High-Speed RMS Converter
Abstract Generated abstract
This paper examines the design of a high-speed converter from alternating voltage to direct voltage proportional to rms value, with emphasis on improving response at low frequencies. It analyzes the role of the averaging filter in an rms converter with feedback, deriving relations between filter time constant, harmonic attenuation, and settling time for multistage filters. The authors argue that passive RC filters provide limited improvement because practical section coupling prevents the ideal equal-pole behavior, while active filters can reduce settling time without sacrificing the specified accuracy. A converter using a sixth-order active filter is described, with reported performance of 0.2 percent error, 0.5 to 5 V dynamic range, 20 Hz to 5 kHz frequency range, and 0.75 s settling time.
Full Text
Proceedings of the Tomsk Polytechnic Institute
of the Order of the October Revolution and the Order of the Red Banner of Labor
named after S. M. Kirov
Vol. 298 1974
UDC 621.314.58
HIGH-SPEED RMS CONVERTER
N. S. BARANOV, Yu. S. OTINOV
(Presented by the scientific seminar of the Department of Radio Engineering)
A converter of alternating voltage into direct voltage is considered, which processes the input signal in accordance with the mathematical relationships for the rms value of the signal. Expressions have been obtained establishing the relationship between the filter time constant, the attenuation coefficient, and the response speed.
It is shown that, in order to increase response speed in the low-frequency region, it is effective to use an active filter.
6 figures, 3 references.
One of the very pressing problems of control-and-measuring technology is the creation of rms converters (RMSCs) with maximum response speed. The block diagram of an idealized RMSC is shown in Fig. 1.
It follows from Fig. 1 that, in order to construct an idealized RMSC, it is necessary to have a squaring device, an averaging filter,
Fig. 1
and a square-rooting unit. All real measuring converters have a finite value of the time constant \(\tau\) (or time constants), and in the block diagram, instead of the integrator, we must include a low-frequency filter (single- or multisection, depending on the number of \(\tau\)’s). At the output we will have a voltage of harmonics whose values are determined by the signal frequency and the characteristics of the filter.
At present there exist many circuit solutions that perform the indicated functions. For the region of medium and low frequencies, one of the most effective solutions is the RMSC considered in [1]. Its block diagram is shown in Fig. 2.
The measured signal is detected by a full-wave detector, whose output signal is fed to a generator of a quadratic dependence. Squaring is implemented by means of a piecewise-linear approximation of the quadratic dependence. The voltage from the output of the squarer is fed to an averaging filter. The inverse-
feedback from the filter output to the squaring device provides the square-root extraction function. The converter considered in [1] is implemented using operational amplifiers, which made it possible to extend the dynamic range in the region of small amplitudes, reduce the temperature error, and increase the linearity of the transfer characteristic of the converter stages.
A distinctive feature of the present work compared with [1] is the increase in the speed of the RMS converter in the low-frequency region while maintaining the achieved accuracy, by replacing the first-order filter with a multistage active filter.
Fig. 2
The presence of feedback in the RMS converter requires that the question of stability not be ignored, since the use of a multistage filter transfers the converter into the class of closed-loop systems of higher order. However, since the transfer coefficient of the feedback circuit is less than unity, there is no danger of instability and, consequently, the restrictions on the order of the filter are removed. This makes it possible to proceed to filter synthesis on the basis of the permissible ripple in the passband and the attainment of the required harmonic suppression. It is true that in this case the RMS converter can be used only in open-loop systems. Since we are interested in high-speed voltage meters, the indicated limitation is not significant.
The expressions given in Fig. 1 are valid for an input voltage with constant amplitude. Of considerable interest is the conversion of an amplitude-modulated signal whose envelope is a function of time. To simplify the discussion, let us consider tonal modulation
\[ U(t)=U_m[1+M\sin\Omega t]\sin\omega t. \]
At the output of the squarer we shall have
\[ U_m^2[1+M\sin\Omega t]^2\frac{1}{2}(1-\cos 2\omega t), \]
where \(M\) is the modulation coefficient.
When choosing the amplitude-frequency characteristic (AFC) for filtering a signal of this kind, it is necessary to ensure:
a) transparency of the filter for the frequency \(\Omega_{\max}\)
\[ \Omega_{\max}\ll\Omega_{\text{cut}}, \]
where \(\Omega_{\max}\) is the maximum frequency of the envelope spectrum;
\(\Omega_{\text{cut}}\) is the cutoff frequency of the filter;
b) attenuation at the lower limiting frequency \(\omega_{\text{l.lim}}\) exceeding the specified value.
At the same time, the shape of the AFC curve must be selected so that the duration of the transient process is as small as possible. In the converter under consideration the filter passband is \(5\ \text{Hz}\). The minimum transient time
of the process can be obtained with sufficiently high accuracy in multistage decoupled or active filters.
Figure 3 shows the circuit of a multistage decoupled passive \(RC\)-filter.
Assuming that the direct-current amplifier (DCA) has infinite gain and finite input resistance, the transfer function will have the form
Fig. 3
\[ \frac{U_2(p)}{U_1(p)} \equiv \frac{U_{n2}(p)}{U_{11}(p)} = \frac{\prod_{i=1}^{n} k_i}{\prod_{i=1}^{n} (1+p\tau_i)} , \tag{1} \]
where \(U_1(p)=U_{11}(p)\), \(U_2(p)=U_{n2}(p)\) are the transforms of the input and output voltages of the filter, respectively; \(n\) is the number of stages.
Since the product \(\prod_{i=1}^{n} k_i\) has no effect on the time and filtering properties of the filter, expression (1) takes the form
\[ \frac{U_2(p)}{U_1(p)} = \frac{1}{\prod_{i=1}^{n} (1+p\tau_i)} . \tag{2} \]
The time constants \(\tau_i=R_{i2}\cdot C_i\) in expressions (1) and (2) are real and positive; therefore the transient response is aperiodic. For the particular case when the time constants are equal, i.e. \(\tau_1=\tau_2=\ldots=\tau_n\), the transfer function takes the form
\[ \frac{U_2(p)}{U_1(p)} = \frac{1}{(1+p\tau)^n}. \tag{3} \]
From expression (3) we find the attenuation coefficient of the second harmonic at the lower operating frequency of the signal at the converter input:
\[ K_{\mathrm{n}}=\left[\sqrt{1+(\tau 2\pi f_{\mathrm{n}})^2}\right]^n , \tag{4} \]
where \(K_{\mathrm{n}}\) is the value of the attenuation coefficient at the frequency \(f_{\mathrm{n}}\); then
\[ \tau=\frac{\sqrt{\sqrt[n]{K_{\mathrm{n}}^2-1}}}{2\pi f_{\mathrm{n}}}. \tag{5} \]
Expression (5) makes it possible to establish the relationship between the filter time constant and the specified attenuation coefficient.
The transient process corresponding to the application of a unit step to the input is described by expression (2)
\[ U_2(t)=1-e^{-\frac{t}{\tau}} \left( 1+\frac{t}{1!\tau} +\frac{t^2}{2!\tau^2} +\ldots+ \frac{t^{n-1}}{(n-1)!\tau^{n-1}} \right). \tag{6} \]
From expression (6) we find the settling time of the filter for the value of \(\tau\) found. By settling time is meant the time during which the output quantity of the filter increases from zero to \(1-\Delta\), where \(\Delta\) is a specified small quantity (in the particular case equal to the basic error), i.e.,
\[ 1-e^{-\frac{t_y}{\tau}}\left(1+\frac{t_y}{1!\tau}+\frac{t_y^2}{2!\tau^2}+\cdots+\frac{t_y^{\,n-1}}{(n-1)!\tau^{\,n-1}}\right)=1-\Delta . \tag{7} \]
On the basis of expressions (5) and (7) one can determine the duration of the transient process for specified values of \(K_{\mathrm{н}}\) and \(\delta\), where \(\delta\) is the basic error.
The duration of the transient process according to the expressions given may be regarded as the minimum possible value for a specified order \(n\). But owing to the fact that the poles of passive \(RC\)-filters are not equal to one another, the duration of the transient process will exceed the minimum expected value.
Fig. 4
Since the use of uncoupled filters leads to complication of the circuit because of the large number of UTCs, the question naturally arises of the possibility of realizing a multistage uncoupled passive \(RC\)-filter, Fig. 4, which would satisfy relations (5) and (7).
Obviously, in order to satisfy the indicated relations it is necessary to observe the condition
\[ \left. \begin{aligned} R_{\mathrm{н}} &\gg R_n \gg R_{n-1}\gg \cdots \gg R_1,\\ C_1 &\gg C_2 \gg C_3 \gg \cdots \gg C_n,\\ \tau_1 &= \tau_2 = \tau_3 = \cdots = \tau_n \end{aligned} \right\} \tag{8} \]
It is practically impossible to satisfy this condition. Therefore, in designing \(RC\)-filters one has to take
\[ \left. \begin{aligned} R_1 &= R_2 = \cdots = R_n,\\ C_1 &= C_2 = \cdots = C_n,\\ \tau_1 &= \tau_2 = \cdots = \tau_n \end{aligned} \right\}. \tag{9} \]
For such a filter, as shown in [3], the filtration coefficient is equal to
\[ K_{\mathrm{н}}\approx \left(\frac{\omega_{\mathrm{н}}R_0C_0}{n^2}\right)^n . \tag{10} \]
It is seen from expression (10) that the dependence of \(K_{\mathrm{н}}\) on the number of filter sections has an extremum. Thus, relation (5) for an uncoupled filter is not attainable. In [3] experimental characteristics are given for a five-section \(RC\)-filter for \(f_{\mathrm{н}}=20\) Hz at \(K_{\mathrm{н}}=3\cdot10^4\); the duration of the transient process is 5.5 sec. Proceeding from the foregoing, the use of a multistage passive \(RC\)-filter gives no gain in settling time in comparison with a coupled \(RC\)-filter.
The presence of an oscillatory process in the filter when an inductance or an active element is introduced into a passive \(RC\)-filter (an active filter is meant) makes it possible to speed up the transient process. A significant gain in this case is possible only in rough measurements. For
For more accurate measurements, the overshoots of the oscillatory transient characteristic must be minimal and the transient-process curve close to aperiodic. Therefore, the gain in the duration of the transient process is small, but, taking into account that implementation of a passive \(RC\) filter is associated with the presence of a large number of amplifiers, the advantages of using active filters in measuring instruments become obvious. In the converter under consideration, in order to increase speed and expand the frequency range in the low-frequency region, a sixth-order active filter is used.
Fig. 5
Fig. 6. — theoretical characteristic;
— experimental characteristic
The converter circuit is shown in Fig. 5, and Fig. 6 gives the experimental and calculated amplitude–frequency characteristics of the filter.
The experimental data for the converter under study are given below.
- Converter error: 0.2%.
- Dynamic voltage range: 0.5–5 V.
- Frequency range: from 20 Hz to 5 kHz.
- Converter settling time: 0.75 s.
REFERENCES
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Ritchie, O. K. A high-speed effective-voltage meter. Elektronika, 1969, No. 20.
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V. A. Ditkin, A. P. Prudnikov. Handbook of Operational Calculus. Moscow, “Vysshaya shkola,” 1965.
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B. I. Shvetskii. Electronic Measuring Instruments with Digital Readout. Kiev, “Tekhnika,” 1970.