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Active and Passive Electronic Components
Volume 2008, Article ID 539618, 6 pages
http://dx.doi.org/10.1155/2008/539618
Research Article

Amplitude and Frequency Control: Stability of Limit Cycles in Phase-Shift and Twin-T Oscillators

1Department of Physics, Faculty of Science, University of Yaoundé-I, P.O. Box 812, Yaoundé, Cameroon
2Department of Physics, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon

Received 1 October 2007; Revised 25 March 2008; Accepted 29 April 2008

Academic Editor: Fahrettin Yakuphanoglu

Copyright © 2008 J. P. Dada et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We show a technique for external direct current (DC) control of the amplitudes of limit cycles both in the Phase-shift and Twin-T oscillators. We have found that amplitudes of the oscillator output voltage depend on the DC control voltage. By varying the total impedance of each oscillator oscillatory network, frequencies of oscillations are controlled using potentiometers. The main advantage of the proposed circuits is that both the amplitude and frequency of the waveforms generated can be independently controlled. Analytical, numerical, and experimental methods are used to determine the boundaries of the states of the oscillators. Equilibrium points, stable limit cycles, and divergent states are found. Analytical results are compared with the numerical and experimental solutions, and a good agreement is obtained.

1. Introduction

In the last decade, there has been a strong interest in controlling the amplitude and frequency of the waveforms generated by oscillators [15]. The interest devoted to the voltage control oscillators (VCOs) is motivated by their technological and fundamental applications. Indeed, the sinusoidal waveforms generated by oscillators are used in measurement, instrumentation, and telecommunications to name a few.

In this paper, we propose a technique for external DC control of the amplitudes of limit cycles in both the phase-shift and twin-T oscillators. The choice of these oscillators is motivated by their capability to generate signals at very low frequencies (VLFs). In the presence of the DC control voltage, the oscillators run at a frequency 𝜔0, where 𝜔0 is the natural frequency determined by the components of the oscillatory network. The schematic diagrams of both the phase-shift and twin-T oscillators are shown, respectively, in Figures 1 and 2. Each oscillator consists of three main parts: the external DC control voltage (𝑣𝑖), the elementary amplifier, and the oscillatory network. The frequency of the waveforms generated is governed by the parameters of the oscillatory network, while the elementary amplifier helps to compensate the damping in the nonlinear oscillator.

539618.fig.001
Figure 1: The phase shift oscillator.
539618.fig.002
Figure 2: The twin-T oscillator.

VCOs have been intensively studied in previous publications; no theoretical expression has been proposed to show how the DC control voltage affects the amplitude of the time evolution of the waveforms generated by oscillators. In VCO circuits, the main goal is also to control frequency of oscillation. The principal aims of this paper are to examine those two aspects, since theoretical results of those circuits may be helpful system designers. We also discover a limit cycle heuristically, and another aim of this paper is to give detailed analysis for the observation of the transition among equilibrium points, stable limit cycles, and divergent solutions, since such a phenomenon never occurs as a codimension-one bifurcation in a dissipative dynamical system.

By applying the Kirchhoff Voltage Law (KVL) and Kirchhoff Current Law (KCL) to the electrical circuits of Figures 1 and 2, the equations describing the motion in the oscillators are obtained. Some mathematical tools are used to derive these equations and obtain the time evolution of the output voltage 𝑣0(𝑡). Some critical values are pointed out to define the transitions in the states of the oscillators. Equilibrium points, stable limit cycles, and divergent states are found. Also, numerical and experimental investigations are carried out to verify the analytical predictions.

2. Analytical Treatment

2.1. Phase-Shift Oscillator
2.1.1. Equation of Motion and Output Voltage

The Phase shift is analyzed based on the linear region. We apply the KCL and KVL on the electrical circuit (Figure 1, interrupter 𝐾 off) to obtain the following equation of motion: 𝑣0+6𝜂11+𝜂1̈𝑣𝑅𝐶0+5𝜂11+𝜂1𝑅2𝐶2̇𝑣0+𝜂11+𝜂1𝑅3𝐶3𝑣0+𝜂21+𝜂1𝑅3𝐶3𝑣𝑖=0,(1)with 𝜂1=𝑅1/(𝑅1+𝑅2) and 𝜂2=1𝜂1.

The time evolution of the output voltage 𝑣0(𝑡) of (1) is expressed as follows:

𝑣0=2𝜆13𝜆1+𝑎1±𝑉sat+𝛼𝑣𝑖1exp2𝜆1+𝑎1𝑡×cosΔ𝑡+𝜆1+𝑎13𝜆1+𝑎1±𝑉sat+𝛼𝑣𝑖𝜆exp1𝑡𝛼𝑣𝑖,(2a) where 𝜆1=𝑎1𝑎26𝑎31𝑎2732+Δ1/3𝑎31+𝑎2732𝑎1𝑎26+Δ1/3𝑎13,(2b)𝑎Δ=234+𝑎13𝑎219𝑎22𝑎3+𝑎22𝑎272𝑎214𝑎,(2c)2𝑎213Δ,(2d)=𝜆21+𝑎1𝜆1+𝑎2𝜆1+𝑎122.(2e)𝑎1𝑎2𝑎3,, 𝛼, 𝑎1=6𝜂11+𝜂1𝑎𝑅𝐶,(3a)2=5𝜂11+𝜂1𝑅2𝐶2𝑎,(3b)3=𝜂11+𝜂1𝑅3𝐶3𝜂,(3c)𝛼=2𝜂1.(3d) and 𝑉sat are defined by 𝑣(𝑡)=𝑣0(𝑡)+𝜉(t),(4)𝜉(𝑡)𝜉(𝑡)𝜉=𝐴1𝜆exp1𝑡+𝐴2𝜆exp2𝑡+𝐴3𝜆exp3𝑡,(5a)𝐴1 is the saturation voltage of the operational amplifiers determined by both the power supplies (static bias) and the internal structure of the operational amplifiers [6].

Equation (2a) predicts oscillations and it is nicer to study the stability of their oscillations.

2.1.2. Stability, DC Amplitude Control of Sinusoidal Oscillations

Using perturbation method, the solution of (1) can be written in the form 𝐴2, where the perturbation parameter 𝐴3 is sufficiently small.

Substituting (4) into (1), 𝜆2𝜆=1+𝑎12+𝑖Δ𝜆,(5b)3𝜆=1+𝑎12𝑖Δ,(5c) can be written in the form 𝜆1 where Δ, 𝑎1 and 𝑎2, are small real constants and 𝑎3𝑎3=𝑎1𝑎2.(6a)𝑎3𝑎1𝑎2(6b) and 𝑎3𝑎1𝑎2.(6c) are defined as above.

From (2b), (2e), and (5a), (5b), (5c), it is clear that the motion of oscillations depends on the critical relations between positive numbers 𝜂1, 𝜂2 and 1/29𝜂15/7. Stable limit cycle is obtained when 𝜂1<1/29, Also, equilibrium points are obtained for 𝜂1=1/29 while divergent solutions deal with 𝜂2=28/29Taking into account these stability conditions, we have found that the motion in the phase-shift oscillator depends on the critical value of 𝑣0(𝑡) (or 𝑣0(𝑡)=±𝑉sat+28𝑣𝑖𝜔Cos0𝑡28𝑣𝑖,(7a)). When 𝜔0=1𝑅𝐶6.(7b), the equilibrium points are obtained; while for 𝐾 we have divergent solutions. Indeed when 𝜔0=𝜂11+𝜂1𝑅𝐶25𝑅+2𝑅3.(8) (i.e., 𝑅3), a stable limit cycle is obtained and the time evolution of the output voltage 𝐾 is expressed as follows: 𝑣0(𝑡) where 𝑣0+2𝑅𝑏𝐶𝑏𝜂2𝐶𝑎𝜂1𝑅𝑎+2𝑅𝑏𝑅𝑎𝑅𝑏𝐶𝑎𝐶𝑏𝜂2̈𝑣0+2𝑅𝑏𝐶𝑏𝜂2𝑅𝑎𝜂1𝐶𝑎+2𝐶𝑏𝑅2𝑎𝑅𝑏𝐶𝑎𝐶2𝑏𝜂2̇𝑣0+1𝑅2𝑎𝑅𝑏𝐶𝑎𝐶2𝑏𝑣01𝑅2𝑎𝑅𝑏𝐶𝑎𝐶2𝑏𝑣𝑖=0.(9)

Equation (7a) clearly shows DC amplitude control of oscillations in the phase-shift oscillator independently on the frequency of oscillations.

2.1.3. Frequency Control

Figure 1 (when the interrupter 𝑣0(𝑡) is on) shows the possibility to control frequency of oscillations by using external potentiometer. By applying the KVL and KCL to this modified electrical circuit, the frequency of oscillations can be written as 𝑎1 It is clear from (8) that we can control the frequency of the oscillator by varying the potentiometer 𝑎2.

2.2. Twin-T Oscillator
2.2.1. Equation of Motion and Output Voltage

Considering the twin-T oscillator (Figure 2, interrupter 𝑎3, off), we have found using KCL and KVL, that the output voltage 𝛼 is solution of the following equation: 𝑎1=2𝑅𝑏𝐶𝑏𝜂2𝐶𝑎𝜂1𝑅𝑎+2𝑅𝑏𝑅𝑎𝑅𝑏𝐶𝑎𝐶𝑏𝜂2𝑎,(10a)2=2𝑅𝑏𝐶𝑏𝜂2𝑅𝑎𝜂1𝐶𝑎+2𝐶𝑏𝑅2𝑎𝑅𝑏𝐶𝑎𝐶2𝑏𝜂2𝑎,(10b)3=1𝑅2𝑎𝑅𝑏𝐶𝑎𝐶2𝑏,(10c)𝛼=1.(10d)

The time evolution of the output voltage 𝛽𝜂1𝜂𝜃,(11a)12𝑅𝑏𝐶𝑏2𝑅𝑏𝐶𝑏+𝑅𝑎𝐶𝑎+2𝑅𝑏𝐶𝑎𝜂,(11b)12𝑅𝑏𝐶𝑏2𝑅𝑏𝐶𝑏+𝑅𝑎𝐶𝑎+2𝑅𝑎𝐶𝑏,(11c) of (9) is given by (2a), (2b), (2c), (2d) where 𝐶𝑎23𝐶𝑏,(11d)𝜃=(4𝑅2𝑏2𝐶2𝑏𝐶𝑎𝐶𝑏𝑅𝑎𝑅𝑏3𝐶2𝑎+2𝐶𝑎𝐶𝑏/4𝑅2𝑏𝐶𝑎𝐶𝑏+2𝑅𝑎𝑅𝑏𝐶2𝑎+8𝑅2𝑏𝐶2𝑎+2𝑅2𝑎𝐶2𝑎+8𝑅2𝑏𝐶2𝑏4𝑅𝑎𝑅𝑏𝐶𝑎𝐶𝑏+(𝑅𝑏𝐶𝑎9𝐶2𝑎𝑅2𝑎12𝐶2𝑏𝑅2𝑎+8𝑅𝑎𝑅𝑏+𝑆)/(4𝑅2𝑏𝐶𝑎𝐶𝑏+2𝑅𝑎𝑅𝑏𝐶2𝑎+8𝑅2𝑏𝐶2𝑎+2𝑅2𝑎𝐶2𝑎+8𝑅2𝑏𝐶2𝑏4𝑅𝑎𝑅𝑏𝐶𝑎𝐶𝑏),(11e)𝛽=4𝑅2𝑏2𝐶2𝑏𝐶𝑎𝐶𝑏𝑅𝑎𝑅𝑏3𝐶2𝑎+2𝐶𝑎𝐶𝑏/4𝑅2𝑏𝐶𝑎𝐶𝑏+2𝑅𝑎𝑅𝑏𝐶2𝑎+8𝑅2𝑏𝐶2𝑎+2𝑅2𝑎𝐶2𝑎+8𝑅2𝑏𝐶2𝑏4𝑅𝑎𝑅𝑏𝐶𝑎𝐶𝑏(𝑅𝑏𝐶𝑎9𝐶2𝑎𝑅2𝑎12𝐶2𝑏𝑅2𝑎+8𝑅𝑎𝑅𝑏/+𝑆)4𝑅2𝑏𝐶𝑎𝐶𝑏+2𝑅𝑎𝑅𝑏𝐶2𝑎+8𝑅2𝑏𝐶2𝑎+2𝑅2𝑎𝐶2𝑎+8𝑅2𝑏𝐶2𝑏4𝑅𝑎𝑅𝑏𝐶𝑎𝐶𝑏,(11f), 𝑆, 12𝐶𝑎𝐶𝑏4𝑅𝑎𝑅𝑏+8𝑅2𝑏+3𝑅2𝑎 and 𝜂1 are redefined as 𝜂1𝜂1=𝑅𝑏𝐶𝑏𝑅𝑎+2𝑅𝑏𝐶𝑎+2𝐶𝑏/4𝑅𝑎+𝑅𝑏𝑅𝑏𝐶2𝑏+2𝑅2𝑎+4𝑅2𝑏+7𝑅𝑎𝑅𝑏𝐶𝑎𝐶𝑏+𝑅𝑎+2𝑅𝑏𝑅𝑎𝐶2𝑎+(4𝑅2𝑎𝑅2𝑏𝐶4𝑏+2𝑅𝑏𝐶2𝑏𝐶2𝑎𝑅3𝑎+2𝑅3𝑏+2𝑅2𝑎𝑅𝑏/4𝑅𝑇)𝑎+𝑅𝑏𝑅𝑏𝐶2𝑏+2𝑅2𝑎+4𝑅2𝑏+7𝑅𝑎𝑅𝑏𝐶𝑎𝐶𝑏+𝑅𝑎+2𝑅𝑏𝑅𝑎𝐶2𝑎,𝜂(12a)1=𝑅𝑏𝐶𝑏𝑅𝑎+2𝑅𝑏𝐶𝑎+2𝐶𝑏/4𝑅𝑎+𝑅𝑏𝑅𝑏𝐶2𝑏+2𝑅2𝑎+4𝑅2𝑏+7𝑅𝑎𝑅𝑏𝐶𝑎𝐶𝑏+𝑅𝑎+2𝑅𝑏𝑅𝑎𝐶2𝑎(4𝑅2𝑎𝑅2𝑏𝐶4𝑏+2𝑅𝑏𝐶2𝑏𝐶2𝑎𝑅3𝑎+2𝑅3𝑏+2𝑅2𝑎𝑅𝑏/4𝑅𝑇)𝑎+𝑅𝑏𝑅𝑏𝐶2𝑏+2𝑅2𝑎+4𝑅2𝑏+7𝑅𝑎𝑅𝑏𝐶𝑎𝐶𝑏+𝑅𝑎+2𝑅𝑏𝑅𝑎𝐶2𝑎,(12b)𝑇8𝑅𝑎𝑅3𝑏𝐶𝑎𝐶3𝑏+𝑅2𝑎𝑅𝑏𝐶3𝑎𝐶𝑏𝑅𝑎+2𝑅𝑏

2.2.2. Stability, DC Amplitude Control of Sinusoidal Oscillations

We can deduce from the stability conditions (6a), (6b), (6c) the following inequalities: 𝜂1=𝜂1𝜂1=𝜂1𝑣0(𝑡) with 𝑣0(𝑡)=±𝑉sat𝑣𝑖𝜔Cos0𝑡+𝑣𝑖,(13a)𝜔0=𝑄+𝑄+𝑅2𝑎𝑅𝑏𝐶2𝑎𝐶3𝑏2𝑅𝑏+𝑅𝑎𝐶𝑎+2𝐶𝑏2𝑅𝑏+𝑅𝑎𝑅2𝑎𝑅𝑏𝐶3𝑏𝐶2𝑎.(13b) where 𝑄 denotes 2𝑅𝑏𝐶2𝑏𝑅𝑎𝐶𝑏𝑅𝑏𝐶𝑎.

With conditions (11a), (11b), (11c), (11d), (11e), (11f), the motion in the twin-T oscillator depends on the critical values of 𝑄 and 4𝑅2𝑏𝐶4𝑏𝑅𝑎𝐶𝑏𝑅𝑏𝐶𝑎2 defined by 𝜂1𝜂1 where 𝜂1𝜂1 denotes 𝜂1𝜂1𝜂1. It appears clearly that a stable limit cycle is obtained for 𝑅𝑏=𝑅𝑎 or 𝐶𝑏=𝐶𝑎. The time evolution of the output voltage 𝜂11/4 is expressed as 1/4𝜂12/5 where 𝜂1=1/4 where 𝜂2=3/4 denotes 𝑣0(𝑡), 𝑣0(𝑡)=±𝑉sat𝑣𝑖𝑡Cos𝑅𝑎𝐶𝑎+𝑣𝑖.(14) denotes 𝐾

In addition, the equilibrium points are obtained when 𝜔0=2𝑅4𝐶2𝑏𝑅𝑎𝐶𝑏𝑅4𝐶𝑎+𝑊2𝑅4+𝑅𝑎𝑅2𝑎𝑅4𝐶3𝑏𝐶2𝑎,(15a) or 𝑊 while we have divergent solutions for 4𝑅24𝐶4𝑏𝑅𝑎𝐶𝑏𝑅4𝐶𝑎2+𝑅2𝑎𝑅4𝐶2𝑎𝐶3𝑏2𝑅4+𝑅𝑎𝐶𝑎+2𝐶𝑏.

We have taken in this study, as an illustration, 𝑅4=𝑅𝑏𝑅3𝑅𝑏+𝑅3.(15b) and 𝑅3. From inequality (11a) and (11b), (11c), (12a), (12b), and (13a), (13b), we have found that when 𝑣𝑖=0.4V the equilibrium points are obtained; while for 𝑅1=1kΩ, we have divergent solutions. A stable limit cycle is obtained when 𝑅𝑎=𝑅𝑏=16KΩ (i.e., 𝐶𝑎=𝐶𝑏=10nF) and the time evolution of the output voltage 1ms/cm is expressed as follows: 1V/cm

Equation (14) clearly shows DC control of the amplitudes of oscillations in the twin-T oscillator.

2.2.3. Frequency Control

Figure 2 (interrupter 𝑣0(𝑡) on) shows the possibility to control frequency of oscillations by using a potentiometer. By applying KVL and KCL to this modified electrical circuit, the frequency of oscillations can be written as where denotes 𝑣𝑖, and 𝜂1=1/29 It is clear from (13a) and (15a), (15b) that we can control the frequency of the oscillator independently of the amplitude, by varying the potentiometer 1/29𝜂15/7.

3. Numerical Computation

The aim of the numerical study is to verify the analytical results established in Section 2. We use the fourth-order Runge-Kutta algorithm [7] (see Figures 3(a), 3(b), 3(c)) and PSpice platform (see Figures 3(d), 3(e), 3(f)). The calculations are performed using real variables and constants in extended mode to obtain good precision on numerical results.

fig3
Figure 3: Waveforms of the Twin-T transient phenomenon with 𝜂1<1/29; 𝜂1=1/29; 𝑣𝑖; 𝑣𝑖=0.535714V; X-input: 𝑣0=15V; Y-input: 𝑣𝑖.

We have computed numerically both the original (1) and (9) to obtain the time evolution of the output voltage 𝜂1=1/4 and to control the amplitudes of the stable limit cycles, respectively, in the phase-shift oscillator and the twin-T oscillator.

Our numerical investigations were focused on the findings of the fundamental parameters (the amplitudes and the frequency) of the stable limit cycles in both oscillators. We have also determined the boundaries defining the transitions (equilibrium points 𝜂1<1/4 stable limit cycle 1/4𝜂12/5 divergent solutions) in the oscillators.

Considering the phase-shift oscillator, we have found (when monitoring the control voltage 𝜂1=1/4) the stable limit cycles when 𝑣𝑖, the stable equilibrium points when 𝑣𝑖=15V and the divergent solutions for 𝑣0=15V. Moreover, we have found that when 𝑣0(𝑡), the limit cycles are obtained for all 𝑣𝑖 and that fora critical value 𝑅1=1KΩ the oscillations are completely damped (i.e., the oscillations vanish), leading to a stable equilibrium point corresponding to a static voltage 𝑅2=28KΩ. We have also found that the frequency of the waveform generated is identical to that from the analytic prediction.

We have also considered the twin-T oscillator. We have found (when monitoring the control voltage 𝑅=6.5KΩ from (9)) the stable limit cycles when 𝐶=10nF (see Figures 3(a) and 3(d)), the stable equilibrium points when 𝑅3=+ (see Figures 3(b) and 3(e)) and the divergent solutions for 𝑉𝑐𝑐=±15V (see Figures 3(c) and 3(f)). Moreover, we have found that when 𝑉𝑐𝑐, the limit cycles are obtained for all 0.528V<𝑣𝑖<0.493V and that for a critical value 𝑣𝑖=0.528V the oscillations vanish, leading to a stable equilibrium point corresponding to a static voltage 𝑣𝑖=0.493V.

We have also found numerically that frequencies of the waveform generated by the two oscillators are identical to those from the analytic predictions (7b), (8), (13b), and (15a), (15b)).

Numerical simulations from (1) give similar figures (Figures 3) for phase-shift oscillator.

Comparing the analytical results with the numerical solutions, we found a good agreement between both methods.

The analytical and numerical predictions show the possibility to obtain stable limit cycles with unbounded values of the amplitudes of oscillations. This is experimentally unrealistic, the dynamics of the oscillators being limited by the static bias of the operational amplifiers. The interest of the experimental study carried out below is then justified, since it helps to obtain the real physical domains in which the limit cycles are obtained.

4. Experimental Analysis

This subsection deals with a direct implementation of the phase-shift and twin-T oscillators. The circuits of Figures 1 and 2 are realized using the operational amplifiers (LM741CN) and the multiturn resistors with a typical error less than 1%. 𝑣0=14.65V is obtained by feeding the output voltage of the operational amplifier to the X-input of an oscilloscope. The offset voltages of the operational amplifiers are cancelled using the method in [8].

We first consider the phase-shift oscillator. In order to control the oscillations by monitoring the DC voltage 𝑣0=13.69V, we set the following values of the circuit components: 𝑅1, 𝑅1=1KΩ, 𝑅2=3KΩ, 𝑅𝑎=𝑅𝑏=𝑅=16KΩ, 𝐶𝑎=𝐶𝑏=𝐶=10nF and 𝑅3=+,. 𝑉𝑐𝑐=±15V is the static bias (power supply) of the operational amplifiers. Our experimental investigations have shown that the limit cycles (that is the oscillations) are obtained when 14.65V<𝑣𝑖<14.72V. When 𝑣𝑖=14.65V (resp., 𝑣𝑖=14.72V), the oscillations vanish, leading to the equilibrium point or static voltage 𝑣0=14.625V (resp., 𝑣0=14.75V). We have also found the extreme sensitivity of the phase-shift oscillator to tiny changes in its components. Indeed, when monitoring the resistor 𝑅1, the stable equilibrium states and divergent states are manifested by a sudden disappearance of the orbit describing the limit cycle.

We also consider the twin-T oscillator. We set the following values of the circuit components: 𝑅3, 𝑣𝑖, , , and . Our experimental investigations have shown that the limit cycles are obtained when . In particular, when (resp., ), the oscillations vanish, leading to a static voltage (resp., ). When monitoring the resistor , the stable equilibrium states and divergent states are also manifested here by a sudden disappearance of the orbit describing the limit cycle. We observe that frequencies are controlled by potentiometer for the two oscillators.

The experimental results (see Figures 3(g), 3(h), 3(i)) are close to the analytical and numerical ones. The experimental investigations confirm that the behavior of the oscillators is limited by the static bias. The experimental boundaries for the occurrence of stable limit cycle are obtained.

5. Conclusion

This paper has proposed a technique for the external DC control of the amplitudes of oscillations in the phase-shift and twin-T oscillators. The time evolution of the waveforms generated by these oscillators is derived, showing that the fundamental characteristics of an oscillation (i.e., the amplitude and the frequency of oscillation) are independently controlled. The stability of the limit cycles has been analyzed and the boundaries defining the states of the oscillators are obtained. Stable equilibrium states, stable limit cycles, and divergent states have been obtained. We have carried out the digital computation to verify the analytic predictions. It is found that the results from both methods are identical. These methods show the existence of limit cycles with unbounded values of the amplitudes of oscillations. The unbounded values of the amplitudes cannot be realized experimentally, the dynamics of the oscillators being limited by the power supply (static bias). The experimental method carried out in this work aims to verify the results obtained from the analytical and numerical methods. This method also helps to determine the physical conditions in which the oscillators can be used. We have found that the amplitudes of limit cycles are bounded when monitoring the DC control voltage . The boundaries of oscillations have been obtained. We have also found both the equilibrium states and divergent states experimentally. The transition to these states passes through a sudden disappearance of the limit cycles. Comparing the experimental results with the analytic predictions, we found a good agreement.

An interesting question was also the control of the frequency of the oscillations in the phase-shift and twin-T oscillators. This was realized by controlling the total impedance of each oscillator oscillatory network.

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