NEW GROUNDED-CAPACITOR SINUSOIDAL OSCILLATORS USING THE CURRENT-FEEDBACK-AMPLIFIER POLE

New current-feedback-operational amplifier (CFOA)-pole-based sinusoidal oscillator circuits are presented. Each circuit uses two CFOAs, two (or three) grounded capacitors and/or resistors. Experimental results are included.


INTRODUCTION
It is well known that oscillator circuit realization using the internal pole of the voltage feedback operational amplifier (VFOA) results in reliable high frequency performance and low component count [1][2][3].The current feedback operational amplifier (CFOA) has a larger bandwidth and higher slew rate compared to the conventional VFOA.Thus the realization of oscillator circuits using the internal pole of the CFOA would result in higher frequencies of operation than their VFOA counterpart.This conjecture was examined and recently a number of oscillator circuits, exploiting to advantage the internal pole of the CFOA, have been presented [4][5][6][7][8][9][10][11][12].In [4] three sinusoidal oscillator circuits using the CFOA pole are presented.While the first oscillator circuit uses two CFOAs and six resistors, the second oscillator circuit uses two CFOAs and eight resistors.While both circuits use no external capacitors, they do not enjoy the attractive feature of independent control of the frequency and the condition of oscillation.Thus, while the frequency of oscillation can be controlled without disturbing the condition of oscillation, the condition of oscillation can not be controlled without disturbing the frequency of oscillation.The third oscillator circuit uses one CFOA, three resistors and one floating capacitor.In this oscillator circuit, using only four passive elements, the frequency of oscillation can not be controlled without disturbing the condition of oscillation.In [5] a multiphase active-R sinusoidal oscillator circuit using the CFOA-pole is presented.Each phase requires one CFOA and two resistors; one of them is floating.Moreover, the frequency of oscillation can not be controlled without disturbing the condition of oscillation.In [6] three single-CFOA-pole-based oscillator circuits are presented.For these oscillators, the frequency of oscillation and the condition of oscillation can be independently controlled.
However, each circuit requires three resistors; one of them floating, and one floating capacitor.Moreover, the control of the frequency of oscillation is achieved by using the floating resistor.The active-R oscillator circuits proposed in [7,8] use two CFOAs and five resistors; three of them are floating.For these oscillators the frequency of oscillation can not be controlled without disturbing the condition of oscillation.The single CFOA-pole-based oscillator circuit proposed in [9] enjoys the independent control ofthe frequency of oscillation and the condition of oscillation.However, it requires two grounded resistors, one floating resistor and one floating capacitor.The CFOA-pole based oscillator circuit proposed in [10] enjoys the independent control of the frequency of oscillation and the condition of oscillation.However, it uses a floating capacitor and three resistors; one of them is floating.The three sinusoidal oscillator circuits proposed in [11] exploit to advantage the internal pole of a single CFOA.Each circuit uses five externally connected passive elements; two (or three) resistors and three (or two) capacitors.In one of the circuits, all the capacitors are grounded.However, the circuit has the disadvantage of interdependent control of the frequency and the condition of oscillation.In two of the circuits, independent control of the frequency and the condition of oscillation can be achieved.However, the implementation of these two circuits requires floating capacitors and floating resistors.The fourth circuit requires floating capacitors and floating resistors and does not enjoy the independent control of the frequency and the condition of oscillation.Finally, in [12] a single-resistance-controlled sinusoidal oscillator is presented.The circuit exploits to advantage the internal poles of two CFOAs and uses three externally connected resistors; two of them are floating.While the frequency of oscillation can be controlled by adjusting a floating resistor without disturbing the condition of oscillation, the condition of oscillation can not be controlled without disturbing the frequency of oscillation.
It appears, therefore, that a sinusoidal oscillator circuit using the CFOA-pole and enjoying the attractive features of (i) independent control of the frequency of oscillation and the condition of oscillation by using grounded elements, (ii) use of externally connected grounded capacitors and resistors and (iii) low output impedance, is not available yet in the literature.It is the major intention of this paper is to present such an oscillator structure.The proposed structure exploits to advantage the internal poles of two CFOAs and uses grounded capacitors and resistors only.

PROPOSED CIRCUIT
The proposed oscillator circuit is shown in Figure 1. Figure 2 shows a simplified equivalent circuit for the CFOA.In this equivalent circuit the parallel combination of Cz and Rz represents the parasitic impedance at node z of the CFOA, the parallel combination of Cy and Ry represents the parasitic impedance at node y of the CFOA a 1--el, lell << represents the current-tracking error of the CFOA, 3 1-2, 1 21 1 represents the input voltage-tracking error, and 1-3, 131 represents the output voltage-tracking error of the CFOA.Using the CFOA equivalent circuit of Figure 2, the equivalent circuit of the proposed oscillator structure of Figure 1, is shown in Figure 3. Routine analysis yields the characteristic equation of the circuit of Figure 1 which can be expressed as (Gr + sCr) (GT + sCr) 0102/1fl2")'12 Y1 Y2 where Gr Gz, -i-Ga-bGy2, GT2 Gz: -t"Gb-l-Gy, Ga--(1/Ra), Gb-- (1/Rb), Gz, (1/Rz,), Gyi (1/Ry,) are the internal conductances at nodes z and y of the ith CFOA, and ai, fli and 7 are the current tracking-error, the input voltage tracking-error and the output voltage tracking-error of the ith CFOA.Using (1) several oscillator circuits can be derived from Figure 1.Table I summarizes the admittances, the frequency and the condition of oscillation of the resulting oscillator circuits using four or five passive elements only.However, for circuits 1-4, the resistors Ra and R b can be removed and the design can account on the internal conductances of the CFOAs.Thus, circuits 1- 4 can be realised using three externally connected passive elements only.
From the Table it can be seen that for the circuits 1-4 the frequency of oscillation can be adjusted by tuning a single element (capacitor or resistor) without disturbing the condition of oscillation, while the condition of oscillation can be adjusted by tuning another element (resistor or capacitor) without disturbing the frequency of oscillation.Thus the circuits 1-4 enjoy independent oscillation and frequency control.For example, for circuit 4, the frequency of oscillation can be adjusted by tuning G1 without disturbing the condition of oscillation, while the condition of oscillation can be adjusted by tuning C1 without disturbing the frequency of oscillation.However, because of the difference terms in the numerator (or denominator) of the oscillation frequency of the circuits 1-4, it is essential to select components to satisfy the condition CT1CT2 > C1 C2 (or GT1GT2 > G1G2).From the Table it can also be seen that circuits 5-6, although using fewer components, their frequency and condition of oscillation are interdependent.These circuits may be useful in applications where single-frequency oscillators are required.The frequency of oscillation of the circuits 5-6 can be adjusted by tuning the resistors R and/or Rb.

SENSETIVITY ANALYSIS
By relating a sensetivity parameter F to the element of variation xi by sF X i_ dF x, F dxi it is easy to show that the Wo-sensetivities of the circuits 5 and Thus, the circuits 5 and 6 enjoy low active and passive Wo-sensetivities.
In the case of circuits 1-4, the passive Wo-sensetivities may be appreciably higher due to the presence of the difference terms in the denominators.For example, for the circuits and 2, the Wo-sense- tivities can be expressed as so =so 2 So So c c (C C2/Cr CT) and so =so =s; =s so=so Thus, while the active Wo-sensetivities of the circuits and 2 are zeros, the passive sensitivities may be appreciably high.However, careful selection of the passive components can yield passive a;o-sensetivities of the same order as those of circuits 5 and 6.
Finally, the tVo-sensetivities of the circuits 3-4 can be expressed as Thus, circuits 3 and 4 has non-zero active o-sensetivities and its passive Wo-sensetivities may be appreciably high.However, careful design may result in active and passive Wo-sensetivities of the same order as those of circuits 5 and 6.

EXPERIMENTAL RESULTS
The proposed sinusoidal oscillator circuits were experimentally tested using the AD844 CFOA.The results obtained from circuits 3-6 are shown in Figure 4.Because of the large values of stray capacitances encountered in a breadboard implementation, it is difficult to successfully obtain oscillations from circuits and 2. Extra care must be taken to minimize the effect of the stray capacitances.Thus, circuits and 2 are more suitable for integrated circuit implementation.
For the AD844 the capacitances Cz -5 pF, Cy---3 pF and the resistances Rz 3 Mf, Ry 10Mf.However, the total parasitic capacitance between terminals Z of the AD844 and the breadboard was measured to be about 25 pF.Thus Crl Cr was taken as 33 pF.
will affect the transfer function especially at high frequencies [13].
Thus, with C1 470 pF for circuit 5 and C2 300 pF for circuit 6, the experimental results are different at high frequencies.As the frequency decreases the results obtained from the two circuits becomes closer to each other.This effect is not observed in circuits 3 and 4.This is attributed to the symmetry between the two circuits regarding the externally connected elements to terminals X.Moreover, circuits 3 and 4 use relatively lower value for capacitances, C1 C2 100 pF.Thus, the effect of Rx would manifest itself at relatively higher frequencies.In order to minimize the effect of the parasitic resistance Rx on the circuit performance, it is recommended here to use CFOAs designed using the power-supply current-sensing technique, rather than the AD844 which is built around the translinear principle [13] and suffers from relatively large value of the parasitic resistance R.

CONCLUSION
New grounded-capacitor grounded-resistor CFOA-pole-based sinu- soidal oscillator circuits have been presented.Each circuit uses two CFOAs, at most five passive elements and exploits, to advantage, the internal poles of the CFOAs.Some of the circuits enjoy independent control of the frequency and the condition of oscillation.The use of grounded capacitors and resistors is an attractive feature for integration.Moreover, each circuit has two low impedance outputs.
Thus, in comparison to similar ideal current-conveyor-based topologies [14], the proposed circuits enjoy the attractive features of using less number of passive components while providing two low- impedance outputs and exploiting, to advantage, the nonidealities of the active devices.

FIGURE 4
FIGURE 4 Measured and calculated results obtained from.(a) Circuits 3, 4 with:

TABLE Frequency and
Condition of Oscillation of the 6 Oscillators