## 2 Methods
### Practical OTA-Based Gyrator
#### 2.2.1 Influence of OTA Terminal Impedances  
Considering the aforementioned OTA parasitic properties, the gyrator schematic from Figure 2 can be modified to the form in Figure 3.  
The parasitic properties of OTA\({}_{1}\) output and OTA\({}_{2}\) input respective to the ground are together modeled with G\({}_{\text{P1}}\) and C\({}_{\text{P1}}\) elements; whereas, the parasities of OTA\({}_{1}\) input and OTA\({}_{2}\) output are modeled with G\({}_{\text{P2}}\) and C\({}_{\text{P2}}\). The relation for input admittance of the gyrator loaded with FO capacitor taking into account the OTA parasitic properties is  
\[Y_{\text{IN}}{}^{\prime}(s)\approx\frac{g_{\text{m1}}g_{\text{m2}}}{s^{\alpha} F+G_{\text{P1}}+sC_{\text{P1}}}+G_{\text{P2}}+sC_{\text{P2}}. \tag{7}\]  
This relation needs to be analyzed to determine the effect of the parasitic properties. To illustrate the meaning of (7), the passive circuit with the identical input admittance \(Y_{\text{IN}}\)'(s) can be depicted in Figure 4.  
Figure 3: Gyrator with OTA parasitic properties. Analyzing (7) and Figure 4, the gyrator transforms the FO capacitor to FO inductor with admittance \(g_{\rm m1}g_{\rm m2}/(s^{\rm F}F)\), which is in agreement with the relation (5) valid for the ideal case. The parasitic properties \(\rm C_{P1}\) and \(\rm G_{P1}\) are also transformed by the gyrator and appear in the network in Figure 4 as an integer-order inductor with admittance \(g_{\rm m1}g_{\rm m2}/(s\rm C_{P1})\) and resistor with conductance \(g_{\rm m1}g_{\rm m2}/G_{\rm P1}\), respectively. The admittance of the parasitic elements \(\rm G_{P2}\) and \(\rm C_{P2}\) is according to (7) additional part of the input admittance \(Y_{\rm IN}\) (s) and therefore, these elements appear at the overall input of the structure in Figure 4.  
Considering the above-mentioned typical values of the OTA parasitic properties, the format of the admittance magnitude and phase frequency characteristics resulting from (7) can be displayed as the asymptotic Bode plots in Figure 5.  
The lower cut-off frequency of the correct gyrator operation is approximately equal to  
\[\omega_{\rm GPI}\approx\left(\frac{G_{\rm PI}}{F}\right)^{\frac{1}{\alpha}}. \tag{8}\]  
At this frequency, the admittance magnitude of the gyrator load \(\mid\omega^{\alpha}F\mid\) equals to \(G_{\rm PI}\). Below this frequency, the gyrator exhibits parasitic behavior with limited input admittance magnitude \(g_{\rm m1}g_{\rm m2}/G_{\rm PI}\) and zero phase. This value represents the highest achievable admittance magnitude at the gyrator input. The value of \(G_{\rm P2}\) has no effect below the frequency \(\omega_{\rm GP1}\), as we assume \(G_{\rm P2}\ll g_{\rm m1}g_{\rm m2}/G_{\rm P1}\). Above the frequency \(\omega_{\rm GP1}\), the operation of the gyrator can be considered correct, i.e., providing the admittance of FO inductor given by (5), which is marked by the blue color in Figure 5. The magnitude of the gyrator input admittance decreases with the roll-off rate \(-20\alpha\) dB/dec, and the constant phase is \(-\alpha\pi/2\). Note that the admittance characteristics of  
Figure 4: Substitute circuit providing the input admittance (7); the relations in the schematic express admittances of the particular elements. Figure 5: Bode plots of (**a**) magnitude and (**b**) phase admittance characteristics according to (7). Blue parts denote ideal function without the influence of parasitics. an ideal FO inductor can be obtained by linearly extending the blue lines in Figure 5 over the entire frequency range. Based on the relationships between the values of the elements in Figure 4, the upper cut-off frequency of the correct behavior of the gyrator may be equal to  
\[Y_{\text{IN}}{}^{\prime}(s)\approx\frac{g_{\text{m1}}g_{\text{m2}}}{s^{\alpha} F+G_{\text{P1}}+sC_{\text{P1}}}+G_{\text{P2}}+sC_{\text{P2}}. \tag{7}\]0  
where the input admittance magnitude \(\ket{Y_{\text{IN}}}^{\prime}\) decreases to the parasitic conductance \(G_{\text{P2}}\).