New CFOA-based sinusoidal oscillators retaining independent control of oscillation frequency even under the influence of parasitic impedances

Abstract

There have been two efforts earlier on evolving CFOA-based fully-uncoupled oscillators i.e. circuits in which none of the resistors controlling the frequency of oscillation (FO) appear in the condition of oscillation and vice versa. However, a non-ideal analysis of the earlier known circuits reveals that due to the effect of the parasitic impedances of the CFOAs, the independent controllability of FO is completely destroyed. The main objective of this paper is to present two new fully-uncoupled oscillators in which the independent controllability of the FO remains intact even under the influence of the non-ideal parameters/parasitics of the CFOAs employed. The workability of the proposed circuits has been confirmed by experimental results using AD844-type CFOAs.

Appendix A

1.1 Analysis of the previously known CFOA-based grounded capacitor SRCOs taking into account the effect of parasitic impedances of the CFOAs

In the earlier literature, there appear to be only two circuits employing CFOAs which belong to the category of fully-uncoupled oscillators, namely, the circuit presented by Soliman [20] and the circuit presented by Bhaskarc [44], which are shown here in Figs. 5 and 6 respectively.

Fig. 5

Fully-uncoupled oscillator proposed by Soliman [20]

Fig. 6

Fully-uncoupled oscillator proposed by Bhaskar [44]

Both these circuits employ exactly the same number of active and passive components as in the circuits presented in this paper. Ideal CO and FO for the circuits of Figs. 5 and 6 are respectively given by:

$$ \begin{aligned} R_{3} &= R_{4}\;\;({\text{for\;Fig.\,5}})\\ R_{1} & = R_{2}\;\;({\text{for\;Fig.\,6}}) \end{aligned} $$

(8)

$$ \begin{aligned} f_{0} &= \frac{1}{2\pi }\sqrt {\frac{1}{{C_{1} C_{2} R_{1} R_{2} }}} \;\;( {\text{for Fig.\,5}}) \\ f_{0} &= \frac{1}{2\pi }\sqrt {\frac{1}{{C_{1} C_{2} R_{3} R_{4} }}} \;\;( {\text{for Fig.\,6}} ) \end{aligned} $$

(9)

From a non-ideal analysis, CE, CO and FO of the oscillator of Fig. 5 are respectively given by:

$$ s^{3} \left( {C_{1}^{'} C_{2}^{'} C_{z2} R_{x3} } \right) + s^{2} \left\{ {C_{1}^{'} C_{2}^{'} \left( {1 + \frac{{R_{x3} }}{{R_{3}^{'} }}} \right) + C_{2}^{'} C_{z2} \left( {\frac{{R_{x3} }}{{R_{z1} }}} \right) + C_{1}^{'} C_{z2} \left( {\frac{{R_{x3} }}{{R_{4}^{'} }} - 1} \right)} \right\} + s\left[ {C_{1}^{'} \left\{ {\left( {1 + \frac{{R_{x3} }}{{R_{3}^{'} }}} \right)\left( {\frac{1}{{R_{4}^{'} }}} \right) - \frac{1}{{R_{3}^{'} }}} \right\} + C_{2}^{'} \left( {1 + \frac{{R_{x3} }}{{R_{3}^{'} }}} \right)\left( {\frac{1}{{R_{z1} }}} \right) + C_{z2} \left( {\frac{{R_{x3} }}{{R_{4}^{'} }} - 1} \right)\left( {\frac{1}{{R_{z1} }}} \right)} \right] + \left[ {\frac{1}{{R_{z1} }}\left\{ {\left( {1 + \frac{{R_{x3} }}{{R_{3}^{'} }}} \right)\left( {\frac{1}{{R_{4}^{'} }}} \right) - \frac{1}{{R_{3}^{'} }}} \right\} + \frac{1}{{R_{1}^{'} R_{2}^{'} }}} \right] = 0\;{\rm where}\;C_{1}^{'} = \left( {C_{1} + C_{z1} } \right);\,C_{2}^{'} = \left( {C_{2} + C_{z2} } \right);\,R_{1}^{'} = \left( {R_{1} + R_{x1} } \right);\;R_{2}^{'} = \left( {R_{2} + R_{x2} } \right);\; R_{3}^{'} = \left( {R_{3} \left\| {R_{z2} } \right.} \right);\;R_{4}^{'} = \left( {R_{4} \left\| {R_{z3} } \right.} \right)$$

(10)

$$ 1 - \frac{{R_{4}^{\prime } }}{{R_{3}^{\prime } }} + \frac{{C_{2}^{\prime } R_{4}^{\prime } R_{x3}^{2} }}{{C_{1}^{\prime } R_{3}^{\prime 2} R_{z1} }} + \frac{{2C_{2}^{\prime } R_{4}^{\prime } R_{x3} }}{{C_{1}^{\prime } R_{3}^{\prime } R_{z1} }} + \frac{{C_{2}^{\prime } R_{4}^{\prime } }}{{C_{1}^{\prime } R_{z1} }} + \frac{{C_{2}^{\prime } R_{4}^{\prime } R_{x3}^{2} C_{z2} }}{{C_{1}^{\prime 2} R_{3}^{\prime } R_{z1}^{2} }} + \frac{{C_{2}^{\prime } R_{4}^{\prime } R_{x3} C_{z2} }}{{C_{1}^{\prime 2} R_{z1}^{\prime } }} - \frac{{2R_{4}^{\prime } R_{x3} C_{z2} }}{{C_{1}^{\prime } R_{3}^{\prime } R_{z1} }} - \frac{{2R_{4}^{\prime } C_{z2} }}{{C_{1}^{1} R_{z1} }} + \frac{{R_{x3}^{2} }}{{R_{3}^{\prime 2} }} + \frac{{2R_{x3} }}{{R_{3}^{\prime } }} + \frac{{2R_{x3}^{2} C_{z2} }}{{C_{1}^{\prime } R_{3}^{\prime } R_{z1} }} + \frac{{2R_{x3} C_{z2} }}{{C_{1}^{\prime } R_{z1} }} + \frac{{R_{x3}^{2} C_{z2} }}{{C_{2}^{\prime } R_{3}^{\prime } R_{4}^{\prime } }} + \frac{{R_{x3} C_{z2} }}{{C_{2}^{\prime } R_{4}^{\prime } }} - \frac{{2R_{x3} C_{z2} }}{{C_{2}^{'} R_{3}^{'} }} - \frac{{C_{z2} }}{{C_{2}^{'} }} - \frac{{R_{4}^{\prime } R_{x3} }}{{R_{3}^{\prime 2} }} + \left( {\frac{{R_{x3} C_{z2} }}{{C_{1}^{\prime } R_{z1} }}} \right)^{2} + \frac{{\left( {R_{x3} C_{z2} } \right)^{2} }}{{C_{1}^{\prime } C_{2}^{\prime } R_{4}^{\prime } R_{z1} }} - \frac{{2R_{x3} C_{z2}^{2} }}{{C_{1}^{\prime } C_{2}^{\prime } R_{z1} }} + \frac{{R_{4}^{\prime } C_{z2} }}{{C_{2}^{\prime } R_{3}^{\prime } }} - \frac{{R_{4}^{\prime } R_{x3} C_{z2}^{2} }}{{C_{1}^{\prime 2} R_{z1}^{2} }} + \frac{{R_{4}^{\prime } C_{z2}^{2} }}{{C_{1}^{'} C_{2}^{'} R_{z1} }} - \frac{{R_{4}^{\prime } R_{x3} C_{z2} }}{{C_{1}^{\prime } R_{1}^{\prime } R_{2}^{\prime } }} = 0 $$

(11)

$$ f_{0}^{'} = \frac{1}{2\pi }\sqrt {\frac{1}{{C_{1} C_{2} R_{1} R_{2} }}} \left[ \frac{{\frac{1}{{\left( {1 + \frac{{R_{x1} }}{{R_{1} }}} \right)\left( {1 + \frac{{R_{x2} }}{{R_{2} }}} \right)}} + \frac{{R_{1} R_{2} }}{{R_{z1} }}\left\{ {\left( {\frac{1}{{R_{4} }} + \frac{1}{{R_{z3} }}} \right)\left( {1 + \frac{{R_{x3} }}{{R_{3} }} + \frac{{R_{x3} }}{{R_{z2} }}} \right) - \frac{1}{{R_{3} }} - \frac{1}{{R_{z2} }}} \right\}}}{{\left( {1 + \frac{{C_{z1} }}{{C_{1} }}} \right)\left( {1 + \frac{{C_{z2} }}{{C_{2} }}} \right)\left( {1 + \frac{{R_{x3} }}{{R_{3} }} + \frac{{R_{x3} }}{{R_{z2} }}} \right)}} + C_{z3} R_{x3} \left\{ {\frac{{\left( {1 + \frac{{C_{z2} }}{{C_{2} }}} \right)}}{{C_{1} R_{z1} }} + \frac{{\left( {1 + \frac{{C_{z1} }}{{C_{1} }}} \right)\left( {\frac{1}{{R_{4} }} + \frac{1}{{R_{z3} }}} \right)}}{{C_{2} }}} \right\} - \frac{{C_{z2} }}{{C_{2} }}\left( {1 + \frac{{C_{z1} }}{{C_{1} }}} \right) \right]^{1/2} $$

(12)

The CE, CO and FO for the oscillator of Fig. 6 are respectively given by:

$$ s^{3} \left( {C_{1}^{'} C_{2}^{'} C_{z3} } \right) + s^{2} \left\{ {C_{1}^{'} C_{2}^{'} \left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{z3} }}} \right) + C_{1}^{'} C_{z3} \left( {\frac{1}{{R_{z} }}} \right) + C_{2}^{'} C_{z3} \left( {\frac{1}{{R_{z1} }}} \right)} \right\} + s\left[ {C_{1}^{'} \left( {\frac{1}{{R_{z} }}} \right)\left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{z3} }}} \right) + C_{2}^{'} \left\{ {\left( {\frac{1}{{R_{z1} }}} \right)\left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{z3} }}} \right) + \left( {\frac{1}{{R_{3}^{'} }}} \right)\left( {\frac{1}{{R_{1} }} - \frac{1}{{R_{2}^{'} }}} \right)} \right\} + C_{z3} \left( {\frac{1}{{R_{3}^{'} R_{4}^{'} }}} \right)} \right] + \left[ {\frac{1}{{R_{z} R_{3}^{'} }}\left( {\frac{1}{{R_{1} }} - \frac{1}{{R_{2}^{'} }}} \right) + \frac{1}{{R_{3}^{'} R_{4}^{'} }}\left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{z3} }}} \right)} \right] = 0\;where\;C_{1}^{'} = \left( {C_{1} + C_{z1} } \right);\,C_{2}^{'} = \left( {C_{2} + C_{z2} + C_{y1} } \right);\,R_{2}^{'} = \left( {R_{2} + R_{x2} } \right);\;R_{3}^{'} = \left( {R_{3} + R_{x1} } \right);\; R_{4}^{'} = \left( {R_{4} + R_{x2} } \right);\;R_{z} = \left( {R_{y1} \left\| {R_{z2} } \right.} \right) $$

(13)

$$ C_{1}^{\prime } C_{2}^{\prime 2} \left( {\frac{1}{{R_{3} + R_{x1} }}} \right)\left( {\frac{1}{{R_{1} }} - \frac{1}{{R_{2} + R_{x3} }}} \right)\left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{z3} }}} \right) + C_{1}^{\prime 2} C_{2}^{\prime } \left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{z3} }}} \right)^{2} \left( {\frac{1}{{R_{y1} }} + \frac{1}{{R_{z2} }}} \right) + C_{1}^{\prime } C_{2}^{\prime 2} \left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{z3} }}} \right)^{2} \left( {\frac{1}{{R_{z1} }}} \right) + C_{1}^{'2} C_{z3} \left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{z3} }}} \right)\left( {\frac{1}{{R_{y1} }} + \frac{1}{{R_{z2} }}} \right)^{2} + 2C_{1}^{\prime } C_{2}^{\prime } C_{z3} \left( {\frac{1}{{R_{y1} }} + \frac{1}{{R_{z2} }}} \right) \left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{z3} }}} \right)\left( {\frac{1}{{R_{z1} }}} \right) + C_{2}^{\prime 2} C_{z3} \left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{z3} }}} \right)\left( {\frac{1}{{R_{z1} }}} \right)^{2} + C_{1}^{\prime } C_{z3}^{2} \left( {\frac{1}{{R_{3} + R_{x1} }}} \right)\left( {\frac{1}{{R_{4} + R_{x2} }}} \right)\left( {\frac{1}{{R_{y1} }} + \frac{1}{{R_{z2} }}} \right) + C_{2}^{\prime } C_{z3}^{2} \left( {\frac{1}{{R_{3} + R_{x1} }}} \right)\left( {\frac{1}{{R_{4} + R_{x2} }}} \right)\left( {\frac{1}{{R_{z1} }}} \right) + C_{2}^{\prime 2} C_{z3} \left( {\frac{1}{{R_{3} + R_{x1} }}} \right)\left( {\frac{1}{{R_{1} }} - \frac{1}{{R_{2} + R_{x3} }}} \right)\left( {\frac{1}{{R_{z1} }}} \right) = 0 $$

(14)

$$ f_{0}^{\prime } = \frac{1}{2\pi }\sqrt {\frac{1}{{C_{1} C_{2} R_{3} R_{4} }}} \left[ {\frac{{\frac{{\left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{z3} }}} \right)}}{{\left( {1 + \frac{{R_{x1} }}{{R_{3} }}} \right)\left( {1 + \frac{{R_{x2} }}{{R_{4} }}} \right)}} + \frac{{R_{4} \left( {\frac{1}{{R_{1} }} - \frac{1}{{R_{2} \left( {1 + \frac{{R_{x3} }}{{R_{2} }}} \right)}}} \right)\left( {\frac{1}{{R_{y1} }} + \frac{1}{{R_{z2} }}} \right)}}{{\left( {1 + \frac{{R_{x1} }}{{R_{3} }}} \right)}}}}{ \left( {1 + \frac{{C_{z1} }}{{C_{1} }}} \right)\left( {1 + \frac{{C_{z2} + C_{y2} }}{{C_{2} }}} \right)\left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{z3} }}} \right) + \left( {1 + \frac{{C_{z1} }}{{C_{1} }}} \right)\left( {\frac{{C_{z3} }}{{C_{2} }}} \right)\left( {\frac{1}{{R_{y1} }} + \frac{1}{{R_{z2} }}} \right) + \left( {1 + \frac{{C_{z2} + C_{y2} }}{{C_{2} }}} \right)\left( {\frac{1}{{R_{z1} }}} \right)\left( {\frac{{C_{z3} }}{{C_{1} }}} \right) }} \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} $$

(15)

From Eqs. (10–12), and (13–15) it may be seen that in both the circuits of Figs. 5 and 6, all the four resistors employed therein are present in the CO as well as in FO. It is, therefore, concluded that in both these circuits the fully-uncoupled nature of CO and FO is completely disturbed when the effect of parasitic of the CFOAs is accounted for.

For the sake of comparison with previously known conventional type of CFOA-based SRCOs, a similar non-ideal analysis has been carried out for an exemplary two-CFOA-two-grounded capacitors (GC) SRCO from [31] shown in Fig. 7. In this context it may be noted that none of the single-CFOA SRCOs known till date employ both grounded capacitors while two-CFOA-based SRCOs do employ both grounded capacitors. However, out of various such two-CFOA-GC SRCOs, the closest to the present class appears to be the one proposed in [31] which also provides control of FO through two resistors (like the circuits proposed in this paper) and hence, the choice.

Fig. 7

An exemplary two-CFOA-GC oscillator proposed by Liu and Tsay [31]

The circuit of Fig. 7 is ideally characterized by the following CO and FO:

$$ {\text{CO}}:\left( {1 + \frac{{C_{2} }}{{C_{1} }}} \right) = \frac{{R_{1} }}{{R_{2} }} $$

$$ f_{0} = \frac{1}{2\Uppi }\sqrt {\frac{{R_{3} }}{{2C_{1} C_{2} R_{1} R_{2} R_{4} }}} $$

However, a re-analysis of this circuit reveals that its non-ideal CE is given by:

$$ s^{3} + s^{2} \left[ {\frac{1}{{C_{1}^{'} }}\left( {\frac{1}{{R_{1} }} + \frac{1}{{R_{y1} }}} \right) + \frac{1}{{C_{2}^{'} }}\left( {\frac{1}{{R_{1} }} + \frac{1}{{R^{'} }} - \frac{1}{{R_{2}^{'} }}} \right) + \frac{1}{{C_{z1} }}\left( {\frac{1}{{R_{z1} }} + \frac{{R_{x1} + 2R_{4} }}{{R_{x1} R_{3} + R_{x1} R_{4} + R_{3} R_{4} }}} \right)} \right] + s\left[ \frac{1}{{C_{1}^{'} C_{2}^{'} }}\left\{ {\frac{1}{{R_{y1} }}\left( {\frac{1}{{R_{1} }} + \frac{1}{{R^{'} }} - \frac{1}{{R_{2}^{'} }}} \right) + \frac{1}{{R_{1} }}\left( {\frac{1}{{R^{'} }} - \frac{1}{{R_{2}^{'} }}} \right)} \right\} + \frac{1}{{C_{z1} C_{2}^{'} }}\left\{ {\frac{1}{{R_{z1} }}\left( {\frac{1}{{R_{1} }} + \frac{1}{{R^{'} }} - \frac{1}{{R_{2}^{'} }}} \right)} \right\} + \frac{1}{{C_{z1} C_{1}^{'} }}\left\{ {\frac{1}{{R_{z1} }}\left( {\frac{1}{{R_{y1} }} + \frac{1}{{R_{1} }}} \right)} \right\} + \left( {\frac{{R_{x1} + 2R_{4} }}{{R_{x1} R_{3} + R_{x1} R_{4} + R_{3} R_{4} }}} \right)\left\{ {\frac{1}{{C_{z1} C_{2}^{'} }}\left( {\frac{1}{{R_{1} }} + \frac{1}{{R^{'} }} - \frac{1}{{R_{2}^{'} }}} \right) + \frac{1}{{C_{z1} C_{1}^{'} }}\left( {\frac{1}{{R_{y1} }} + \frac{1}{{R_{1} }}} \right)} \right\} \right] + \frac{1}{{C_{1}^{'} C_{2}^{'} C_{z1} }}\left[ \left( {\frac{1}{{R_{z1} }} + \frac{1}{{R_{3} }}} \right)\left\{ {\frac{1}{{R_{y1} }}\left( {\frac{1}{{R_{1} }} + \frac{1}{{R^{'} }} - \frac{1}{{R_{2}^{'} }}} \right) + \frac{1}{{R_{1} }}\left( {\frac{1}{{R^{'} }} - \frac{1}{{R_{2}^{'} }}} \right)} \right\} + \frac{{\left( {\frac{1}{{R_{3} }} - \frac{1}{{R_{x1} }}} \right)\left( {\frac{1}{{R_{1} }}\frac{1}{{R_{x1} }}\frac{1}{{R_{2}^{'} }}} \right) - \frac{1}{{R_{3} }}\left( {\frac{1}{{R_{y1} }} + \frac{1}{{R_{1} }}} \right)\left( {\frac{1}{{R_{1} }} + \frac{1}{{R^{'} }} - \frac{1}{{R_{2}^{'} }}} \right)}}{{\frac{1}{{R_{x1} }} + \frac{1}{{R_{3} }} + \frac{1}{{R_{4} }}}}\right]$$

where $R^{'} = R_{y2} \left\| {R_{z2} } \right.,\,\,\,C_{2}^{'} = C_{2} + C_{z2} ,\,\,C_{1}^{'} = C_{1} + C_{y1} ,\,\,\,R_{2}^{'} = R_{w1} + R_{2} + R_{x2}$.

From above equation it is, thus, seen that, as expected, in this circuit also, when the various parasitic non-ideal effects of the CFOAs are accounted for, both the frequency-controlling resistors R ₃ and R ₄ creep into all the coefficients of the CE and hence, also in the CO.