**2. General Configuration of the VCII-Based Oscillator**

The symbolic representation and internal structure of VCII are shown in Figure 1. In this block, *Y* is a low-impedance (ideally zero) current input terminal. The current entering into *Y* node is transferred to *X* terminal which is a high-impedance (ideally infinity) current output port. The voltage produced at *X* terminal is transferred to *Z* terminal which is a low-impedance (ideally zero) voltage output terminal. The relationship between port currents and voltages are given by: *vZ* = *αvX*, *iX* = *βiY* and *vY* = 0. In the ideal case we have *<sup>α</sup>* = 1 and *<sup>β</sup>* <sup>=</sup> ± 1. If *<sup>β</sup>* = 1 we are considering a VCII+, whereas if *<sup>β</sup>* <sup>=</sup> −1 we have a VCII−.

Using the approach presented in [5,9], the general configuration of an RC-active oscillator based on a single VCII is shown in Figure 2, where NGC represents 4-terminal network consisting of only capacitors and conductances.

**Figure 1.** VCII: (**a**) symbol; (**b**) internal structure.

**Figure 2.** General configuration of RC-VCII oscillator.

The characteristic equation (CE) of the whole system can be calculated replacing, in the circuit of Figure 2, the equivalent model of a VCII of Figure 1b and considering a fictitious input at the *Y* node (of course, no input signal will be present in an actual oscillator circuit), as shown in Figure 3a at the building block level and Figure 3b in more detail.

**Figure 3.** Positive feedback system: (**a**) general schematic; (**b**) positive feedback in the VCII-based oscillator.

The configurations in Figures 2 and 3 can hence be seen as a positive feedback system for which the current transfer function (TF) is given by:

$$T\_I(s) = \frac{i\_{out}(s)}{i\_{in}(s)} = \frac{A(s)}{1 - A(s)\beta(s)}.\tag{1}$$

Since *A*(*s*) = ±1 and *β*(*s*) = *if*(*s*)/*iout*(*s*), (1) becomes:

$$T\_I(\mathbf{s}) = \frac{\pm 1}{1 \mp i\_f(\mathbf{s}) / i\_{\rm out}(\mathbf{s})}. \tag{2}$$

However, since from Figure 3b *iout* = −*iX* and in an oscillator circuit there is no input (*iin* = 0), we have *if* = *iY* and the TF is given by:

$$T\_I(\mathbf{s}) = \frac{\pm 1}{1 \mp i\_Y(\mathbf{s}) / i\_{\rm out}(\mathbf{s})} = \frac{\pm i\_X(\mathbf{s})}{i\_X(\mathbf{s}) \pm i\_Y(\mathbf{s})}.\tag{3}$$

From (3), we can derive the condition of existence (CE) as:

$$i\_X(s) \pm i\_Y(s) = 0.\tag{4}$$

By assuming *vZ* = *vX*, *vY* = 0, the transconductance functions of the passive network in Figure 2 can be expressed by a rational expression as:

$$\frac{i\_X(s)}{v\_Z(s)} = \frac{N\_X(s)}{D(s)}\tag{5}$$

$$\frac{i\_Y(s)}{v\_Z(s)} = \frac{N\_Y(s)}{D(s)}\tag{6}$$

where *NX*(*s*) and *NY*(*s*) are the numerators at *X* and *Z* nodes, respectively, while *D*(*s*) is a common denominator. Using (5) and (6) in (4), the CE becomes:

$$N\_X(s) \pm N\_Y(s) = 0\tag{7}$$

In (7), the plus and minus signs are for VCII<sup>−</sup> and VCII+ respectively. To ensure a pure sinusoidal oscillation, the CE in (7) should be a second-order polynomial with purely imaginary roots. This requires the network NGC to include at least two capacitors. It has to be noted that, in Figure 2, by using a VCII+ rather than a VCII−, at least three capacitors are required to provide a phase shift to generate a positive feedback loop. Therefore, no canonic oscillator is possible using VCII+, and for the following, we will consider the VCII in Figure 2 as a VCII−. By then assuming a network with only two capacitors, Equation (7) will be in the form:

$$as^2 + bs + c = 0.\tag{8}$$

In order to start the oscillation, the following commonly known criteria must be satisfied:

$$b = 0\tag{9}$$

$$\frac{c}{a} > 0\tag{10}$$

with *c* = 0, *a* = 0, so that, according to the Barkhausen criterion, purely imaginary poles for the closed-loop transfer function are obtained. The oscillation frequency is:

$$
\omega\_o = \sqrt{\frac{c}{a}}.\tag{11}
$$
