On the Stability of Oscillations in Self-Oscillators with Inertial Nonlinearity
Abstract Generated abstract
This paper analyzes amplitude stability and self-modulation in self-excited oscillators that use inertial nonlinear elements such as incandescent lamps or thermistors for amplitude limiting. It proposes a large-signal mathematical model relating the dynamic resistance of the inertial element to current, derives shortened envelope equations for several LC, RC, and RL oscillator configurations, and examines stationary regimes by linearizing the resulting systems. The analysis refines the instability boundary, showing that a linear active-element characteristic is not the limiting case for stable operation and that a positive slope in the active-element characteristic can be permissible if a derived inequality is satisfied. Numerical integration and experimental oscillator mock-ups are reported to confirm the theoretical stability conditions.
Full Text
On the Stability of Oscillations of Self-Excited Oscillators with Inertial Nonlinearity
V. P. Zibalov, M. S. Roitman
(Presented by the scientific seminar of the Department of Radio Engineering)
Obtaining sinusoidal oscillations with a low harmonic content and a stable amplitude is at present a highly topical problem. To solve it, in most cases an inertial nonlinear element (INE) is introduced into self-excited oscillators—a low-power incandescent lamp or a thermistor. Such an element is almost linear for instantaneous values of the alternating current, provided that its time constant is much greater than the time constant of the oscillator’s oscillatory circuit. At the same time, the INE is a nonlinear element for the effective values of the current and performs the role of a limiter of the oscillation amplitude. This makes it possible to use the most linear section of the characteristics of the active elements. The latter ensures a low harmonic content in the generated oscillations.
In the practical implementation of \(LC\)-, \(RC\)-, and \(RL\)-self-excited oscillators with an INE, both in tube and transistor versions, automodulation of the oscillation amplitude may occur. Several works [1–3] are devoted to elucidating the causes of this undesirable phenomenon. The conclusions of the authors of the cited works reduce to the following:
a) in the case of linear characteristics of the active elements, such self-excited oscillators must be absolutely stable;
b) automodulation (instability of the envelope of the output signal) arises as a consequence of an increase, at least over some portion, in the slope of the active element as a function of the oscillation amplitude.
In the present work an analysis is given of the stability of oscillations in self-excited oscillators with an INE, and the boundaries of unstable regimes are refined.
Mathematical Model of an Inertial Nonlinear Element
When studying the stability of oscillations and transient processes in self-excited oscillators, it is necessary to have a mathematical description of the INE that would relate the resistance of such an element to the current flowing through it. As such a mathematical model, suitable for large signals, an equation of the following form is proposed:
\[ \tau_{\mathrm{л}}(Z)\frac{dZ}{dt}+Z=R_{\mathrm{л}}(I_{\mathrm{л}}), \tag{1} \]
where
\(Z\) is the dynamic resistance of the INE;
\(I_{\mathrm{л}}\) is the current flowing through the INE;
$\tau_\ell(Z)$ is a function expressing the dependence of the time constant of the INE on the magnitude of the dynamic resistance;
$R_\ell(I_\ell)$ is the static ohm-ampere characteristic of the INE.
The indicated functions are determined experimentally. The dependence $\tau_\ell(Z)$ is determined from transient or frequency characteristics taken at small perturbations of the current $I_\ell$. It should be noted here that $I_\ell$ and $R_\ell$ are interdependent quantities. This makes it possible to carry out various recalculations. To check the proposed mathematical model of the INE, calculations were performed from equation (1) of transient processes for large changes in the input quantity $I_\ell$. The experimental and calculated curves of the change of dynamic resistance in time $Z(t)$ are shown in Fig. 1. The calculated and experimental curves differ by no more than 10%. The latter can be attributed mainly to the errors of the apparatus used to record the transient processes. As is seen from Fig. 1, the qualitative picture of the transient processes is reproduced quite well. This was facilitated by the introduction into equation (1) of the indicated nonlinear dependences, determined experimentally.
Fig. 1. Transient processes in the INE (NSM6×20 incandescent lamp). The upper curves correspond to a current change from 4 to 8 mA, the lower ones—from 8 to 4 mA.
Equations of a self-excited oscillator with inertial nonlinearity
Table 1 gives the block diagrams of the investigated self-excited oscillators of various types. The so-called “shortened” equations describing the behavior of the system in terms of the envelope of the output signal were obtained by the known method [4]. As an example, let us compose the equation of an $LC$ oscillator with a circuit in the anode branch, made according to block diagram 1. The differential equation of such an oscillator for the output voltage has the following form:
\[ \frac{d^2 u}{dt^2} + \left( \frac{r_{\mathrm{e}}}{L} - \frac{S k_{\mathrm{oc}}\dfrac{R}{R+Z}}{C} \right) \frac{du}{dt} + \omega_0^2 u = 0, \tag{2} \]
where $r_{\mathrm{e}}$ is the equivalent resistance of the circuit losses;
$k_{\mathrm{oc}}$ is the feedback coefficient;
\[ \omega_0=\frac{1}{\sqrt{LC}} \]
is the carrier frequency of the oscillator oscillations;
$S$ is the tube transconductance (for the time being we take $S=\mathrm{const}$).
The remaining notation is clear from the diagrams of Table 1.
We assume the solution of equation (2) in the following form:
\[ u=U(t)\cos\omega_0 t, \tag{3} \]
where $U(t)$ is a function slowly varying in time.
We substitute expression (3) into equation (2), first finding
\[ \frac{du}{dt} \quad \text{and} \quad \frac{d^2u}{dt^2}. \]
After carrying out all the transformations required by the method of slowly varying amplitudes, equation (2) can be represented in the form:
\[ \frac{dU}{dt}+\frac{1}{2}\left(\frac{r_{\mathrm{e}}}{L}-\frac{S k_{0.\mathrm{s}}\dfrac{R}{R+Z}}{C}\right)U=0. \tag{4} \]
Taking into account that \(\dfrac{2L}{r_{\mathrm{e}}}=\tau_{\mathrm{k}}\) and \(\dfrac{L}{r_{\mathrm{e}}C}=R_{0.\mathrm{e}}\), we rewrite (4) in the following form:
\[ \frac{dU}{dt}=\frac{1}{\tau_{\mathrm{k}}} \left[ S R_{0.\mathrm{e}} k_{0.\mathrm{s}}\frac{R}{R+Z}-1 \right]U. \tag{5} \]
Equations for the envelope have been obtained analogously for the other circuits of Table 1. For self-excited oscillators constructed according to block diagram 3, \(Q\) denotes the damping coefficient of a selective circuit of the \(RL\) or \(RC\) type.
Stability of Stationary Regimes
Since in self-excited oscillators only stationary solutions are of interest, for the selected example of an \(LC\) oscillator we shall consider the existence and stability of stationary regimes. The latter are determined as the roots of the system of equations
\[ \frac{dU}{dt}=0,\qquad \frac{dZ}{dt}=0, \]
in which there are two pairs of roots (for monotonic \(S(U)\)). The first pair of roots will be
\[ U_n=0,\qquad Z_n=R_{\mathrm{l.o.}}. \]
The second pair of roots can be obtained by solving the system of equations
\[ \begin{cases} S(U)R_{0.\mathrm{e}}k_{0.\mathrm{s}}\dfrac{R}{R+Z}-1=0;\\ R_{\mathrm{l}}(I_{\mathrm{l}})-Z=0, \end{cases} \tag{6} \tag{7} \]
where
\[ I_{\mathrm{l}}=\frac{U}{R+Z}. \tag{8} \]
Equation (6) is nonlinear and is easily solved graphically. Such calculations for specified parameter values are given in Fig. 2. From the found \(U_n=U^0\) we find \(Z_n=R_{\mathrm{l}}^{0}\), using equations (7) and (8).
Now let us investigate the stability of the possible stationary regimes. We expand the right-hand sides of the envelope equations in a Taylor series and retain terms with first-degree deviations. We obtain the system of equations in deviations
\[ \left\{ \begin{aligned} \frac{d\Delta U}{dt} &= \frac{1}{\tau_{\mathrm{k}}} \left\{ \left[ S(U_n)+U_n\frac{\partial S}{\partial U} \right] R_{0.\mathrm{e}}k_{0.\mathrm{s}}\frac{R}{Z_n+R} -1 \right\}\Delta U \\ &\quad -\frac{1}{\tau_{\mathrm{k}}} \frac{S(U_n)R_{0.\mathrm{e}}k_{0.\mathrm{s}}RU_n}{(Z_n+R)^2}\Delta Z; \\[6pt] \frac{d\Delta Z}{dt} &= \frac{1}{\tau_{\mathrm{l}}(Z_n)} \cdot \frac{1}{R+Z_n} \cdot \frac{\partial R_{\mathrm{l}}}{\partial I_{\mathrm{l}}}\Delta U - \frac{1}{\tau_{\mathrm{l}}(Z_n)} \left[ 1+ \frac{U_n}{(Z_n+R)^2} \frac{\partial R_{\mathrm{l}}}{\partial I_{\mathrm{l}}} \right]\Delta Z. \end{aligned} \right. \tag{9} \]
For the first stationary regime \((U_n=0,\ Z_n=R_{\ell.o})\), the system in deviations will be
\[ \begin{cases} \dfrac{d\Delta U}{dt} = \dfrac{1}{\tau_k} \left[ S(0)\cdot R_{o.e}\cdot \kappa_{0.c}\dfrac{R}{R_{\ell.o}+R}-1 \right]\Delta U; \\[1.2em] \dfrac{d\Delta Z}{dt} = -\dfrac{1}{\tau_\ell(R_{\ell.o})}\cdot \dfrac{1}{R_{\ell.o}+R}\cdot \dfrac{\partial R_\ell}{\partial I_\ell}\cdot \Delta U - \dfrac{1}{\tau_\ell(R_{\ell.o})}\Delta Z . \end{cases} \]
Fig. 2
The characteristic determinant of the system for this regime is
\[ D(\lambda)\equiv \left| \begin{array}{cc} \lambda-\dfrac{1}{\tau_k} \left[ S(0)\cdot R_{o.e}\cdot \kappa_{0.c}\dfrac{R}{R_{\ell.o}+R}-1 \right] & 0 \\[1.2em] -\dfrac{1}{\tau_\ell(R_{\ell.o})}\cdot \dfrac{R}{R_{\ell.o}+R}\cdot \dfrac{\partial R_\ell}{\partial I_\ell} & \lambda+\dfrac{1}{\tau_\ell(R_{\ell.o})} \end{array} \right| = \]
\[ = \lambda^2+ \left\{ \dfrac{1}{\tau_\ell(R_{\ell.o})} - \dfrac{1}{\tau_k} \left[ S(0)\cdot R_{o.e}\cdot \kappa_{0.c}\dfrac{R}{R_{\ell.o}+R}-1 \right] \right\}\lambda - \]
\[ - \dfrac{1}{\tau_k\tau_\ell(R_{\ell.o})} \left[ S(0)\cdot R_{o.e}\cdot \kappa_{0.c}\dfrac{R}{R_{\ell.o}+R}-1 \right] =0. \]
If the condition for excitation of the generator is satisfied,
\[ \kappa_{0.c}> \dfrac{1}{S(0)R_{o.e}}\cdot \dfrac{R_{\ell.o}+R}{R}, \]
then this regime will be unstable. For the second stationary regime \((U_n=U^0,\ Z_n=R_\ell^0)\), the characteristic equation of the system will be
\[ \lambda^2+a_1\lambda+a_2=0, \]
where
\[ a_1= \dfrac{1}{\tau_\ell(R_\ell^0)} \left[ 1+\dfrac{U^0}{(R_\ell^0+R)^2} \dfrac{\partial R_\ell}{\partial I_\ell} \right] - \]
\[ - \dfrac{1}{\tau_k} \left\{ \left[ S(U^0)+U^0\dfrac{\partial S}{\partial U} \right] R_{o.e}\cdot \kappa_{0,c}\dfrac{R}{R_\ell^0+R} -1 \right\}; \]
\[ \begin{aligned} a_2={}&-\frac{1}{\tau_{\text{k.l.}}\left(R_{\text{l}}^0\right)} \cdot \frac{S\left(U^0\right)R_{\text{o.e.}}\kappa_{\text{o.s.}}RU^0}{\left(R_{\text{l}}^0+R\right)^3} \frac{\partial R_{\text{l}}}{\partial I_{\text{l}}} \\ &-\frac{1}{\tau_{\text{k.l.}}\left(R_{\text{l}}^0\right)} \cdot \left[ 1+\frac{U^0}{\left(R_{\text{l}}^0+R\right)^2} \frac{\partial R_{\text{l}}}{\partial I_{\text{l}}} \right]\times \\ &\times \left\{ \left[ S\left(U^0\right)+U^0\frac{\partial S}{\partial U} \right] R_{\text{o.e.}}\kappa_{\text{o.s.}}\frac{R}{R_{\text{l}}^0+R}-1 \right\}. \end{aligned} \]
Since the constant term \(a_2\) of the characteristic equation is always positive,\(^1\) then for \(\dfrac{\partial S}{\partial U}\ll 0\) the stability of the regime is ensured by the positivity of the coefficient of \(\lambda\), i.e., \(a_1>0\). For \(\dfrac{\partial S}{\partial U}>0\) the system may be either stable or unstable. In the case when
\[ \frac{\partial S}{\partial U} < \left.\frac{\partial S}{\partial U}\right|_{\text{crit}} = \frac{ \displaystyle \frac{\tau_{\text{k}}}{\tau_{\text{l}}\left(R_{\text{l}}^0\right)} \left[ 1+\frac{U^0}{\left(R_{\text{l}}^0+R\right)^2} \frac{\partial R_{\text{l}}}{\partial I_{\text{l}}} \right] - S\left(U^0\right)R_{\text{o.e.}}\kappa_{\text{o.s.}}\frac{R}{R_{\text{l}}^0+R} +1 }{ \displaystyle U^0R_{\text{o.e.}}\kappa_{\text{o.s.}}\frac{R}{R_{\text{l}}^0+R} }, \tag{11} \]
the self-oscillator will be stable with respect to the envelope; otherwise it will be unstable.
Fig. 3. Transient process in self-oscillator 1 with respect to the envelope of the output voltage for
\[
\frac{dS}{dU}>
\left.\frac{dS}{dU}\right|_{\text{crit}}
\]
The analysis was also carried out by numerical methods. The original nonlinear system of equations for the envelope was integrated by the Runge–Kutta method on a BESM-4 electronic digital computer with various functions \(S(U)\). On the working segment of the characteristic \(S(U)\), the slope was varied (different values of \(\dfrac{\partial S}{\partial U}>0\) were specified). The results of the solution for self-oscillator 1 are shown in Figs. 3 and 4. In Fig. 3, self-modulation of the oscillations of the output voltage is clearly visible; this solution was obtained for
\[ \frac{\partial S}{\partial U}> \left.\frac{\partial S}{\partial U}\right|_{\text{crit}} . \]
From Fig. 4 it is seen that the transient process decays, mo-
\(^1\) It is assumed that the INE has a positive temperature coefficient of resistance. For example, for an incandescent lamp with a metallic filament, \(\dfrac{\partial R_{\text{l}}}{\partial I_{\text{l}}}>0\).
Table 1
| No. | Block diagram of the self-oscillator | System of equations for the envelope |
|---|---|---|
| 1 | Diagram with an \(LC\)-type selective circuit, nonlinear element \(Z\), resistance \(R\), and a tube. | \[\frac{dU}{dt}=\frac{1}{\tau_k}\left[S(U)R_{oe}K_{oc}\frac{R}{R+Z}-1\right]U;\] \[\frac{dZ}{dt}=\frac{1}{\tau_\lambda(Z)}\left[R_\lambda(I_\lambda)-Z\right],\] \[\text{where}\quad I_\lambda=\frac{U}{R+Z}\] |
| 2 | Diagram with an \(LC\)-type selective circuit, nonlinear element \(Z\), resistance \(R\), and a tube. | \[\frac{dI}{dt}=\frac{1}{\tau_k}\left[\frac{S(I)R_{oe}K_{oc}}{1+S(I)R_e}-1\right]I,\] \[\frac{dZ}{dt}=\frac{1}{\tau_\lambda(Z)}\left[R_\lambda(I_\lambda)-Z\right],\] \[\text{where}\quad I_\lambda=\frac{R}{R+Z}I,\qquad R_e=\frac{RZ}{R+Z}\] |
| 3 | Diagram with an \(RC\)- or \(RL\)-type selective circuit, nonlinear element \(Z\), resistance \(R\), and amplifier \(K(U)\). | \[\frac{dU}{dt}=\frac{1}{2\tau}\left[\frac{K(U)(R+Z)}{R+Z+ZK(U)}-Q\right]U,\] \[\frac{dZ}{dt}=\frac{1}{\tau_\lambda(Z)}\left[R_\lambda(I_\lambda)=Z\right],\] \[\text{where}\quad I_\lambda=\frac{U}{R+Z}\] |
modulation disappears; this solution was obtained for \(\dfrac{\partial S}{\partial U} < \left.\dfrac{\partial S}{\partial U}\right|_{\mathrm{crit}}\), but with
\[ \dfrac{\partial S}{\partial U} > 0. \]
Similar results were obtained in an experimental verification of the derived conditions on mock-ups of self-excited oscillators constructed according to the block diagrams presented. The calculations and the experiment carried out fully confirmed the theoretical analysis.
In the figure: vertical axis \(u(t)\,[\mathrm{V}]\); horizontal axis \(t\,[\mathrm{s}]\).
Fig. 4. Transient process in self-excited oscillator 1 according to the envelope
of the output voltage for \(\dfrac{dS}{dU} < \left.\dfrac{dS}{dU}\right|_{\mathrm{crit}}\)
Conclusions
-
A linear characteristic of the active element (\(S=\mathrm{const}\)) is not the boundary of stable operating regimes of self-excited oscillators with INE.
-
A rise of the characteristic of the active element in the operating region is quite permissible so long as inequality (11)\(^{2}\) is satisfied.
-
The amplitude stability margin (for a linear characteristic \(S=\mathrm{const}\)) is determined mainly by the steepness of the ohm-ampere characteristic of the INE.
\(^{2}\) Similar inequalities defining the conditions for the onset of self-modulation have also been obtained for self-excited oscillators 2 and 3.
References
-
I. I. Fedorov. On certain parasitic processes in oscillation generators with “inertial nonlinearity” in the oscillatory circuit. — Radiotekhnika, vol. 9, 1954, No. 6.
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A. A. Lvovich. Transient processes and self-modulation in amplitude-stable generators with thermoresistances. — Elektrosvyaz, 1959, No. 5.
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V. N. Laskavyi. On amplitude instability of generators with inertial nonlinearity. — Pribory i sistemy upravleniya, 1969, No. 6.
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I. S. Gonorovskii. Radio Engineering Circuits and Signals. Moscow, “Sovetskoe radio,” 1964.