Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Vol. 231 1971

On the Stability of Amplitude-Stable Self-Oscillators with Inertial Controlled Elements

M. S. Roitman

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

In [1, 2], generators were proposed in which the voltage at the output is stabilized by the action of controlled inertial elements (CIE) on the feedback (FB) circuit of the master oscillator. The structural diagram of such a generator with the combined action of negative and positive feedback is shown in Fig. 1, where

Fig. 1.

1 — controlled inertial element, linear for instantaneous voltage values;
2 — circuits of positive \((\beta_{\mathrm{p}})\) and negative \((\beta_{0})\) feedback;
3 — amplifier of the master oscillator;
4 — power amplifier.

The relative influence of the instability of the \(m\)-th node of the generator on the output voltage can be determined by the formula

\[ G_{\lambda_{\mathrm{fb}}}^{U_{\mathrm{out}}} = \frac{ G_{\lambda}^{U_{\mathrm{out}}} }{ 1 - G_{U_{\mathrm{in}}}^{U_{\mathrm{out}}} } = \frac{ \displaystyle \prod_{i=m}^{n} G_i }{ \displaystyle 1 - \prod_{i=1}^{n} G_i } = \frac{ G_{\lambda}^{U_{\mathrm{out}}} }{ G_{\mathrm{st}} }, \tag{1} \]

where

\[ G_{\lambda}^{U_{\mathrm{out}}} = \frac{\Delta U_{\mathrm{out}}}{U_{\mathrm{out}}} \bigg/ \frac{\Delta \lambda}{\lambda} \]

is the influence of the relative instability of the \(m\)-th node in the open-loop system (opened at points \(a\)—\(b\));

\[ G_{\mathrm{st}} \]

is the stabilization coefficient in the closed-loop system;

\[ G_{U_{\mathrm{in}}}^{U_{\mathrm{out}}} = \frac{\dfrac{\Delta U_{\mathrm{out}}}{U_{\mathrm{out}}}} {\dfrac{\Delta U_{\mathrm{av}}}{U_{\mathrm{av}}}} \quad \text{is the relative transfer coefficient of the open-loop system;} \]

\[ G_i \quad \text{is the relative transfer coefficient of the \(i\)-th unit.} \]

It follows from (1) that, for \(G_{\mathrm{st}}\gg 1\), the first unit has the greatest influence on the output voltage. The influence of the subsequent units is weakened by a factor of \(G_1\). Therefore it is necessary to strive to make the transfer coefficient of the first unit and its “interference immunity” maximal. Generators satisfying the requirements indicated in [3] are classified as ideal. When highly stable CEEs with \(G_1\gg 1\) are used, what has been said applies entirely to generators made according to the block diagram of Fig. 1.

Unfortunately, the practical realization of such generators is hindered by the stability problem that arises. Moreover, the higher the requirements imposed on the purity of the waveform of the generated voltage, the greater the contradictions one has to face. In view of the importance of the issues touched upon, let us consider them in somewhat more detail and try to find acceptable solutions.

Let us open the system shown in Fig. 1 at points \(a—b\) and apply to the input a voltage modulated according to the envelope

\[ U_{\mathrm{in}}(t)=U[1+a\sin\Omega t]\sin\omega t, \]

where

\[ a \text{ is the relative modulation depth,} \]
\[ \Omega \text{ is the modulation frequency.} \]

The relative transfer coefficient of the open-loop system (which is a function of the frequency \(\Omega\)) is equal to\(^*\)

\[ G_{U_{\mathrm{in}}}^{U_{\mathrm{out}}}(\Omega) = G_{U_{\mathrm{in}}}^{R_\phi}(\Omega) G_{R_\phi}^{\beta}(\Omega) G_{\beta}^{k}(\Omega) G_{k}^{U'_{\mathrm{out}}}(\Omega) G_{U'_{\mathrm{out}}}^{U_{\mathrm{out}}}(\Omega). \tag{2} \]

\[ G_{U_{\mathrm{in}}}^{R_\phi}(\Omega) = \frac{dR_\phi/R_\phi}{dU_{\mathrm{in}}/U_{\mathrm{in}}(\Omega)} = \frac{-G_\phi}{(1+j\Omega\tau_l)(1+j\Omega\tau_\phi)}, \]

where

\[ G_\phi \text{ is the static transfer coefficient of the lamp–photoresistor pair;} \]
\[ \tau_l \text{ is the time constant of the lamp;} \]
\[ \tau_\phi \text{ is the time constant of the photoresistor;} \]

\[ G_{R_\phi}^{\beta}(\Omega) = \frac{d\beta_0/\beta_0}{dR_\phi/R_\phi} \simeq -(1-\beta_0); \]

\[ G_{\beta}^{k}(\Omega) = \frac{dk/k}{d\beta_0/\beta_0} \simeq (K\beta_n-1) = K\beta_0. \]

The quantity \(G_{k}^{U'_{\mathrm{out}}}\) may be taken equal to 1 only when a non-inertial nonlinear element is included in the feedback circuit. If, however, an inertial nonlinear element or a CEE is used, then the indicated assumption is completely unjustified. This extremely important circumstance, unfortunately, was not taken into account by everyone, including A. A. L’vovich.

\(^*\) We assume everywhere that the slope of the phase–frequency characteristic of all adjustable links at the carrier frequency is negligibly small. In other words, none of the links is sharply selective. Otherwise, additional instabilities may arise. Their detailed analysis is given in [3].

Indeed, if the $\beta$-circuits are linear for instantaneous voltage values, then any small and relatively rapid change of some amplifier parameter will not lead to breakdown of oscillation only in the case of nonlinearity of the amplifier itself.

Let us adopt an approximation of the amplifier transfer coefficient when a voltage $U_m \sin \omega t$ is applied to its input in the form

\[ K[u(t)] = K_0 - K''(U_m \sin \omega t)^2 = K_0 \left[1 - \frac{K''}{K_0}(U_m \sin \omega t)^2\right]. \]

The linearized (averaged) value $K(u)$ is equal to*

\[ K(u) = K_0 \left[1 - \frac{3}{4}\cdot \frac{K''}{K_0} U_m^2\right] \]

and the coefficient of the amplifier’s 3rd harmonic is equal to

\[ k_{3\mathrm{h}} = \frac{1}{4}\cdot \frac{K''}{K_0}, \]

\[ \frac{dK(u)}{K} \approx -\frac{3}{2}\left(\frac{K''}{K_0}U_m^2\right)\frac{dU_m}{U_m} \]

and

\[ G^{U'_{\mathrm{out}}}_{k}(\Omega) \approx -\frac{1}{\dfrac{3}{2}\left(\dfrac{K''}{K_0}U_m^2\right)} \cdot \frac{1}{1+j\Omega\tau_k}, \]

where $\tau_k$ is the equivalent time constant of the self-oscillator.

The power amplifier may be considered linear $\left(G^{U_{\mathrm{out}}}_{U'_{\mathrm{out}}}=1\right)$, and therefore,

\[ G^{U_{\mathrm{out}}}_{U_{\mathrm{in}}}(\Omega) \approx -\frac{K\beta_0(1-\beta_0)G_{\phi}} {\dfrac{3}{2}\left(\dfrac{K''}{K_0}U_m^2\right)} \cdot \frac{1}{(1+j\Omega\tau_k)(1+j\Omega\tau_l)(1+j\Omega\tau_{\phi})}. \tag{3} \]

At the output we obtain a voltage with envelope

\[ a\left|G^{U_{\mathrm{out}}}_{U_{\mathrm{in}}}(\Omega)\right| \sin\left[\Omega t+\varphi_G(\Omega)\right]. \]

It is obvious that if
$\left|G^{U_{\mathrm{out}}}_{U_{\mathrm{in}}}(\Omega)\right|\ge 1$
for $\varphi_G(\Omega)=2n\pi$ $(n=0,1,2\ldots)$, then the closed-loop system is unstable.

The closed-loop system will be stable if

\[ G_{\mathrm{st}}(\Omega)=1-G^{U_{\mathrm{out}}}_{U_{\mathrm{in}}}(\Omega)>0 \]

for all values of $\Omega$.

The characteristic equation of the system is equal to

\[ G_{\mathrm{st}}(p)= \frac{K\beta_0(1-\beta_0)G_{\phi}} {\dfrac{3}{2}\left(\dfrac{K''}{K_0}U_m^2\right)} \cdot \frac{1}{(1+p\tau_k)(1+p\tau_l)(1+p\tau_{\phi})} +1. \tag{4} \]

Let us note that the static influence coefficient of node 3, calculated from (1 and 3), has the same value both when the amplifier nonlinearity is taken into account and when it is not.

\[ \overline{\phantom{xxxxxxxx}} \]

* In the case of an even nonlinearity, a refined solution must be sought [4].

For example, in the first case

\[ G^{U'_{\text{out}}}_{K(\text{fb})} = \frac{G^{U'_{\text{out}}}_{K}}{G_{\text{st}}} = \frac{\dfrac{3}{2}\left(\dfrac{K''}{K_0}U_m^2\right)} {\dfrac{K\beta_0(1-\beta_0)G_\phi}{\dfrac{3}{2}\left(\dfrac{K''}{K_0}U_m^2\right)}} = \frac{1}{K\beta_0(1-\beta_0)G_\phi}; \]

in the second,

\[ G^{U'_{\text{out}}}_{K(\text{fb})} = \frac{G^{U'_{\text{out}}}_{K}}{G_{\text{st}}} = \frac{1}{K\beta_0(1-\beta_0)G_\phi}. \]

Apparently, this circumstance was the main reason why amplifier distortions were not taken into account in considering the stability of the system.

It follows directly from (4) that the nonlinearity of the amplifier has a decisive influence on stability. There is a contradiction: the lower the distortion of the oscillator, the lower the critical value of the loop amplifier, i.e., the lower the stability of the output voltage.

The results obtained can also be extended to oscillators in which the action is exerted on the PFB circuit [5, 6, 7], while an NFB circuit is present or absent.

For an amplifier with negative feedback,

\[ \frac{dK_{\text{fb}}}{K_{\text{fb}}} = \frac{dK/K}{1+K\beta_0} \quad \text{and} \quad \left(\frac{K''}{K_0}U_m^2\right)_{\text{fb}} \approx \left(\frac{K''}{K_0}U_m^2\right)\cdot \frac{1}{1+K\beta_0}, \]

expression (3) takes the form

\[ G^{U_{\text{out}}}_{U_{\text{in}}}}(\Omega) \approx \frac{-G_\phi\cdot G^\beta_{K\phi}} {\dfrac{3}{2}\left(\dfrac{K''}{K_{\text{c}}}U_m^2\right)_{\text{fb}}} \cdot \frac{1}{(1+j\Omega\tau_K)(1+j\Omega\tau_n)(1+j\Omega\tau_\phi)}, \tag{5} \]

where

\[ \left(\frac{K''}{K_0}U_m^2\right)_{\text{fb}} = \frac{\left(\dfrac{K''}{K_0}U_m^2\right)}{1+K\beta_0} \]

is the nonlinearity of the amplifier with allowance for NFB (if it is present).

Let us show that, for small \(\kappa_f\), the system is generally inoperative.

Take \(K=300\), \(\beta=\dfrac{1}{3}\), \(\kappa_{\text{3g}}=5\%\) (for an amplifier not covered by NFB). For the same amplifier covered by NFB, we have

\[ \kappa_{\text{3g.fb}} \approx \frac{\kappa_{\text{3g}}}{1+K\beta} \sim 0.05\%, \]

and the third-harmonic coefficient of the entire oscillator (the ICE is assumed ideal)

\[ \kappa_{\text{3g.gen}} \approx \frac{\kappa_{\text{3g.fb}}} {\left|1-\dfrac{\dot{\beta}_n(3\omega)}{\dot{\beta}_n(\omega)}\right|} \sim 0.08\%. \]

Proceeding from the relationship between \(K(u)\) and \(\kappa_{\text{3g}}\) (for the adopted approximation), we have

\[ \frac{3}{2}\frac{K''}{K_0}U_m^2 = 0.0015. \]

If the static coefficient of influence of amplifier 3 is taken equal to 1, then even then, for the stability of generation, the condition must be satisfied

\[ \frac{100}{0.15} < \frac{\tau_1+\tau_2}{\tau_3} + \frac{\tau_2+\tau_3}{\tau_1} + \frac{\tau_1+\tau_2}{\tau_3}, \]

where \(\tau_1, \tau_2, \tau_3\) are the time constants of the system \((\tau_{\mathrm{k}}, \tau_{\mathrm{l}}, \tau_{\phi})\), in descending order.

Relaxing the requirement that \(k_{f\,\mathrm{gen}}\) be small, one can obtain high stability of \(U_{\mathrm{out}}\) without any refinements. For example, for \(k_{f\,\mathrm{gen}}\simeq 5\%\) and the same ratios of \(\tau\),

\[ G^{U'_{\mathrm{out}}}_{K(\mathrm{fb})}=\frac{1}{62}! \]

The interesting fact is that, for power amplifier 4, the coefficient of influence is equal to

\[ G^{U_{\mathrm{out}}}_{U_{\mathrm{out\,coupling}}(\Omega\to 0)} = \frac{ G^{U_{\mathrm{out}}}_{U'_{\mathrm{out}}} }{ 1+ \frac{K\beta_0(1-\beta_0)G_{\phi}} {\dfrac{3}{2}\left(\dfrac{K''}{K_0}U_m^2\right)} }. \]

For it, the self-oscillator is essentially a critical component.

Analysis of expression (4) shows that, in order to increase stability, nonlinear elements should be introduced into the self-oscillator. The use of inertia-free nonlinear elements leads to an increase in nonlinear distortions, which is undesirable. Consequently, it is necessary to introduce an inertial nonlinear element (INE).

Let us verify the effectiveness of the proposed measure. Let us include an INE (a low-power incandescent lamp) in amplifier 3.

In this case

\[ G^{U'_{\mathrm{out}}}_{K}(p) = \frac{-1} {\left[ G_{\mathrm{kl}}\dfrac{1}{p\tau_{\mathrm{n}}+1} + \dfrac{3}{2}\left(\dfrac{K''}{K_0}U_m^2\right) \right]} \cdot \frac{1}{p\tau_{\mathrm{k}}+1} \]

and

\[ G_{\mathrm{st}}(p)\simeq \frac{ K\beta_0(1-\beta_0)G_{\phi} }{ \left[ G_{\mathrm{kl}}\dfrac{1}{p\tau_{\mathrm{n}}+1} + \dfrac{3}{2}\dfrac{K''}{K_0}U_m^2 \right] } \cdot \frac{1}{p\tau_{\mathrm{k}}+1} \cdot \frac{1}{p\tau_{\mathrm{l}}+1} \cdot \frac{1}{p\tau_{\phi}+1} +1, \tag{6} \]

where

\[ G_{\mathrm{kl}}=G^{R_{\mathrm{NE}}}_{U_{\mathrm{v}}}\cdot G^{K}_{R_{\mathrm{NE}}}. \]

Comparison of (4) and (6) shows that, when the INE has large inertia, the stability of the system does not increase; if, however, the INE has little inertia, the stability increases substantially. We used this possibility to construct high-frequency amplitude-stable self-oscillators with controlled inertial elements [6].

Thus, in principle, it is possible to create high-quality and comparatively simple highly stable self-oscillators.

In practical terms, however, implementation is often impeded by two obstacles:

1. At present we do not have inertial nonlinear elements with a small time constant and characteristics that make it easy to couple an INE to the circuit*.

2. It is necessary to apply special measures to protect the INE against overload, which in most cases leads to complication of the circuit.

It should also be taken into account that, because of the smallness of $\tau_{\mathrm{n}}$, the distortions of the self-oscillator at low frequencies increase.

Fig. 2.

From the analysis carried out it follows that, in order to realize the greater advantages of amplitude-stable self-oscillators with a controlled inertial element, a search for new solutions is necessary.

One such solution is an oscillator whose functional diagram is given in Fig. 2.

Amplifier 1, with transfer coefficient $K$, is encompassed by frequency-dependent positive 2 and negative 3 feedbacks, forming selective amplifier 4. Selective amplifier 4 is encompassed by positive frequency-independent feedback containing noninertial nonlinear elements (semiconductor diodes). Stabilization of the voltage at the output of power amplifier 6 is achieved by the fact that lamp 7, connected at the output, illuminates photoresistor 8 in the negative-feedback circuit of amplifier 4, forming an autostabilization loop. The meaning of the described solution is as follows.

1. Since we cannot create an amplitude-stable oscillator with very small nonlinear distortions on the basis of a wideband amplifier and inertial NEs (because of loss of stability), by introducing noninertial nonlinear elements we increase the distortions in the $\beta_{\mathrm{p}}$ circuit and thereby the permissible stabilization coefficient.

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

3. To stabilize $U_{\mathrm{out}}$, we apply automatic regulation of the transfer 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 problems: stabilization

* The lowest-power incandescent lamps among those produced in series are NSM $6.3\ \mathrm{V}\times20\ \mathrm{mA}$.

$U_{\text{out}}$ and minimizing $K_f$. But we tried to do this within a single system, combining the functions performed by certain nodes*.

The splitting of functions substantially facilitates the solution of the problem posed, but in the final analysis inevitably leads to a significant complication of the system, which is by no means always justified.

REFERENCES

1. M. S. Roitman. Some questions of improving the accuracy of a strain-gauge installation. Proceedings of the Tomsk City Scientific-Technical Conference. Tomsk, Tomsk State University Press, 1959.

2. M. S. Roitman. The use of photoresistors for voltage stabilization. Proceedings of the Second Conference on Avtometry. Novosibirsk, Publishing House of the Siberian Branch of the USSR Academy of Sciences, 1962.

3. A. A. Lvovich. Questions of the theory and practice of amplitude stabilization of high-frequency electrical oscillations. Doctoral dissertation, Leningrad, LEIS, 1967.

4. M. S. Roitman. On one method for finding the stationary solution for self-oscillators. In the present collection.

5. M. S. Roitman. Amplitude-stable generators with small nonlinear distortions. Izvestiya Vuzov SSSR. Radioelektronika, No. 12, 1969.

6. M. S. Roitman. Amplitude-stable electronic generators. Avtometriya, No. 1, 1969.

7. M. S. Roitman. Amplitude-stable self-oscillator. Author’s certificate No. 270831.

* Although the particular solution considered is not optimal, the strategy proved effective and enabled us to create efficiently operating self-oscillators.