Proceedings of the Tomsk Polytechnic Institute, awarded the Order of the October Revolution and the Order of the Red Banner of Labor, named after S. M. Kirov

Vol. 298
1974

UDC 621.317.765

THEORY OF AMPLITUDE-STABLE LOW-FREQUENCY OSCILLATORS WITH SMALL NONLINEAR DISTORTIONS

M. S. Roitman

The interrelations among nonlinear distortions, the stabilization coefficient, stability, and the duration of transient processes in low-frequency amplitude-stable oscillators are analyzed. Ways of overcoming this interrelation and of creating oscillators with stable oscillation amplitude and a low level of higher harmonics are considered.
9 figures, 15 references.

An oscillator with a single nonlinear frequency-independent element can be reduced to the block diagram shown in Fig. 1, where 1 and 3 are linear elements with transfer coefficients \(\beta_n(\omega)\), \(K_y(\omega)\), and 2 is a nonlinear element, the linearized (averaged) value of whose transfer coefficient is equal to

\[ K_{\mathrm n}(U)=\frac{U_{m\text{out}}\sin\omega t}{U_m\sin\omega t} = \frac{ \left\{ \frac{1}{2\pi} \int_0^{2\pi} K_{\mathrm n}\,[U_m\sin\omega t]\sin\omega t\,d\omega t \right\} U_m\sin\omega t }{ U_m\sin\omega t }. \tag{1} \]

The system will oscillate at frequencies \(\omega\) for which the well-known phase-balance conditions are satisfied,

\[ \varphi_y(\omega)+\varphi_\beta(\omega)=2n\pi\quad(n=0,1,2\ldots), \tag{2} \]

and the amplitude condition

\[ K_{\mathrm n}(U)\cdot K_y(\omega)\,\beta_n(\omega)-1=0. \tag{3} \]

Fig. 1

If conditions (2), (3) are satisfied only at one frequency, then the oscillations will be quasi-sinusoidal. The oscillation frequency \((\omega_0)\), in the first approximation, is determined from (2), and the oscillation amplitude from (3). It is easy to see that, for stability of the stationary regime, the condition

\[ \frac{dK_{\mathrm n}(U)}{dU}<0 \tag{4} \]

must be satisfied.

Formulas (1), (2), (3) are valid regardless of where the nonlinear element is located (provided, of course, that the nonlinearity of the input and output resistances of this element has little effect on the system, as is usually sought in practice). Considering the oscillator as

amplifier with feedback, we can, in the first approximation, determine the harmonic coefficients [1, 2]

\[ \dot K_{nг}= \frac{K_n \dot K_y(n\omega)/\dot K_y(\omega)} {1-K_{\mathrm{н}}(U)\dot\beta_n(n\omega)\dot K_y(n\omega)}, \tag{5} \]

where \(K_n\) is the coefficient of the \(n\)-th harmonic at the output of the nonlinear element (NE); \(\dot K_y(n\omega)\), \(\dot\beta_n(n\omega)\) are the transfer coefficients for the \(n\)-th harmonic.

Expression (5) is valid only if \(\dfrac{K_y(\omega)}{K_y(n\omega)}>1\), or if the NE has no sections on which \(K_{\mathrm{н}}[u(t)]\ll K_{\mathrm{н}}(U)\). Otherwise, in (5) one should use the instantaneous value of the transfer coefficient \(K_{\mathrm{н}}[U(t)]\). Neglecting this important factor may lead to a substantial error. As an example, let us consider an oscillator with a Wien bridge, whose circuit is shown in Fig. 2. Let us assume that an ideal two-sided limiter is used as the NE and that the clipping angle is equal to

\[ \gamma=0.2\left(\gamma\approx \sqrt{\frac{2\delta U}{U_m}}\right), \]

Fig. 2

\(U_m\) is the amplitude of the voltages at the limiter input, \(\delta U\) is the value of the clipped “top,” and \(K_y=300\). The averaged value of the NE transfer coefficient is

\[ K_{\mathrm{н}}(U)\approx 1-\frac{8\gamma}{3\pi}\frac{\delta U}{U_m} =1-0.0034; \]

the instantaneous value of the transfer coefficient on the limiting section is

\[ K_{\mathrm{н}}[U(t)]=0 \]

(for an ideal limiter) and \(K[U(t)]=1\) outside this section; the coefficient of the 3rd harmonic at the NE output is

\[ K_3\approx 1.2\sqrt{\left(\frac{\delta U}{U_m}\right)^3}\approx 0.34\%. \]

The coefficient of the \(n\)-th harmonic for the oscillator structure under consideration is, in general form, equal to

\[ \dot K_{nг}\approx \frac{K_n} {1+K_y\dot\beta_0(n\omega) \left[ 1-\frac{\dot\beta_n(n\omega)}{\dot\beta_0(n\omega)} \right]\big|_{K_{\mathrm{н}}[U(t)]=1}} + \frac{K_n} {\left[ 1-\frac{\dot\beta_n(n\omega)}{\dot\beta_0(n\omega)} \right]\big|_{K_{\mathrm{н}}[U(t)]=0}}. \tag{6} \]

From this we obtain \(K_{3г}\approx 1.5\), \(K_3\approx 0.5\%\).

If instead we used the linearized transfer coefficient \(K_{\mathrm{н}}(U)\), we would obtain

\[ \dot K_{nг}= \frac{K_n} {1+K_{\mathrm{н}}(U)K_y\dot\beta_0(n\omega) \left[ 1-\frac{\dot\beta_n(n\omega)}{\dot\beta_0(n\omega)} \right]} \tag{7} \]

and \(K_{3г}<0.01\%\), which is substantially smaller than the real value.

In most cases of analysis of amplitude-stable oscillators, the first approximation is quite sufficient. But when the nonlinearity

of the volt-ampere characteristic of the nonlinear element is even, finding a refined solution is fundamentally necessary. A rough criterion for the obligatoriness of the refined solution may be the relation \(K_{2\mathrm{g}}^{2}>K_{3\mathrm{g}}\). The order of estimating the corrections to the amplitude and frequency of the fundamental harmonic of the oscillations, as well as to the magnitude of the 3rd harmonic, is given in [3] and is not considered here.

It follows from (3) that variation of the transmission coefficient of any stage leads to a change in the amplitude of the oscillations. For example, the relative instability of the gain \(\Delta K_{\mathrm{y}}/K_{\mathrm{y}}\) will cause a voltage deviation equal to

\[ \frac{\Delta U}{U} = \frac{\Delta K_{\mathrm{y}}/K_{\mathrm{y}}} {\dfrac{dK_{\mathrm{n}}(U)}{K_{\mathrm{n}}(U)} \left|\dfrac{dU}{U}\right|} = \frac{\Delta K_{\mathrm{y}}}{K_{\mathrm{y}}}\cdot \frac{1}{G_{\mathrm{st.y}}}, \tag{8} \]

where \(G_{\mathrm{st.y}}\) is the stabilization coefficient. To ensure constancy of the voltage at the generator output, one must strive to use stages with minimal relative instability and to obtain as large a value as possible of \(G_{\mathrm{st}}\). The practical realization of the latter obvious requirement is associated with very great difficulties because of the interrelations among \(G_{\mathrm{st}}\), the coefficient of nonlinear distortions, stability, and the duration of transient processes. Let us try to clarify these interrelations and indicate ways of overcoming them.

Directly from (3) and (4) there follows the well-known truth that stable generation without the presence of a nonlinear stage is fundamentally impossible. The transmission coefficient of a nonlinear element may be a function of instantaneous or integral (mean, rms, etc.) voltage values. In the first case we are dealing with inertia-free nonlinear elements (usually they are simply called nonlinear elements), and in the second case with inertial nonlinear elements (INE). The transmission coefficient of a nonlinear element \(K[u(t)]\) may be represented in the form of a Maclaurin series:

\[ K[u(t)] = K + \frac{u(t)}{1!}\frac{dK}{dU} + \frac{u^{2}(t)}{2!}\frac{d^{2}K}{dU^{2}} + \ldots + \frac{[u(t)]^{i}}{i!}\frac{d^{i}K}{dU^{i}} . \tag{9} \]

For electronic elements, the series (9) is usually rapidly convergent, and in a qualitative analysis of self-oscillators it is permissible to restrict oneself to the first three terms, which give a picture of the nonlinear element (constant, even, and odd components).

Thus, approximating the transmission coefficient of the nonlinear stage in the form of the series (9) and applying \(U_{m}\sin\omega t\) to its input, it is easy to find the linearized value \(K_{\mathrm{n}}(U)\).

\[ K_{\mathrm{n}}(U)=K_{\mathrm{n}}-\frac{3}{8}\cdot \frac{d^{2}K_{\mathrm{n}}[u(0)]}{dU^{2}}\,U_{m}^{2}. \tag{10} \]

The terms of the series (9) with odd derivatives do not contribute to the averaged value of the transmission coefficient (in an open-loop system), but only lead to the appearance of even harmonics at the generator output. Hence it follows that it is expedient to use nonlinear elements whose characteristics have only even derivatives, and these are negative (for satisfying the stability condition (4)).

Suppose that we have fulfilled these requirements. Then we obtain

\[ K_{\mathrm{n}}(U)=K_{\mathrm{n}}-\frac{3}{8} \frac{d^{2}K_{\mathrm{n}}[U(0)]}{dU^{2}}\,U_{m}^{2}. \tag{11} \]

The coefficient of the 3rd harmonic at the output of the NE is

\[ K_3 \approx \frac{1}{8K_{\mathrm n}}\cdot \frac{d^2 K_{\mathrm n}[u(0)]}{dU^2}\,U_m^2 \tag{12} \]

and at the generator output

\[ \dot K_{3\mathrm g}= \frac{[1/8\,K_{\mathrm n}]\,\dfrac{d^2K_{\mathrm n}}{dU^2}\,[U(0)]\,[\dot K_y(n\omega)/\dot K_y(\omega)]\,U_m^2} {1-K_{\mathrm n}(U)\dot K_y(n\omega)\dot\beta_n(n\omega)}; \tag{13} \]

\[ G_{\mathrm{st}}= \frac{dK_{\mathrm n}(U)/K_{\mathrm n}(U)}{dU/U} = \frac{3}{4}\cdot \frac{\{d^2K_{\mathrm n}[u(0)]/dU^2\}\,U_m^2}{K_{\mathrm n}}. \tag{14} \]

In self-oscillators with inertialess NEs, if relation (9) is satisfied, there is no danger of instability, since, in terms of the oscillation envelope, such a self-oscillator is a first-order inertial element with a certain equivalent time constant \(\tau_{\mathrm e}\). For a nonstationary regime the generated voltage is equal to
\(u_{\mathrm{out}}(t)\simeq U_{m0}e^{\alpha_{\mathrm e}t}\sin[\omega_0t-\varphi_0(t)]\),

where

\[ \alpha_{\mathrm e}=[K_{\mathrm n}(U)K_y(\omega)\beta_n(\omega)-1]\frac{1}{\tau_{\mathrm e}} =\frac{1}{\tau_{\mathrm e}^{1}}. \tag{15} \]

In the steady-state regime,

\[ \alpha_{\mathrm e y}=[K_{\mathrm n}(U_0)K_y(\omega)\beta(\omega)-1]\frac{1}{\tau_{\mathrm e}}=0. \]

And, consequently, (15) can be rewritten in the form

\[ \alpha_{\mathrm e}=[K_{\mathrm n}(U)-K_{\mathrm n}(U_0)]\, \frac{K_y(\omega)\beta(\omega)}{\tau_{\mathrm e}}. \tag{16} \]

In the absence of oscillations \(K_{\mathrm n}(0)-K_{\mathrm n}(U_0)>0\), and the ratio

\[ \frac{K_{\mathrm n}(0)-K_{\mathrm n}(U_0)}{K_{\mathrm n}(0)} \tag{17} \]

characterizes the self-excitation margin of the self-oscillator and, for a given value of \(\tau_{\mathrm e}\), the duration of the process of oscillation build-up. Substituting into (17) the value of \(K_{\mathrm n}(U)\) from (11), we obtain

\[ \frac{K_{\mathrm n}(0)-K_{\mathrm n}(U_0)}{K_{\mathrm n}(0)} = \frac{3}{8}\, \frac{\{dK_{\mathrm n}^{2}[U(0)]/dU^2\}\cdot U_{m0}^{2}}{K_{\mathrm n}}. \tag{18} \]

Comparison of (12), (13), (14), (15), and (18) clearly shows that in self-oscillators based on wideband amplifiers \((K_y(\omega)\simeq \mathrm{const})\) and inertialess nonlinear elements, the stabilization coefficient, nonlinear distortions, and the duration of transient processes are interrelated. Taking this interrelation into account, for a comparative assessment of the quality of various implementations of self-oscillators, we introduce the coefficient

\[ \lambda=K_{3\mathrm g}\frac{\Delta_{\mathrm e}U_{\mathrm{out}}}{U_{\mathrm{out}}} = K_{3\mathrm g}\left( \frac{1}{G_{\mathrm{st}.y}}\cdot\frac{\Delta K_y}{K_y} + \frac{1}{G_{\mathrm{st}.\beta}}\frac{\Delta\beta}{\beta} \right). \tag{19} \]

For a self-oscillator assembled according to the structural scheme shown in Fig. 2, with an NE of the relay type,

\[ \lambda \approx \frac{ \dfrac{1}{8K_{\mathrm n}}\, \dfrac{d^2K_{\mathrm n}[U(0)]}{dU^2}\, U_m^2 }{ 1+K_{\mathrm n}(U)\cdot K_y\cdot\beta_0 \left[ 1-\dfrac{\dot\beta_n(3\omega)}{\beta_0} \right] } \times \]

\[ K_{\mathrm{н}} \frac{ \left( \dfrac{\Delta K_y}{K_y} + \dfrac{\Delta \beta}{\beta} K_y \beta_0 K_{\mathrm{н}}(U) \right) }{ \frac{3}{4}\dfrac{d^2 K_{\mathrm{н}}}{dU^2}\,\left|U(0)\right| \cdot U_m^2 } \approx \frac{ \dfrac{\Delta K_y}{K_y} + \dfrac{\Delta \beta}{\beta} K_y \beta_0 K_{\mathrm{н}}(U) }{ K_{\mathrm{н}}(U)\,6K_y\beta_0 \left|1-\dfrac{\dot{\beta}_n(3\omega)}{\beta_0}\right| } = \frac{ \dfrac{\Delta K_{y.\mathrm{feedback}}}{K_{y.\mathrm{feedback}}} + \dfrac{\Delta \beta}{\beta} }{ 6\left|1-\dfrac{\beta_n(3\omega)}{\beta_0}\right| }. \tag{20} \]

For the Wien bridge \(\beta_0=\dfrac{1}{3}\) and

\[ \dot{\beta}_n(\omega)= \frac{\omega/\omega_0}{3\omega/\omega_0+\left(\dfrac{\omega}{\omega_0}\right)^2 j-j}. \]

We would obtain exactly the same result if an NE were introduced into the feedback circuit. \(\lambda\) can be reduced by lowering the instability of the amplifier (provided that the parameters of the feedback circuits are stable) and by synthesizing such a frequency-dependent circuit that would make it possible to obtain positive feedback at the quasi-resonance frequency and, possibly, deeper negative feedback at the harmonics. In other words, it is desirable that at the harmonic frequencies the phase shift of the circuit \(\dot{\beta}_n(\omega)\) change by \(180^\circ\). Unfortunately, when the potential possibilities are implemented in self-oscillators, the practical gain from synthesizing the indicated circuit is small (at most by a factor of \(2\div 3\)). This is explained by the fact that if in the amplifier we have already introduced the maximum permissible negative feedback \((K_y\beta_0=F_{\mathrm{perm}})\), then a further increase in the depth of the overall feedback at the higher harmonics will lead to instability of the amplifier. The indicated limitation would be removed by synthesizing a positive-feedback circuit satisfying the conditions \(\beta_n(n\omega)\gg 1\) and \(\varphi_n(n\omega)\sim 180^\circ\) for \(n=2,3,\ldots,7\), and \(\beta_n(n\omega)/n_{>8}\to 0\). We do not yet have such circuits.

Fig. 3

In the self-oscillator shown in Fig. 3, generation will occur in the case of nonzero tuning of the TT-\(RC\) bridge, when the phase shift of the feedback circuit is \(\varphi_\beta \simeq 180^\circ\).

\[ \dot{\beta}(\omega)= \frac{ \left(\dfrac{\omega_0}{\omega}-\dfrac{\omega}{\omega_0}\right) + j\left(n+\dfrac{n}{m}-1\right) }{ \left(\dfrac{\omega_0}{\omega}-\dfrac{\omega}{\omega_0}\right) + j\left(n-\dfrac{1}{m}+\dfrac{1}{m}+\dfrac{1}{n}\right) }, \]

\[ m=\frac{R_2}{R_1}=\frac{C_1}{C_2}, \qquad n=\frac{R_3}{R_1}=\frac{C_1}{C_3}, \]

and the coefficient

\[ \lambda \approx \left| \frac{ \dfrac{\Delta K_y}{K_y} + \dfrac{\Delta\beta}{\beta}K_yK_{\mathrm{н}} }{ 1+6\dfrac{K_yK_{\mathrm{н}}}{1-j\dfrac{4}{\dfrac{\omega}{\omega_0}-\dfrac{\omega_0}{\omega}}} } \right| \approx \left( \frac{\dfrac{\Delta K_y}{K_y}}{6K_yK_{\mathrm{н}}} + \frac{\Delta\beta}{\beta} \right)1.8. \tag{21} \]

Depending on the character of the nonlinear element (NE) used, \(\lambda\) changes somewhat. For example, by including in the \(\beta_0\) circuit of the self-oscillator with a Wien bridge a two-sided limiter in parallel with the resistor \(R_1\), we obtain \(K_{\text{zg}}\approx\)

\[ \approx 1.8 \left|\sqrt{\left(\frac{\delta U}{U_m}\right)^3}\right|,\quad G_{\text{st.u}}\approx K_y\beta_0(U)\frac{4\sqrt{2}}{\pi}\left|\sqrt{\frac{\delta U}{U_m}}\right| \]

\[ \text{and}\quad \lambda=\frac{\delta U}{U_m} \left( \frac{\Delta K_y}{K_y}\cdot\frac{1}{K_y\beta_0(U)} +\frac{\Delta\beta}{\beta} \right). \tag{22} \]

Comparison of (22) and (20) shows that, at small clipping angles, a relay-type NE makes it possible to obtain better results. But the gain cannot be large, since the minimum value of \(\frac{\delta U}{U_m}\) is limited by the condition

\[ \frac{\delta U}{U_m}> \frac{ \displaystyle \frac{\Delta K_y}{K_y}\cdot\frac{1}{K_y\beta_0} +\frac{\Delta\beta}{\beta} }{ \displaystyle \frac{4\sqrt{2}}{\pi}\left|\sqrt{\frac{\delta U}{U}}\right| }. \tag{23} \]

Small nonlinear distortions can be obtained if the amplifier has a stable transmission coefficient at the oscillation frequency and filtering capabilities at the harmonics. Attenuation of the harmonics can be carried out in two essentially different ways:

a) use a highly selective amplifier, at whose quasi-resonance frequency generation takes place (there are no other frequency-dependent links in the system);

b) use a low-pass filter which simultaneously serves to obtain the phase-balance condition.

Fig. 4

Known selective \(RC\)-amplifiers with feedbacks do not satisfy the stated conditions. First, even the very realization of a large harmonic suppression

\[ \frac{K(\omega)}{K(3\omega)}>1000 \]

is substantially complicated by possible instability. Second, special measures have to be taken to reduce the amplifier’s own nonlinear distortions \((K_{fy})\). Third, an increase in \(Q_e\) inevitably leads to an increase in the instability of the transmission coefficient at the quasi-resonance frequency. To confirm what has been said, let us consider the self-oscillator with a selective amplifier shown in Fig. 4. The coefficient

\[ \lambda\approx \frac{ \displaystyle \frac{1}{8K_n}\cdot \frac{d^2K_n[U(0)]\,U_m^2}{dU^2} \cdot \frac{ K_{\gamma1}\{1+K[\beta_0-\beta_n(\omega)]\} }{ K_{\gamma2}\{1+K[\beta_0-\dot{\beta}_n(3\omega)]\} } }{ \displaystyle [1-K_n(U)\cdot K_y(3\omega)]\cdot \frac{3}{4K_n}\cdot \frac{d^2K_n[U(0)]\,U_m^2}{dU^2} } \times \]

\[ \times \frac{ \displaystyle \frac{\Delta K}{K}+\frac{\Delta\beta}{\beta}K\beta }{ \displaystyle 1+K[\beta_0-\beta_n(\omega)] } \approx \]

\[ \approx \frac{1}{6}\, \frac{1}{\left|1+K\beta_0\left[1-\frac{\dot{\beta}_n(3\omega)}{\beta_0}\right]\right|} \left|\frac{\Delta K}{K}+\frac{\Delta\beta}{\beta}K\beta_0\right| \approx \]

\[ \approx \frac{1}{6}\, \frac{\dfrac{\Delta K_{\mathrm{fb}}}{K_{\mathrm{fb}}}+\dfrac{\Delta\beta}{\beta}} {\left|1-\dfrac{\dot{\beta}_n(3\omega)}{\beta_0}\right|} \]

did not change in comparison with \(\lambda\) for the oscillator shown in Fig. 2. This is natural, since a decrease in \(K_{\mathrm{fb}}\) as the selective properties of the selective amplifier increase is accompanied by the same increase in the influence of instability of the networks. The indicated difficulties are largely overcome when constructing the selective amplifier on the basis of a harmonic-compensation network [4]. In such amplifiers (Fig. 5) values

\[ \frac{K(\omega)}{K(3\omega)} > 3000 \]

are achieved, and the small nonlinear distortions may be neglected [5]. The broadband amplifier \(У\) is enclosed in feedback through an active rejector filter, ARF, forming a selective amplifier with a certain equivalent \(Q\). But since the ARF introduces small frequency-phase distortions of the higher harmonic components, the signal from its output is also used to compensate harmonics in the summing junction \(\Sigma_2\) at the system output. The harmonic-compensation loop is open and therefore does not reduce the stability of the entire system. But even with such a construction the instability

\[ \frac{\Delta_\Sigma K}{K} \]

is too large. Thus, for example, when a distributed \(RC\) line made of manganin microwire is used as the null circuit in an amplifier [6], at

\[ \frac{K(\omega)}{K(3\omega)} = 1000 \]

the instability \(\Delta_\Sigma K/K\) reached \(5\%\), and

\[ \lambda=\frac{\delta U}{U}\frac{K(3\omega)}{K(\omega)}\cdot\frac{\Delta_\Sigma K}{K}. \]

In accordance with (23), taking \(\delta U/U=0.15\), we obtain the specific value

\[ \lambda=0.15\cdot 5\cdot 10^{-5}=0.75\cdot 10^{-5}. \]

To ensure the same \(\lambda\) in the oscillator shown in Fig. 4 is possible only under the condition

\[ \left(\frac{\Delta K_{\mathrm{fb}}}{K_{\mathrm{fb}}}+\frac{\Delta\beta}{\beta}\right)\leqslant 3\cdot 10^{-5}, \]

which is practically unrealizable.

Fig. 5

For a further improvement in the quality of the oscillator with a harmonic-compensation network, it is necessary to insert a phase-correcting network (PC) between the ARF and the output summing junction \(\Sigma_2\). This makes it possible, at the same \(Q\) (and, consequently, at the same instability \(\Delta_\Sigma K/K\)), to increase the suppression of the 3rd harmonic [5]. We note in particular that even if it were possible to create a selective amplifier ideally satisfying the stated requirements, then even in that case we would encounter a contradiction between the equivalent quality factor \(Q_\mathrm{e}\) and inertia, owing to the well-known relation

\[ \tau_{\mathrm{eq}}=Q_\mathrm{e}\frac{2}{\omega}. \tag{24} \]

For example, at 20 Hz for \(Q_{\mathrm{e}}=250\) we obtain \(\tau_{\mathrm{e}}\simeq 4\) sec. The oscillator variant with a low-pass filter has a certain advantage in that all the LPF sections can comparatively easily be made stable. But the attenuation of harmonics is considerably smaller than when a selective amplifier is used. For the oscillator shown in Fig. 6, the filtering coefficient of the 3rd harmonic

Fig. 6

is equal to \(\dfrac{K(\omega)}{K(3\omega)}=18.5\), and, taking the instabilities of all three amplifiers to be identical, we obtain

\[ \lambda=\frac{\delta U}{U}\,\frac{3}{18.5}\,\frac{\Delta_{\mathrm{e}}K}{K} \simeq \frac{\delta U}{U}\,\frac{1}{6}\cdot\frac{\Delta_{\mathrm{e}}K}{K}. \]

Practically achievable is \(\dfrac{\Delta_{\mathrm{e}}K}{K}\simeq 0.1\%\), which, for \(\dfrac{\delta U}{U}=0.05\), gives \(K_{3\mathrm{g}}\simeq 0.07\%\); \(\dfrac{\Delta U_{\mathrm{out}}}{U_{\mathrm{out}}}\simeq 0.25\%\) and \(\lambda=0.84\cdot 10^{-5}\).

Thus, using in an oscillator only non-inertial nonlinear elements, it is in principle impossible simultaneously to obtain high stability of the oscillation amplitude and very small nonlinear distortions. It is still more impossible to combine the indicated requirements with a short duration of transient processes.

Let us proceed to consider oscillators with inertial nonlinear elements. Any INE may be represented as a combination of a certain voltage converter and a controlled element. If a voltage \(U\sqrt{2}\sin\omega t\) is applied to the input of converters of the rms and mean values, then at their outputs, respectively, we obtain

\[ U_{\mathrm{out,rms}}(t) = A_1U \left\{ 1-\frac{1}{2}\, \frac{\cos(2\omega t+\varphi)} {\sqrt{\displaystyle\prod_{i=1}^{n}\left[(2\omega\tau_i)^2+1\right]}} \right\}; \tag{25} \]

\[ U_{\mathrm{out,avg}}(t) = A_2U \left\{ 1-\frac{2}{3}\, \frac{\cos(2\omega t+\varphi)} {\sqrt{\displaystyle\prod_{i=1}^{n}\left[(2\omega\tau_i)^2+1\right]}} +\ldots \right\}, \tag{26} \]

where \(n\) is the number of filtering sections in the converters. Under the condition that \(\omega\tau\gg 1\), for the relative changes of the resistance of controlled elements of both types in statics (i.e., for changes slow in time) we obtain

\[ \frac{\Delta R}{R}=G_U^R\frac{\Delta U}{U}. \]

At low frequency, at the output of the converter we shall have a voltage of the second harmonic,

the value of which is determined by the signal frequency, by the number of filtering elements \(n\), and by their time constants \(\tau_i\).

\[ U_{\text{out}}(t)=A(U)\left\{ \frac{b\cos(2\omega t+\varphi)} {\sqrt{\displaystyle\prod_{i=1}^{n}\left[(2\omega\tau_i)^2+1\right]}} \right\}, \tag{27} \]

\(b=\dfrac{1}{2}\) for a measuring transducer of the rms value and \(b=\dfrac{2}{3}\) for a measuring transducer of the mean value of the voltage. In the general case, various intermediate elements may be included between the measuring transducer (MT) and the controlled resistance in the self-oscillator, and the second harmonic from the MT output will lead to relative modulation of the resistance:

\[ \frac{\Delta R(t)}{R} = G_U^R \frac{ b\sqrt{\displaystyle\prod_{l=1}^{q_1}\left[(2\omega\tau_l')^2+1\right]} }{ \sqrt{\displaystyle\prod_{i=1}^{n}\left[(2\omega\tau_i)^2+1\right] \prod_{l=1}^{q}\left[(2\omega\tau_l)^2+1\right]}} \cos(2\omega t+\varphi), \tag{28} \]

where \(q_1\) is the number of lead-type elements with time constants \(\tau_l'\); \(q\) is the number of inertial elements, taking into account the dynamic properties of the controlled resistance; \(G_U^R=\dfrac{\Delta R}{R}\big/\dfrac{\Delta U}{U}\).

Modulation of the resistance at twice the oscillation frequency leads to the appearance at the output of the self-oscillator of a nonlinear “bias” voltage of the first harmonic and a voltage of the third harmonic. For example, in the self-oscillator whose circuit is shown in Fig. 7, the relative nonlinear voltage bias is equal to

\[ \frac{\Delta U'_{\text{out}}}{U_{\text{out}}} \approx G_U^R G_R^K\frac{b}{2} \sqrt{ \frac{ \displaystyle\prod_{l=1}^{q_1}\left[(2\omega\tau_l')^2+1\right] }{ \displaystyle\prod_{i=1}^{n+q}\left[(2\omega\tau_i)^2+1\right] } }, \tag{29} \]

and the coefficient of the 3rd harmonic is equal to

\[ K_{3\text{г}} \approx G_U^R G_R^K\frac{b}{2} \sqrt{ \frac{ \displaystyle\prod_{l=1}^{q_1}\left[(2\omega\tau_l')^2+1\right] }{ \displaystyle\prod_{i=1}^{n+q}\left[(2\omega\tau_i)^2+1\right] } } \cdot \frac{1}{ \left[ 1-\dfrac{\beta_n(3\omega)}{\beta(\omega)} \right] }. \tag{30} \]

But the first two factors characterize the stabilization coefficient in the system. Expressions (29, 30) are valid both for two-terminal inertial nonlinear elements (semiconductor thermistors, incandescent lamps) and for four-terminal controlled resistances (indirectly heated thermistors, photoelectric transducers, Gauss elements, rectifiers in combination with optocouplers or with field-effect transistors, etc.). Of course, in each specific case, in (29, 30) one must substitute the transfer function of the actually adopted circuit. In the case of an incandescent lamp, in the first approximation we have a first-order inertial element and \(K_3\) is determined by formula (9) of work [7], while in the case of a photoelectric transducer the element is of the second order and \(K_3\) is found from (19), [8]. Directly from (30) one can draw several general conclusions.

1. Nonlinear distortions in the region of low frequencies are directly proportional to the stabilization coefficient. In other words, in principle it is impossible to require fulfillment of the conditions \(G_{\mathrm{st}}\to B\to\infty\) and \(K_{\mathrm{zg}}\to \varepsilon \to 0\).

Fig. 7

1. The introduction of a forcing element (elements) leads to an increase in \(K_{\mathrm{zg}}\).

2. Increasing the time constants \(\tau_i\) simultaneously leads to an increase in the inertia of the system.

3. Increasing the number of inertial elements, all other conditions being equal, makes it possible to reduce \(K_{\mathrm{zg}}\).

Unfortunately, for \(n+q>1\) we immediately encounter the problem of stability of the envelope generator. In view of the special importance of the question touched upon, let us consider it in more detail. Let us open the system shown in Fig. 7 at point \(a\). The relative transfer coefficient of the open-loop system for modulation of the envelope with frequency \(\Omega\) is equal to

\[ G_{U_{\mathrm{in}}}^{U_{\mathrm{out}}}(\Omega) = G_{U_{\mathrm{in}}}^{R}(\Omega)\, G_{K}^{K_y}\, G_{K_y}^{U_{\mathrm{out}}}(\Omega). \tag{31} \]

Any real amplifier possesses certain nonlinear distortions. Suppose that its linearized transfer coefficient is described by an equation analogous to (10).

Then

\[ G_{K_y}^{U_{\mathrm{out}}}(\Omega) = \frac{\dfrac{\Delta U_{\mathrm{out}}(\Omega)}{U_{\mathrm{out}}}} {\dfrac{\Delta K(\Omega)}{K}} = \frac{1}{\dfrac{3}{4}\dfrac{d^2K_y}{dU^2}\dfrac{U^2 m}{K_y}} \cdot \frac{1+\dfrac{3}{4}\dfrac{d^2K_y}{dU^2}\cdot j\Omega\tau_э} {1+j\Omega\tau_{\mathrm{eq}}} \approx \]

\[ \approx \frac{1}{\dfrac{3}{4}\dfrac{d^2K_y}{dU^2}\dfrac{U^2 m}{K_y}} \cdot \frac{1}{1+j\Omega\tau_{\mathrm{eq}}}, \]

where \(\tau_{\mathrm{eq}}\) is the equivalent time constant of the envelope self-oscillator. Let us take \(n+q=2\), and expression (31) assumes the form

\[ G_{U_{\mathrm{in}}}^{U_{\mathrm{out}}}(\Omega) \approx \frac{G_{U_{\mathrm{in}}}^{R}\cdot G_R^{K_y}} {\dfrac{3}{4}\dfrac{d^2K_y}{dU}\cdot \dfrac{U^2 m}{K_y}} \cdot \frac{1}{1+j\Omega\tau_1} \cdot \frac{1}{1+j\Omega\tau_2} \cdot \frac{1}{1+j\Omega\tau_{\mathrm{eq}}}. \tag{32} \]

In a closed system the stabilization coefficient is equal to \(G_{\mathrm{st}}(\Omega)=1-G_{U_{\mathrm{in}}}^{U_{\mathrm{out}}}(\Omega)\), and generation is stable if \(G_{\mathrm{st}}(\Omega)>0\) for any values of \(\Omega\). The characteristic equation of the system is

\[ \left| \frac{G_{U_{\mathrm{in}}}^{R}\cdot G_R^{K_y}} {\dfrac{3}{4}\dfrac{d^2K_y}{dU}\dfrac{U^2 m}{K_y}} \right| \cdot \frac{1}{(1+p\tau_1)(1+p\tau_2)(1+p\tau_{\mathrm{eq}})} +1=0. \tag{33} \]

It is not difficult to note that if the amplifier were ideal (\(K_{\mathrm{zg}}\to 0\)), then the system would be completely inoperative (for more detail see [9]). The same is also true for control circuits with a number of inertial elements greater than two. Therefore, in such generators it is impossible

simultaneously provide very high stability of \(U_{\text{out}}\) and \(K_f \to 0\).

In the case of using a first-order nonlinear element or a combination of an amplitude converter with a single-link filter and an inertialess controlled element at medium frequencies, constancy of \(U_{\text{out}}\) and \(K_f \to 0\) is in principle attainable. In practice, variations of \(U_{\text{out}}\) are determined by the instability of the nonlinear element, and \(K_f\) by the amplifier’s own nonlinear distortions [10]. In the \(RC\)-oscillator described in [10], in the region of medium frequencies,

\[ K_f \simeq 0.002\% \quad \text{at } U_{\text{out}} = 30\ \text{V and } \frac{\Delta U_{\text{out}}}{U_{\text{out}}} < 0.1\%, \]

which gives \(\lambda \leqslant 2 \cdot 10^{-8}\).

But in accordance with (30), nonlinear distortions increase sharply at low frequencies. A second extremely important drawback of these oscillators is the very long duration of transient processes as \(K_{\text{3g}} \to 0\). The occurrence of self-modulation [11, 12] is also possible if

\[ \frac{dK_y(U)}{dU} > 0. \]

Thus, so far none of the solutions considered gives the desired results. It is precisely for this reason that many regard attempts at “intrusion” into the oscillator itself as unjustified, and consider it expedient to build precision sources of alternating voltages in the form of an oscillator on which the basic requirement is imposed of small \(K_f\), and of a degree of voltage stabilization. When the durations of periodic processes are not of decisive importance, such an approach may also prove rational. Nevertheless, the search for such solutions is certainly justified when, by maximally combining functions, it is possible, on the basis of an oscillator, to obtain exceedingly simple precision sources of sinusoidal voltage. This is even more important if one takes into account the continuously increasing requirements on permissible voltage fluctuations and on the durations of transient processes.

What, then, are the strategic directions in the search for solutions to the posed problem? Let us consider them in order. The meaning of the first direction is as follows.

1. Since we cannot create an amplitude-stable generator with very small nonlinear distortions on the basis of a wideband amplifier and inertial controlled elements (because of loss of stability), by introducing inertialess nonlinear elements we increase the distortions in the \(\beta_n\)-circuit and thereby the permissible stabilization coefficient (33).

2. In order to ensure small distortions at the generator output, we make the amplifier selective.

3. To stabilize \(U_{\text{out}}\), we apply automatic regulation of the transmission coefficient of the selective amplifier.

It is easy to see that, in essence, we have split the problem of obtaining a stable sinusoidal voltage into two tasks: stabilization of \(U_{\text{out}}\) and minimization of \(K_f\), but we have tried to do this within one system, combining the functions performed by certain units.

An effective solution in this direction is an oscillator whose functional diagram is shown in Fig. 8, where IP is a measuring converter, CC is a comparison circuit, \(K_2\) is a mismatch-signal amplifier, and RZ is a link with regulated transmission coefficient. When the characteristic of NE is relay-type, expression (33) takes the form

\[ \frac{G_{U_{\text{out}}}^{R} \cdot G_{ky}^{K}}{(1+p\tau_1)(1+p\tau_2)(1+p\tau_y)} + 1 = 0 \]

and the desired stabilization coefficient can easily be obtained. But in doing so, the requirements imposed on the selective element become stringent. In the case of deep two-sided voltage limiting and the need to have at the output of the entire generator \(K_{3\Gamma} \leqslant 0.01\%\), the coefficient of suppression of the third harmonic by the selective element must be

\[ \frac{K(\omega)}{K(3\omega)} \geqslant 3000, \]

which is practically attainable only in an amplifier with a harmonic-compensation element (Fig. 5). Such high selectivity makes it possible to create precision sources according to the scheme of Fig. 8, both by the action of the measuring transducer on the element with an adjustable transmission coefficient (RZ), and by the action of the measuring transducer on the limiter NZ itself (the RZ element becomes unnecessary).

Fig. 8

Amplitude-stable generators with highly selective amplifiers possess the following important advantages.

1. In connection with the increase in the permissible cut-off angle, the value of the attenuation coefficient of the influence of instability of the self-oscillator elements at which the system is still stable (33) increases sharply, i.e., the possibility of stabilization at a specified level of nonlinear distortions is facilitated.

2. The influence of the finite value of the measuring-transducer inertia on nonlinear distortions in the low-frequency region is almost completely eliminated, since

\[ K_{3\Gamma} \approx G_{\mathrm{st}}\frac{b}{2} \sqrt{ \frac{ \displaystyle\prod_{i=1}^{q_{1}}\left[(2\omega\tau_i)^2+1\right] }{ \displaystyle\prod_{i=1}^{n+q}\left[(2\omega\tau_i)^2+1\right] } } \cdot \frac{1}{ \sqrt{ 1+Q_{\mathrm{eqv}}^{2}\left(3-\frac{1}{3}\right)^{2} } }. \tag{34} \]

1. The requirement on the linearity of the controlled element is greatly weakened, which leads to simplification of the generator circuit and to the possibility of using inertia-free regulating elements. A consequence of this is a reduction in the order of the automatic-control system and an increase in its stability.

2. Fluctuation phenomena are substantially reduced.

In the self-oscillators under consideration, a coefficient \(\lambda = 5\cdot 10^{-9}\) is realizable at substantially lower frequencies than in self-oscillators based on wideband amplifiers and INE. Unfortunately, these generators are operable only at fixed frequencies. As a result, relation (24) remains in force, as does the contradiction between the inertia of the self-oscillator and the value of \(K_{3\Gamma}\). It is true that the introduction of an automatic-control loop for \(U_{\mathrm{out}}\) makes it possible to reduce the duration of transient processes, but only to a certain value limited by the permissible frequency-dependent change in the level of the stabilized voltage due to the nonlinear “displacement” (29). In practice, the duration of the transients is

\[ t_n > \frac{10\cdot 2\pi}{\omega}. \]

The second strategic direction is also the combined use of inertial and inertia-free nonlinear (controllable) elements, but with the fundamentally important difference that for

to obtain small nonlinear distortions, not the high-selectivity properties of the amplifier are used, but such a construction of the inertialess regulation circuit in which its action in the steady-state regime would be minimal and would increase very sharply in the nonsteady-state regime.

The synthesis of an inertialess nonlinear circuit with transfer coefficient

\[ K_{\text{n}}(U)=K_0\left[1-\frac{1}{K_0}\frac{d^2K_{\text{n}}}{dU^2}(U_0-U_m)^2\right] \]

and

\[ \frac{1}{K_0}\frac{d^2K_{\text{n}}}{dU^2}U_0^2 \gg 1 \]

makes it possible to obtain a short duration of transient processes in the self-oscillator (see (16)) and low nonlinear distortions in the steady-state regime, since at the amplitude of the established oscillations \(U_{\text{t.ust}}=U_0\), \(K_{\text{n}}(U)=K_0\). Because of variations in the parameters of the self-oscillator due to the influence of ambient temperature, frequency switching, aging of elements, and other internal and external disturbances, the condition \(U_{\text{t.ust}}\ne U_0\) may be violated (all the more so because in the steady-state regime \(G_{\text{st}}\) of the nonlinear element is very small), the consequence of which is an inadmissible increase in nonlinear distortions and also instability of \(U_{\text{out}}\). Therefore it is necessary to introduce a second, inertial, automatic-regulation circuit for \(U_{\text{out}}\). A good illustration of the practical effectiveness of this approach is provided by the Krohn—Hite type 4100A programmable generator [14]. Unfortunately, this approach also has substantial shortcomings. First, when the permissible variations of \(U_{\text{out}}\) do not exceed hundredths of a percent, the durations of the transients are comparatively large and are determined by the dynamic characteristics of the inertial automatic-regulation circuit. Second, because of the presence of the inertial AGC, in the low-frequency region the relations (29, 30) remain in force, leading to the consequences noted above. A third approach is the use of amplitude strobe detectors in the automatic-regulation circuit.

Let us imagine the following hypothetical case: we have synthesized a nonlinear circuit whose transfer coefficient is a function of the extremal value of the voltage and does not depend on the instantaneous value. Then \(K_{\text{n}}(U)=K_{\text{n}}[U_m]\), and throughout the entire period

\[ K_{\text{n}}[U(t)]=\text{const}. \]

Let us also satisfy the condition

\[ G_{\text{st}}(\Omega)=\frac{dK_{\text{n}}(U)}{K_{\text{n}}(U)}\cdot \frac{U_m}{dU(\Omega)}=\text{const}\gg 1. \]

It is easy to see that in the self-oscillator there is no longer a strict interrelation between \(G_{\text{st}}\) and \(K_{3\text{g}}\) in the low-frequency region, which makes it possible to create amplitude-stable self-oscillators with low nonlinear distortions even at infralow frequencies and with short transient durations. Unfortunately, despite the fact that, with respect to the envelope, the self-oscillator is a first-order system, owing to the relay nature of the regulating action the restriction on the permissible value of \(G_{\text{st}}\) from the stability condition is preserved. Therefore, if one wishes to increase \(G_{\text{st}}\), it is necessary to introduce an additional inertial element into the stabilization circuit and, in fact, to reduce the speed of response of the system. The realization of an NE with the indicated characteristics means that during a very short interval of time (“instantaneously”) it is necessary to obtain information about the extremal value of the voltage and to store this information unchanged throughout the entire period. The transfer coefficient

CS must change in a relay fashion at the instant when the instantaneous value of the input voltage is equal to zero, and it \((K_{\mathrm{n}}[U(t)])\) is unchanged over the period.

By further logical reasoning we arrive at a controlled amplitude detector. The latter means that at the instant when the function reaches its extremal value, a control pulse (strobe pulse) is applied to the electronic switch, which opens, and the value of the function is stored for the duration of the period. A similar solution was first set forth in patent [14] and used in a generator by Optimation Inc. However, the indicated generators, although they do represent a substantial step forward, still do not fully realize the capabilities of amplitude-stable generators with strobe detectors.

Fig. 9

The fourth direction was proposed by us and consists in using, as the first link of the voltage stabilization system, a frequency-independent sharpening unit in combination with an amplitude detector. We shall call a unit “sharpening” if its relative transmission coefficient

\[ G_g=\frac{\Delta U_{\mathrm{out.d}}/U_{\mathrm{out.d}}}{\Delta U_{\mathrm{in.d}}/U_{\mathrm{in.d}}}>1. \]

Frequency independence of the transmission coefficient means that the unit is made of inertia-free nonlinear elements. Let us apply to an amplitude detector with a negligibly small charge time constant and a large discharge time constant a sinusoidal voltage (see curve 1 in Fig. 9). The voltage at the detector output will have the form shown by curve 2. The relative coefficients of the harmonics at the output of the IP are, for the first harmonic,

\[ K_{1\mathrm{r}}=\frac{T}{\tau_{\mathrm{dis}}}\cdot\frac{2}{\pi} \]

and for the second,

\[ K_{2\mathrm{r}}=\frac{T}{\tau_{\mathrm{dis}}}\cdot\frac{1}{\pi}. \]

Subsequent harmonics may be neglected.

In the closed-loop system, at the output of the self-oscillator we obtain, respectively, the second and third harmonics equal to

\[ K_{2\mathrm{r}}\approx G_{\mathrm{st}}\frac{T}{\tau_{\mathrm{dis}}}\cdot\frac{1}{\pi} \]

and

\[ K_{3\mathrm{r}}\approx G_{\mathrm{st}}\frac{T}{\tau_{\mathrm{dis}}}\cdot\frac{1}{2\pi}. \]

It is seen that increasing the stabilization coefficient leads to an increase in the harmonics. Even with \(G_{\mathrm{st}}=1\), in order that \(K_{3\mathrm{r}}<0.05\%\) at \(20\ \text{Hz}\), it is necessary that \(\tau_{\mathrm{dis}}>16\ \text{s}\).

As the sharpening unit we introduce a two-sided limiter, and at the output we obtain the curve of Fig. 9, b. When this curve is applied, after amplification by \(U_m/\delta U\) times, to the input of the amplitude detector (see

curve 1 in Fig. 9,c) at its output we shall have almost the same as when a sinusoid is applied (see curve 2 in Fig. 9,c).

The overall stabilization coefficient in the system increases by

\[ G_g=\frac{3}{2}\frac{U_m}{\delta U} \]

times, while the nonlinear distortions practically do not increase. Thus, the introduction of a preliminary voltage-limiting operation and an amplitude detector makes it possible, to a considerable degree, to overcome the contradiction existing in low-frequency self-oscillators between the level of nonlinear distortions and the stability of the output voltage. It is precisely this path that we adopted in implementing the master oscillator in the GK-12A calibrator [15].

The brief analysis presented shows that there exist effective ways of reducing the interdependence between the attainable values of \(K_f\), \(G_{\mathrm{st}}\), and \(\tau_e\).

Combined solutions appear especially promising. In this connection, questions of the “elegance” of the technical implementations of amplitude-stable self-oscillators acquire extraordinary importance.

REFERENCES

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