Instability of an oscillator amplitude control system

T.H. O’Dell

Abstract: A closed-loop amplitude control system must be included in the design of an electronicoscillator when a sinusoidal output of very low distortion is required. The instability of sucha control system is considered and an experimental oscillator is used to check the theoreticalpredictions.

1 Introduction

Sinusoidal sources with a very low distortion have beenneeded for many years, particularly by audio engineers.The need for a very low distortion means that thesimplest kind of oscillator system, shown here in Fig. 1,will usually be unacceptable. In the system shown inFig. 1, the oscillator involves a feedback loop in whichF(s) is a linear amplifier, s being the complex variable ofthe Laplace transform and F(s) the amplifier transferfunction, and GðvoÞ is an inverting gain block which hasa frequency independent gain, GðvoÞ; just above unity atlow input levels, but falling as vo increases: a limitingdevice. F(s) thus determines the frequency of oscillationwhereas GðvoÞ determines the amplitude. Sidorowicz [1]has given an interesting theoretical review of oscillatorsof this kind.

A sinusoidal source with a really low distortion calls foran oscillator system in which the inverting gain block has itsgain properly controlled by means of a separate control-loop. In the simplest kind of amplitude control system, therectified output of the oscillator would be compared withsome external reference, vref ; and the error used to controlthe gain. Such a simple system can have a very poortransient response to a change in vref and may be unstable.These problems have led designers to adopt fast amplitudecontrol systems for their oscillators: either based uponmultiphase rectifiers that do not involve filters [2–6] orsystems that use sampling techniques [7, 8]. All these fastsystems are useful, but the fact remains that the stabilityproblems of the simple rectifier plus low pass filter systemremain unexplained: quite recently such a system could bedescribed as ‘a complicated dynamical system which defiessimple analytical solutions’ by [9].

We are concerned with the instability problem of thesimple kind of amplitude control system and seek a solutionto the highly non-linear differential equation that governs it.The frequency of the instability oscillation and itsdependence on the oscillator output level are predictedand compared with measurements made on an experimentaloscillator. A simple method of stabilisation is proposed,which may well be as satisfactory as the fast control systemsreferred to above.

2 Theory

A second-order sinusoidal oscillator using all-pass networkswill be chosen for this work. This means that the transferfunction, F(s), of the network that determines the frequencyof oscillation will be given by:

FðsÞ ¼ ð1 � sC1R1Þ2=ð1 þ sC1R1Þ2 ð1Þ

involving two capacitors, both with a capacitance of C1; andtwo resistors, both with a resistance of R1: When the loopshown in Fig. 1 is closed by means of the inverting gainblock, GðvoÞ; and GðvoÞ ¼ �1; the frequency of oscillationwill be given by:

fo ¼ 1=2pC1R1 ð2Þ

the frequency at which FðsÞ ¼ �1:Oscillators using all-pass networks have been considered

by Sidorowicz [10], and his earlier experimental work[11, 12] demonstrated the wide tuning range that can beachieved with simple variable components. All-passoscillators do have a disadvantage, however, simplybecause of their all-pass nature: any distortion that mightbe introduced by a non-linear element, such as the GðvoÞblock shown in Fig. 1, will not be attenuated before reachingthe output, vo in Fig. 1, if F(s) is all-pass. Oscillators of theall-pass variety really do need a proper control-loop todetermine the output level if a very low distortion sinewaveis called for at the output.

In view of this, we consider an oscillator of the kindshown in Fig. 2, where F(s) is given by (1) and theoscillator-loop is closed via the inverting gain block, GðvgÞ:The oscillator-loop then has an open-loop transfer function�GðvgÞð1 � sC1R1Þ2=ð1 þ sC1R1Þ2 which, for an oscil-lator, must equal unity at the oscillation frequency. Settings ¼ d=dt; this gives the second-order differential equationwhich governs the oscillator-loop as:

d2vo

dt2þ

2ð1 � GðvgÞÞC1R1ð1 þ GðvgÞÞ

dvodt

þ vo

ðC1R1Þ2¼ 0 ð3Þ

showing that the oscillations will grow exponentially whenGðvgÞ > 1 and decay exponentially when GðvgÞ < 1: GðvgÞis a controlled gain, being controlled by the amplitudecontrol-loop seen at the top of Fig. 2. The control-loop takesthe oscillator output, vo; into an ideal full wave rectifier(IFWR) and compares the output of this with vref to providean error input to the integrator. Because the integrator timeconstant, C2R2; will be made much greater than C1R1

(to ensure negligible ripple on the integrator output, vgÞthe output from the IFWR can be taken at its average value,2vvo=p; and it is when vvo ¼ �pvref=2 that the total input

q IEE, 2004

IEE Proceedings online no. 20040104

doi: 10.1049/ip-cta:20040104

The author is at 44 Westbourne Gardens, London W2 5NS, UK

Paper first received 4th July and in revised form 21st November 2003

IEE Proc.-Control Theory Appl., Vol. 151, No. 2, March 2004194

current to the integrator virtual earth sums to zero and vgshould equal the value that makes GðvgÞ ¼ 1: This is thethinking behind this kind of oscillator, but several authors[3, 6, 7, 9] have pointed out that there is a problem with thisdesign: when the frequency is changed, or the amplitudesetting, vref ; is changed, the output level will fluctuate atsome frequency well below fo; and may continue to fluctuatefor some time. This is a great disadvantage in low frequencyaudio oscillators.

To understand the cause of this problem, the differentialequation for the control-loop must be found and combinedwith (3) to give the equation that governs the entire system.To do this, voðtÞ will be represented by:

voðtÞ ¼ vvoð1 þ mðtÞÞ expð jwotÞ ð4Þ

where wo ¼ 2pfo: In (4), it is m(t) which represents theexpected slow fluctuations in the amplitude of the oscillatoroutput: fluctuations in the amplitude level set by vref : Thethree currents flowing into the virtual earth of the integratorshown in Fig. 2 are thus vref=R2; 2vvoð1 þ mðtÞÞ=pR2 andC2dvg=dt; the first being a constant and the last two varyingat a frequency well below fo: These three currents must allsum to zero, and because vref will be set to �2vvo=p; theresult of this summation will be simply:

dvg=dt þ ð2vv0=pC2R2ÞmðtÞ ¼ 0 ð5Þ

The problem now is to couple (3) and (5), using (4) and theequation defining GðvgÞ: When a modern device is used forthis gain block, a linear relationship:

GðvgÞ ¼ vg=V ð6Þ

where V is a constant, can be obtained very closely inpractice.

The first step is to substitute (4) into (3) and set both thereal and imaginary parts of the result to zero. This gives twonew differential equations:

d2m

dt2þ 2ð1 � GÞC1R1ð1 þ GÞ

dm

dt¼ 0 ð7Þ

and

dm

dtþ ð1 � GÞð1 þ mÞ

C1R1ð1 þ GÞ ¼ 0 ð8Þ

where, for brevity, m(t) and GðvgÞ are represented by simplym and G.

It is (8) that takes us further because, when differentiatedwith respect to time, it yields:

d2m

dt2þ ð1 � GÞC1R1ð1 þ GÞ

dm

dt� 2ð1 þ mÞC1R1ð1 þ GÞ2

dG

dt¼ 0 ð9Þ

and, when (6) is used to replace the term dG=dt withV�1dvg=dt; (5) can be used to turn (9) into:

d2m

dt2þ ð1 � GÞC1R1ð1 þ GÞ

dm

dtþ 4vvoð1 þ mÞpVC1R1C2R2ð1 þ GÞ2

m ¼ 0

ð10ÞEquation (10) may look formidable, G being a function oftime and m occurring as m2 in the last term, but a greatsimplification results when it is assumed that m � 1: that is,the fluctuations in the oscillator output level are very smallcompared to the level set by vref : Then, ð1 þ mÞ � 1 andð1 þ GÞ � 2; reducing (10) to simply:

d2m

dt2þ ð1 � GÞ

2C1R1

dm

dtþ vv0

pVC1R1C2R2

m ¼ 0 ð11Þ

which shows that m should be a very lightly dampedsinusoidal oscillation at a frequency, fm; given by:

fm ¼ 1

2pvvo

pVC1R1C2R2

� �12

ð12Þ

The oscillations are lightly damped because, when theiramplitude is small, G � 1 and the coefficient of dm=dt in(11) will tend to vanish, leaving a description of simpleundamped harmonic motion at the frequency given by (12).

3 Experiment

In order to check the theory given above, an oscillator of thetype shown in Fig. 2 was built, and a simplified circuitdiagram of this is shown in Fig. 3. In Fig. 3, the oscillatorloop, which was made up from F(s) and GðvgÞ in Fig. 2, nowconsists of four OP77 op-amps, connected to give two all-pass networks in cascade, and a voltage controlled variable

Fig. 1 An oscillator where the output amplitude, vvo; is controlled by a voltage-dependent gain, GðvoÞ

Fig. 2 An oscillator where a separate control-loop sets vvo equalto jpvref=2j

IEE Proc.-Control Theory Appl., Vol. 151, No. 2, March 2004 195

gain device: an EL4451. This device only became availablea few years ago [13] and provides a 70 MHz signalbandwidth between input (pins 4 and 5) and output (pin14) with a gain, G, that can be controlled by means of thevoltage applied between pins 7 and 8. The EL4451 thusprovides the gain block described by (6). The constant, V, isvery close to 1 V and 0 < G < 2. The variation in G withcontrol voltage is very linear within these limits. Theoscillator-loop has been simplified in Fig. 3 only by omittingone compensation capacitor needed to correct the OP77gains at high frequency, and by omitting the offsetcorrection components needed for the EL4451.

The control-loop, which was shown in Fig. 2 as beingmade up by an IFWR and a simple integrator, is now seen inFig. 3 as a practical realisation of the same idea. The IFWRis a well known circuit, discussed by Peyton and Walsh [14].It appears here as the two AD817 op-amps, the five identicalresistors, and the two Schotty diodes at the bottom of Fig. 3.The integrator is the OP16 op-amp, with the sameconnection of two resistors, both with resistances of R2;and a capacitor with a capacitance of C2; shown in Fig. 2.The additional resistor R3 will be discussed below. Thereare also two additional buffer amplifiers providing twooutputs: vs; the sinewave output, and vc; the cosinewaveoutput. The sinewave output is used as the input to thecontrol-loop now, in contrast to vo which was the input inFig. 2, and because the AD817 buffer amplifiers have a gainof two, the expression for fm becomes:

fm ¼ 1

2pvvs

pVC1R1C2R2

� �12

ð13Þ

where vvs is the amplitude of the sinewave output.Values of C1 and R1 were selected by a 12 position switch

to give 12 output frequencies, nominally spaced in octavesbetween 12.5 Hz and 25.6 kHz. The integrator had acapacitor C2 with a capacitance of 4.7 mF and a resistor R2

with a resistance of 100 kO: The other resistors in Fig. 3were eight with a resistance of 2.2 kO; all marked R4; eightwith a resistance of 8.2 kO; all marked R5; and five with

a resistance of 1 kO; all marked R6: The resistor, R7; feedingthe 4.7 V Zener diode, which protects pin 7 of the EL4451,has a resistance of 3.9 kO:

The oscillator gave an excellent output waveform at all 12frequencies and, while the output amplitude set by vref didoscillate about the desired value, as predicted by (12), it dideventually settle down provided vref was set high enough tomake vvs > 1V: The reason for this damping at high levelwas the small drop in gain around the oscillator-loop, bothin the OP77 s and in the EL4451, as the level increased. Thismeant that some kind of damping had to be introduced ifaccurate measurements of fm were to be made and comparedwith (13). This was done by simply adding a resistor, R3; inseries with the integrating capacitor C2: This change caneasily be shown to make the differential equation governingthe system become:

d2m

dt2þ R3vvspVC1R1R2

dm

dtþ vvspVC1R1C2R2

m ¼ 0 ð14Þ

when G ¼ 1:Equation (14) shows the same resonant frequency, fm; for

the system that was given by (13), when G ¼ 1; but now thesystem is damped when R3 is non-zero. Note, however,that this damping depends upon the amplitude, vvs; that canbe set by means of the input vref :

Damping was only introduced into this work to enableexperimental measurements of fm to be made at very lowlevels of vvs: This was done by choosing a value of R3 wellbelow the value:

R3ðcritÞ ¼ 2ðpVC1R1R2=vvsC2Þ12 ð15Þ

which, as (14) shows, would make the system criticallydamped. The actual value of R3 chosen was not criticalbecause it was only needed to stabilise the system while thelevel of vvs was being set. Then, a very small sinusoidalsignal close to the expected frequency, fm; was injected via athird resistor, with a resistance equal to R2; connected to theOP16 virtual earth. The resistor R3 was then short-circuited

Fig. 3 A simplified circuit diagram of the experimental oscillator

IEE Proc.-Control Theory Appl., Vol. 151, No. 2, March 2004196

and the frequency of the injected signal adjusted to find thesharp resonance of the system at fm:

Figure 4 shows some experimental results obtained withthe system shown in Fig. 3. In Fig. 4, measurements of fm asa function of vvs are shown for two values of the oscillationfrequency: fo ¼ 120 Hz and fo ¼ 1 kHz: These results arecompared in Fig. 4 with (13), which is a straight line of slopeone-half on this log-log plot, and it can be seen thatagreement is excellent.

4 Conclusions

It has been shown that the stability problem of an oscillatoramplitude control-loop system can have a simple analyticalsolution when the oscillator-loop obeys a second-orderdifferential equation (see (3)), and the control-loop obeys afirst-order (see (5)). Such simplifications can only be madefor oscillators working at low frequencies where the transferfunction of the frequency determining network (F(s) here) issimply second-order and other frequency determiningeffects can be ignored. For example, really good agreementbetween the theoretical results of Section 2 and theexperiments described in Section 3 could only be obtainedat frequencies well below the maximum frequency of25.6 kHz at which the experimental oscillator worked. Thiswas because of the small phase shift that the OP77 op-ampsintroduce at frequencies approaching their gain-bandwidthproduct of 400 kHz. Because there are four OP77 s incascade in this experimental oscillator, one of the four musthave its gain-bandwidth product reduced anyway in order toensure that the oscillator does not oscillate at two differentfrequencies simultaneously: one determined by C1R1 andthe other, higher, one by the four OP77 s in cascade. All thismeans that the simple expression for fo; given here by (2), isonly accurate at the lower frequencies.

At even lower frequencies, below 10 Hz, the theory againbegins to break down, when applied to the experimentsdescribed here. This is because our assumption that C2R2 C1R1 is no longer valid when C2R2 is left constant whileC1R1 is increased.

The introduction of damping by means of a resistor, R3; inseries with the integrating capacitor, C2; means that theOP16, shown in Fig. 3, now has a gain R3=R2 at frequenciesabove 1=2pC2R3: Because R3 � R2; a small ripple at amultiple of the oscillator frequency will now be added to thegain control voltage, vg: This is just what is happening in thefast amplitude control systems referred to in Section 1, andis also suggested by the work of Linares-Barranco et al. [15]who discuss an oscillator system which has a peak detectorfollowed by an integrator in its control-loop.

This may suggest that the introduction of the resistor R3;suitably switched in value to satisfy (15) as the oscillatorfrequency range is switched, could well be just as good amethod of curing the instability problems of sinusoidaloscillators as the fast control-loop methods. What may benearer the truth is that the work given here shows moreclearly what the problems of these fast control-loops maybe: the gain in the control-loop, GðvgÞ in Fig. 2 and theEL4451 in Fig. 3, is now modulated about the required value(unity) at a frequency which is a multiple of fo: Equation (3)shows that this modulation causes a small growth in theoscillator output for half a cycle of f0 followed by a smallfall in output for the other half. These halves may be madeup of several smaller fractions of a cycle depending uponwhether the ripple is at 2fo; as it is here, or at a highermultiple as it is in many of the fast control-loop designs. Forexample the ripple may be at both 4fo and 8fo [3] or only at8fo [6]. This modulation of GðvgÞ; and the consequentmodulation of the oscillator output about the required level,must, of course, introduce a small harmonic content to theoutput of, what is intended to be, a pure sinewave oscillator.

5 References

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2 Porter, S.N.: ‘Signal generators with rapid automatic amplitudestabilization’. 1968, U.S. Patent No. 3419815

3 Vannerson, E., and Smith, K.C.: ‘Fast amplitude stabilization of an RCoscillator’, IEEE J. Solid-State Circuits, 1974, 9, (4), pp. 176–179

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5 Mikhael, W.B., and Tu, S.: ‘Continuous and switched capacitormultiphase oscillators’, IEEE Trans. Circuits Syst., 1984, 31,pp. 280–293

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10 Sidorowicz, R.S.: ‘An abundance of sinusoidal RC oscillators’, Proc.Inst. Electr. Eng., 1972, 119, (3), pp. 283–293

11 Sidorowicz, R.S.: ‘Some novel RC oscillators for radio frequencies’,Electron. Eng., 1967, 39, pp. 498–502

12 Sidorowicz, R.S.: ‘Some novel RC oscillators for radio frequencies’,Electron. Eng., 1967, 39, pp. 560–564

13 Harvey, B.: ‘Application Note No. 5: High-purity sinewave oscillatorswith amplitude stabilization’ (Elantec Inc., Milpitas, CA, USA, 1997)

14 Peyton, A.J., and Walsh, V.: ‘Analog electronics with op-amps’(Cambridge University Press, Cambridge, England, 1993), pp. 248–255

15 Linares-Barranco, B., Rodriguez-Vazquez, A., Sanchez-Sinencio, E.,and Huertas, J.L.: ‘Generation, design and tuning of OTA-C high-frequency sinusoidal oscillators’, IEE Proc., Circuits, Devices Syst.,1992, (5), pp. 557–568

Fig. 4 Measurements of the frequency of the instability, fm; as afunction of oscillator output, vvs; are compared with (13) for twovalues of fo

IEE Proc.-Control Theory Appl., Vol. 151, No. 2, March 2004 197