*4.1. Resistive Port Impedances*

If only the resistive parasitics *Gx*, *Ry* and *Rz* in (39)–(41) are considered, the oscillation condition for the Type I canonic oscillator becomes:

$$\frac{\alpha \; |\beta \; |s(R\_5 || R\_x) \mathbf{C}\_3|}{\left[1 + s(R\_5 || R\_x) \mathbf{C}\_5\right] \left[1 + s(R\_1 + R\_y + R\_z) \mathbf{C}\_3\right]} = 1. \tag{42}$$

It is evident from (42) that the effect of the port impedances is limited, since they are simply summed to the ones from the NGC network (that have to be chosen as much larger than the corresponding parasitics to make them negligible). The oscillation frequency in Table 1 is modified as follows:

$$
\omega\_0 \prime = \sqrt{\frac{G\mathfrak{g}'}{\mathbb{C}\_5 \mathbb{C}\_3 \mathbb{R}\_1}} = \omega\_0 \cdot \sqrt{\frac{1 + \mathcal{G}\_\mathfrak{x} / \mathcal{G}\_\mathfrak{5}}{1 + \mathcal{G}\_\mathfrak{1} \left(\mathcal{R}\_\mathcal{Y} + \mathcal{R}\_\mathfrak{z}\right)}}\tag{43}
$$

where *R*<sup>1</sup> = *R*<sup>1</sup> + *Ry* + *Rz* and *G*<sup>5</sup> = 1/*R*<sup>5</sup> = *G*<sup>5</sup> + *Gx*, and the oscillation condition becomes *<sup>α</sup>* <sup>|</sup>*β*<sup>|</sup>

$$\frac{\mathbb{1}\left|\mathcal{P}\right|}{\frac{R\_1}{R\_5}\left(1+\frac{R\_5}{R\_x}\right)\left(1+\frac{R\_y+R\_z}{R\_1}\right)+\frac{C\_5}{C\_3}}=1.\tag{44}$$

If the parasitic capacitance *Cx* at the X terminal is also considered, Equations (43) and (44) have to be slightly modified by considering *G*<sup>5</sup> = *C*<sup>5</sup> + *Cx* instead of *C*5. Inductances *Ly* and *Lz* can be neglected in several applications and have not been considered in the following. However, for the sake of completeness, we report below the expression for the oscillation frequency when inductive parasitics are also considered:

$$
\omega\_0 \prime = \omega\_0 \cdot \sqrt{\frac{1 + \mathcal{G}\_x / \mathcal{G}\_5}{\left(L\_y + L\_z\right)\left(1 + \mathcal{G}\_x / \mathcal{G}\_5\right) \frac{\mathcal{G}\_5 \mathcal{G}\_1}{\mathcal{G}\_5} + \left(1 + \mathcal{G}\_x / \mathcal{G}\_5\right)\left[1 + \mathcal{G}\_1 \left(R\_y + R\_z\right)\right]}}} \tag{45}
$$

#### *4.2. Single-Pole Transfer Functions*

If the non-ideal transfer functions in (36) and (37) are also considered in addition to the terminal resistive parasitics in (38)–(40), the denominator of the oscillation condition in (34) becomes of fourth degree:

$$\frac{\alpha \, |\beta| \, sG\_5^{\prime} R\_5^{\prime}{}^{\prime}}{as^4 + bs^3 + cs^2 + ds + \varepsilon} = 1\tag{46}$$

Prime variables are considered for *R*<sup>5</sup> , *G*<sup>5</sup> and *R*<sup>1</sup> to account for parasitic resistances *Ry* and *Rz* and admittance *Yx*, as in the previous subsection, and we have

$$a = R\_5 \, ^\prime \mathbb{C}\_5 \, ^\prime R\_1 \, ^\prime \mathbb{C}\_3 \, \tau\_3 \, \tau\_2 \tag{47a}$$

$$b = R\mathfrak{s}'\mathbb{C}\mathfrak{s}'R\_1'^\prime\mathbb{C}\mathfrak{s}\cdot(\pi\_{\mathfrak{x}}+\pi\_{\mathfrak{z}}) + \pi\_{\mathfrak{x}}\pi\_{\mathfrak{z}}\cdot(R\mathfrak{s}'\mathbb{C}\mathfrak{s}' + R\_1^\prime\mathbb{C}\mathfrak{s})\tag{47b}$$

$$\mathbf{c} = \mathbf{R}\_{\mathsf{S}} \mathbf{'} \mathbf{C}\_{\mathsf{S}} ^{\prime} \mathbf{R}\_{1} ^{\prime} \mathbf{C}\_{\mathsf{S}} + \boldsymbol{\tau}\_{\mathsf{x}} \boldsymbol{\tau}\_{\mathsf{z}} + \left( \mathbf{R}\_{\mathsf{S}} ^{\prime} \mathbf{C}\_{\mathsf{S}} ^{\prime} + \mathbf{R}\_{1} ^{\prime} \mathbf{C}\_{\mathsf{S}} \right) \cdot \left( \boldsymbol{\tau}\_{\mathsf{x}} + \boldsymbol{\tau}\_{\mathsf{z}} \right) \tag{47c}$$

$$d = R\mathfrak{s}'\mathbb{C}\mathfrak{s}' + R\_1\mathfrak{s}'\mathbb{C}\mathfrak{s} + \mathfrak{r}\_\mathfrak{x} + \mathfrak{r}\_\mathfrak{z} \tag{47d}$$

$$
\mathbf{e} = \mathbf{1} \tag{47\mathbf{e}}
$$

A real value is obtained for the left-hand side, under the hypothesis of a purely imaginary denominator. By equating to zero the real part of the denominator at *ω* = *ω*0, we get:

$$
\omega\_0 \prime^2 = \frac{c}{2a} \left( 1 - \sqrt{1 - \frac{4a}{c^2}} \right) \cong \frac{c}{2a} \left( \frac{1}{2} \frac{4a}{c^2} \right) = \frac{1}{c} \tag{48}
$$

where *c* is given by (47c). The approximation <sup>4</sup>*<sup>a</sup> <sup>c</sup>*<sup>2</sup> << 1 is justified under the hypothesis that the parasitic time constants *τ<sup>x</sup>* and *τ<sup>z</sup>* are significantly lower than the time constants *τ<sup>A</sup>* = *R*<sup>5</sup> *C*5 and *τ<sup>B</sup>* = *R*<sup>1</sup> *C*3. Finally, the oscillation frequency *ω*<sup>0</sup> can be expressed in terms of the ideal value *ω*0, by using the expression of coefficient *c*:

$$
\omega\_0' \cong \frac{1}{\sqrt{c}} = \omega\_0 \frac{1}{\sqrt{1 + \frac{\mathbf{r}\_x \mathbf{r}\_z + (\mathbf{r}\_A + \mathbf{r}\_B)(\mathbf{r}\_x + \mathbf{r}\_z)}{\mathbf{r}\_A \mathbf{r}\_B}}} \tag{49}
$$

Under the simplifying assumptions *τ<sup>x</sup>* = *τ<sup>y</sup>* = *τpar* and *τ<sup>A</sup>* = *τ<sup>B</sup>* = *τ*, the relative error on the oscillation frequency (1 − *ω<sup>o</sup>* /*ωo*) can be readily expressed as a function of the ratio *τpar*/*τ*, thus providing a design guideline for the bandwidth of the VCII transfer functions. The graph in Figure 8 shows that errors lower than 10% can be obtained if the time constant ratio is lower than 0.06.

**Figure 8.** Relative error on the oscillation frequency vs. the time constant ratio *τpar*/*τ*.
