## 2 Methods
### Practical OTA-Based Gyrator
#### 2.2.1 Influence of OTA Terminal Impedances
Above this frequency, the input admittance has the magnitude value \(G_{\text{P2}}\) and is purely real. Further, above the frequency  
\[Y_{\text{IN}}{}^{\prime}(s)\approx\frac{g_{\text{m1}}g_{\text{m2}}}{s^{\alpha} F+G_{\text{P1}}+sC_{\text{P1}}}+G_{\text{P2}}+sC_{\text{P2}}. \tag{7}\]1  
the effect of the parasitic capacitance \(C_{\text{P2}}\) prevails, and the input admittance \(Y_{\text{IN}}\)' has purely capacitive character, which even more differs from the required FO inductor. As demonstrated in Figure 5 by the dashed part of the characteristic, when the conductance \(G_{\text{P2}}\) is lower than the higher value of the two admittances \(g_{\text{m1}}g_{\text{m2}}/(\omega^{n}F)\) and \(\omega_{\text{CP2}}\), the upper cut-off frequency of the correct gyrator behavior is  
\[Y_{\text{IN}}{}^{\prime}(s)\approx\frac{g_{\text{m1}}g_{\text{m2}}}{s^{\alpha} F+G_{\text{P1}}+sC_{\text{P1}}}+G_{\text{P2}}+sC_{\text{P2}}. \tag{7}\]2  
In this case, the FO inductance appearing at the input port of the gyrator and the parasitic capacitance \(C_{\text{P2}}\) create a parallel resonant circuit, which can be a potential source of instability. Analyzing the relation (7), the real part of \(Y_{\text{IN}}\)' cannot be lower than \(G_{\text{P2}}\), which means that the parasitic conductance \(G_{\text{P2}}\) ensures damping of the resonance. The value of \(\alpha\) also has an impact on the gyrator stability. The more it is less than one, the more the resonance is damped. The standard inductor in Figure 4 resulting from immittance inversion of the parasitic capacitance \(C_{\text{P1}}\) affects the input admittance of the gyrator negligibly. Its effect would be in decreasing the input admittance above a cut-off frequency that is higher than \(\omega_{\text{CP2}}\); however, in this frequency band, the input admittance grows, due to the admittance of \(C_{\text{P2}}\), which outweighs. The following recommendations can be drawn from the previous analysis to improve the performance of the gyrator mainly as far as operation bandwidth, and range of obtainable admittance magnitude are concerned:  
* If there is a possibility to modify the fractance \(F\), increase the values of both (\(g_{\text{m1}}g_{\text{m2}}\)) and \(F\) such that the ratio (\(g_{\text{m1}}g_{\text{m2}}\))/\(F\) remains constant. This decreases the lower cut-off frequency \(\omega_{\text{GPI}}\), and thus, expands the band of correct operation of the gyrator to lower frequencies without affecting the FO inductance. Moreover, the maximum obtainable magnitude of input admittance \(g_{\text{m1}}g_{\text{m2}}/G_{\text{P1}}\) increases. * From the OTA parasitic properties, the main effort should be to decrease \(G_{\text{P1}}\), which also brings a decrease of \(\omega_{\text{GPI}}\) and an increase of maximum input admittance magnitude. Adherence to the recommendation on decreasing \(G_{\text{P1}}\) will also efficiently increase \(\omega_{\text{GP2}}\), as due to the gyrator topology, we may assume \(G_{\text{P1}}\approx G_{\text{P2}}\). This can be achieved by choosing an OTA structure that offers low parasitic conductance at input and output terminals. An alternative way to reduce a parasitic conductance in the gyrator is connecting a negative conductance in parallel as described in Section 2.2.2. To further optimize the gyrator operation at high frequencies in terms of increasing \(\omega_{\text{CP2}}\), it is recommended to choose an OTA structure with low \(C_{\text{P2}}\). It can also be observed that for lower values of \(\alpha\), the bandwidth of the correct gyrator operation is increased, which is related to the lower slope of the admittance magnitude characteristic. Thus, a gyrator implementing a FO inductor provides higher bandwidth than a gyrator implementing a classic inductor.