Amplitude-Phase Harmonic Analyzer
Abstract Generated abstract
This paper addresses selective amplitude and phase analysis of harmonic components in electrical signals from nonlinear circuits, focusing on the generator of selecting signals and the synchronous detector as the critical elements of an amplitude-phase harmonic analyzer. It compares three approaches to producing selecting signals, including harmonic multiplication with filters, synthesis from a limited reference signal, and synchronized relaxation frequency multiplication, emphasizing their phase stability, frequency range, and susceptibility to measurement error. The authors describe a transistor-based analyzer using a synchronized multivibrator with symmetry control, operating at 50 and 400 Hz and measuring Fourier coefficients through the sixth harmonic over a 0.25 to 250 V signal range. Example measurements for a half-wave rectified sinusoid are presented to illustrate the analyzer’s performance.
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AMPLITUDE-PHASE HARMONIC ANALYZER
B. A. Solomin, E. I. Goldshtein, V. N. Stanevko
(Presented by the scientific-technical seminar of the Department of Instruments and Devices of Automation Systems)
The problem of studying the harmonic composition of electrical signals in nonlinear circuits requires the creation of amplitude-phase harmonic analyzers (APHA). To evaluate the harmonic composition of the indicated signals, it is sufficient to obtain their coefficients of expansion in a Fourier series. The determination of these coefficients can be carried out by successive selective measurement of individual harmonics in the spectrum of the signal under study. For brevity, we shall call this way of solving the stated problem the method of selective analysis.
The general block diagram of a selective harmonic analyzer is shown in Fig. 1.
Here a sinusoidal signal from the reference-voltage source (RVS), having frequency \(f_0\) and amplitude \(U_0\), is applied simultaneously to the investigated four-terminal network (IFN) and to the generator of selecting signals (GSS), which provides a sinusoidal or rectangular voltage with frequency \(f=n\cdot f_0\) (\(n\) is the number of the harmonic being investigated) and with two strictly defined phases \((\varphi_n\) and \(\varphi'_n)\), satisfying the condition
\[ \varphi_n=\varphi'_n=\frac{\pi}{2}\quad \cap\quad f=f_n. \]
Fig. 1
The signals from the GSS and IFN are fed to a synchronous detector (SD), which provides selective measurement of the required harmonic.
From the obtained measurement results for each harmonic component \((A_n, A'_n)\) of the \(n\)-th harmonic, the Fourier-series coefficients for this harmonic can be determined.
In the circuit in Fig. 1, the most important elements (from the design and metrological point of view) are the GSS and SD, since the RVS and IFN are external elements of the analyzer. Let us consider the principal requirements imposed on the GSS and SD.
The GSS must provide the production of selecting signals with frequency \(f\), a multiple of \(f_0\), in whose spectrum there is no component with
with frequency $f_0$. In practice, the selecting voltage contains a certain amount of first-harmonic voltage. This leads to the appearance of a certain measurement error $\Delta A_{f0}$.
In [1] it is indicated that the rectification coefficient of the first harmonic by a synchronous detector when commutating with higher harmonics is equal to
\[ \frac{\sqrt{2}}{\pi}\cdot \cos \varphi, \]
$q$ is the ratio of the first-harmonic voltage to the commutation voltage;
$\varphi$ is the phase shift between the first harmonics in the measuring and selecting circuits.
The HSS must also ensure high phase stability $(\varphi_n, \varphi'_n)$ of the selecting signals. The SD must ensure minimal inherent nonlinear distortions, have a stable zero, and operate in a vector-measuring mode.
Fig. 2. 1 — reference-voltage source;
2 — $\pi/4$ phase shifter; 3 — $\pi/2$ phase shifter; 4, 5 — limiters
At present a large number of circuit solutions for the SD are known (for example [2]) that satisfy the stated requirements to a considerable degree. The more difficult task is the selection of the HSS.
Three basic methods of constructing an HSS are possible:
a) multiplication of $f_0$ on nonlinear elements, followed by selective extraction of the harmonic $f_n$;
b) synthesis of a rectangular selecting signal $f_n$ from a limited reference voltage $f_0$;
c) use of relaxation pulse frequency multipliers (a brief description of such multipliers is given in [3]).
The first method was used in the analyzer considered in [1]. The principal disadvantages of this method consist in the use of a cumbersome system of resonant filters, which, on the one hand, leads to difficulty in expanding the frequency range of the analyzer and, on the other hand, causes a significant phase error even with a slight drift of the frequency $f_0$ of the reference source.
The essence of the second method is illustrated by the diagrams in Fig. 2,a, where the voltages at various points are shown using as an example a generator of the selecting signal of the second harmonic. The circuit of the generator is shown in Fig. 2,b.
By eliminating the dc component in the voltage \(U_\tau\), we obtain a rectangular selecting signal of the second harmonic, rigidly tied to the phase of the reference voltage. An analogous method can be used to obtain the other harmonics of the selecting voltage as well.
An obvious advantage of this method of creating the HSS is the possibility of rigidly locking the initial phase to the phase of the reference voltage over a frequency range, with acceptable switching of the shaping circuits, and, in addition, a small phase error with a small drift of the frequency \(f_0\). The main requirements in constructing such HSS units are to ensure the symmetry and equality of the limited signals \((U_a, U_b)\), and also to ensure the specified phase shifts with high accuracy.
Fig. 3. 1 — reference-voltage source,
2 — phase shifter \(\pi/2\), 3 — limiter,
4 — differentiating circuit, 5 — synchronized multivibrator
To satisfy these requirements, stabilization of the amplitude of the reference voltage and careful adjustment of the system elements are necessary. In addition, the system is acceptable only for low numbers “\(U\)” of the harmonics under investigation.
The block diagram of an HSS operating according to the third method is shown in Fig. 3,a.
A sinusoidal signal of the reference voltage with a specified phase shift \((\varphi_n\) or \(\varphi'_n)\) is applied to a symmetrical limiter. The rectangular pulses at the limiter output are shortened by a differentiating circuit and are applied as stabilizing pulses to a multivibrator operating in the self-oscillating mode at a frequency close to the frequency of the required harmonic \(f_n\). Under the action of these pulses, the multivibrator oscillations are locked in phase to the harmonic frequency \(f_n\). The synchronization process is illustrated by the diagrams in Fig. 3,b. Here \(U_1\) is the voltage at the limiter output, \(U_2\) is the voltage at the output of the differentiating circuit, and \(U_3\) is the voltage at the output of the multivibrator.
If the natural frequency of the multivibrator \(f_{\mathrm{m}}\) is less than the harmonic frequency \(f_n\), i.e. \(f_{\mathrm{m}}<f_n\), then each \(n\)-th successive negative or positive pulse of the multivibrator will be shortened by an amount \(\Delta t\). It is easy to show that \(\Delta t=n\left(\frac{1}{f_{\mathrm{m}}}-\frac{1}{f_n}\right)\). The shortening of certain multivibrator pulses, which disturbs the symmetry of the output voltage, is equivalent to the appearance in the composition of the selec-
...of the switching voltage of the reference voltage of the fundamental frequency \(f_0\), which can cause an error in measurement.
When a sufficiently stable multivibrator is used, it is possible to eliminate the measurement error arising from the slow drift of the frequency of the RVO. In the proposed circuit, such an error is eliminated by feeding the signal from the multivibrator output to a symmetry meter. In the case of asymmetry of the output voltage, the operator adjusts the multivibrator frequency within small limits, achieving minimum readings of the asymmetry meter.
The advantages of an SSS implemented according to the third method are as follows:
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The possibility of obtaining high harmonic numbers without substantially complicating the circuit.
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The possibility of analyzer operation over a wide set of frequencies \(f_0\).
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Small phase error when the frequency \(f_0\) drifts.
We have developed an amplitude–phase harmonic analyzer with an SSS implemented according to the third method. The general block diagram of this analyzer is shown in Fig. 4.
Fig. 4
The SSS in our analyzer includes a step attenuator \(D_1\) to provide the required amplitude range of the reference voltage, a calibrated phase shifter ensuring a phase shift \(\varphi'_n=\dfrac{\pi}{2}\), a limiter amplifier LA, a synchronized multivibrator MV, a symmetry meter SM, and a shaping amplifier SA, necessary for providing isolation and obtaining the required parameters of the switching signal. The signal channel, in addition to the synchronous detector SD, also includes a calibrated step attenuator \(D_2\).
Table 1
| Fourier-series coefficients | \(a_1\) | \(a_2\) | \(a_3\) | \(a_4\) | \(a_5\) | \(a_6\) | \(b_1\) | \(b_2\) | \(b_3\) | \(b_4\) | \(b_5\) | \(b_6\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Calculated values of the coefficients | 0 | −0.212 | 0 | −0.0421 | 0 | 0.0182 | 0.5 | 0 | 0 | 0 | 0 | 0 |
| Values of the coefficients obtained with the APHA | 0 | −0.2 | 0 | −0.055 | 0 | −0.04 | 0.5 | 0 | 0 | 0 | 0 | 0 |
The analyzer is built with transistors and operates at reference-signal frequencies of 50 and 400 Hz, providing measurement of the expansion coefficients of the first six harmonics of the reference frequency. The amplitude of the reference voltage and of the useful signal can vary in the range 0.25 V–250 V. The synchronous detector is implemented according to the circuit described in [2].
The quality of operation of the harmonic analyzer attained at present, at a reference-voltage frequency of 50 Hz, is illustrated
Table 1 contains the values of the Fourier-series coefficients from the 1st through the 6th harmonic for the curve of half-wave rectification of a sinusoidal voltage.
REFERENCES
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N. S. Roitman, E. I. Tsimbalist, N. P. Fefelov, V. Ya. Supyan. Low-frequency amplitude-phase analyzer for studying relationships between the character of the spectrum and the hardness of a ferromagnetic material. Izv. TPI, vol. 141, pp. 163–165, 1966.
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M. S. Roitman. A simple synchronous detector with a stable zero. Izv. TPI, vol. 141, pp. 163–165, 1966.
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V. F. Samoilov, V. G. Makoveev. Pulse Engineering. Moscow, “Svyaz,” 1964.