New Very-Low-Frequency Third-Order Quadrature Sinusoidal Oscillators Using CFOAs

Abstract

In this paper, through a systematic approach, four new very-low-frequency third-order quadrature sinusoidal oscillators (TOQSO), especially suitable for generating very low oscillation frequencies in the sub-audio range (< 20 Hz and further down), have been derived. Each of the four proposed TOQSOs employs three current feedback op-amps (CFOA), five resistors and three grounded capacitors, as preferred for integrated circuit (IC) implementation. All the oscillators provide independent control of the condition of oscillation (CO) and the frequency of oscillation (FO). The workability and utility of the proposed TOQSOs have been corroborated by simulation results and the hardware implementation results employing AD844-type IC CFOAs.

Appendix 1

Let us consider the characteristic equation of the third order system as given in Eq. (1):

$$ a_{3} s^{3} + a_{2} s^{2} + a_{1} s + a_{0} \, = \,0 $$

(29)

where $a_{3} = C_{3} R_{3}$,$a_{2} = \frac{{C_{3} R_{3} }}{{C_{2} R_{2} }}$, $a_{1} = \frac{1}{{C_{2} R_{4} }}$ and $a_{0} = \frac{1}{{C_{1} C_{2} R_{1} R_{4} }}$ (as in Eq. (18)).

The Routh array can be constructed from Eq. (29) as follows [23]:

s3	a3	a1
s2	a2	a0
s1	$\frac{{a_{2} a_{1} - a_{3} a_{0} }}{{a_{2} }}$	0
s0	a0	0

As per the Routh-Hurwitz criterion [23], the characteristic equation will have a pair of imaginary axis roots (corresponding to imaginary axis poles), if any row in the Routh array, corresponding to odd power of s, e.g., s³, s⁵.. contains all ‘0’s. The row just above the so-called all zero row is used to create the auxiliary equation whose solution gives the value of the imaginary axis roots. From the above array, it is noted that in the row corresponding to s¹ element, there is only single nonzero entry, indicating the possibility that if this entry becomes conditionally zero, then the characeristic equation will have a pair of imaginary conjugate roots. Applying this criteria, the condition of oscillation (CO) is given as:

$$ {\text{CO}}:\frac{{a_{2} a_{1} - a_{3} a_{0} }}{{a_{2} }} = 0\,which\,implies\,\frac{{a_{1} }}{{a_{3} }} = \frac{{a_{0} }}{{a_{2} }} $$

(30)

The auxalliary equation (AE) is formed as:

$$ a_{2} s^{2} + a_{0} = 0 $$

(31)

whose solution gives the frequency of oscillation (FO) as:

$$ {\text{FO}}:\omega_{0} = \sqrt {\frac{{a_{0} }}{{a_{2} }}} \, $$

(32)

Using the cofficients of Eq. (29), CO will turn out to be $C_{2} R_{2} = C_{1} R_{1}$ and FO as $\sqrt {\frac{1}{{C_{2} C_{3} R_{3} R_{4} }}}$.

Alternatively, for representing a sinusoidal oscillator, Eq. (29) should have a pair of imaginary conjugate roots (responsible for constant-amplitude sustained sinusoidal oscillations) with the third pole lying on the negative real axis (to ensure stability).

Thus, Eq. (29) should be factorable as:

$$ \left( {s^{2} + \omega_{0}^{2} } \right)\left( {s + \alpha } \right) = 0 $$

(33)

yeilding roots as s =  ± jω₀ and s = -α.

On comparing the cofficients of Eqs. (33) and (29), it turns out that for the roots of the Eq. (29) to be of type Eq. (33), the required condition is:

$$ \frac{{a_{1} }}{{a_{3} }} = \frac{{a_{0} }}{{a_{2} }} $$

(34)

And the value of ω₀ from the quoted comparison turns out to be

$$ \omega_{0} = \sqrt {\frac{{a_{0} }}{{a_{2} }}} \,\, $$

(35)