In this case, the automatic-control system is substantially simplified.

The systematic error in determining the phase in the analyzer is given by the formula

\[ \Delta \varphi = \pm \arcsin \delta, \]

where \(\delta\) is the static error of the automatic-control system; and, at \(\delta = 1\%\), \(\Delta \varphi\) is only \(0^\circ 35'\). The example given shows the advantage of such an analyzer over the one considered above.

REFERENCES

1. M. S. Roitman, Yu. K. Rybin, Yu. G. Svinolupov. Low-frequency amplitude-phase spectrum analyzer. In: Nondestructive Testing of Ferromagnetic Materials and Products by the Method of Higher Harmonics. Tomsk, 1971.

2. L. M. Rakhovich. Scheme for analyzing the spectrum of periodic and nearly periodic signals. Proceedings of the NTC LEIS, L., LEIS Publishing House, 1962.

3. V. G. Kriksunov, A. S. Tetelbaum. On one method for determining Fourier coefficients. Izvestiya Vuzov, “Radioelectronics,” 1969, No. 12.

4. A. S. Tetelbaum. Investigation of methods of complex spectral analysis based on the use of computing technology. Dissertation, Kiev, 1971.

5. M. S. Roitman, Yu. K. Rybin. Broadband high-selectivity phase detector. In: Electronic Technology in Automation. M., issue 4, 1973.

6. Yu. K. Rybin. Synchronous detector. Positive decision on application No. 1635302/26-9 of April 27, 1972.

7. A. S. Tetelbaum. Resonatorless spectrum analyzer of electrical signals. USSR Author’s Certificate No. 321768, class 6/01 r 23/16.