Identification of two-amplifier active-/? sinusoidaloscillators

M.T. Abuelma'atti, BSc, PhDW.A. Almansoury

Indexing terms: Oscillators, Amplifiers, Circuit theory and design

Abstract: Using a unified representation for the 2-amplifier active-/? oscillators, new circuits of thisclass of oscillators can be discovered systemati-cally. The experimental results obtained from sixnew circuits are presented in this paper.

1 Introduction

Over the past few years, a number of schemes have beendeveloped for realising active-/? sinusoidal oscillators[1-11]. These capacitor-less schemes exploit the gain-frequency characteristics of a couple of operationalamplifiers. All of the papers available on this subject,with few exceptions, are concerned only with one newcircuit in each paper. This is because these circuits werediscovered either by accident or with the help of intuitionand various hit-and-miss techniques.

Our objective in this paper is, therefore, to provide aunified treatment for the 2-amplifier active-/? sinusoidal-oscillator circuits. Apart from the obvious advantage ofunifying the analysis of this class of oscillators, this treat-ment leads to the systematic discovery of numerous new2-amplifier active-/? sinusoidal oscillators. Owing tospace limitations, the performance of only six new cir-cuits will be considered in this paper.

2 Analysis

Fig. 1 shows the generalised representation of a 2-amplifier active-/? oscillator. The transfer function of thecircuit with the feedback loop disconnected at M is givenby

\ mA

^

R 10

RbRe

R2R7

Paper 5298G (E10), first received 16th May 1986 and in revised form21st January 1987Dr. Abuelma'atti is and Miss Almansoury was formerly, with theDepartment of Electrical Engineering and Computer Science, Faculty ofEngineering, University of Bahrain, PO Box 32038, Isa Town, BahrainMiss Almansoury is now with the Department of Electrical and Elec-tronic Engineers, University of Liverpool, PO Box 147, Liverpool L693BX, United Kingdom

Fig. 1 Generalised representation of 2-amplifier active-R sinusoidaloscillator

where

/?fl = /?1||/?3||/?7

Rh = R2\\R4\\Rl0

Rc = R5\\R6\\(R8 + R9)

Rd = K6||/?10

m = R9/(R8 + R9)

Now, let the open-loop gain of the operational amplifiersbe represented by the 2-pole model which is given by

^s) = A2(s) =A0wacob

(S + (DO)(S + COh)(2)

where Ao is the DC gain, wa is the first corner frequency,ojb is the second corner frequency and B = Aoa)a is thegain-bandwidth product of the operational amplifier.

By closing the loop and substituting eqn. 2 into eqn. 1,the characteristic equation of the system, T(s) — 1 = 0 ,becomes

a(s + wa)2(s + a>b)

2 y)(s + coa)(s + a)b)A0 a)acob

+ dA20(D

2aa>2

b=0 (3)

(1) where

1 Ra =

10

RaRbRcRd RaRcRl0 RaRbR6

RbRcRdR3

1

IEE PROCEEDINGS, Vol. 134, Pt. G, No. 3, JUNE 1987 137

mRV =

5 =

10 m R 10

R2R5R7

m

condition of oscillation can be shown to be approx-imately

An> —y)

a (P(5)

Eqn. 5 can be easily satisfied in practice by appropriateselection of the resistances.

Obviously, numerous oscillator circuits can be derivedfrom the circuit of Fig. 1 using different combinations ofresistors. A few of these circuits are already known (see,for example, references 1-3), but the majoirity of the cir-cuits are new. In the following Section a selected cata-logue of some of the new circuits will be presented.

3 Experimental results

Fig. 2 shows a catalogue of a selected number of newoscillator circuits derived from the generalised circuit ofFig. 1 by using appropriate combinations of resistors.Each circuit was built and tested using the low cost 741operational amplifiers and fairly good sinusoidal oscil-lations were obtained in each case. Fig. 3 shows typicaloutput waveforms taken from the selected new oscillatorcircuits and Table 1 summarises the performance of eachcircuit.

* MR9

Fig. 2 Six new 2-amplifier active-R sinusoidal oscillators derived fromthe general circuit of Fig. 1

By equating the imaginary part of eqn. 3 to zero, the fre-quency of oscillation can be shown to be

co0 =1/2

(4)

By equating the real part of eqn. 3 to zero, and using eqn.4 and noting that for most practical cases cob P coa, the

4 Concluding remarks

By analysing a generalised circuit employing two oper-ational amplifiers and ten resistors, a unified treatmentfor a class of active-i? oscillator circuits has been pre-sented. By using an appropriate combination of resistors,numerous new 2-amplifier active-K oscillators can be sys-tematically discovered. Six such circuits have been pre-sented in this paper. This is in sharp contrast withprevious papers on this subject which, with few excep-tions, contained only one new circuit in each paper.Depending on the application, one circuit may be pre-ferred to another.

It is not our objective, in this paper, to evaluate whichof the numerous circuits that can be systematicallyderived from the circuit of Fig. 1 are better. Such anevaluation would depend not only on the applications inmind, but also on the IC technology. Indeed, some con-figurations are clearly superior for voltage controlledoscillators. Some use a smaller number of resistors and

Table 1: Summary of the performance of the six new circuits

Circuit Frequency ofoscillation

Approximate conditionof oscillation

Frequencyrange

Resistancevariation

Remarks

Fig. 2b

Fig. 2c

Fig. Id

Fig. 2e

Fig. 2f

1 +~

uit 1 +4n+n2-/77

a;.. 2n6t/_ 4m - n 2

f 1 / 1 1 \ 1 w hm ~ n

I / W J 1 + - ]An\ - *

[ 1 / 7 7 - 1 I w. 2(m+n-1+ Ao\ - * —

2/77+n-1 J d/a /77-12m

O^ 4/77 + 4/7 -/T72

Wfi ( 1 + * )

K, 2 (2+ /77+ / f ) (1

180 kHz-1.05 MHz

31 kHz-310 kHz

13kHz- i60kHz

294 kHz-500 kHz

460 kHz-870 kHz

35 kHz-530 kHz

ft,=3.3kQft7 = 33 kQft8 = 22 Q - ooft9 = 220 kQft, = 50 0 - ooft2 = 0.5 k Q - ooft3 = 0 - 1 kQft3 = 10kQft6 = 9 k Q - ooft7 = 0 - 9 kQ

ft,0 = 0 - 7 kQft3 = 70 Q - 47 kQft5 = 75 Qff6 = 9 kQft, =39 kQft3 = 39 kQft7 = 15 kQ - 54 kQft2 = ft4 = 9 kQ

ft! = 0^-100kQ

m = ft7/ft,n = ft8/ft9

m = R,/R3n = ft(/ft2

ft( = ft,||ft2||ft3m = ft /ftn = ft 3/ft7°

m = ft3/ft6n = ft3/ft*

m = RJR3n = ft,/ft2

ftf = ft,||ft3||ft7m = fte/ft9n — D IfD 4- I

— S' * Bk = RJR.

138

* R5 can be the output resistance of the operational amplifier plus any added external resistance

IEE PROCEEDINGS, Vol. 134, Pt. G, No. 3, JUNE 1987

Fig. 3 Typical output waveform of the circuits of Figs. 2a-f respectively

Direct bias voltage = ± 10 V

/o* 7

Rq

/o« 3

/o

= 1 MHz= 33 Kn= 220 kn= 108 kHz= 47 kn= 120 kHz

R8

K.« 5

* 3

= 3.3 kn= 220n

= 46n= 398n= Rb = R, = 9 kn

de

f

/ 0 = 294 kHz/ 0 = 870 kHz

R7 = 54 kn/ 0 = 420 kHz

, /K, = 1 ( « 8 •

R3/?,

R2/R4

+ «9)/«5

= i?6

= «3

= 1= 18

= 91= 39

hence consume less power. Others are clearly better for awide frequency range of operation. Obviously the oscil-lator circuits which can be derived from the generalisedcircuit of Fig. 1 can be classified into sets; the members ofeach set use equal number of resistors. This is the reasonwhy we included a selected catalogue of potentially usefulprototype 2-amplifier active-/? oscillator configurations.

5 References

1 ABUELMA'ATTI, M.T., and ALSAWAFY, A.H.: 'Two new active-R oscillators', Int. J. Electron., 1986, 60, pp. 167-173

2 BHATTACHARYYA, B.B., and RAUT, R.: 'A simple wide-rangeactive-R oscillator', AEU, 1981, 35, pp. 457-459

3 SHAH, N.A., and BHAT, C.K.: 'Operational amplifier-based volt-age-controlled oscillator', Int. J. Electron., 1985, 59, pp. 649-652

4 DRAKE, J.M., and MICHELL, J.A.: 'Continuously tunable sinus-oidal /?-oscillator with a minimum number of elements', ibid., 1981,50, pp. 141-147

5 LI, M.K., LI, C.W., and CHOI, K.C.: 'On the synthesis of second-order active-R filters', ibid., 1979, 47, pp. 483-489

6 MONI, R.S., and RAO, K.R.: 'Large signal instability in active-/?and a class of compensated active-RC filters', Proc. IEEE, 1981, 69,pp. 1366-1367

7 NANDI, R.: 'Active-/? realization of bilinear RL impedance andtheir applications in a high-Q parallel resonator and external capac-itorless oscillator', ibid., 1978, 66, pp. 1666-1668

8 NEDUNGADI, A., and VENKATESRAWANI, S.: 'Generalizedsecond-order active-/? filters', Int. J. Circuit Theory & Appi, 1982,10, pp. 311-322

9 PYARA, V.P., and JAMUAR, S.S.: 'Single element controlled oscil-

IEE PROCEEDINGS, Vol. 134, Pt. G, No. 3, JUNE 1987 139

lator without external capacitors', Electron. Lett., 1980, 16, pp.607-608

10 SIDDIQI, M.A., and AHMED, M.T.: 'A low component oscillatorwith operational amplifiers and resistors'. 12th ASILOMAR Con-ference on Circuits, Systems and Computers, 1978, pp. 12-13

11 VENKATARMANI, Y., and VENKATESRAWANI, S.: 'Active-/?multifunction circuit synthesis', Electron. Lett., 1982,18, pp. 96-97

Muhammad Taher Abuelma'atti was bornin Cairo, Egypt, in 1942. He received theBSc degree in electrical engineering in1963 from Cairo University and the PhDdegree in 1979 from Bradford University,UK. From 1963 to 1967, he worked at theMilitary Technical College in Cairo as ateaching assistant. He was with the Ironand Steel Company in Helwan, Cairo,from 1967 to 1973 as a senior electricalengineer. From 1973 to 1976, he was with

the College of Engineering, University of Riyadh, Saudi Arabia,as a teaching assistant. From 1980 to 1981, he worked with theFaculty of Engineering, University of Khartoum, Sudan, as a

lecturer, and from 1981 to 1982 he was with the College ofEngineering, University of King Saud, Saudi Arabia, as anassistant professor. Since September 1982, he has been with theGulf Polytechnic in Bahrain, where he is currently involved inteaching and research. His research interests include problemsrelated to analysis and synthesis of nonlinear electronic circuitsand systems. He is the author or co-author of over 50 journalarticles and technical presentations.

Wafa Abdulrahman Almansoury was bornin Manama, Bahrain, in 1965. Shereceived the BSc degree in electricalengineering in June 1986 from the Uni-versity of Bahrain. Since October 1986she has been with the Department ofElectrical Engineering and Electronics atthe University of Liverpool, UK, whereshe is working towards the Master ofScience degree in the area of microelec-tronic systems and telecommunications.

140 1EE PROCEEDINGS, Vol. 134, Pt. G, No. 3, JUNE 1987