Generation of second-order single-OTA RC oscillators

Y. Tao J.K.Fidler

Indexing terms: RC Oscillators, Boolean method

Abstract: A direct and exhaustive method of generating canonic single-amplifier RC oscillators is introduced. Using this method, the ‘complete categories’ of the bandpass-based single-OTA three- and four-node RC canonic oscillators have been generated. The condition of oscillation of the resulting structures can be adjusted independently of the frequency of oscillation (FO) by the active transconductance parameter of the OTA. All the resulting oscillators enjoy low component counts. The three-node oscillators each contain only two capacitors and two resistors, and the four-node oscillators each contain only two capacitors and three resistors. The sensitivities (absolute values) of the FO to the active parameter are zero and the sensitivities to the passive elements are all no greater than 1/2.

1 Introduction

A great deal of research has been carried out into sin- gle active element canonic filters and oscillators employing current mode (CM) devices such as opera- tional transconductance amplifiers (OTAs), current conveyors (CCs), current feedback amplifiers (CFAs), and four-terminal floating nullors (FTFNs), owing to the simplicity of the circuits and much improved prop- erties over their conventional op-amp counterparts.

Efforts have been made to generate exhaustively some types of canonic single active device filters/oscilla- tors [1-61 from which preferable structures can then be found by choosing the ones with desirable properties, such as low sensitivities and ease of tuning, from the ‘complete category’. In this paper however, a method of directly generating a class of oscillators enjoying the desired properties is introduced. The criteria for gener- ating the oscillators are discussed in Section 3 and all the three- and four-nlode single-OTA RC oscillators which comply with the criteria are exhaustively gener- ated in Sections 4 and 5 .

The Boolean methold [7-91 is a systematic way to exhaust some classes of circuits and is applied in this paper to the generation of oscillator structures. As has 0 IEE, 1998 ZEE Proceedings online no. 19981872 Paper fmt received 7th July 1997 and in revised form 5th January 1998 The authors are with the Department of Electronics, University of York, Heslington, York, YO1 5DD, UK

been reported, the method is most straightforward for the generation of networks with all-positive transfer function coefficients. However, in this paper its appli- cation is expanded to the generation of single OTA cir- cuits, in which case negative coefficients may also appear in the system functions.

Practical oscillator circuits require a nonlinear mech- anism in order to stabilise the waveform amplitude. This may be provided explicitly, or through the nonlin- ear characteristics of the active device. Detailed consid- erations of such techniques is not included in this paper.

2 Boolean circuit synthesis method

The circuit generation method proposed by Bhattach- aryya and Darkani [lo] is very popular. It enables one to find all the circuits of a certain type with the desired system function. It first generates a fully-connected net- work with each branch admittance being Yi. The net- work function of interest is then expressed in terms of Y, which may take the value of 0, sci, gi or sci + g,, respectively, for no connection, a capacitor, a conduct- ance or a capacitor and a conductance connected in parallel. The decision on what kind of connection Yj should represent relies greatly on the researchers’ expe- rience and may involve a great deal of redundant com- putation.

The Boolean circuit synthesis method is a symbolic topological synthesis method which utilises graph theo- retic and logic minimisation techniques to provide new network structures. Its advantage is that it allows the circuits to be generated in a systematic way and no ‘guess work’ based on experience or redundant compu- tation is involved.

To develop new circuit topologies, first a fully-con- nected network is generated according to the generic descriptions of the network and analysed to give the transfer function in terms of the branch admittance Yi. Assuming each branch to be a capacitor and a con- ductance in parallel, i.e. = sCi + gi, the transfer func- tion is expanded into a function of conductance and capacitor variables [Note 11. Then Boolean minimisa- tion is applied to this function and all the topological networks which meet the requirements on the form of the transfer function are obtained. Details of the Boolean method and examples of generating passive Note 1: In this paper, a ‘branch‘ is considered to be the connection (a component, or components connected in parallel) between two nodes. However, the case in which a conductance and a capacitor are connected in serial is considered to be two branches in serial and composes three nodes.

IEE Proc.-Circuits Devices Syst., Vol. 145, No. 4, August 1998 271

RC filters, single-opamp RC filters and oscillators using this method can be found in [7-91.

3 General considerations

In this Section, the criteria for generating the oscilla- tors will be discussed and it will be shown that a band- pass-based oscillator provides the least sensitivity of the FO to changes in amplifier transfer ratio, as well as the best dynamic arrangement for maintenance of the oscil- lation criterion. This is achieved by investigating both first- and second-order sensitivities [Note 21.

I

structure in Fig. 1: (i) K is the parameter controlling the CO, thus it is essential to minimise the sensitivity of the FO to K. In the ideal case, FO is not a function of K. (ii) When K is adjusted downwards, the roots of the CE should migrate into the left half of the s-plane (LHP), and the oscillation diminish. When K is adjusted upwards, the roots should migrate into the right half of the s-plane (RHP), and the oscillation increase, to achieve easy and intuitive control of the oscillation. To generate oscillators satisfying both criteria, it has been suggested in [SI that the passive subnetwork of the oscillator should be in the form of a bandpass filter (BPF):

’ I i.e. in eqn. 1,

a2 = a0 = 0 The frequency of oscillation is then

(10)

- Fig. 1 Single-mpl$er RC oscillator

Consider a second-order single-amplifier oscillator with the general structure as shown in Fig. 1. Without loss of generality, the transfer function from node 1 to node 2 can be written as

a 2 2 + a1s + a0

b2s2 + bls + bo T ( s ) =

where a, and b, ( i = 0, I , 2) are expressions of the pas- sive component variables. For the circuit to oscillate, it must satisfy [I 11:

for s = jw,, where w, is the FO and K is the transfer ratio of the active device. Substituting eqn. 1 into eqn. 2, we have the characteristic equation (CE):

which should have its roots on the imaginary axis. The roots of the CE can be solved from eqn. 3 as

where the negative of the real part of the roots is

1 - K ’ T ( s ) = 0 (2)

(bz - Ka2)s2 + ( b l - K u ~ ) s + (bo - KUO) = 0 ( 3 )

41,2 = -cl * jw bl - Kal

2(bz - K a z )

(4)

(5) cT=

and the square of the imaginary part is

w = 2 bo - Kao - [ b1 - Kal 1’ (6) bz - KCQ 2(bz - K a z )

The frequency of oscillation (FO) is bo - Kao

= b2 - Kaz > O (7)

and the condition of oscillation (CO) is

bl = Kal (8) bl i.e. K = - a1

Two criteria [8] have been considered in this paper for generating single-amplifier RC oscillators of the Note 2: The frst- and second-order derivatives of the (FO)’ to the ampli- fier transfer ratio are considered instead of the strict sensitivities for sim- plicity. This does not make any difference to the conclusion.

bo b2 -

The question is: Is an RC network taking the form of a BPF the optimal solution for oscillators as shown in Fig. 1 which need to satisfy the two criteria? The answer is yes. A mathematical proof is given as fol- lows:

The first criterion is examined on both RC BPF and non-BPF subnetworks in an attempt to find out if there exist any RC non-BPF circuits which lead to oscillators whose FO sensitivity to K is less than sensitivity of the BPF oscillators.

From eqn. 6, one may obtain

dw2 azbo - aob2 1 (bl - Kal) (azb l - Q b z ) dK (b2 - Kaz)2 2 ( b ~ - Ka2)3

- - - - -

(12) Substituting eqn. 8 into eqn. 12, i.e. when the CO is satisfied, eqn. 12 becomes

dw2 a2bo -aobz - a2b0 - aobz - - - - dK (bz - K u z ) ~ (bz - a z b i / a i ) 2

(13) - a:(a;lbo - aob2)

(aibz - azbi)’ -

It can be easily shown that d d l d K = 0 for a BPF oscil- lator under the CO, by substituting eqn. 10 into eqn. 13.

To make the FO of the non-BPF oscillator as insen- sitive to the changes of the active parameter K as the BPF, eqn. 13 must be equal to zero. The constraint on the circuit is thus

a2bo - a062 = 0 i.e. a2 = aobz/bo (14) al = 0 is another possibility, however, if al = 0, from eqn. 8, in order to satisfy the CO, it must hold that b1 = Kal = 0. This conflicts with the fact that the transfer functions of passive RC networks cannot miss interme- diate coefficients, according to Hakimi and Mayeda, as cited in [7]. So this latter assumption is seen to be impractical and is discarded. That is, when eqn. 14 is satisfied, d d i d K of the RC non-BPF oscillator is zero under CO, which is as good as, but no better than the RC BPF case. The second-order derivative is examined next.

For the non-BPF case, .from eqn. 12, with eqn. 14 satisfied, the second-order derivative can be written as

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( 1 5 ) Substituting eqn. 8 into eqn. 15, i.e. under the CO, eqn. 15 simplifies to

&w2 1 (2: 1 4 -- ---- = -- - 2

dK2 2 (alba -- ~2b1)’ 2 b: (1 - K f )

Eqn. 16 gives the second-order derivative of the FO with respect to the active parameter K for an RC oscil- lator under the CO with eqn. 14 satisfied.

On the other hand, f he second-order derivative of the oscillator with the RC: BPF subnetwork can be easily obtained by substituting eqn. 10 into eqn. 15:

(16)

into the RHP and the oscillation is increased instead of being decreased, which does not comply with the sec- ond criterion.

For the BPF transfer function, substitution of eqn. 10 into eqn. 5 leads to

and it is seen that (r increases as K decreases and the poles of the CE move (further) into the LHP under all circumstances, causing the oscillation to decrease, which complies with the second criterion.

In conclusion, to generate second-order single-ampli- fier RC active oscillators with a structure as shown in Fig. 1, it is better to choose the RC subnetwork in the oscillator to be in the form of a BPF, as far as the two criteria are concerned.

Thus, the oscillator generation method applied in this paper first synthesises the passive RC BPFs using the Boolean method and then connects an OTA to each of the BPFs to obtain the desired oscillators. Comparing eqn. 16 with eqn. 17, assuming K and a.

positive, it can be seen that when (1 - Kao/bo)2 > 1, eqn. 16 is less than eqn. 17, which seems to indicate that under this condition, the oscillator employing an RC BPF subnetwork does not work as well as the one employing an RC non-BPF subnetwork.

1

0 1 2

Comparison of seconu-order derivative of non-BPF and BPF KadbO

Fig.2

It can be shown, liowever, that the RC non-BPF oscillators under this condition fail to satisfy the sec- ond criterion. From Fig. 2, it is seen that the condition for this to be obtained is

K ~ o / b o > 2 (18) To investigate this, lei Kao/bo = 2 + 6, with 6 > 0. For the non-BPF oscillators, when eqn. 14 is satisfied, eqn. 5 becomes

bl - l Y U l - bl - K u ~ 2 ( b Z - K @ ) 2b2 1 - KLQ

(7=---

( b J

(19) K u ~ - bl - bl - K u ~ - ----

2b2 (1 - K C ) 2b2(1 6)

The denominator of eqn. 19 is positive since it is always true for passive networks that b, > 0 (i = 0, 1, 2). So the negative of the real part of the CE poles, o, decreases as K decreases and the poles move (further)

IEE Proc -Circuits Devices Sysi , Vol 145, No 4, August 1998

a

b Fig.3 Generation of three-node oscillators a General structure b Model for deriving passive RC B P h

4 Single-OTA three-node RC oscillators

Using the method introduced in Section 3 to generate second-order oscillators with a structure as shown in Fig. 3a, it is only necessary to synthesise the three- node passive BPFs, which can be derived using the Boolean method. Using the model in Fig. 3b, the transfer function of the passive network can be written as

213

No. Components FO (a,’) CO

1 c2 C ? gs g4 gi R d C 2 ci gmlc3=(g4+g r)/c 7+g4Icz

~~~~

Fig. 4 Three-node oscillator structures

Network

Assuming that into the form of eqn. 1, where

= sc, + g,, eqn. 21 can be expanded

a2 = 0 a1 = c2 a0 = 92 b2 = CZC~ + ~ 2 ~ 4 + ~ 3 ~ 4

b l = e293 + c2g4 + c3g4 + c3g2 + c4g2 + c4g3 bo = 9293 + ~ 2 ~ 4 + 9394 (22)

The corresponding Boolean variables A, and B, can be expressed with the logical variables C, and G,, each of which can be either ‘true’ if the component exists or ‘false’ if it is absent from the circuits. Thus we have

A2 = f a l s e AI = C2 A0 = G 2

B 2 = Cz . C 3 + Cz . C 4 + C3 . Cq

B 1 = Cz . G 3 + Cz . Gq + C 3 . G4 + C3 . G 2

+ Cq . G:! + C, a G3 Bo = G 2 * G3 i- G 2 . Gg i- G3 . Gg (23 )

where ‘.’ and ‘+’ denote the Boolean ‘AND’ and ‘OR’, respectively. To generate second-order BPFs, it is required that

F = x . A l z . B a - B o = l (24) Note that B1 has been excluded from eqn. 24 to reduce the calculation. This does not affect the results since passive networks ensure that there can be no missing intermediate terms in their system functions.

Eqn. 24 is then expanded using eqn. 23 and logically minimised to give all the passive BP filters

F = C2.C3’G3’Gg.G2+C2.C4.G3.G4.G2= 1 (25) The two terms in eqn. 25 each represents a canonic BPF structure. In each term, the uncomplemented vari- ables represent the components which must exist in the circuit and the complemented variables represent the ones which must not. Those variables which do not appear in a term represent the ‘optional’ components, whose existence or absence have no effect on the form of the transfer function, which, in this case, is the sec- ond-order bandpass transfer function.

Thus two oscillators are derived by simply connect- ing an OTA to each of the BP filters (see structures Nos. 1 and 2 in Fig. 4). These two oscillators have been derived by other researchers using a different method [2].

Two more three-node structures can be obtained from the degeneration Note 31 while deriving the four- node structures using the same method. See structures Nos. 3 and 4 in Fig. 4. They are listed in this Section for the convenience of classification. The component Note 3 ‘Degenerakon’ 111 ths paper refers to the degradakon of a c r m t to a lower node-numbered circut. For example, degenerahon occurs when the exlstence of a conductance m a cucut is detemned by the Boolean method to be compulsory, although its value does not affect the form of the investigated system function Its value can then be chosen arbitranly In our case, choosing the conductance to be mfimte corre- sponds to replacing the component wth short mcut, whxh wdl reduce the node number by one

numbering refers to Fig. 5. These two structures have been proposed in [ 3 ] .

It is easy to see that the four structures form the complete category of the Fig. 1-type single-OTA three- node RC oscillators which satisfy the two criteria. The four structures each employ only two capacitors and two resistors.

Structures marked ‘GC’ in the following tables are the ones employing grounded capacitors.

5 Single-OTA four-node RC oscillators

To generate exhaustively the Fig. 1-type four-node oscillators which satisfy the two criteria, first we enu- merate all the possible ways [Note 41 in which the OTA is connected to the RC network and then generate the oscillators of each class using the method in Section 3. The fully-connected RC network used for the genera- tion of the three classes of oscillators is shown in Fig. 5.

3

Fig.5

3 1

+ - Fig. 6 General structure of class 1 oscillators

5.1 Class 7 In this class of structure, the three terminals of the OTA, i.e. the inverting and non-inverting input

Note 4: Although some other ways of connection may also be possible for generating oscillators such as the negative resistance oscillators, they are not of the structure shown in Fig. 1 and thus not considered in this paper, the same for three-node structures.

214 IEE Proc.-Circuits Devices Syst., Vol. 14S, No. 4, August 1998

Fig. 7 Class 1 of four-node o,scillator structures

terminals and the output terminal are not connected either to each other or to the ground, as shown in Fig. 6.

This class of structure uses the differential input of the OTA, i.e. neither of the OTA's input terminals is grounded. Thus, the transfer function of the passive RC network from the output port to the differential input port of the OTA (;an be expressed as

k=O

(26) where ab bj and dk are expressions of cis and gis and each includes only positive terms. Routine analysis yields

Hd(s) = (y2y3 - y4;fl)/

(y6y5y4 + &y5y3 + y 6 y 5 f i + ySY5Y1 + y6 y4 y 2 -k y6 y4 Y1 + YS Y3 Y2 + Y6 Y3 y1 + U5 Y4 Y3 41 Y5 U4 Yl + Y5 Y3 Y2 + Ys Y2 Y1 + Y4Y3Y2 + Y4Y3Yl + Y4Y2Y1 + Y3Y2Y1)

(27) Assuming 5 = sci + gi (i = 1, 2 ... 6), eqn. 27 can be expanded into the form of eqn. 26. The numerator coefficients of eqn. 27 in this case, are

a2 = C2C3 a1 I= C2g3 + C3g2 a0 = g2g3 b2 = c1c4 b l = clg4 + e491 bo = 9194 (28)

The corresponding Boolean variables Ai and Bj can then be expressed with the logical variables C, and G,

A2 = C 2 . G Ai = 1?2*G3 + C3*G2 A0 Gn*G3 B2 = Ci.C4 Bi = C1.Gg + C4eG1 Bo = Gl'G4

(29)

Similarly, the denominator coefficients Di (i = 0 ... 3) are also expressed in Boolean variables Ci and Gi.

To generate second-order BP filters, it is required that

--- - - F = A2 + B2 * A1 * A0 e Bo * 0 3 e D2 * Do = 1 (30)

Note that D1 has been excluded from eqn. 30 for the reason given in Section 4.

After the Boolean minimisation all the passive BPFs are generated, from which the desired oscillators can be obtained by connecting an OTA to each of the BP fil- ters. The results are given in Fig. 7.

Z,,(S) =BPF B .1+- 6

Fig.8 General structure of class 2 oscillators

5.2 Class 2 For this class, the cases in which one or more of the OTA's three terminals are connected to the ground are considered. With a little consideration, it can be seen that the structure in Fig. 8 is the only possible case for realising the oscillators of concern.

To generate four-node oscillators having the struc- ture shown in Fig. 8, the method introduced in Section 3 is again applied. Z,&) is the transfer impedance of the passive RC network (Fig. 5) from node 3 to node 1. Seven novel canonic four-node oscillators have been derived and are listed in Fig. 9.

IEE Proc.-Circuits Devices Syst., Vol. 14.5. No. 4. August 1998 215

Fig. 9 cluss 2 offour-node oscillator structures

5.3 Class3 For this class, the cases are considered in which there exist(s) common node(s) among the OTA‘s three tenni- nals. Among these cases the only possible structure for realising oscillators is shown in Fig. 10, where Z3,(s) and &(s) are the passive network impedances. Using the method in Section 3, six oscillators have been gen- erated. See Fig. 11.

No. 1

- Fig. 10 General structure of class 3 oscillators

Components cZcSg3g5g6

1 1 1 CZc3g4g5g6

CZ c3 g3 gS g6

c4 c6 g2 g3 g5

Fig. 11 class 3 offour-node oscillator structures

6 Conclusion

In this paper, a method of directly generating all the Fig. 1 -structured canonic single-amplifier RC oscilla- tors having the desired features is introduced. This is a general method for generating all the single amplifier oscillators having the structure shown in Fig. 1, although only single-OTA RC oscillators have been discussed. The complete categories of canonic single- OTA three- and four-node RC oscillators which satisfy the two criteria discussed in Section 3 have been gener- ated. The workability of all the oscillators has been verified by PSpice 6.0 using the ideal OTA model.

All the oscillators derived enjoy a low component count: the three-node oscillators contain only two capacitors and two resistors and the four-node oscilla- tors contain only two capacitors and three resistors.

The oscillators also enjoy low FO sensitivities. The sensitivity of the FO is zero to the active parameter K and no greater than 1/2 to the passive elements. It can be observed from the Tables that the FO expressions of all the oscillators derived can be classified into three forms and the sensitivity to passive elements of each form can be calculated as shown in Fig. 12.

The CO of the structures can be adjusted independ- ently of the oscillation frequencies. Furthermore, when two ganged equal-valued capacitors are used, the FO of all the oscillators can be adjusted linearly by tuning the capacitors without interfering with the CO.

Among the single-OTA RC oscillators derived, struc- tures 3 in Fig. 4 and 3, 6, 7 in Fig. 7 are particularly attractive because they also employ two grounded

216 IEE Proc -Circuits Devices Syst., Vol 145, No. 4, August 1998

Fig. 12 Sensitivity calculation

capacitors (GC). The alpplication of a GC is desirable 4 CELMA, s., MARTINEZ7 P.A., and CARLOSENA, A,: ‘Approach to the synthesis of canonic RC-active oscillators using CCII’, IEE Proc Circuits Devices Syst., 1994, 141, (6), pp. 493- for both monolithic IC technology and thin-film fabri- 497

the capacitors easily accounted for or tuned out as they 5 HOU, C.-L., YEAN, R., and CHANG, C.-K.: ‘Singie-element are now in parallel with the grounded capacitors. It is controlled oscillators using single FTFN, Electron. Lett,, 1996,

32, (22), pp. 2032-2033

when variable-frequency oscillators (VFO) are realised resistance-controlled-oscillators using a single current-feedback- by ganging the two capacitors [l]. amplifier’, ZEE Proc. Circuits Devices Syst., 1996, 143, (11, pp.

cation, and makes the parasitic capacitors surrounding

to have the capacitors grounded 6 SENANI, R,, and SINGH, V,K,: ‘Synthesis of canonic single-

71-72

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3 ABUELMA‘ATTI, M.T., and KHAN, M.H.: ‘Grounded capaci- tor oscillators using a single operational transconductance ampli- fier’, Frequenz, 1996, 50, (11-12), pp, 194-197

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