UDC 621.316.722.1.025

PRECISION HIGH-SPEED STABILIZERS OF LOW- AND INFRALOW-FREQUENCY VOLTAGES

M. S. ROITMAN, Yu. A. BULATOV, E. S. LITVAK

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

Variants of the construction of voltage stabilizers for low and infralow frequencies with various measuring transducers are considered, as well as the interrelation between the stabilization coefficient, the nonlinear-distortion coefficient, and speed of response.

In order to ensure the basic metrological characteristics of stabilizers of low- and infralow-frequency voltages, it is proposed to use a controlled amplitude detector as the measuring transducer.

Illustrations 8, bibliography 2.

To solve an ever wider range of problems in automation and measurement technology, precision high-speed stabilizers of low- and infralow-frequency voltages are required. The creation of such stabilizers is associated with the need to overcome a number of fundamental difficulties. In the present article an attempt is made to consider these difficulties, to outline ways of overcoming them, and to set forth briefly the practical results of the solutions found.

In the overwhelming majority of cases, because load resistances may vary over wide limits, the stabilizer (at least its output stage) is made in the form of a closed automatic-control system. The output voltage is monitored by a measuring transducer, and the error signal, after amplification, acts on the regulating element.

In all implementations of such stabilizers known in world practice, there is an unambiguous relationship between the coefficients of nonlinear distortion, stabilization, and the time constants of the elements of the device. This prevents obtaining large stabilization coefficients within a single loop and makes it necessary to use elements with large time constants, which sharply reduces the speed of response of the system and increases fluctuation phenomena.

Let us analyze the indicated interrelation. Idealized “linear” transducers of instantaneous and mean values may be represented by the structural diagrams shown in Fig. 1, a, b. All real measuring transducers (MT) have a finite value of the time constant \(\tau\) (or time constants), and in the structural diagram we must, instead of an integrator, include a low-frequency filter (single- or multistage, depending on the number of \(\tau\)). It is obvious that at the output of the MT we shall have a voltage of harmonics whose values are determined by the frequency of the signal, the number of filtering elements, and their time constants. These harmonics, passing through the channel dicta: mid-sentence ends here on the visible page.

autoregulation, act on the regulating element (the multiplying element) in a system with parametric feedback, leading to the appearance of voltage harmonics at the stabilizer output. The multiplying element transposes the spectrum of the control signal (for example, the second harmonic in the spectrum of the control signal produces a third harmonic in the spectrum of the output signal). An analogous situation occurs in compensating-type stabilizers as well, since in them

Fig. 1

a) \(U_m \sin \omega t \rightarrow \dfrac{1}{2}U_m^2(1-\cos 2\omega t) \rightarrow \dfrac{1}{2}U_m^2 \rightarrow \dfrac{U_m}{\sqrt{2}}\)

b) \(U_m \sin \omega t \rightarrow \dfrac{2U_m}{\pi}\left(1-\dfrac{2}{3}\cos 2\omega t-\dfrac{2}{15}\cos 4\omega t\right) \rightarrow \dfrac{2U_m}{\pi}\)

a transformation must in principle be performed: the stabilizing voltage is a parameter of the comparison circuit. To clarify what has been said, let us consider a static system of automatic stabilization of the output voltage with parametric feedback, consisting of a “linear” rms-value converter 1 with a single-section filter 2, a square-root element 3, a comparison circuit 4, a frequency-dependent circuit 5, an amplifier 6, a linear multiplying element 7, and a power amplifier 8 (Fig. 2).

Let us open the system at point \(a\) and apply a voltage \(U_m \sin \omega t\), close in magnitude to the nominal output voltage of the stabilizer. All transformations are shown in the structural diagram (Fig. 2). The comparison circuit is assumed to be unbalanced; the value of the unbalance voltage must satisfy the condition

\[ K_1 \Delta U_0 = U_{\text{bias}}, \]

where \(U_{\text{bias}}\) is the required bias level for normal operation of the multiplying element (for selecting the nominal operating point). The voltage at the stabilizer output is equal to

\[ \begin{aligned} U_{\text{out}}(t)={}& K_m U_{\text{in}} K_2 \left( 1-K_1\frac{\Delta U}{\sqrt{2}U_{\text{bias}}} \right)\sin \omega t + \\ &+K_m U_{\text{in}}K_2 \frac{K_1U_m}{4\sqrt{2}U_{\text{bias}}} \sqrt{ \frac{ \displaystyle\prod_{1}^{q_1}\left[4(\tau_n')^2\omega^2+1\right] }{ \displaystyle(1+4\tau_0^2\omega^2)\prod_{1}^{q_2}(4\tau_n^2\omega^2+1) } } \cdot \sin(\omega t+\varphi_1) - \\ &-K_m U_{\text{in}}K_2 \frac{K_1U_m}{4\sqrt{2}U_{\text{bias}}} \sqrt{ \frac{ \displaystyle\prod_{1}^{q_1}\left[4(\tau_n')^2\omega^2+1\right] }{ \displaystyle(1+4\tau_0^2\omega^2)\prod_{1}^{q_2}(4\tau_n^2\omega^2+1) } } \cdot \sin(3\omega t+\varphi_3). \end{aligned} \tag{1} \]

Fig. 2.

Labels and formulas appearing in the diagram:

\[ u_{\mathrm{ex}}=u_m\sin\omega t \]

\[ u_{\mathrm{out}}=k_r u_1(t) \]

\[ u_1(t) \]

\[ u_2(t) \]

\[ u_3(t) \]

\[ u_4(t) \]

\[ u_5(t) \]

\[ u_6(t) \]

\[ u_7(t) \]

\[ \frac{D(j\omega)}{1+j2\omega\tau}\sin\omega t \]

\[ \frac{k_r u_m}{\sqrt{2}\,U_{cm}} \]

\[ \frac{\cos 2\omega t}{2(1+j2\omega\tau)}D(j\omega) \]

\[ \frac{1}{\sqrt{2}}u_m \left( 1-\frac{\cos 2\omega t}{1+j2\omega\tau} \right) \]

\[ \frac{1}{2}u_m^2\left(1-\cos 2\omega t\right) \]

\[ \frac{1}{2}u_m^2 \left( 1-\frac{\cos 2\omega t}{1+j2\omega\tau_0} \right) \]

\[ \frac{1}{\sqrt{2}}u_m \left( 1-\frac{1}{2}\frac{\cos 2\omega t}{1+j2\omega\tau_0} \right) \]

\[ u_\varphi=\frac{1}{\sqrt{2}}u_m+\Delta u_\varphi \]

\[ u_\varphi+\frac{1}{\sqrt{2}}u_m \frac{\cos 2\omega t}{1+j2\omega\tau_0} \]

\[ \frac{5D(j\omega)}{u_m^2(1+j\omega\tau_1)\,u_m^2(1+j\omega\tau_n)} \]

\[ -\frac{\Delta u}{\sqrt{2}}+\Delta u_\varphi \]

\[ -\frac{1}{\sqrt{2}}u_m \frac{\cos 2\omega t}{2(1+j2\omega\tau)} D(j\omega) \]

\[ k_1\left[ u_0-\frac{\Delta u}{\sqrt{2}} -\frac{u_m}{\sqrt{2}}\frac{1}{2} \frac{\cos 2\omega t}{1+j2\omega\tau_0} \right] \]

\[ k_2\sin\omega t \left[ 1-k_1\frac{\Delta u}{u_{cm}} +\frac{k_1u_m}{u_{cm}}\frac{D(j\omega)}{1+j2\omega\tau} \sin\omega t +\frac{k_1u_m}{u_{cm}}\frac{D(j\omega)}{1+j2\omega\tau} \sin 3\omega t \right] \]

\[ u_0=\frac{u_m}{\sqrt{2}} \frac{\cos 2\omega t}{1+j2\omega\tau_0} \]

\[ \frac{1}{2}u_m^2 \left( 1-\frac{\cos 2\omega t}{1+j2\omega\tau_0} \right) \]

\[ \frac{1}{\sqrt{2}}u_m \left( 1-\frac{1}{2} \frac{\cos 2\omega t}{1+j2\omega\tau_0} \right) \]

\[ \frac{1}{\sqrt{2}}u_m \]

\[ \frac{1}{1+j2\omega\tau_0} \]

\[ \sqrt{\phantom{x}} \]

Block labels:

\[ k_2 \]

\[ k_1 \]

\[ \frac{1}{1+j2\omega\tau_0} \]

\[ 7 \]

\[ 6 \]

\[ 5 \]

\[ 4 \]

\[ 3 \]

\[ 2 \]

\[ 1 \]

The coefficient of nonlinear distortion in the third harmonic is equal to

\[ |\dot K_{r3}| \simeq \frac{K_1 U_m}{4\sqrt{2}\,U_{\mathrm{cm}}} \sqrt{ \frac{\displaystyle \prod_{1}^{q_1}\left[4(\tau_n')^2\omega^2+1\right]} {\displaystyle (1+4\tau_1^2\omega^2)\prod_{1}^{q_2}(4\tau_n^2\omega^2+1)} } = \]

\[ = \frac{G_{\mathrm{st}}}{4} \sqrt{ \frac{\displaystyle \prod_{1}^{q_1}\left[4(\tau_n')^2\omega^2+1\right]} {\displaystyle (1+4\tau_1^2\omega^2)\prod_{1}^{q_2}(4\tau_n^2\omega^2+1)} }, \tag{2} \]

where \(G_{\mathrm{st}}\) is the stabilization coefficient in the system.

We would obtain this expression also with zero tuning, as well as in compensating-type stabilizers.

In the case of using an average-value measuring transducer in the stabilizer, expression (2) takes the form

\[ |\dot K_{r3}| \simeq \frac{G_{\mathrm{st}}}{3} \sqrt{ \frac{\displaystyle \prod_{1}^{q_1}\left[4(\tau_n')^2\omega^2+1\right]} {\displaystyle (1+4\tau_0^2\omega^2)\prod_{1}^{q_2}(4\tau_n^2\omega^2+1)} }. \tag{3} \]

It follows directly from expressions (2) and (3) that, in stabilizers with a closed-loop action chain, for a given \(G_{\mathrm{st}}\), nonlinear distortions depend on the values of the time constants and on the number of filtering elements. The more inertial elements there are, the smaller, other things being equal, is \(K_{r3}\). But if the number of elements is greater than two, we immediately encounter the problem of stability. On the basis of stability conditions, one should strive to use systems of no higher than second order. If \(G_{\mathrm{st}}=200\), \(K_{r3}\leq 0.02\%\) at a frequency of \(20\ \mathrm{Hz}\), then for a first-order system with an average-value measuring transducer the required time constant is

\[ \tau \simeq \frac{G_{\mathrm{st}}}{3K_{r3}\,2\omega}=1325\ \mathrm{s}\ (!), \]

and for a second-order system with equal time constants

\[ \tau_1=\tau_2=\frac{1}{2\omega}\sqrt{\frac{G_{\mathrm{st}}}{3K_{r3}}}=2.5\ \mathrm{s}. \]

It is thus seen that the inertia of the stabilizer is very large and increases still more as the frequency of the voltage being stabilized decreases. The situation does not change if a traditional amplitude-value measuring transducer is used. It has a small charging time constant, but a large discharging time constant.

Consequently, the inertia of all traditional automatic-stabilization systems with small nonlinear distortions sharply worsens their dynamic properties. This circumstance is especially undesirable in the construction of automated monitoring and measuring systems, where questions of response speed acquire primary importance.

Let us consider a stabilizer with an open-loop action chain, the block diagram of which is shown in Fig. 3. The system is invariant to changes in \(U_{\mathrm{in}}\) only on the condition that the sum of the transfer functions for the voltage envelope of the forward path \(K_1(p)\cdot K_{z1}(p)\) and the path

compensations \(K_{\mathrm{IP}}(p) \cdot K_2(p)K_{\Sigma 2}(p)\) is equal to zero. In practice, only partial invariance can be achieved for the steady-state mode, i.e., for slow changes in time of \(U_{\mathrm{in}}\), since it is not possible to equalize the dynamic properties of the channels. The static voltage-stabilization coefficient can be sufficiently large and is limited by the mutual instability of the moduli of the channel transfer coefficients.

The coefficient of nonlinear distortion at the third harmonic in the circuit (Fig. 3), when an rms-value MT is used and \(K_1(p)=1\) (which is in fact the case in all existing implementations), in the low-frequency region is equal to

Fig. 3

\[ K_{\Gamma 3} \simeq \frac{1}{4} \sqrt{ \frac{ \displaystyle \prod_{1}^{q_1}\left[4(\tau_n')^2\omega^2+1\right] }{ \displaystyle (1+4\tau_0^2\omega^2)\prod_{1}^{q_2}\left(4\tau_n^2\omega^2+1\right) } }. \tag{4} \]

From a comparison of (4) with (2) and (3), it is clear that stabilizers with an open action circuit have a great advantage with respect to nonlinear distortion in the low-frequency region and with respect to the stabilization coefficient that is achievable in principle. But such stabilizers are inoperable under variations of the load resistance and, like closed systems, have limited speed of response.

Can the speed of response or the stabilization coefficient be increased without increasing nonlinear distortion? Yes, it can. Let us connect, at the output of the stabilizer, as the first element of the voltage-stabilization system, a frequency-independent sharpening unit, i.e., a unit with relative transfer coefficient equal to

\[ G_g(\Omega)= \frac{dU_{\mathrm{out}}(\Omega)/U_{\mathrm{out}}} {dU_{\mathrm{in}}(\Omega)/U_{\mathrm{in}}} =\mathrm{const}\gg 1. \]

The frequency independence of the transfer coefficient means that the unit is implemented from inertia-free nonlinear elements.

To the output of sharpening unit 1 of the two-sided voltage limiter we connect amplifier 2 and amplitude detector 3, Fig. 4 (4 is the comparison circuit, 5 is the multiplier element). With such a system configuration, as shown in [1], the overall stabilization coefficient of the system increases by \(G_g=\dfrac{3}{2}\dfrac{U_m}{\Delta U}\) times, while the nonlinear distortions practically do not increase. Thus, the introduction of voltage-limiting and amplitude-detection operations makes it possible to overcome to a significant extent the contradiction existing in low-frequency stabilizers between the achievable level of nonlinear distortion and the stability of the output voltage.

The use of a sharpening unit also makes it possible to substantially improve the characteristics of stabilizers with an open-loop control circuit. This concerns not only nonlinear distortions and dynamic properties, but also the fundamentally attainable stabilization coefficient \(G_{\mathrm{st}}(0)\) for slow voltage variations.

Fig. 4

The block diagram of the stabilizer is presented in Fig. 5. The system will be invariant in the quasi-stationary mode when the relation

\[ G_1 - G_g G_{\mathrm{ip}} G_p = 0. \]

is satisfied.

The instability of \(U_{\mathrm{out}}\) is determined by the instability of the elements and is equal to

\[ \frac{\Delta U_{\mathrm{out}}}{U_{\mathrm{out}}} = \frac{\Delta G_1}{G_1} - \left( \frac{\Delta G_g}{G_g} + \frac{1}{G_g}\frac{\Delta G_{\mathrm{ip}}}{G_{\mathrm{ip}}} + \frac{1}{G_g}\frac{\Delta K}{K} + \frac{1}{G_g G_{\mathrm{ip}}}\frac{\Delta G_p}{G_p} \right). \tag{5} \]

It is seen from (5) that, for \(G_g \gg 1\), the influence of the instability of the amplifier, the amplitude detector, and the regulating element is sharply weakened;

Fig. 5

the constancy of the transfer coefficients of the direct channel and of the sharpening unit becomes decisive.

The latter can comparatively easily be made stable. Since the load resistance may fluctuate over wide limits, the advantages of the stabilizing stage with an open-loop control circuit can be realized only in the presence of a very high-quality matching amplifier \((K_c)\) with an output resistance close to zero. The advances achieved in the field of measuring amplifiers, above all through the rational use of

structural methods [2], make it possible to regard this problem as practically solvable.

In order to eliminate the interrelation between \(G_{\text{st}}\) and \(K_{\text{g}}\) in low-frequency stabilizers with a closed-loop action circuit, it is necessary to solve two problems:

1. The change in the transmission coefficient of the controlled element must occur by relay action at the moment when the instantaneous value of the input voltage is equal to zero, and must remain unchanged during the transmission period.

2. Information on the value of the input signal must be obtained in a time shorter than one period.

Transducers of the rms and mean value are, by their very nature, unsuitable for this purpose (at very low frequencies one may use the sampling method and a computing device, but such a complex solution is so far unjustified). Ordinary amplitude detectors are also unsuitable. By further logical reasoning we arrive at a controlled amplitude detector.

As a result of the investigations carried out, we succeeded in solving the indicated problems and in constructing precision low-frequency stabilizers with a closed-loop action circuit (Fig. 6), with an open-loop action circuit (Fig. 7), and of a combined type. As the frequency-independent sharpening unit 1, a two-sided limiter on diodes and a temperature-compensated Zener diode is used. The rectifier bridge serves to prevent recharging of the Zener diode capacitances, which makes it possible substantially to raise the upper limiting frequency of the stabilized voltage. If, however, the stabilizer is intended for operation in the region of low and infralow frequencies, the need for the diode bridge disappears, and as a two-sided limiter it is expedient simply to use two temperature-compensated Zener diodes connected back-to-back in parallel. Such a construction of the sharpening unit makes it possible to obtain a very high stability of the relative transmission coefficient \(G_{\text{g}}\), since at present we know of no more stable nonlinear elements.

Fig. 6

At the resistor \(R\) we obtain the clipped peaks of the sinusoid. After amplification by amplifier 2 the pulses enter the input of the amplitude-

…detector, consisting of diode 3 and capacitor 4. The detector responds only to pulses of positive polarity and must have a small charging time constant and a very large discharging time constant (\(\tau\) times). In order for the amplitude detector to be able to respond to each pulse, capacitor 4 is preliminarily discharged through the briefly opened switch on a field-effect transistor with an insulated gate, 5.

Fig. 7

Figure 8 gives voltage diagrams: \(A\)—at the input of unit 1, \(B\)—after amplifier 2, \(C\)—across capacitor 4. The voltage at the detector output has dips during \(t_{\mathrm{av}}\), \(t_{ef}\), etc., which can lead to distortion of the signal shape at the output of the stabilizer. To eliminate this danger, a filter can be connected after amplifier 6. But such a solution is undesirable, since it leads to a reduction in speed. Therefore, to eliminate the “dips,” a rewriting detector is used, consisting of controlled electronic switch 7 and storage capacitor 8. At the instants \(t_d\), \(t_n\), etc., information about the extreme value of \(U_{\text{out}}(t)\) is “rewritten” (Fig. 8, \(D\)). The “stored” voltage, through amplifier 9, acts on regulating element 10, stabilizing the voltage at the output of the power amplifier (11). Control circuit 12, at the instants \(t_a\), \(t_e\), etc., briefly opens switch 5 and, at \(t_d\), \(t_h\), etc., switch 7. The maximum delay in the system is less than one period. It is easy to see that in the stabilizer there is no longer a strict relationship between \(G_{\mathrm{st}}\) and \(K_{\mathrm{f}}\) in the low-frequency region, which makes it possible to create stabilizers with low nonlinear distortion even at infralow frequencies.

Unfortunately, owing to the presence of delay and the relay nature of the regulating action, the limitation on the permissible value of \(G_{\mathrm{st}}\) imposed by the stability condition remains. Therefore, if one wishes to increase \(G_{\mathrm{st}}\), it is necessary to introduce an additional inertial element into the stabilization circuit and in effect reduce the speed of the system. True, even in this case substantial progress is achieved in comparison with traditional solutions.

The use of a sharpening element in combination with an amplitude-controlled detector and a “rewriting” circuit makes it possible to improve significantly the metrological characteristics of stabilizers with an open action chain (Fig. 7). In principle, such a stabilizer may introduce no additional nonlinear distortions even in the region of very low frequencies, and the delay in the compensating channel does not exceed the duration of one period. From a comparison of Fig. 6 and Fig. 7 it is seen that the greater part of the units coincides and,

therefore, a combined solution in the form of series-connected stabilization stages with a closed action loop and with an open action loop proves especially effective.

Fig. 8

In this case, without greatly increasing the complexity of the system and practically at the same level of nonlinear distortion, it is possible sharply to increase \(G_{\mathrm{st}} = G_{\mathrm{st}1}\cdot G_{\mathrm{st}2}\).

However, it should be noted that, in the stabilizers under consideration, the constancy of \(U_{\mathrm{out}}\) depends on the invariability of the level of the harmonics. Therefore the potential capabilities of such stabilizers can be fully realized only when the nonlinear distortions in the input voltage \(U_{\mathrm{in}}(t)\) and those introduced by the controlled dividers and power amplifiers are negligibly small.

In conclusion, let us consider the device of a precision stabilizer with a guaranteed value of the magnitude of the output voltage. If, in a combined-type stabilizer with a sharpening unit and a controlled amplitude detector, the above-mentioned requirement of small nonlinear distortions is satisfied, and the amplifiers in the direct-transmission path constitute a high-quality DC amplifier with low zero drift, then \(U_{\mathrm{out}}\) will practically not vary with frequency in the low- and infralow-frequency range. Let us connect to the output a comparator for the rms value of the voltage \(U_{\mathrm{out}}\) and, consequently, obtain the possibility, at the upper frequencies where the comparator is operable,

guarantee the value of \(U_{\text{out}}\) itself. And this also makes it possible, at infralow frequencies (where the comparator is already inoperative), to guarantee the value of \(U_{\text{out}}\) with an error not exceeding the frequency distortions of the direct-transmission channel and variations in the influence of nonlinear distortions. Both of these factors can be reduced to values of less than fractions of a percent.

REFERENCES

1. M. S. Roitman. Theory of amplitude-stable low-frequency autogenerators with small nonlinear distortions. Present collection.

2. M. S. Roitman, V. M. Sergeev, V. A. Butenko. Some questions of the theory and practical implementation of structural methods for improving the accuracy of measuring amplifiers. Izvestiya TPI, vol. 270, Tomsk, TSU Publishing House, 1973.