Hindawi Publishing CorporationActive and Passive Electronic ComponentsVolume 2008, Article ID 539618, 6 pagesdoi:10.1155/2008/539618
Research ArticleAmplitude and Frequency Control: Stability of Limit Cycles inPhase-Shift and Twin-T Oscillators
J. P. Dada,1 J. C. Chedjou,2 and S. Domngang1
1 Department of Physics, Faculty of Science, University of Yaounde-I, P.O. Box 812, Yaounde, Cameroon2 Department of Physics, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon
Correspondence should be addressed to J. P. Dada, dada [email protected]
Received 1 October 2007; Revised 25 March 2008; Accepted 29 April 2008
Recommended by Fahrettin Yakuphanoglu
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 thetotal impedance of each oscillator oscillatory network, frequencies of oscillations are controlled using potentiometers. The mainadvantage of the proposed circuits is that both the amplitude and frequency of the waveforms generated can be independentlycontrolled. 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 andexperimental solutions, and a good agreement is obtained.
In the last decade, there has been a strong interest incontrolling the amplitude and frequency of the waveformsgenerated by oscillators [1–5]. The interest devoted to thevoltage control oscillators (VCOs) is motivated by theirtechnological and fundamental applications. Indeed, thesinusoidal waveforms generated by oscillators are used inmeasurement, instrumentation, and telecommunications toname a few.
In this paper, we propose a technique for external DCcontrol of the amplitudes of limit cycles in both the phase-shift and twin-T oscillators. The choice of these oscillatorsis motivated by their capability to generate signals at verylow frequencies (VLFs). In the presence of the DC controlvoltage, the oscillators run at a frequency ω0, where ω0 isthe natural frequency determined by the components of theoscillatory network. The schematic diagrams of both thephase-shift and twin-T oscillators are shown, respectively,in Figures 1 and 2. Each oscillator consists of three mainparts: the external DC control voltage (vi), the elementaryamplifier, and the oscillatory network. The frequency of thewaveforms generated is governed by the parameters of theoscillatory network, while the elementary amplifier helps tocompensate the damping in the nonlinear oscillator.
VCOs have been intensively studied in previous publica-tions; no theoretical expression has been proposed to showhow the DC control voltage affects the amplitude of thetime evolution of the waveforms generated by oscillators. InVCO circuits, the main goal is also to control frequency ofoscillation. The principal aims of this paper are to examinethose two aspects, since theoretical results of those circuitsmay be helpful system designers. We also discover a limitcycle heuristically, and another aim of this paper is togive detailed analysis for the observation of the transitionamong equilibrium points, stable limit cycles, and divergentsolutions, since such a phenomenon never occurs as acodimension-one bifurcation in a dissipative dynamicalsystem.
By applying the Kirchhoff Voltage Law (KVL) andKirchhoff Current Law (KCL) to the electrical circuits ofFigures 1 and 2, the equations describing the motionin the oscillators are obtained. Some mathematical toolsare used to derive these equations and obtain the timeevolution of the output voltage v0(t). Some critical valuesare pointed out to define the transitions in the states ofthe oscillators. Equilibrium points, stable limit cycles, anddivergent states are found. Also, numerical and experimen-tal investigations are carried out to verify the analyticalpredictions.
The Phase shift is analyzed based on the linear region. Weapply the KCL and KVL on the electrical circuit (Figure 1,interrupter K off) to obtain the following equation ofmotion:
...v 0 +
6η1(1 + η1
)RC
v0 +5η1(
1 + η1)R2C2
v0
+η1(
1 + η1)R3C3
v0 +η2(
1 + η1)R3C3
vi = 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 = 2λ1
3λ1 + a1
(±Vsat + αvi)
exp[− 1
2
(λ1 + a1
)t]
× cos(√Δ∗t
)+λ1 + a1
3λ1 + a1
(±Vsat + αvi)
exp(λ1t)− αvi,
(2a)
where
λ1 =(a1a2
6− a3
1
27− a3
2+√Δ)1/3
−(a3
1
27+a3
2− a1a2
6+√Δ)1/3
− a1
3,
(2b)
Δ = a23
4+a1
3
(a2
1
9− a2
2
)a3 +
a22
27
(a2 − a2
1
4
), (2c)
a2 � a21
3, (2d)
Δ∗ = λ21 + a1λ1 + a2 −
(λ1 + a1
2
)2
. (2e)
a1, a2, a3, and α are defined by
a1 =6η1(
1 + η1)RC
, (3a)
a2 =5η1(
1 + η1)R2C2
, (3b)
a3 =η1(
1 + η1)R3C3
, (3c)
α = η2
η1. (3d)
Vsat is the saturation voltage of the operational amplifiersdetermined by both the power supplies (static bias) and theinternal structure of the operational amplifiers [6].
Equation (2a) predicts oscillations and it is nicer to studythe stability of their oscillations.
2.1.2. Stability, DC amplitude controlof sinusoidal oscillations
Using perturbation method, the solution of (1) can bewritten in the form
v(t) = v0(t) + ξ(t), (4)
where the perturbation parameter ξ(t) is sufficiently small.Substituting (4) into (1), ξ(t) can be written in the form
ξ = A1 exp(λ1t)
+ A2 exp(λ2t)
+ A3 exp(λ3t), (5a)
where A1, A2, and A3 are small real constants and
λ2 = −λ1 + a1
2+ i√Δ∗, (5b)
λ3 = −λ1 + a1
2− i√Δ∗, (5c)
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 relationsbetween positive numbers a1, a2, and a3. Stable limit cycle isobtained when
a3 = a1a2. (6a)
Also, equilibrium points are obtained for
a3 ≺ a1a2 (6b)
while divergent solutions deal with
a3 � a1a2. (6c)
Taking into account these stability conditions, we have foundthat the motion in the phase-shift oscillator depends on thecritical value of η1 (or η2). When 1/29 ≺ η1 ≺ 5/7, theequilibrium points are obtained; while for η1 < 1/29, wehave divergent solutions. Indeed when η1 = 1/29 (i.e., η2 =28/29), a stable limit cycle is obtained and the time evolutionof the output voltage v0(t) is expressed as follows:
v0(t) = (±Vsat + 28vi)
Cos(ω0t)− 28vi, (7a)
J. P. Dada et al. 3
where
ω0 = 1RC√
6. (7b)
Equation (7a) clearly shows DC amplitude control ofoscillations in the phase-shift oscillator independently on thefrequency of oscillations.
2.1.3. Frequency control
Figure 1 (when the interrupter K is on) shows the possibilityto control frequency of oscillations by using external poten-tiometer. By applying the KVL and KCL to this modifiedelectrical circuit, the frequency of oscillations can be writtenas
ω0 =√
η1(1 + η1
)RC2
(5R
+2R3
). (8)
It is clear from (8) that we can control the frequency of theoscillator by varying the potentiometer R3.
2.2. Twin-T oscillator
2.2.1. Equation of motion and output voltage
Considering the twin-T oscillator (Figure 2, interrupter Koff), we have found using KCL and KVL, that the outputvoltage v0(t) is solution of the following equation:
...v 0 +
2RbCbη2 − Caη1(Ra + 2Rb
)
RaRbCaCbη2v0
+2RbCbη2 − Raη1
(Ca + 2Cb
)
R2aRbCaC
2bη2
v0
+1
R2aRbCaC
2b
v0 − 1R2aRbCaC
2b
vi = 0.
(9)
The time evolution of the output voltage v0(t) of (9) isgiven by (2a), (2b), (2c), (2d) where a1, a2, a3, and α areredefined as
a1 =2RbCbη2 − Caη1
(Ra + 2Rb
)
RaRbCaCbη2, (10a)
a2 =2RbCbη2 − Raη1
(Ca + 2Cb
)
R2aRbCaC
2bη2
, (10b)
a3 = 1R2aRbCaC
2b
, (10c)
α = −1. (10d)
2.2.2. Stability, DC amplitude controlof sinusoidal oscillations
We can deduce from the stability conditions (6a), (6b), (6c)the following inequalities:
β ≺ η1 ≺ θ, (11a)
η1 ≺ 2RbCb2RbCb + RaCa + 2RbCa
, (11b)
η1 ≺ 2RbCb2RbCb + RaCa + 2RaCb
, (11c)
with
Ca � 23Cb, (11d)
θ = (4R2b
(2C2
b − CaCb)− RaRb
(3C2
a + 2CaCb))
/(4R2
bCaCb + 2RaRbC2a + 8R2
bC2a + 2R2
aC2a + 8R2
bC2b
− 4RaRbCaCb)
+(RbCa
√9C2
aR2a − 12C2
b
(R2a + 8RaRb
)+ S)
/(4R2bCaCb + 2RaRbC2
a + 8R2bC
2a + 2R2
aC2a + 8R2
bC2b
− 4RaRbCaCb),(11e)
β = (4R2b
(2C2
b − CaCb)− RaRb
(3C2
a + 2CaCb))
/(4R2
bCaCb + 2RaRbC2a + 8R2
bC2a + 2R2
aC2a + 8R2
bC2b
− 4RaRbCaCb)
−(RbCa
√9C2
aR2a − 12C2
b
(R2a + 8RaRb
)+ S)
/(4R2
bCaCb + 2RaRbC2a + 8R2
bC2a + 2R2
aC2a + 8R2
bC2b
− 4RaRbCaCb),
(11f)
where S denotes 12CaCb(4RaRb + 8R2
b + 3R2a
).
With conditions (11a), (11b), (11c), (11d), (11e), (11f),the motion in the twin-T oscillator depends on the criticalvalues of η∗1 and η∗∗1 defined by
η∗1 =(RbCb
(Ra + 2Rb
)(Ca + 2Cb
))
/(4(Ra + Rb
)RbC
2b +
(2R2
a + 4R2b + 7RaRb
)CaCb
+(Ra + 2Rb
)RaC
2a
)
+(√
4R2aR
2bC
4b + 2RbC2
bC2a
(R3a + 2R3
b + 2R2aRb)− T
)
/(4(Ra + Rb
)RbC
2b +
(2R2
a + 4R2b + 7RaRb
)CaCb
+(Ra + 2Rb)RaC2
a
),
(12a)
η∗∗1 = (RbCb(Ra + 2Rb
)(Ca + 2Cb
))
/(4(Ra + Rb
)RbC
2b +
(2R2
a + 4R2b + 7RaRb
)CaCb
+(Ra + 2Rb
)RaC
2a
)
−(√
4R2aR
2bC
4b + 2RbC2
bC2a
(R3a + 2R3
b + 2R2aRb)− T
)
/(4(Ra + Rb
)RbC
2b +
(2R2
a + 4R2b + 7RaRb
)CaCb
+(Ra + 2Rb
)RaC
2a
),
(12b)
where T denotes 8RaR3bCaC
3b + R2
aRbC3aCb
(Ra + 2Rb
). It
appears clearly that a stable limit cycle is obtained for η1 = η∗1
4 Active and Passive Electronic Components
or η1 = η∗∗1 . The time evolution of the output voltage v0(t)is expressed as
v0(t) = (±Vsat − vi)
Cos(ω0t)
+ vi, (13a)
where
ω0 =
√√√√Q +
√Q′ + R2
aRbC2aC
3b
(2Rb + Ra
)(Ca + 2Cb
)
(2Rb + Ra
)R2aRbC
3bC
2a
.
(13b)
where Q denotes −2RbC2b
(RaCb − RbCa
), Q′ denotes
4R2bC
4b
(RaCb − RbCa
)2
In addition, the equilibrium points are obtained whenη1 ≺ η∗∗1 or η1 � η∗1 while we have divergent solutions forη∗∗1 ≺ η1 ≺ η∗1 .
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 whenη1 ≺ 1/4 the equilibrium points are obtained; while for1/4 ≺ η1 ≺ 2/5, we have divergent solutions. A stable limitcycle is obtained when η1 = 1/4 (i.e., η2 = 3/4) and the timeevolution of the output voltage v0(t) is expressed as follows:
v0(t) = (±Vsat − vi)
Cos(
t
RaCa
)+ vi. (14)
Equation (14) clearly shows DC control of the ampli-tudes of oscillations in the twin-T oscillator.
2.2.3. Frequency control
Figure 2 (interrupter K on) shows the possibility to controlfrequency of oscillations by using a potentiometer. Byapplying KVL and KCL to this modified electrical circuit, thefrequency of oscillations can be written as
ω0 =√√√√−2R4C
2b
(RaCb − R4Ca
)+√W
(2R4 + Ra
)R2aR4C
3bC
2a
, (15a)
where W denotes 4R24C
4b
(RaCb − R4Ca
)2+ R2
aR4C2aC
3b
(2R4 +
Ra)(Ca + 2Cb
), and
R4 = RbR3
Rb + R3. (15b)
It is clear from (13a) and (15a), (15b) that we can control thefrequency 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 analyticalresults established in Section 2. We use the fourth-orderRunge-Kutta algorithm [7] (see Figures 3(a), 3(b), 3(c))and PSpice platform (see Figures 3(d), 3(e), 3(f)). Thecalculations are performed using real variables and constantsin extended mode to obtain good precision on numericalresults.
R1
R2
OA−+ V0(t)
vi
K
R3Rb
Ca
Cb Cb
Ra Ra
Figure 2: The twin-T oscillator.
We have computed numerically both the original (1)and (9) to obtain the time evolution of the output voltagev0(t) and to control the amplitudes of the stable limit cycles,respectively, in the phase-shift oscillator and the twin-Toscillator.
Our numerical investigations were focused on the find-ings of the fundamental parameters (the amplitudes and thefrequency) of the stable limit cycles in both oscillators. Wehave also determined the boundaries defining the transitions(equilibrium points → stable limit cycle → divergent solu-tions) in the oscillators.
Considering the phase-shift oscillator, we have found(when monitoring the control voltage vi) the stable limitcycles when η1 = 1/29, the stable equilibrium points when1/29 ≺ η1 ≺ 5/7 and the divergent solutions for η1 < 1/29.Moreover, we have found that when η1 = 1/29, the limitcycles are obtained for all vi and that fora critical valuevi = −0.535714 V the oscillations are completely damped(i.e., the oscillations vanish), leading to a stable equilibriumpoint corresponding to a static voltage v0 = 15 V. We havealso found that the frequency of the waveform generated isidentical to that from the analytic prediction.
We have also considered the twin-T oscillator. We havefound (when monitoring the control voltage vi from (9)) thestable 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 (seeFigures 3(c) and 3(f)). Moreover, we have found that whenη1 = 1/4, the limit cycles are obtained for all vi and that fora critical value vi = 15 V the oscillations vanish, leading toa stable equilibrium point corresponding to a static voltagev0 = 15 V.
We have also found numerically that frequencies of thewaveform generated by the two oscillators are identical tothose 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 numericalsolutions, we found a good agreement between both meth-ods.
The analytical and numerical predictions show the pos-sibility to obtain stable limit cycles with unbounded values
J. P. Dada et al. 5
0 2 4 6 8 1010
11
12
13
14
15
16η1 = 1.028/4 η1 = 1.02/4
(a) (d) (g)
0 2 4 6 8 1010
11
12
13
14
15
16η1 = 0.99/4 η1 = 0.95/4
(b) (e) (h)
0 2 4 6 8 109
11
13
15
17
19η1 = 1.3/4 η1 = 1.14/4
(c) (f) (i)
Figure 3: Waveforms of the Twin-T transient phenomenon with vi = 0.4 V; R1 = 1 kΩ; Ra = Rb = 16 KΩ; Ca = Cb = 10 nF; X-input:1 ms/cm; Y-input: 1 V/cm.
of the amplitudes of oscillations. This is experimentallyunrealistic, the dynamics of the oscillators being limited bythe static bias of the operational amplifiers. The interest ofthe experimental study carried out below is then justified,since it helps to obtain the real physical domains in whichthe limit cycles are obtained.
4. EXPERIMENTAL ANALYSIS
This subsection deals with a direct implementation of thephase-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 errorless than 1%. v0(t) is obtained by feeding the output voltageof the operational amplifier to the X-input of an oscilloscope.The offset voltages of the operational amplifiers are cancelledusing the method in [8].
We first consider the phase-shift oscillator. In orderto control the oscillations by monitoring the DC voltagevi, we set the following values of the circuit components:R1 = 1 KΩ, R2 = 28 KΩ, R = 6.5 KΩ, C = 10 nF,R3 = +∞ and Vcc = ±15 V. Vcc is the static bias (powersupply) of the operational amplifiers. Our experimentalinvestigations have shown that the limit cycles (that is theoscillations) are obtained when −0.528 V < vi < 0.493 V.
When vi = −0.528 V (resp., vi = 0.493 V), the oscillationsvanish, leading to the equilibrium point or static voltagev0 = 14.65 V (resp., v0 = −13.69 V). We have also foundthe extreme sensitivity of the phase-shift oscillator to tinychanges in its components. Indeed, when monitoring theresistor R1, the stable equilibrium states and divergent statesare manifested by a sudden disappearance of the orbitdescribing the limit cycle.
We also consider the twin-T oscillator. We set thefollowing values of the circuit components: R1 = 1 KΩ, R2 =3 KΩ, Ra = Rb = R = 16 KΩ, Ca = Cb = C = 10 nF, R3 =+∞, and Vcc = ±15 V. Our experimental investigations haveshown that the limit cycles are obtained when −14.65 V <vi < 14.72 V. In particular, when vi = −14.65 V (resp., vi =14.72 V ), the oscillations vanish, leading to a static voltagev0 = −14.625 V (resp., v0 = 14.75 V). When monitoring theresistor R1, the stable equilibrium states and divergent statesare also manifested here by a sudden disappearance of theorbit describing the limit cycle. We observe that frequenciesare controlled by potentiometer R3 for the two oscillators.
The experimental results (see Figures 3(g), 3(h), 3(i)) areclose to the analytical and numerical ones. The experimentalinvestigations confirm that the behavior of the oscillators islimited by the static bias. The experimental boundaries forthe occurrence of stable limit cycle are obtained.
6 Active and Passive Electronic Components
5. CONCLUSION
This paper has proposed a technique for the external DCcontrol of the amplitudes of oscillations in the phase-shiftand twin-T oscillators. The time evolution of the waveformsgenerated by these oscillators is derived, showing that thefundamental characteristics of an oscillation (i.e., the ampli-tude and the frequency of oscillation) are independentlycontrolled. The stability of the limit cycles has been analyzedand the boundaries defining the states of the oscillators areobtained. Stable equilibrium states, stable limit cycles, anddivergent states have been obtained. We have carried outthe digital computation to verify the analytic predictions. Itis found that the results from both methods are identical.These methods show the existence of limit cycles withunbounded values of the amplitudes of oscillations. Theunbounded values of the amplitudes cannot be realizedexperimentally, the dynamics of the oscillators being limitedby the power supply (static bias). The experimental methodcarried out in this work aims to verify the results obtainedfrom the analytical and numerical methods. This methodalso helps to determine the physical conditions in which theoscillators can be used. We have found that the amplitudesof limit cycles are bounded when monitoring the DCcontrol voltage vi. The boundaries of oscillations have beenobtained. We have also found both the equilibrium statesand divergent states experimentally. The transition to thesestates passes through a sudden disappearance of the limitcycles. Comparing the experimental results with the analyticpredictions, we found a good agreement.
An interesting question was also the control of thefrequency of the oscillations in the phase-shift and twin-T oscillators. This was realized by controlling the totalimpedance of each oscillator oscillatory network.
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