## 3 Simulation Results and Discussion
### Inductance Simulator for Fractional-Order Respiratory Model  
Fractional-order electrical circuit models are useful in modeling the mechanical impedance of the respiratory system [11; 12]. Mechanical impedance of the human respiratory system can be represented by an equivalent model given in Figure 14.  
Figure 12: Simulated magnitude (**a**) and phase (**b**) characteristics of gyrator input admittance for \(\alpha=0.75\). Figure 13: Simulated magnitude (**a**) and phase (**b**) characteristics of gyrator input admittance for \(\alpha=1\). \begin{table}
\begin{tabular}{c c c c c} \hline
**Set** & \(\alpha\) = 0.25 & \(\alpha\) = 0.5 & \(\alpha\) = 0.75 & \(\alpha\) = 1 \\ \hline
1 & 62 Hz–395 kHz & 950–1 MHz & 1.44 kHz–226 kHz & 1.45 kHz–72 kHz \\
2 & 0.4 mHz–2.43 MHz & 2.4 Hz–24.6 kHz & 26 Hz–173 kHz & 71 Hz–148 kHz \\ \hline \end{tabular}
\end{table} 
Table 3: Comparison of frequency ranges of FO inductance simulators for maximum phase error \(\pm 2\) degrees. The impedance of this system is expressed as  
\[Z(s)=s^{\alpha}L+R+\frac{1}{s^{\beta}C}. \tag{15}\]  
The experimentally extracted values of (15) for human respiratory system are \(L=0.0193\) H-sec\({}^{-0.5313}\), \(\alpha=0.4687\), \(R=0.0147\)\(\Omega\), \(C=0.9921\) F-sec\({}^{-0.2729}\), \(\beta=0.7271\) and the frequency range of the model validity is 4 Hz to 48 Hz [12]. The FO inductor with impedance \(s^{\alpha}L\) included in this model will be implemented by the discussed OTA-based gyrator in this section as an example of its practical utilization. The grounded version of the model will be demonstrated, as the floating version requires OTA with differential output (see Figure 8), which is not the case of the used integrated circuit LT1228. The value of \(F_{\mathrm{IN}}\) in accordance with (5) and (15) is equal to \(1/L=51.81\) H\({}^{-1}\)-sec\({}^{0.5313}\). This value represents a very high admittance magnitude of 11.43 S at the corner frequency of 4 Hz, and the implementation of the inductor is therefore challenging. Rules A, in Section 2.3, will be used to design the gyrator element parameters. The transconductances are chosen \(g_{\mathrm{m1}}=g_{\mathrm{m2}}=20\) mS again as the maximum possible values for LT1228. To obtain the required \(\mid\)\(\gamma_{\mathrm{IN}}\)\(\mid\) = 11.43 S at 4 Hz, the magnitude of loading admittance is \(\mid\)\(Y\)\(\mid\) = 34.987 \(\mu\)S at 4 Hz, its order is \(\alpha=0.4687\), and \(F=7.72\)\(\mu\)F-sec\({}^{-0.5313}\). Thanks to the narrow frequency band (4 Hz to 48 Hz) compared to the simulations in the previous subsection and to implement it by available components, the FO capacitor loading the gyrator was emulated by the Valsa structure [29] containing standard resistors and capacitors, as shown in Figure 15. Another reason to employ this RC emulator is that the GFREQ part cannot be used in transient analysis, which will demonstrate dynamic optimization of the gyrator at the end of this section. The element values of the Valsa RC circuit computed by the method described in Reference [29] for the mentioned values \(\alpha\), \(F\), maximum phase error \(\pm 0.6\) degrees, and the frequency range 0.8 Hz to 240 Hz (to have sufficient margin) are listed in Table 4.  
The admittance magnitude and phase characteristics of the emulated FO capacitor (solid black lines) and the ideal values (dashed blue lines) are presented in Figure 16. It can be observed that the circuit operates according to the above-mentioned required parameters. Figure 14: FO impedance model of the human respiratory system [11]. Figure 15: Valsa RC circuit to emulate FO capacitor. The gyrator lower cut-off frequency computed by (8) is \(f_{\rm GPI}=37.6\) Hz, which is not sufficiently low, as the FO inductor should operate from 4 Hz. This limitation is caused by the parasitic conductance \(G_{\rm PI}=100\) uS, which will be decreased by the circuit with negative admittance. Thus, now the compensation circuit in the dashed box in Figure 9 is replaced and connected in parallel with the FO capacitor loading the gyrator.