Sinusoidal oscillators with explicit current output

INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS Int. J. Circ. Theor. Appl. 2010; 38:131–147 Published online 7 August 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cta.531

Sinusoidal oscillators with explicit current output employing current-feedback op-amps S. S. Gupta1 , R. K. Sharma2 , D. R. Bhaskar3 and R. Senani4, ∗, † 1 Department

of Industrial Policy and Promotion, Ministry of Industry, Udyog Bhawan, New Delhi 110 011, India 2 Department of Electronics and Communication Engineering, Guru Gobind Singh Indraprastha University, Ambedkar Institute of Technology, Shakarpur, Delhi, India 3 Faculty of Engineering and Technology, Department of Electronics and Communication Engineering, Jamia Millia Islamia, New Delhi 110 025, India 4 Analog Signal Processing Research Laboratory, Electronics and Communication Engineering Department, Netaji Subhas Institute of Technology (formerly, Delhi Institute of Technology), Azad Hind Fauj Marg, Sector 3, Dwarka, New Delhi 110 078, India

SUMMARY Employing a state-variable synthesis, a number of new current-mode oscillators with explicit current output have been derived, which can be practically implemented from commercially available current-feedback op-amps (CFOA). The workability of the proposed structures has been confirmed by experimental results using AD844-type CFOAs and some sample results have been presented. Copyright q 2008 John Wiley & Sons, Ltd. Received 11 August 2007; Revised 21 June 2008; Accepted 5 July 2008 KEY WORDS:

current-mode oscillators; current-feedback op-amps; current-mode circuits

1. INTRODUCTION Sinusoidal oscillators are important building blocks in several instrumentation, electronic and communication systems. In view of the extensive research on current-mode filters and other signal processing circuits [1–39], the design of oscillators providing an explicit current output (ECO) from a high output impedance node has also become important. Sinusoidal oscillators with ECO would also be useful as signal generators to test various current-mode circuits. Although there have been a number of investigations [1–6] on realizing oscillators with ECO using other ∗ Correspondence

to: R. Senani, Analog Signal Processing Research Laboratory, Electronics and Communication Engineering Department, Netaji Subhas Institute of Technology (formerly, Delhi Institute of Technology), Azad Hind Fauj Marg, Sector 3, Dwarka, New Delhi 110 078, India. † E-mail: [email protected]

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building blocks, such as first generation current conveyors [1, 2], differential difference current conveyors [3], differential difference complementary current-feedback amplifiers [4], four terminal floating nullors [5], unity-gain voltage and current followers [6], none of these building blocks are commercially available yet. Owing to the well-known advantages of the current-feedback op-amps (CFOA) as compared with the conventional voltage-mode op-amps (VOA), namely the availability of nearly constant bandwidth for moderate values of gain, higher slew rates (typically in excess of 2000 V/s), relatively higher operational frequency range, possibility of designing various signal processing/signal generation circuits with the least possible number of passive components without requiring any component-matching conditions or realization constraints, CFOAs with an externally accessible compensation pin (z-terminal) such as AD844 from analog devices have been found to be particularly attractive building blocks for various signal processing and signal generation applications. Although there have been numerous studies on realizing sinusoidal oscillators using CFOA, most of them have dealt with circuits providing a voltage-mode signal available from a low-outputimpedance node (for instance, see [7–19] and the references cited therein). In this paper, we show how the state-variable approach of the synthesis of oscillators [8] can be extended to systematically synthesize current-mode sinusoidal oscillators with ECO using only two AD844 type CFOAs.‡

2. SYNTHESIS OF THE NEW OSCILLATORS The state-variable methodology employed here was introduced by Senani and Gupta [8] and has subsequently been found to be a powerful tool to evolve single resistance controlled oscillators (SRCO) using a variety of active building blocks [2, 4, 6, 20, 22–25]. In the present work, this methodology has been tailored to suit the evolution of the SRCOs with ECO. A canonic second-order (i.e. employing only two capacitors) oscillator can, in general, be characterized by the following autonomous state equation: ⎡•⎤ a11 a12 x1 x1 • ⎣ ⎦= (1) or [ x] = [A][x] • a21 a22 x2 x2 where x1 and x2 are the state variables representing the voltages across the two capacitors employed in the oscillator. From the above, the characteristic equation (CE) is given by s 2 −(a11 +a22 )s +(a11 a22 − a12 a21 ) = 0 that gives the condition of oscillation (CO) and frequency of oscillation (FO) as CO: (a11 +a22 ) = 0 FO: 0 = (a11 a22 −a12 a21 ) ‡

(2) with a11 a22 >a12 a21

(3)

Of course, several of the earlier known oscillators based on CCII+/CCII− can also be implemented by AD844s; however, versions of such oscillators using exclusively CCII+ and providing ECO would typically require three or more CCII+ and hence as many CFOAs, whereas the circuits derived here require no more than two CFOAs.

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Int. J. Circ. Theor. Appl. 2010; 38:131–147 DOI: 10.1002/cta

SINUSOIDAL OSCILLATORS WITH EXPLICIT CURRENT OUTPUT

133

The proposed methodology consists of the following steps: (i) selection of the parameters ai j , i = 1, 2; j = 1, 2, in accordance with the required features (e.g. non-interacting controls for FO and CO through separate resistors, which require that f 0 should be controllable by a resistor that does not feature in the CO and the CO should also be independently adjustable by another resistor that does not appear in the expression of FO), (ii) conversion of the resulting state equations into node equations (NE) and finally (iii) constitution of a physical circuit from these NE. Different circuits are expected to be generated by making different choices of the parameters: a11 , a12 , a21 and a22 . For non-interactive controls of CO and FO, let us assume that CO is to be controlled by a resistor R1 (independent of some resistor R2 ) and that FO is to be controlled by the resistor R2 (independent of the resistor R1 ), with a third resistor R3 featuring in both CO and FO. These conditions lead to the following requirements: (a) The expression of (a11 +a22 ) should either not have terms containing R2 or they should be cancelled out. Thus, in (a11 +a22 ), there should be two terms left with opposite signs involving R1 and R3 . (b) Similarly, to have FO independent of R1 , the expression (a11 a22 −a12 a21 ) should either not have the terms containing R1 or they should be cancelled out. Thus, FO should be a function of resistors R2 and R3 only (along with C1 and C2 ). On the basis of the above consideration, as an example, we can construct the required [A] matrix of Equation (1) by choosing a11 = 1/C1 R1 , a22 = −1/C2 R3 , which satisfy the requirement (a). Now by choosing a12 = −(1/C1 )(1/R1 +1/R2 ), a21 = 1/C2 R3 , we can satisfy the requirement (b). An exemplary [A] matrix of Equation (1), thus, takes the following form:

⎤ ⎡ 1 1 1 1 − + ⎢ C 1 R1 C 1 R1 R2 ⎥ ⎥ ⎢ (4) [A]1 = ⎢ ⎥ ⎦ ⎣ 1 1 − C 2 R3 C 2 R3 which results in the following CO and FO: C2 R3 C1

(5)

1 0 = √ C 1 C 2 R2 R3

(6)

CO: R1 =

On the other hand, substitution of Equation (4) into Equation (1) gives the following NE:

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C1

dx1 x1 − x2 x2 = − dt R1 R2

(7)

C2

dx2 x1 − x2 = dt R3

(8)



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For meeting the specific objective of having an explicit current-mode output, we consider implementing Equations (7) and (8) using CFOAs characterized by the following matrix equation: ⎤⎡ ⎤ ⎡ ⎤ ⎡ 0 iy 0 0 0 0 ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢ v x ⎥ ⎢ 0 1 0 0⎥ ⎢ v y ⎥ ⎥⎢ ⎥ ⎢ ⎥=⎢ (9) ⎥⎢ ⎥ ⎢ ⎥ ⎢ 0 0 1 0 i ⎦ ⎣ ix ⎦ ⎣ z⎦ ⎣ vw

0 0 0 1

vz

From the above matrix characterization, it is seen that unlike a VOA, CFOA exhibits a high-input impedance at only one input terminal (y) whereas the x input terminal has a low input impedance, and similar to a VOA, the input voltages vx and v y are equal. The current inputed at terminal x is conveyed to port z with unity current transfer ratio and if i z is terminated into a load, the voltage created therein (vz ) is available at terminal w with a low-output impedance. Thus, a buffered output is available at the w-terminal of the CFOA. To synthesize the required circuit, we have to keep in mind that the z-terminal of at least one of the CFOAs has to be left unutilized to facilitate the current output available from a high-output impedance node. The circuit thus formulated from Equations (7) and (8) is shown here as Circuit-1 in Figure 1 where the mechanism of constructing the circuit can be understood by following the various current segments of Equations (7) and (8) marked therein. It may be seen that as intended, the ECO is available from the z-terminal of the first CFOA. It may be noted that in the Circuit-1 of Figure 1, R2 is connected between the terminals x and w of CFOA2. This R2 can also be connected between the x-terminal of CFOA1 and the x-terminal of CFOA2 without affecting the resulting equation. This results in Circuit-2 of Figure 1, which retains the same CO and FO as that of Circuit-1 of Figure 1. In a similar manner, three more variants of the basic circuit are possible with alternative realizations of the NE (7) and (8), which are shown as Circuits-3–5 in Figure 1. It may also be noted that in Circuit-1 the resistor R2 can also be connected between the terminals x and z of CFOA2; however, this reduces the effective value of R2 to half of its earlier value. The resulting configuration is shown as Circuit-6 in Figure 1. In this operation, the FO of the circuit accordingly changes to

2 0 = (10) C 1 C 2 R2 R3 Nevertheless, the nature of the circuit does not change and it still remains an oscillator with essentially the same properties. A slightly different variant of Circuit-6 is shown as Circuit-7 in Figure 1. It is now clear that by forming different [A] matrices in accordance with the defined objectives, various other circuits can be generated. For instance, consider the following two alternative [A] matrices (both of which can be seen to satisfy the objectives (a) and (b) outlined earlier):

⎡ 1 1 1 1 1 1 1 ⎤ − − − + ⎢ C 1 R1 R3 C1 R1 R2 R3 ⎥ ⎥ [A]2 = ⎢ (11) ⎦ ⎣ 1 1 − C 2 R3 C 2 R3 Copyright q




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Figure 1. New CFOA-based current-mode oscillators synthesized from matrix [A]1 .

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⎡

1 ⎢− C R 1 3 ⎢ [A]3 = ⎢ ⎣ 1 C 2 R3

⎤ 1 1 1 1 − − + C1 R1 R2 R3 ⎥ ⎥ ⎥

⎦ 1 1 1 − C 2 R1 R3

(12)

Following the methodology explained above, the circuits derived corresponding to the matrices [A]2 and [A]3 are shown as Circuits-8–11 in Figure 2 and Circuit-12 in Figure 3. These circuits have the CO and FO given by the following equations:

C2 CO: R1 = C1 +C2

R3

for Circuits-8–11

R3

for Circuit-12

(13)

and

CO: R1 =

C1 C1 +C2

1 FO: 0 = √ C 1 C 2 R2 R3

(14) for Circuits-8–10 and 12

and

0 =

2 C 1 C 2 R2 R3

for Circuit-11

Although a two-CFOA SRCO with an ECO has been known earlier in [9], the circuit therein requires one additional equality condition of C1 = C2 for realizing the oscillator and, therefore, does not belong to the class considered here, which is free from any such constraint. On the other hand, the Circuits-1, 8, 12 were reported in a piecemeal manner earlier in [20]; all the remaining circuits are completely new. It may be noted that in all the circuits, an additional explicit voltage-mode output is available from the w-terminal of CFOA-1, after the current-mode output is taken from the z-terminal of CFOA-1 (assuming z-terminal to be terminated into some load impedance Z L ). Finally, it may be observed that in Circuit-5 of Figure 1, the resistor R3 is connected at a virtual ground node created at the x-terminal of CFOA1. If we connect other grounded elements, namely C1 and R1 or different combinations of R1 , R3 and C1 at this virtual ground node, the resulting circuit will still remain SRCO with the same CO and FO. This process would generate six additional variants of Circuit-5. In a similar manner, six additional variants of Circuit-7 and seven additional variants of each for the Circuits 10 and 11 can also be obtained.§

§ The

proposed method can also be used (see Appendix A) to generate oscillators in which two-resistors (providing independent control of oscillation frequency) can be ganged-tuned to make the frequency linear in conductance, which normally is readily adjusted by an external control voltage (when such resistors are replaced by two JFETs/MOSFETs derived by a common-gate control-voltage).

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R2

x

1

w

z

y

C2

R3 C2

i out y

y

z x

x

y

2 C1

z

x

R3

w

w

1

R1

Circuit-8

z

x

C2

z

x

C2

i out

w i out

y

2

R2

1

R3

w

y

R1

C1

y

1

x

w

Circuit-9

y

R3

z

R2

i out

R1

2

z

2

w

x

R1

C1

Circuit-10

z

w

R2 C1

Circuit-11


R1 x w

1 z

x

R2

y

w

z

2 y

C2

iout C1

R3

Circuit-12


Although all the 12 circuits employ exactly the same number of active and passive components, a careful comparison between them reveals the following: (1) Except Circuit-12 of Figure 3, the parasitic compensating capacitor at the z-pin of CFOA-2 can be easily absorbed by the capacitor C1 in all the circuits and thus, the non-ideal CE remains second-order in all the cases. (2) Circuits-1–11 have the condition of setting resistor R1 grounded, which is advantageous from the view point of the ease of incorporation of automatic amplitude stabilization schemes [31, 32]. Copyright q



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Figure 4. Variation of frequency with R2 for the oscillator of Circuit-1 in Figure 1 (C1 = C2 = 1 nF, R3 = 1 k, R1 = 100 +1 k pot).

(3) The parasitic input resistance Rx of both the CFOAs can be easily accommodated in R1 in Circuit-12; on the other hand, in Circuits-4 and 8, R x of CFOA1 can be absorbed in R2 and that of CFOA2 in R1 . Furthermore, a non-ideal analysis of the circuits (see Appendix B) reveals that the influence of parasitics R x , R p and C p is minimal in these cases (None of the other circuits can absorb Rx of both the CFOAs). (4) The non-ideal expressions of Appendix B also reveal that, of all the oscillators presented here, those of Circuit-4 and 8 are the only ones in which the frequency controlling resistor R2 does not disturb the CO even non-ideally (the corresponding expressions for CO are independent of R2 ). Thus, Circuits-4 and 8 appear to be the best circuits of the entire class.

3. HARDWARE IMPLEMENTATION AND EXPERIMENTAL RESULTS To verify the workability of the newly synthesized oscillators, they were constructed from AD844type CFOAs biased with ±15 V DC power supplies and were found to work satisfactorily in accordance with the theory. Some sample experimental results (with a termination of 1 k at z-terminal) are shown in Figures 4 and 5. Figure 4 shows the variation of oscillation frequency with R2 for the oscillator of Circuit-1 of Figure 1, whereas Figure 5 shows a typical waveform obtained from the same circuit. The percentage THD at amplitude of 4.4 V (p–p), over the observed frequency range, was found to be smaller than 0.59%. Similar to traditional VOA circuits, the mechanism for settling the amplitude¶ is also based on the saturation characteristic of AD844. ¶ Additional

feedback circuit will be needed to stabilize or control the amplitude as prevalent in traditional op-ampbased oscillator circuits.

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Figure 5. A typical waveform generated from the oscillator of Circuit-1 in Figure 1 (99.79 kHz).

The non-ideal analysis of the circuits (Appendix B) reveals that the CE of the circuits remains a second-order in spite of the influence of various parasitic impedances of the CFOAs. The non-ideal expressions for CO and FO for all the circuits have been worked out incorporating CFOA parasitics Rx , R p and C p . From Table I, it is clear that as resistor R1 is variable, in all cases its value can be set such that the CO can always be fulfilled even non-ideally and none of the circuits, thus, have any start-up difficulty. How fast or slow the build-up of oscillations takes place eventually depends upon to what extent the roots of the non-ideal CE are located on the right half of the s-plane which is ultimately decided by the value of the resistor R1 . As an example, for the Circuit-1 of Figure 1, we show the build-up of oscillations for two different values of R1 in Figure 6, which clearly substantiates this assertion. Furthermore, from the experimental studies, it has been confirmed that in all the cases, R1 can always be adjusted such that the circuits start-up building oscillations without any difficulty. In conclusion, none of the proposed circuits have any start-up difficulty. On the other hand, from Table II we can see that the parasitics of the CFOA make the non-ideal expression frequency to be different from their ideal counterparts. From these non-ideal expressions, the percentage errors in the Circuits-1–12, ideally designed for f 0 = 159 kHz for Circuits-1–5, 8–10, 12, f 0 = 225 kHz for Circuits-6,7,11, with component values C1 = C2 = 100 pF, R2 = R3 = 10 k, have been found to be −3.46, −3.46, −3.45, −2.5, −3.46, −3.28, −3.28, −2.66, −4.08, −3.93, −3.6 and −4.06, respectively, which are not very large. However, the parasitics would limit the operation of the oscillations at higher frequencies due to increased errors in the value of the oscillation frequencies. However, with a judicious choice of component values (i.e. keeping R1 , R2 , R3 much greater than Rx and keeping C1 , C2 much greater than C p ) oscillations around 1 MHz range are easily attainable. As an example, Figure 7 shows a typical waveform (1.06 MHz, 2.3 V (p–p)) obtained from Circuit-8.

4. CONCLUDING REMARKS It was shown how a state-variable methodology can be employed to derive systematically new sinusoidal oscillators providing explicit-current-mode-output using commercially available CFOAs. Copyright q



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Table I. The non-ideal condition of oscillations for the derived new oscillators. Circuit

Non-ideal CO

Ideal CO

1

2

3 4

1 − 1 − Rx + Rx 1 + 1 R1 R p R p R3 +Rx R1 R2 (R3 +Rx ) (C1 +C p ) − 0 C2 Rx Rx 1+ R + R 1 2 ⎫ ⎡⎧ ⎤ R3 ⎪ ⎪ ⎨ ⎬ (R3 +Rx ) ⎢ R1 ⎥ − R3 ⎣ ⎦ R x R3 ⎪ ⎪ ⎩ 1+ Rx + Rx + ⎭ Rp R1 R2 R1 R2 (Rx +R3 ) (C1 +C p ) ⎤ 0 − ⎡ R R R R (R +R ) C2 x x 3 x 2 1 R2 + R2 (Rx +R3 ) ⎢ 1+ R1 + R2 R1 ⎥ R1 ⎣ −R ⎦ 2 Rx2 R3 R R 1+ Rx + Rx + R R (R 1 2 1 2 x +R3 )

1 − 1 R 3 R1 R p (C1 +C p ) 0 − R C2 Rx x 1+ R + R +R (C1 +C p ) − R3 C2

1

x

2

1 1 R1 +Rx − R p

R3 C1 C2 − R1 0

R3 C1 C2 − R1 0

R3 C1 C2 − R1 0 R3 C1 C2 − R1 0

0

⎤

Rx 1 + 1 − 1 + 1 1+ R1 R2 Rp ⎥ (C1 +C p ) ⎢ R2 R p −⎣ ⎦ (R3 + R x )0 C2 R R 1+ Rx + Rx 1 2 ⎡ ⎤ 1 + 1

R R R 1 1 1 1 1 2 x ⎢ R +R R +R − R +R ⎥ x p p ⎥ 1 2 2 ⎢ (C1 +C p ) ⎢ 1+ R3 ⎥ −⎢ ⎥ (R3 + R x )0 C2 R R ⎢ ⎥ 1+ Rx + Rx ⎣ ⎦ 1 2

⎡

5

6

7

8

9

10

11

12

⎡ ⎤ Rx Rx Rx 1 1 (C1 +C p ) ⎣ R1 1− R2 − R p 1+ R1 + R2 ⎦ − (R3 + R x )0 R R C2 1+ Rx + Rx 1 2

1 − 1 − 1 R +R R R x p 1 3 C2 0 R3 (C1 +C p ) +

Rx Rx 1 + 1 1 + 1 − 1− − R1 R2 +Rx R p R3 R2 +Rx R3 C2 0 R Rx (C1 +C p ) + R3 1+ Rx + R +R x 1 2

1 − 1 − 1 − Rx + Rx 1 + 1 R R +R R R R +R R R x p p x 1 3 3 1 2 C2 0 R R (C1 +C p ) + (R3 +Rx ) 1+ Rx + Rx 1 2 ⎤ ⎡ 1 R

1− Rx R 1 1 1 2 ⎦ ⎣ R R − R p + (R3 +Rx ) 1+ Rx + Rx C2 1 2 0 (C1 +C p ) + (R3 +Rx )

R +2R R +2R 1− 1 R x − 1 R x p 3 C2

0 C1 − R1 +2Rx R1 +2Rx C p R1 +2Rx + R +C R R 3

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p

2

R3 C1 C2 − R1 0

R3 C1 C2 − R1 0

R3 C1 C2 − R1 0

R3 C1 C2 + 1− R1 R3 C1 C2 + 1− R1

R3 C1 C2 + 1− R1

R3 C1 C2 + 1− R1

R3 C2 C1 + 1− R1

0 0

0

0

0

2



141

(b)

(a)

Figure 6. Building up of oscillations for the Circuit-1 in Figure 1 for two values of resistor R1 : (a) R2 = 1 k, R3 = 1 k, C1 = 1 nF, C2 = 1 nF; With R1 = 430 , Vc1(0) = Vc2(0) = 0 V, observed settling-time = 200 S and (b) R2 = 1 k, R3 = 1 k, C1 = 1 nF, C2 = 1 nF; with R1 = 420 , Vc1(0) = Vc2(0) = 0 V, observed settling-time = 130 S.

A family of 12 such circuits has been systematically derived. Practicability of the new circuits has been demonstrated through some sample experimental and simulation results based upon AD844-type CFOAs. It has to be noted that at present the AD844 is the only CFOA in which the high-impedance node z is externally accessible and available at the output. Therefore, limitations of this component (for example, the gain bandwidth product limited at 60 MHz and the minimum supply of ±4.5 V) prevent the proposed circuits to be used in low-voltage and in high-frequency systems. However, in view of a number of recent attempts in evolving low-voltage, low-power CFOAs for high frequency applications [27–29], we believe that with new improved CFOA ICs available in the future, the frequency range of operation of the proposed circuits can be extended beyond the limitations of the AD844. It is, therefore, expected that the ongoing search for newer topologies of SRCOs with ECO using CFOAs may lead to circuits that might be useful as test oscillators for verifying various currentmode signal processing circuits such as current-mode filters, current-mode precision rectifiers, etc. to which the proposed kind of circuits can be interfaced with out any additional hardware. An interesting question is whether or not the oscillators of the kind synthesized in this paper can be realized with a single CFOA? In this context, it is worth pointing out that indeed two circuits employing only a single CFOA and providing ECO have been known earlier [21]; however, they are both third-order oscillators and do not possess single element controllability of CO and FO as available in the presented circuits. In view of this, therefore, the problem of searching any second-order SRCOs with an ECO, realizable with only a single CFOA (as an improvement over the circuits presented here) appears to be very interesting (but challenging) and is open to investigation.

Recent

criticism not withstanding [30], the current-mode techniques have given a way to a number of important analog signal processing/signal generation circuits as is evident from a vast amount of literature on current-mode circuits and techniques published over the past 35 years.

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6

5

4

3

2

1

Circuit

1 C 1 C 2 R2 R3 ⎡

⎤⎡

1

2

R R R R 1+ R 2 + R x + R x R2 p p p 1 R R C1 C2 R2 (R3 +Rx ) 1+ Rx + Rx

Non-ideal FO

⎤

⎢ ⎢ ⎢ ⎢ ⎣

⎤⎡

1

C1 C2 R3

1 + 1 Rp R2

1

2

R R 1+ Rx + Rx 1 2 2 Rx Rx Rx 1 R + R R + R p 1+ R + R 1 2 1 2 2 R R C1 C2 R2 (R3 +Rx ) 1+ Rx + Rx

C1 C2 R3

Rx 1 1 1 1 1 1 R2 + R p − 1− R p R1 + R2 + R1 + R2

R3 (Rx +R2 ) R3 Rx Rx ⎢ 1+ R1 + R2 1+ R1 (Rx +R3 ) ⎥ ⎢ R32 R3 +Rx

⎦⎣ +⎣

R1 R p + Rx2 R3 Rx2 R3 R R R R 1+ Rx + Rx + R R (R 1+ Rx + Rx + R R (R 1 2 1 2 1 2 x +R3 ) 1 2 x +R3 )

1 + 1 R2 R p

R R C1 C2 R3 1+ Rx + x R2 1

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎤⎥ ⎥ ⎥⎥ − R 1 ⎦⎥ Rp ⎦

R (R +R ) R3 R R 1+ Rx + Rx 1+ R3 (Rx +R2 ) ⎥ ⎢⎢ ⎥⎢ R3 +Rx R x 1 2 1 3 1 − ⎦

⎢⎣

R p ⎦ ⎣1− Rx2 R3 Rx2 R3 ⎢ R R R R 1+ Rx + Rx + R R (R 1+ Rx + Rx + R R (R ⎢ +R ) +R ) 1 2 1 2 1 2 x 3 1 2 x 3 ⎢

⎡⎡

Table II. Non-ideal expression for frequency of oscillation for the derived new oscillators.

2 C 1 C 2 R2 R3

1 C 1 C 2 R2 R3

1 C 1 C 2 R2 R3

1 C 1 C 2 R2 R3

1 C 1 C 2 R2 R3

1 C 1 C 2 R2 R3

Ideal FO

142 S. S. GUPTA ET AL.


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12

11

10

9

8

7

Circuit

1

C1 C2 R2 R3

1

!

2

1 + 1 R3 R p R2 [C1 C2 +C p (C1 +C2 )]

Rx 2 R2+ R1 R2 R R C1 C2 R3 1+ Rx + Rx 1 2

C1 C2 R3 1+ R + R2 1

Rx 1 1 1 1 R2 + R p + R p R1 + R2 R R C1 C2 R3 1+ Rx + Rx

1 + Rx 1 + 1 R R p R1 R 2 2

Rx Rx

R R C1 C2 R2 (R3 +Rx ) 1+ Rx + Rx 1 2

Non-ideal FO 2 R + RRRx + R1p 1+ RRx + RRx 2 1 2 1 2

Table II. Continued.

1 C 1 C 2 R2 R3

2 C 1 C 2 R2 R3

1 C 1 C 2 R2 R3

1 C 1 C 2 R2 R3

1 C 1 C 2 R2 R3

2 C 1 C 2 R2 R3

Ideal FO


143


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S. S. GUPTA ET AL.

Figure 7. A typical waveform generated from the oscillator of Circuit-8 in Figure 2 (1.06 MHz, 2.3 Vpp). Component values: C1 = C2 = 100 pF, R1 = 404 , R2 = R3 = 1 k.

APPENDIX A: DERIVATION OF OSCILLATORS PROVIDING OSCILLATION FREQUENCY LINEARLY VARIABLE IN CONDUCTANCE THROUGH GANGED VARIABLE RESISTOR Here, we illustrate how the proposed method can also be used to generate oscillators in which two resistors (providing independent control of oscillation frequency) can be ganged-tuned to make the frequency linear in conductance that normally is readily adjusted by an external control voltage (when such resistors are replaced by two JFETs/MOSFETs derived by a common-gate control-voltage). An exemplary [A] matrix leading to such an oscillator is the following: ⎡ ⎤

1 1 Rb − ⎢ C R ⎥ C 1 Ra R1 1 1 ⎢ ⎥ [A] = ⎢ (A1)

⎥ ⎣ 1 1 1 Rb ⎦ − − + C 2 R1 C 2 R2 C 2 Ra R1 This leads to the following NE: C1

dx1 1 Rb = − x1 + x2 dt R1 Ra R1

(A2)

C2

dx2 1 1 Rb = − x1 − x2 + x2 dt R1 R2 Ra R1

(A3)

A physical implementation of Equation (A2) and (A3) leads to the following circuit given in Figure A1 with CO and FO as CO :

Ra R1 C 2 = + Rb R2 C 1

and

FO: 20 =

1 C 1 C 2 R1 R2

(A4)

Enumeration of an entire catalogue of such oscillators is out side the scope of the present paper. Copyright q




+

+

1

Ra

145

2

-

Rb

R1

R2 C2

C1

Figure A1. An exemplary oscillator circuit providing frequency control through ganged variable resistances.

y

1 1

Ix

Rp

Cp

w z

Rx x

Ix

Figure B1. Non-ideal equivalent circuit of the CFOA.

APPENDIX B: NON-IDEAL ANALYSIS OF THE NEW OSCILLATORS CONSIDERING THE CFOA PARASITIC IMPEDANCES R x , R p AND C p Assuming the non-zero input resistance R x (normally around 50–100 ) at terminal x and non-ideal z-terminal output impedance modeled by a resistor R p (around 3 M) in parallel with a capacitance C p (normally between 3.5–5.5 pF) (Figure B1) the derived new circuits have been re-analyzed. It has been found that the inclusion and consideration of parasitic impedance does not increase the order of the non-ideal CE of the circuits that are all found to be of degree 2. Consequently, none of the circuits are likely to have any start-up difficulties or multi-mode oscillations as prevalent in the case of the other types of oscillators [36]. The non-ideal CO and FO for the various circuits are shown in Tables I and II, respectively.

APPENDIX C: ILLUSTRATION OF THE DESIGN METHOD USED FOR OSCILLATION START-UP CONDITIONS USING BARKHAUSEN’S CONDITION For the Circuit-3 in Figure 1, the loop gain is found to be

s 1 1 + V0 C 1 R1 R2

T (s) = = 1 1 1 Vin 2 + s +s + C 2 R3 C 1 R2 C 1 C 2 R2 R3 Copyright q


(C1)


146

S. S. GUPTA ET AL.

From which

"

#1/2 1 2 1 + R1 R2 |T (j)| = "

2

2 #1/2 1 1 1 1 −2 +2 + C 1 C 2 R2 R3 C 2 R3 C 1 R2 2 C12

(C2)

Now, according to Barkhausen’s criteria |T (j)|1 and angle of {T (j)} = 0 results at 1 = √ C 1 C 2 R2 R3 At this frequency, |T (j)|1 simplifies to R1 C1 R3 C2 . Thus, the above condition represents the limit on parameter choices of the resistors and capacitors in the circuit for ensuring start-up of oscillations. ACKNOWLEDGEMENTS

This work was performed at the Analog Signal Processing Research Lab. of Netaji Subhas Institute of Technology (formerly, Delhi Institute of Technology), New Delhi, India. The authors are thankful to Professor Marco Gilli and anonymous reviewers for their constructive suggestions. REFERENCES 1. Bhaskar DR, Prasad D, Imam SA. Grounded-capacitor SRCOs realized through a general scheme. Frequenz 2004; 58(7/8):175–177. 2. Senani R, Gupta SS. Novel SRCOs using first generation current conveyor. International Journal of Electronics 2000; 87(10):1187–1192. 3. Kilinc S, Jain V, Aggarwal V, Cam U. Catalogue of variable frequency and single resistance controlled oscillators employing a single differential-difference-complementary-current-conveyor. Frequenz 2006; 60:142–146. 4. Gupta SS, Senani R. Grounded-capacitor SRCOs using a single differential-difference-complementary-currentfeedback-amplifier. IEE Proceedings—Circuits Devices and Systems 2005; 152(1):38–48. 5. Senani R. On equivalent forms of single op-amp sinusoidal RC oscillators. IEEE Transactions on Circuits and Systems I 1994; 41(10):617–624. 6. Gupta SS, Senani R. New single resistance controlled oscillator configurations using unity-gain cells. Analog Integrated Circuits and Signal Processing 2006; 46:111–119. 7. Senani R, Singh VK. Synthesis of canonic single-resistance-controlled-oscillators using a single current-feedbackamplifier. IEE Proceedings—Circuits Devices and Systems 1996; 143(1):71–72. 8. Senani R, Gupta SS. Synthesis of single-resistance-controlled oscillators using CFOAs: simple state-variable approach. IEE Proceedings—Circuits Devices and Systems 1997; 144(2):104–110. 9. Gupta SS, Senani R. State variable synthesis of single-resistance-controlled grounded capacitor oscillators using only two CFOAs: additional new realisations. IEE Proceedings—Circuits Devices and Systems 1998; 145(2): 415–418. 10. Toker A, Cicekoglu O, Kuntman H. On the oscillator implementations using a single current feedback op-amp. Computers and Electrical Engineering 2002; 28:375–389. 11. Celma S, Martinez PA, Carlosena A. Current feedback amplifiers based sinusoidal oscillators. IEEE Transactions on Circuits and Systems I 1994; 41(12):906–908. 12. Wu DS, Liu SI, Hwang YS, Wu YP. Multiphase sinusoidal oscillator using the CFOA pole. IEE Proceedings— Circuits Devices and Systems 1995; 142(1):37–40. 13. Abuelmaatti MT, Farooqi AA, Alshahrani SM. Novel RC oscillators using the current-feedback operational amplifier. IEEE Transactions on Circuits and Systems I 1996; 43(2):155–157. Copyright q




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Sinusoidal oscillators with explicit current output

Sinusoidal oscillators with explicit current output

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