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

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 [1–5]. 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 (𝑣_𝑖), 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.

Figure 1 

The phase shift oscillator.

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: 𝑣⃛₀+6𝜂¹1+𝜂¹𝑅𝐶𝑣̈⁰+5𝜂¹1+𝜂¹𝑅²𝐶²𝑣̇₀+𝜂¹1+𝜂¹𝑅³𝐶³𝑣₀+𝜂²1+𝜂¹𝑅³𝐶³𝑣_𝑖=0,(1)with 𝜂₁=𝑅₁/(𝑅₁+𝑅₂) and 𝜂₂=1−𝜂₁.

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

𝑣₀=2𝜆¹3𝜆₁+𝑎₁±𝑉^sat+𝛼𝑣_𝑖exp−12𝜆¹×cos√+𝑎₁𝑡Δ^∗𝑡+𝜆¹+𝑎₁3𝜆₁+𝑎₁±𝑉^sat+𝛼𝑣_𝑖exp𝜆¹𝑡−𝛼𝑣_𝑖,(2a) where 𝜆₁=𝑎¹𝑎₂6−2𝑎7³¹−𝑎2³+√Δ^1/3−𝑎³¹27+𝑎2³−𝑎¹𝑎₂6+√Δ^1/3−𝑎¹3,(2b)Δ=𝑎²³4+𝑎3¹𝑎9²¹−𝑎2²𝑎³+𝑎²²27𝑎²−𝑎²¹𝑎4,(2c)₂≻𝑎²¹Δ3,(2d)_∗=𝜆²¹+𝑎₁𝜆₁+𝑎₂−𝜆¹+𝑎₁2².(2e)𝑎₁𝑎₂𝑎₃,, 𝛼, 𝑎₁=6𝜂¹𝑎1+𝜂¹𝑅𝐶,(3a)₂=5𝜂¹1+𝜂¹𝑅²𝐶²𝑎,(3b)₃=𝜂¹1+𝜂¹𝑅³𝐶³𝛼=𝜂₂,(3c)𝜂₁.(3d) and 𝑉_sat are defined by 𝑣(𝑡)=𝑣₀(𝑡)+𝜉(t),(4)𝜉(𝑡)𝜉(𝑡)𝜉=𝐴₁exp𝜆₁𝑡+𝐴₂exp𝜆₂𝑡+𝐴₃exp𝜆₃𝑡,(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), 𝜆₂=−𝜆¹+𝑎₁2+𝑖√𝜆Δ^∗,(5b)₃=−𝜆¹+𝑎₁2−𝑖√Δ^∗,(5c) can be written in the form 𝜆₁ where Δ^∗, 𝑎₁ and 𝑎₂, are small real constants and 𝑎₃𝑎₃=𝑎₁𝑎₂.(6a)𝑎₃≺𝑎₁𝑎₂(6b) and 𝑎₃≻𝑎₁𝑎₂.(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 1/29≺𝜂₁≺5/7. 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)=±𝑉_sat+28𝑣_𝑖Cos𝜔⁰𝑡−28𝑣_𝑖,(7a)). When 𝜔₀=1𝑅𝐶√6.(7b), the equilibrium points are obtained; while for 𝐾 we have divergent solutions. Indeed when 𝜔₀=⎷𝜂₁1+𝜂¹𝑅𝐶²𝑅5+𝑅2₃.(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 𝑣⃛₀+2𝑅^𝑏𝐶_𝑏𝜂₂−𝐶_𝑎𝜂₁𝑅^𝑎+2𝑅_𝑏𝑅_𝑎𝑅_𝑏𝐶_𝑎𝐶_𝑏𝜂₂𝑣̈₀+2𝑅^𝑏𝐶_𝑏𝜂₂−𝑅_𝑎𝜂₁𝐶^𝑎+2𝐶_𝑏𝑅^2𝑎𝑅_𝑏𝐶_𝑎𝐶^2𝑏𝜂₂𝑣̇₀+1𝑅^2𝑎𝑅_𝑏𝐶_𝑎𝐶^2𝑏𝑣₀−1𝑅^2𝑎𝑅_𝑏𝐶_𝑎𝐶^2𝑏𝑣_𝑖=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=2𝑅^𝑏𝐶_𝑏𝜂₂−𝐶_𝑎𝜂₁𝑅^𝑎+2𝑅_𝑏𝑅_𝑎𝑅_𝑏𝐶_𝑎𝐶_𝑏𝜂₂𝑎,(10a)₂=2𝑅^𝑏𝐶_𝑏𝜂₂−𝑅_𝑎𝜂₁𝐶^𝑎+2𝐶_𝑏𝑅^2𝑎𝑅_𝑏𝐶_𝑎𝐶^2𝑏𝜂₂𝑎,(10b)₃=1𝑅^2𝑎𝑅_𝑏𝐶_𝑎𝐶^2𝑏𝛼=−1.,((1100dc))

The time evolution of the output voltage 𝜂𝛽≺𝜂₁≺𝜃,(11a)₁≺2𝑅^𝑏𝐶_𝑏2𝑅_𝑏𝐶_𝑏+𝑅_𝑎𝐶_𝑎+2𝑅_𝑏𝐶_𝑎𝜂,(11b)₁≺2𝑅^𝑏𝐶_𝑏2𝑅_𝑏𝐶_𝑏+𝑅_𝑎𝐶_𝑎+2𝑅_𝑎𝐶_𝑏,(11c) of (9) is given by (2a), (2b), (2c), (2d) where 𝐶_𝑎≻23𝐶^𝑏𝜃,=(4𝑅_2𝑏(11d)/4𝑅_2𝑏2𝐶^2𝑏−𝐶_𝑎𝐶_𝑏−𝑅^𝑎𝑅_𝑏3𝐶^2𝑎+2𝐶_𝑎𝐶_𝑏𝐶_𝑎𝐶_𝑏+2𝑅_𝑎𝑅_𝑏𝐶^2𝑎+8𝑅^2𝑏𝐶^2𝑎+2𝑅^2𝑎𝐶^2𝑎+8𝑅^2𝑏𝐶^2𝑏−+4(𝑅𝑅_𝑎𝑏𝑅_𝑏𝐶_𝑎𝐶_𝑏/(4𝑅_2𝑏𝐶_𝑎9𝐶^2𝑎𝑅^2𝑎−12𝐶^2𝑏𝑅^2𝑎+8𝑅_𝑎𝑅_𝑏+𝑆)𝐶_𝑎𝐶_𝑏+2𝑅_𝑎𝑅_𝑏𝐶^2𝑎+8𝑅^2𝑏𝐶^2𝑎+2𝑅^2𝑎𝐶^2𝑎+8𝑅^2𝑏𝐶^2𝑏𝛽=−4𝑅4_2𝑏𝑅_𝑎𝑅_𝑏𝐶_𝑎𝐶_𝑏),(11e)/4𝑅_2𝑏2𝐶^2𝑏−𝐶_𝑎𝐶_𝑏−𝑅^𝑎𝑅_𝑏3𝐶^2𝑎+2𝐶_𝑎𝐶_𝑏𝐶_𝑎𝐶_𝑏+2𝑅_𝑎𝑅_𝑏𝐶^2𝑎+8𝑅^2𝑏𝐶^2𝑎+2𝑅^2𝑎𝐶^2𝑎+8𝑅^2𝑏𝐶^2𝑏−−(𝑅4_𝑏𝑅_𝑎𝑅_𝑏𝐶_𝑎𝐶_𝑏/4𝑅_2𝑏𝐶_𝑎9𝐶^2𝑎𝑅^2𝑎−12𝐶^2𝑏𝑅^2𝑎+8𝑅_𝑎𝑅_𝑏+𝑆)𝐶_𝑎𝐶_𝑏+2𝑅_𝑎𝑅_𝑏𝐶^2𝑎+8𝑅^2𝑏𝐶^2𝑎+2𝑅^2𝑎𝐶^2𝑎+8𝑅^2𝑏𝐶^2𝑏−4𝑅_𝑎𝑅_𝑏𝐶_𝑎𝐶_𝑏,(11f), 𝑆, 12𝐶_𝑎𝐶_𝑏4𝑅^𝑎𝑅_𝑏+8𝑅^2𝑏+3𝑅^2𝑎 and 𝜂^∗1 are redefined as 𝜂^1∗∗𝜂^∗1=/4𝑅^𝑏𝑅𝐶_𝑏𝑅^𝑎+2𝑅_𝑏𝐶^𝑎+2𝐶_𝑏_𝑎+𝑅_𝑏𝑅^𝑏𝐶^2𝑏+2𝑅^2𝑎+4𝑅^2𝑏+7𝑅_𝑎𝑅_𝑏𝐶^𝑎𝐶_𝑏++(𝑅_𝑎+2𝑅_𝑏𝑅^𝑎𝐶^2𝑎/4𝑅4𝑅^2𝑎𝑅^2𝑏𝐶^4𝑏+2𝑅_𝑏𝐶^2𝑏𝐶^2𝑎𝑅^3𝑎+2𝑅^3𝑏+2𝑅^2𝑎𝑅_𝑏−𝑇)_𝑎+𝑅_𝑏𝑅^𝑏𝐶^2𝑏+2𝑅^2𝑎+4𝑅^2𝑏+7𝑅_𝑎𝑅_𝑏𝐶^𝑎𝐶_𝑏𝜂+𝑅_𝑎+2𝑅_𝑏𝑅_𝑎𝐶^2𝑎,(12a)_1∗∗=/4𝑅^𝑏𝑅𝐶_𝑏𝑅^𝑎+2𝑅_𝑏𝐶^𝑎+2𝐶_𝑏_𝑎+𝑅_𝑏𝑅^𝑏𝐶^2𝑏+2𝑅^2𝑎+4𝑅^2𝑏+7𝑅_𝑎𝑅_𝑏𝐶^𝑎𝐶_𝑏−+(𝑅_𝑎+2𝑅_𝑏𝑅^𝑎𝐶^2𝑎/4𝑅4𝑅^2𝑎𝑅^2𝑏𝐶^4𝑏+2𝑅_𝑏𝐶^2𝑏𝐶^2𝑎𝑅^3𝑎+2𝑅^3𝑏+2𝑅^2𝑎𝑅_𝑏−𝑇)_𝑎+𝑅_𝑏𝑅^𝑏𝐶^2𝑏+2𝑅^2𝑎+4𝑅^2𝑏+7𝑅_𝑎𝑅_𝑏𝐶^𝑎𝐶_𝑏+𝑅_𝑎+2𝑅_𝑏𝑅^𝑎𝐶^2𝑎,(12b)𝑇8𝑅_𝑎𝑅^3𝑏𝐶_𝑎𝐶^3𝑏+𝑅^2𝑎𝑅_𝑏𝐶^3𝑎𝐶_𝑏𝑅^𝑎+2𝑅_𝑏

2.2.2. Stability, DC Amplitude Control of Sinusoidal Oscillations

We can deduce from the stability conditions (6a), (6b), (6c) the following inequalities: 𝜂₁=𝜂^∗1𝜂₁=𝜂^1∗∗𝑣₀(𝑡) with 𝑣₀(𝑡)=±𝑉_sat−𝑣_𝑖Cos𝜔⁰𝑡+𝑣_𝑖,(13a)𝜔₀=⎷𝑄+𝑄^+𝑅^2𝑎𝑅_𝑏𝐶^2𝑎𝐶^3𝑏2𝑅^𝑏+𝑅_𝑎𝐶^𝑎+2𝐶_𝑏2𝑅^𝑏+𝑅_𝑎𝑅^2𝑎𝑅_𝑏𝐶^3𝑏𝐶^2𝑎.(13b) where 𝑄 denotes −2𝑅_𝑏𝐶^2𝑏𝑅^𝑎𝐶_𝑏−𝑅_𝑏𝐶_𝑎.

With conditions (11a), (11b), (11c), (11d), (11e), (11f), the motion in the twin-T oscillator depends on the critical values of 𝑄^ and 4𝑅^2𝑏𝐶^4𝑏𝑅^𝑎𝐶_𝑏−𝑅_𝑏𝐶_𝑎² defined by 𝜂₁≺𝜂^1∗∗ where 𝜂₁≻𝜂^∗1 denotes 𝜂^1∗∗≺𝜂₁≺𝜂^∗1. It appears clearly that a stable limit cycle is obtained for 𝑅_𝑏=𝑅_𝑎 or 𝐶_𝑏=𝐶_𝑎. The time evolution of the output voltage 𝜂₁≺1/4 is expressed as 1/4≺𝜂₁≺2/5 where 𝜂₁=1/4 where 𝜂₂=3/4 denotes 𝑣₀(𝑡), 𝑣₀(𝑡)=±𝑉_sat−𝑣_𝑖Cos𝑡𝑅_𝑎𝐶_𝑎+𝑣^𝑖.(14) denotes 𝐾

In addition, the equilibrium points are obtained when 𝜔₀=⎷−2𝑅₄𝐶^2𝑏𝑅^𝑎𝐶_𝑏−𝑅₄𝐶_𝑎+√𝑊2𝑅⁴+𝑅_𝑎𝑅^2𝑎𝑅₄𝐶^3𝑏𝐶^2𝑎,(15a) or 𝑊 while we have divergent solutions for 4𝑅²⁴𝐶^4𝑏𝑅^𝑎𝐶_𝑏−𝑅₄𝐶_𝑎²+𝑅^2𝑎𝑅₄𝐶^2𝑎𝐶^3𝑏2𝑅⁴+𝑅_𝑎𝐶^𝑎+2𝐶_𝑏.

We have taken in this study, as an illustration, 𝑅₄=𝑅^𝑏𝑅₃𝑅_𝑏+𝑅₃.(15b) and 𝑅₃. From inequality (11a) and (11b), (11c), (12a), (12b), and (13a), (13b), we have found that when 𝑣_𝑖=0.4V the equilibrium points are obtained; while for 𝑅₁=1kΩ, we have divergent solutions. A stable limit cycle is obtained when 𝑅_𝑎=𝑅_𝑏=16KΩ (i.e., 𝐶_𝑎=𝐶_𝑏=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 𝑣_𝑖, 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 1/29≺𝜂₁≺5/7.

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.

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 1/4≺𝜂₁≺2/5 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 𝑣_𝑖, the stable equilibrium points when 𝑣_𝑖=15V and the divergent solutions for 𝑣₀=15V. Moreover, we have found that when 𝑣₀(𝑡), the limit cycles are obtained for all 𝑣_𝑖 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 𝑉_𝑐𝑐=±15V (see Figures 3(c) and 3(f)). Moreover, we have found that when 𝑉_𝑐𝑐, the limit cycles are obtained for all −0.528V<𝑣_𝑖<0.493V and that for a critical value 𝑣_𝑖=−0.528V the oscillations vanish, leading to a stable equilibrium point corresponding to a static voltage 𝑣_𝑖=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, 𝑅_𝑎=𝑅_𝑏=𝑅=16KΩ, 𝐶_𝑎=𝐶_𝑏=𝐶=10nF and 𝑅₃=+∞,. 𝑉_𝑐𝑐=±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<𝑣_𝑖<14.72V. When 𝑣_𝑖=−14.65V (resp., 𝑣_𝑖=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, 𝑣_𝑖, , , 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.