GENERATION OF THE MINIMUM COMPONENT OSCILLATORS FROM SALLEN KEY FILTERS

Transcription

1 Journal of Circuits, Systems, and Computers Vol. 0, No. 6 (0) 65 8 #.c World Scienti c Publishing Company DOI: 0.4/S GENEATION OF THE MINIMUM COMPONENT OSCILLATOS FOM SALLEN KE FILTES AHMED M. SOLIMAN Electronics and Communication Engineering Department, Faculty of Engineering, Cairo University, Egypt 6 asoliman@ieee.org eceived 5 March 0 Accepted 8 April 0 Two new minimum passive component oscillators using inverting current conveyor (ICCII ) acting as a voltage negative impedance converter are generated from the Sallen Key low-pass and high-pass lters. It is also shown that the Sallen Key low-pass, high-pass, and band-pass lters are the origin of the three minimum component oscillators using the current conveyor acting as a current negative impedance converter. In addition, it is also shown that the Sallen Key high-pass and band-pass lters are the origin of the two minimum component oscillators using single input single output transconductance ampli er as the active element. Although this paper is considered partially a review paper it includes new generation methods and new minimum component oscillator circuit realizations. Simulation results for the new oscillators using ICCII are included. Keywords: Minimum component oscillators; Op Amp; CCII; ICCII; VNIC; CNIC.. Introduction Several oscillators are available in the literature using the operational ampli er (Op Amp), current conveyor (CCII), inverting current conveyor (ICCII), current feedback operational ampli er (CFOA), and the transconductance ampli er (TA) as the basic building block. Most recently there has been interest in nding the source circuit employing the Op Amp that can be transformed to generate the newly reported oscillators using other active elements., It is proved in this paper that the Sallen Key lowpass, high-pass, and band-pass lters 9 are very powerful circuits and are the origin of many oscillators that are available in the literature 0 as well as to new oscillators. Although minimum passive component oscillators using two resistors and two capacitors cannot be realized using a single Op Amp, they can be realized using a single CFOA or a single ICCII or a single CCIIþ 5 or a single input single output TA also known as the voltage controlled current source (VCCS). 0 *This paper was recommended by egional Editor Piero Malcovati. 65

2 66 A. M. Soliman It is shown in this paper that the Sallen Key low-pass, high-pass, and band-pass lters are the origin of the three minimum component oscillators using the CCIIþ acting as a current negative impedence converter (CNIC). It is also shown that the Sallen Key low-pass, high-pass, and band-pass lters are the origin of three minimum component oscillators using the ICCII acting as a VNIC. Two of the three generated circuits using ICCII are new. In addition it is also shown that the Sallen Key high-pass and band-pass lters are the origin of the two minimum component oscillators using the single input single output TA reported in ef. 0.. Sallen Key Filters The second-order Sallen Key low-pass, high-pass, and band-pass lter circuits employing a single noninverting ampli er also known as voltage controlled voltage source (VCVS) of gain K are shown in Fig.. Electronics textbooks that have included this family of the Sallen Key lters are many and only few of them are referenced here as in efs The VCVS of gain K larger than one is realized using an Op Amp and two resistors. The transfer functions of the three lter circuits shown in Fig. are given by T BP ðsþ ¼ s þ s T LP ðsþ ¼ s þ s T HP ðsþ ¼ s þ s K C þ þ K C C s K þ þ K C C þ þ sk C þ þ þ þ K C C C ; ðþ C ; ðþ C þ þ : C ðþ These three circuits, although introduced long time ago, have remain as very powerful source circuits as will be demonstrated next.. KC Oscillators The rst generation method of minimum passive component active C oscillators namely two resistors and two capacitors and using a single controlled source was introduced in ef. 0. It is proved in ef. 0 that there are a total of 6 oscillators using two resistors, two capacitors, and a single controlled source. The generation method is based on taking all possible second-order two resistor and two capacitor

3 Generation of the Minimum Component Oscillators from Sallen Key Filters 67 C V IN K (a) VIN C K (b) V IN K C (c) Fig.. (a) Sallen Key low-pass lter. (b) Sallen Key high-pass lter. (c) Sallen Key band-pass lter. circuits in a closed loop with the controlled source and derive the practical oscillator circuits using the Op Amp as the active element. It is found that among the 6 canonic oscillator circuits given in ef. 0, there are only three oscillator circuits using the VCVS of gain K as shown in Fig.. It is interesting to report here that these three oscillators can also be generated from the Sallen Key lters by setting V IN equal to zero that is ground the input node

4 68 A. M. Soliman C K C circuit (a) K C C circuit (b) K C circuit C (c) Fig.. Three well-known oscillators using VCVS of gain K. 9,0 and set equal to in nity in the circuit of Fig. (c) resulting in the oscillator circuits of Fig.. The characteristic equations for these three oscillators are obtained from Eqs. () () by setting the denominator equal to zero and set equal to in nity in Eq. () and are given by s þ s þ þ K C C s þ s þ þ K C C þ þ C ¼ 0 ; C ¼ 0 ; ð4þ ð5þ

5 Generation of the Minimum Component Oscillators from Sallen Key Filters 69 s þ s þ þ K þ ¼ 0 : ð6þ C C C From the above three equations the necessary condition of oscillation for each of the three circuits is given respectively as follows: For the oscillator circuit of Fig. (a): K ¼ þ C þ : ð7þ C For the oscillator circuit of Fig. (b): K ¼ þ þ C : ð8þ For the oscillator circuit of Fig. (c): K ¼ þ þ C : ð9þ The radian frequency of oscillation is the same for each of the three oscillator circuits and is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! o ¼ : ð0þ C It should be noted that real world oscillators are nonlinear circuits. The above analysis is based on assuming linear oscillators and is used as a starting point for oscillator design. The steady-state oscillations, however, are achieved by varying or controlling the s term in Eqs. (4) (6). It is worth noting that in the oscillator of Fig. (c) by replacing the VCVS by an Op Amp and two resistors results in the well-known Wien bridge oscillator. 4 8 The e ect of the nite and frequency dependent gain of the Op Amp in the Wien bridge oscillator has been studied in ef. 6. Additionally, detailed study of the nite and frequency dependent gain of the Op Amp in the three oscillator circuits together with active compensation methods was given in ef Minimum Component NIC Oscillators There are two types of the two resistor plus two capacitor NIC oscillators, one is the VNIC oscillator circuits and the second is the CNIC oscillators using CCIIþ and already known in the literature. 4,5 The generation of the three VNIC oscillator circuits from the circuits of Fig. is given in detail in the following section. 4.. VNIC oscillators using ICCII From Fig. (a) and realizing the VCVS of gain K by an Op Amp and two resistors results in Fig. (a). eplacing the Op Amp in Fig. (a) by its nullator norator 8

6 70 A. M. Soliman C C circuit + A _ 4 (K-) (a) C C circuit 4 I G (b) C VNIC 4 (c) Fig.. (a) Oscillator of Fig. (a) realized using an Op Amp. (b) Pathological realization of (a) for K ¼. (c) ealization of (b) after interchanging ground terminal.

7 Generation of the Minimum Component Oscillators from Sallen Key Filters 7 Table. Summary of the de nitions and symbols of the pathological elements. Pathological element De nition Symbol Nullator 8 V ¼ I ¼ 0 I + V _ Norator 8 V and I are arbitrary I + V _ Voltage mirror (VM) 9,0 V ¼ V I ¼ I ¼ 0 I I + + _ V V _ Current mirror (CM) 9,0 V and V are arbitrary I ¼ I, and they are also arbitrary + I I + V V model, for convenience to the reader Table is included summarizing the de nition of the four pathological elements namely nullator and norator, 8 voltage mirror (VM), and current mirror (CM) 9 to be used in this paper. Setting K ¼ in the circuit of Fig. (a) results in the oscillator circuit of Fig. (b) in which the grounded current I G equal to zero. Interchanging the ground terminal from node to node, the circuit shown in Fig. (c) is obtained. Examining the pathological circuit in the dotted box, it is seen that it realizes a VNIC as given in Table. It is worth noting in an earlier work in ef. this pathological circuit appeared in setting the VCVS equivalent nullor circuit and it was stated that: this network

8 7 A. M. Soliman Table. Summary of the de nitions and pathological realizations of the VNIC and CNIC. Active element De nition Pathological realization VNIC 9 V ¼ V I ¼ I V V I I CNIC 5,9 V ¼ V I ¼ I V V I I cannot be implemented using current conveyors. It is a correct statement at the time of publication of ef., but now after the ICCII was introduced in ef. 0,it is the VNIC circuit realizable by ICCII as given in Table. Figure 4(a) represents the equivalent oscillator circuit to Fig. (c) using the ICCII as a VNIC. From Eq. (7) and setting K ¼ results in the following condition of oscillation for the circuit of Fig. 4(a). C ¼ þ : ðþ Following similar steps, the circuits of Figs. (b) and (c) can be transformed to obtain the oscillator circuits of Figs. 4(b) and 4(c), respectively. The conditions of oscillation are obtained from Eqs. (8) and (9) by setting K ¼ resulting in the following conditions: ¼ þ C ; ðþ ¼ þ C ; ðþ

9 Generation of the Minimum Component Oscillators from Sallen Key Filters 7 Table. Summary of the de nitions and pathological realizations of the TA. Active element De nition Pathological realization TA or VCCS I ¼ 0 I ¼ GmV V arbitrary V I I Gm V ealization I V V I I Gm ealization II V I Gm I V ealization III The radian frequency of oscillation for each of the three circuits of Fig. 4 is the same as given by Eq. (0). It should be noted that the two circuits of Figs. 4(a) and 4(b) are new. The circuit of Fig. 4(c) was reported before in ef.. It is worth noting that the parasitic element a ecting the circuit of Fig. 4(a) is ; on the other hand C Z can be absorbed in. The parasitic element a ecting the circuit of Fig. 4(b) is C Z ; on the other hand can be absorbed in. The circuit of Fig. 4(c), however, is not a ected by parasitic and C Z and can absorb the e ect of in and C Z in C. 4.. CNIC oscillators using CCII+ It can also be shown that the oscillators of Fig. can be transformed to realize the minimum passive component oscillators using CCIIþ acting as CNIC. Consider the

10 74 A. M. Soliman C ICCII Z- (a) C C ICCII Z- (b) C ICCII Z- (c) Fig. 4. Three oscillators using ICCII as a VNIC. circuit of Fig. (c) and apply the adjoint circuit theorem to it by replacing nullator by norator and vice versa results in the circuit of Fig. 5. Figure 6(a) represents the equivalent oscillator circuit to Fig. 5 using the CCIIþ as a CNIC as given in Table. It should be noted that the VNIC and the CNIC are adjoint to each other. Similarly the circuits of Figs. (b) and (c) can be transformed to obtain the oscillator circuits of Figs. 6(b) and 6(c), respectively. The conditions of oscillation for the three circuits of Fig. 6 are given by Eqs. () (), respectively, with the radian frequency of oscillation as given by Eq. (0). The parasitic element e ects are the same as the adjoint circuits of Fig. 4 and the circuit of Fig. 6(c) can absorb the e ects of and C Z.

11 Generation of the Minimum Component Oscillators from Sallen Key Filters 75 C CNIC 4 Fig. 5. Adjoint circuit to Fig. (c). C CCII Z+ (a) C C CCII Z+ (b) C CCII Z+ (c) Fig. 6. Three known oscillators using CCIIþ as a CNIC. 5

12 76 A. M. Soliman 5. TA or VCCS Oscillators The generation method introduced in ef. 0 resulted in two minimum component oscillators using a single VCCS and shown in Fig. 7 of ef. 0. It is proved in this section that these two oscillators can be generated from the Sallen Key high-pass and band-pass lters, respectively. Consider the oscillator circuit of Fig. (b) and take to be the output resistance of the VCVS of gain K as shown in Fig. 7(a). ealizing the VCVS by a CCIIþ of gain VCVS with O = K C (a) C VCVS with O = CCII Z+ /K (b) VCVS with O = C ICCII Z- /K (c) Fig. 7. (a) Oscillator circuit of Fig. (b); (b) Equivalent circuit to (a) using CCIIþ; (c) Equivalent circuit to (a) using ICCII.

13 Generation of the Minimum Component Oscillators from Sallen Key Filters 77 K results in the circuit shown in Fig. 7(b); similarly the VCVS can be realized by an ICCII of gain K which results in the circuit shown in Fig. 7(c). Similarly from the oscillator circuit of Fig. (c), the circuits shown in Fig. 8 can be obtained. It should be noted that the circuit of Fig. 8(b) has been reported before in ef. 4. From Table 4, the two equivalent circuits of Figs. 7(b) and 7(c) are realizable by a TA having Gm equal to K/ resulting in the well-known circuit shown in Fig. 9(a). 0 K VCVS with O = C (a) C VCVS with O = CCII Z+ /K (b) C VCVS with O = ICCII Z- /K (c) Fig. 8. (a) Oscillator circuit of Fig. (c); (b) Equivalent circuit to (a) using CCIIþ; (c) Equivalent circuit to (a) using ICCII.

14 78 A. M. Soliman Table 4. ICCII or CCIIþ realizations of VNIC, CNIC, and TA. Active element VNIC ICCII or CCII realization ICCII- Z CNIC CCII+ Z ICCII- Z Gm TA or VCCS CCII+ Z Gm

15 Generation of the Minimum Component Oscillators from Sallen Key Filters 79 V V Gm = K/ C (a) V V Gm =K/ C (b) Fig. 9. Two minimum component oscillators using TA. 0 It should be noted that realizations I and II in Table uses grounded conductance and are adjoint to each other. ealizations III in Table uses oating conductance and is self adjoint. The condition of oscillation is derived from Eq. (8) by setting K ¼ Gm resulting in Gm ¼ G þ G þ C : ð4þ The radian frequency of oscillation is given by Eq. (0). It is seen the Gm controls the condition of oscillation without a ecting the frequency of oscillation. Similarly, the two equivalent oscillator circuits of Figs. 8(b) and 8(c) are realizable by a TA having Gm equal to K/ resulting in the well-known circuit shown in Fig. 9(b). This circuit was rst generated in ef. 0 by a systematic method of using minimum passive components C circuits with the VCCS as the active element. The condition of oscillation is derived from Eq. (9) by setting K ¼ Gm resulting in Eq. (4). Of course since the two circuits of Fig. 9 are adjoint to each other they must have the same condition of oscillation and the same expression for the radian frequency of oscillation given by Eq. (0).

16 80 A. M. Soliman Due to the attractive features of the oscillator of Fig. 9(a), 0 it has been most recently generalized to a fully di erential version and used in the generation of a new F oscillator Simulation esults The oscillator circuit shown in Fig. 4(a) is simulated using the di erential voltage current conveyor (DVCC) 6 which realizes the ICCII as special case; the DVCCS is biased with.5 V. The oscillator is designed for oscillation frequency equal to.6 MHz by taking C ¼ ¼ 0 pf, ¼ ¼ 0 k. Figure 0(a) represents the output voltage waveform. Figure 0(b) represents the output voltage waveform of the oscillator circuit of Fig. 4(b) using C ¼ ¼ 0 pf and ¼ ¼ 0 k to realize an oscillation frequency equal to 56 khz. Figure 0(c) represents the output voltage waveform of the oscillator circuit of Fig. 4(c) using ¼ C ¼ 0 pf and ¼ ¼ 0 k to realize an oscillation frequency equal to 796 khz. The total power dissipation for each of circuits, and in Fig. 4 is given by mw. (a) Fig. 0. (a) Simulation results for the circuit of Fig. 4(a); (b) Simulation results for the circuit of Fig. 4(b); (c) Simulation results for the circuit of Fig. 4(c).

17 Generation of the Minimum Component Oscillators from Sallen Key Filters 8 (b) (c) Fig. 0. (Continued )

18 8 A. M. Soliman 7. Conclusion It is shown that the Sallen Key low-pass, high-pass, and band-pass lters are the source circuits of the three minimum component oscillators using CCIIþ acting as a CNIC. Three minimum passive component oscillators using ICCII acting as VNIC are also generated from the Sallen Key low-pass, high-pass, and band-pass lters, two of them are new. In addition, it is also shown that the Sallen Key highpass and band-pass lters are the origin of the two minimum component oscillators using single input single output TA as the active element. Although this paper is considered partially as a review paper it includes new generation methods and new minimum component oscillator realizations using ICCII. The three ICCII minimum component oscillators are the adjoint of the three well-known minimum components CCIIþ oscillators.,4 Simulation results for the new oscillators using ICCII are included. The intention of this paper is not only to generate some new ICCII minimum component oscillators but also to review many of the valuable work in efs. and 0 and to show the link between the circuits known since long time and the new CCIIþ and ICCII oscillators based on new approaches of current mode circuits. 7 It is worth noting that the oscillators reported in ef. 8 do not belong to the single input single output TA reported in this paper and shown in Fig. 9 and they employ a two input TA with positive feedback to realize a negative resistor in series with a bu er at the inverting TA input. eferences. A. M. Soliman, Transformation of oscillators using Op Amps, unity gain cells and CFOA, Analog Integr. Circuits Signal Process. 65 (00) A. M. Soliman, On the generation of CCII and ICCII oscillators from three Op Amps Oscillator, Microelectron. J. 64 (00) P. Sallen and E. L. Key, A practical method of designing C active lters, IE Trans. Circuits Theory (955) A. S. Sedra and K. C. Smith, Microelectronic Circuits, 4th edn. (Oxford University Press, 998). 5. L. T. Bruton, C Active Circuits, Theory and Design (Prentice Hall, 980), pp M. E. Van Valkenburg, Analog Filter Design (Holt inehart and Winston, 98). 7. A. Budak, Passive and Active Network Analysis and Synthesis (Houghton Mi²in, 974). 8. W. Stanley, Operational Ampli er with Linear Integrated Circuits (Merrill Publishing Company, 984). 9. S. K. Mitra, Analysis and Synthesis of Linear Active Networks (Wiley, 969). 0. B. B. Bhattacharyya, M. Sundaramurthy and M. N. S. Swamy, Systematic generation of canonic sinusoidal C active oscillators, IEE Proc. Electron. Circuits Syst. 8 (98) A. M. Soliman, Current feedback operational ampli er based oscillators, Analog Integr. Circuits Signal Process. (000) A. M. Soliman, Generation of oscillators based on grounded capacitor current conveyors with minimum passive components, J. Circuits Syst. Comput. 8 (009)

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