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

Authors: J. P. Dada; J. C. Chedjou; S. Domngang
Journal: Active and Passive Electronic Components
 (2008)

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## 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.


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## Body

## 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 frequencyω0, 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.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 Figures1 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 (Figure1, interrupter K off) to obtain the following equation of motion:  (1)v⃛0+6η1(1+η1)RCv¨0+5η1(1+η1)R2C2v˙0+η1(1+η1)R3C3v0+η2(1+η1)R3C3vi=0,with η1=R1/(R1+R2) and η2=1−η1.The time
evolution of the output voltagev0(t) of (1) is expressed as follows:(2a)v0=2λ13λ1+a1(±Vsat+αvi)exp[−12(λ1+a1)t]×cos(Δ*t)+λ1+a13λ1+a1(±Vsat+αvi)exp(λ1t)−αvi,
where
(2b)λ1=(a1a26−a1327−a32+Δ)1/3−(a1327+a32−a1a26+Δ)1/3−a13,(2c)Δ=a324+a13(a129−a22)a3+a2227(a2−a124),(2d)a2≻a123,(2e)Δ*=λ12+a1λ1+a2−(λ1+a12)2.a1, a2, a3, and α are defined by
(3a)a1=6η1(1+η1)RC,(3b)a2=5η1(1+η1)R2C2,(3c)a3=η1(1+η1)R3C3,(3d)α=η2η1.Vsat 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
(4)v(t)=v0(t)+ξ(t), where the perturbation parameter ξ(t) is sufficiently small.Substituting (4)
into (1), ξ(t) can be written in the form
(5a)ξ=A1exp(λ1t)+A2exp(λ2t)+A3exp(λ3t), where A1, A2, and A3 are small real constants and
(5b)λ2=−λ1+a12+iΔ*,(5c)λ3=−λ1+a12−iΔ*,λ1 and Δ* 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 a1, a2, and a3.
Stable limit cycle is obtained when
(6a)a3=a1a2. Also,
equilibrium points are obtained for
(6b)a3≺a1a2 while
divergent solutions deal with
(6c)a3≻a1a2.Taking into
account these stability conditions, we have found that the motion in the phase-shift
oscillator depends on the critical value of η1 (or η2). When 1/29≺η1≺5/7,
the equilibrium
points are obtained; while for η1<1/29, we have divergent solutions. Indeed when η1=1/29
(i.e., η2=28/29), a stable limit cycle is
obtained and the time evolution of the output voltage v0(t) is expressed as follows:
(7a)v0(t)=(±Vsat+28vi)Cos(ω0t)−28vi, where
(7b)ω0=1RC6.Equation (7a) clearly shows DC amplitude control of oscillations in the
phase-shift oscillator independently on the frequency of oscillations.


## 2.1.2. Stability, DC
amplitude control  of sinusoidal oscillations

Using perturbation
method, the solution of (1) can be written in the form
(4)v(t)=v0(t)+ξ(t), where the perturbation parameter ξ(t) is sufficiently small.Substituting (4)
into (1), ξ(t) can be written in the form
(5a)ξ=A1exp(λ1t)+A2exp(λ2t)+A3exp(λ3t), where A1, A2, and A3 are small real constants and
(5b)λ2=−λ1+a12+iΔ*,(5c)λ3=−λ1+a12−iΔ*,λ1 and Δ* 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 a1, a2, and a3.
Stable limit cycle is obtained when
(6a)a3=a1a2. Also,
equilibrium points are obtained for
(6b)a3≺a1a2 while
divergent solutions deal with
(6c)a3≻a1a2.Taking into
account these stability conditions, we have found that the motion in the phase-shift
oscillator depends on the critical value of η1 (or η2). When 1/29≺η1≺5/7,
the equilibrium
points are obtained; while for η1<1/29, we have divergent solutions. Indeed when η1=1/29
(i.e., η2=28/29), a stable limit cycle is
obtained and the time evolution of the output voltage v0(t) is expressed as follows:
(7a)v0(t)=(±Vsat+28vi)Cos(ω0t)−28vi, where
(7b)ω0=1RC6.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

Figure1 (when the interrupter K 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
(8)ω0=η1(1+η1)RC2(5R+2R3). It is clear from (8) that we can control the frequency of the oscillator by varying the potentiometer R3.


## 2.2. Twin-T oscillator

## 2.2.1. Equation of motion and output voltage

Considering the twin-T oscillator (Figure2, interrupter K off), we have found using KCL and KVL, that the output voltage v0(t) is solution of the following equation:
(9)v⃛0+2RbCbη2−Caη1(Ra+2Rb)RaRbCaCbη2v¨0+2RbCbη2−Raη1(Ca+2Cb)Ra2RbCaCb2η2v˙0+1Ra2RbCaCb2v0−1Ra2RbCaCb2vi=0.The time evolution
of the output voltagev0(t) of (9) is given by (2a), (2b), (2c), (2d) where a1, a2, a3, and α are redefined as
(10a)a1=2RbCbη2−Caη1(Ra+2Rb)RaRbCaCbη2,(10b)a2=2RbCbη2−Raη1(Ca+2Cb)Ra2RbCaCb2η2,(10c)a3=1Ra2RbCaCb2,(10d)α=−1.


## 2.2.2. Stability, DC amplitude control  of sinusoidal oscillations

We  can deduce from
the stability conditions (6a), (6b), (6c) the following inequalities:
(11a)β≺η1≺θ,(11b)η1≺2RbCb2RbCb+RaCa+2RbCa,(11c)η1≺2RbCb2RbCb+RaCa+2RaCb, with
(11d)Ca≻23Cb,(11e)θ=(4Rb2(2Cb2−CaCb)−RaRb(3Ca2+2CaCb))/(4Rb2CaCb+2RaRbCa2+8Rb2Ca2+2Ra2Ca2+8Rb2Cb2−4RaRbCaCb)+(RbCa9Ca2Ra2−12Cb2(Ra2+8RaRb)+S)/(4Rb2CaCb+2RaRbCa2+8Rb2Ca2+2Ra2Ca2+8Rb2Cb2−4RaRbCaCb),(11f)β=(4Rb2(2Cb2−CaCb)−RaRb(3Ca2+2CaCb))/(4Rb2CaCb+2RaRbCa2+8Rb2Ca2+2Ra2Ca2+8Rb2Cb2−4RaRbCaCb)−(RbCa9Ca2Ra2−12Cb2(Ra2+8RaRb)+S)/(4Rb2CaCb+2RaRbCa2+8Rb2Ca2+2Ra2Ca2+8Rb2Cb2−4RaRbCaCb), where S denotes 12CaCb(4RaRb+8Rb2+3Ra2).With conditions (11a), (11b), (11c), (11d), (11e), (11f), the motion in
the twin-T oscillator depends on the critical values of η1* and η1** defined by
(12a)η1*=(RbCb(Ra+2Rb)(Ca+2Cb))/(4(Ra+Rb)RbCb2+(2Ra2+4Rb2+7RaRb)CaCb+(Ra+2Rb)RaCa2)+(4Ra2Rb2Cb4+2RbCb2Ca2(Ra3+2Rb3+2Ra2Rb)−T)/(4(Ra+Rb)RbCb2+(2Ra2+4Rb2+7RaRb)CaCb+(Ra+2Rb)RaCa2),(12b)η1**=(RbCb(Ra+2Rb)(Ca+2Cb))/(4(Ra+Rb)RbCb2+(2Ra2+4Rb2+7RaRb)CaCb+(Ra+2Rb)RaCa2)−(4Ra2Rb2Cb4+2RbCb2Ca2(Ra3+2Rb3+2Ra2Rb)−T)/(4(Ra+Rb)RbCb2+(2Ra2+4Rb2+7RaRb)CaCb+(Ra+2Rb)RaCa2), where T denotes
8RaRb3CaCb3+Ra2RbCa3Cb(Ra+2Rb). It appears
clearly that a stable limit cycle is obtained for η1=η1* or η1=η1**.
The time evolution of the output voltage v0(t) is expressed as
(13a)v0(t)=(±Vsat−vi)Cos(ω0t)+vi, where
(13b)ω0=Q+Q′+Ra2RbCa2Cb3(2Rb+Ra)(Ca+2Cb)(2Rb+Ra)Ra2RbCb3Ca2. where Q denotes  −2RbCb2(RaCb−RbCa), Q′ denotes 4Rb2Cb4(RaCb−RbCa)2In addition, the equilibrium points
are obtained whenη1≺η1** or η1≻η1* while we have divergent solutions for η1**≺η1≺η1*.We have taken in
this study, as an illustration,Rb=Ra and Cb=Ca.
From inequality (11a) and (11b), (11c), (12a), (12b), and (13a), (13b), we have found that
when η1≺1/4 the equilibrium points are obtained; while for 1/4≺η1≺2/5,
we have divergent solutions. A stable
limit cycle is obtained when η1=1/4
(i.e., η2=3/4) and the time evolution of the output voltage v0(t) is expressed as follows:
(14)v0(t)=(±Vsat−vi)Cos(tRaCa)+vi.Equation (14)
clearly shows DC control of the amplitudes of oscillations in the twin-T
oscillator.


## 2.2.3. Frequency control

Figure2 (interrupter K 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
(15a)ω0=−2R4Cb2(RaCb−R4Ca)+W(2R4+Ra)Ra2R4Cb3Ca2, where W denotes 4R42Cb4(RaCb−R4Ca)2+Ra2R4Ca2Cb3(2R4+Ra)(Ca+2Cb), and
(15b)R4=RbR3Rb+R3. 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 R3.


## 3. Numerical computation

The aim of the numerical study is to verify the analytical results
established in Section2. 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.Waveforms of the Twin-T transient phenomenon withvi=0.4V; R1=1kΩ;
Ra=Rb=16KΩ; Ca=Cb=10nF; X-input: 1ms/cm; Y-input: 1V/cm.We have computed
numerically both the original (1) and (9) to obtain the time evolution of the
output voltage v0(t) 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→ stable limit cycle
→ divergent solutions) in the oscillators.Considering the phase-shift
oscillator, we have found (when monitoring the control voltagevi) the stable limit cycles when η1=1/29, the stable equilibrium points when
1/29≺η1≺5/7
and the divergent solutions for η1<1/29.
Moreover, we have found that when η1=1/29,
the limit cycles are obtained for all
vi and that fora critical value
vi=−0.535714V
the oscillations are completely damped (i.e.,
the oscillations vanish), leading to a stable equilibrium point corresponding
to a static voltage
v0=15V. 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
voltagevi from
(9)) the stable limit cycles when
η1=1/4  (see Figures 3(a) and 3(d)), the stable
equilibrium points when η1<1/4 (see Figures 3(b) and 3(e)) and the divergent
solutions for
1/4≺η1≺2/5
(see Figures 3(c) and 3(f)). Moreover, we
have found that when η1=1/4,
the limit cycles are obtained for all
vi and that for a critical value vi=15V
the oscillations vanish, leading to a
stable equilibrium point corresponding to a static voltage v0=15V.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 Figures1 and 2 are realized using the operational amplifiers (LM741CN) and the
multiturn resistors with a typical error less than 1%. v0(t) 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 voltagevi, we set the following values of the circuit components:
R1=1KΩ, R2=28KΩ, R=6.5KΩ, C=10nF, R3=+∞ and
Vcc=±15V. Vcc 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 −0.528V<vi<0.493V. When
vi=−0.528V (resp., vi=0.493V), the oscillations vanish, leading to the
equilibrium point or static voltage v0=14.65V (resp.,
v0=−13.69V). 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:R1=1KΩ, R2=3KΩ, Ra=Rb=R=16KΩ, Ca=Cb=C=10nF, R3=+∞, and
Vcc=±15V. Our experimental investigations have shown that the limit cycles are
obtained when
−14.65V<vi<14.72V. In particular, when vi=−14.65V
(resp., vi=14.72V
), the oscillations vanish, leading to a
static voltage v0=−14.625V (resp.,
v0=14.75V). When monitoring the resistor R1,
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 R3 for the two oscillators.The experimental results (see Figures3(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 voltagevi.
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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*Source: 539618-2008-08-03.xml*