## 3 Simulation Results and Discussion
### Inductance Simulator for Fractional-Order Respiratory Model
The transconductance \(g_{\rm mC}\) is 97 uS, and it leads to \(G_{\rm PI}\approx 3.5\) uS. The expected cut-off frequency after compensation is \(f_{\rm GPI}\approx 0.03\) Hz and maximum obtainable input admittance \(g_{\rm mI}g_{\rm m2}/G_{\rm PI}\approx 115\) S, which is ten times higher than the required admittance magnitude of the implemented FO inductor at 4 Hz. Note that these values are valid for the gyrator loaded with an ideal FO capacitor. It can be expected that the results will be degraded after connecting the emulator of FO capacitor, which has limited accuracy of its admittance. The simulated magnitude and phase frequency characteristics of the gyrator input admittance are shown in Figure 17. The solid black lines represent simulated results, and the blue lines indicate ideal values with the solid part denoting the frequency band of interest, i.e., 4 Hz to 48 Hz. In this frequency band, the maximum absolute magnitude error is 1.5 S, and phase error is 3.1 degrees when comparing the simulated results with the ideal ones. It is apparent that the error increases at the lower bound of the frequency band. The magnitude error can be easily corrected by fine increasing the product \(g_{\rm mI}g_{\rm m2}\), which can be set electronically. The phase deviation can be reduced by fine-tuning the transconductance \(g_{\rm mC}\); however, the demands on the accuracy of setting \(g_{\rm mC}\) increase, and it is important to prevent overcompensating \(G_{\rm PI}\) to a negative value. The dynamic optimization was also performed based on the conditions introduced in Section 2.2.3. When following these conditions, only allowed voltages and currents  
Figure 16: Admittance magnitude (**a**) and phase (**b**) characteristics of the approximated FO capacitor from Figure 15.  
will appear in the gyrator structure, and the amplifiers will operate in linear range without overloading. The maximum input voltage amplitude of both OTAs for their linear operation is considered \(V_{\text{OTA,MAX}}=50\) mV. The maximum input voltage of the gyrator can be determined as the maximum value of \(V_{\text{IN}}\) that meets both conditions (13) and (14) simultaneously. The maximum input current can then be computed using \(I_{\text{IN}}=V_{\text{IN}}\,|\,Y_{\text{IN}}\,|\). The following Table 5 shows the maximum allowed amplitudes of voltages and currents at the input (\(V_{\text{IN}}\), \(I_{\text{IN}}\)) of the gyrator, ensuring linear operation of both OTAs for three frequency points, in the beginning, inside (at 25 Hz), and at the end of the operating band. The computed amplitudes of voltages and currents at the gyrator load (\(V_{\text{LOAD}}\), \(I_{\text{LOAD}}\)) are also stated. Applying the rules for dynamic optimization, the transient simulation with the maximum allowed input current amplitude \(I_{\text{IN}}=1\) mA and frequency 25 Hz was carried out, and the resulting current and voltage waveforms are presented in \[Z(s)=s^{\alpha}L+R+\frac{1}{s^{\beta}C}. \tag{15}\]0 The voltages are relative to ground, and currents are considered positive when flowing into the gyrator (\(I_{\text{IN}}\)) and into the loading FO capacitor (\(I_{\text{LOAD}}\)). \[Z(s)=s^{\alpha}L+R+\frac{1}{s^{\beta}C}. \tag{15}\]1 Simulated magnitude (**a**) and phase (**b**) characteristics of gyrator input admittance implementing FO inductor for the respiratory model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(f\) (Hz) & \(V_{\text{IN}}\) (\(\mu\)V) & \(I_{\text{IN}}\) (mA) & \(V_{\text{LOAD}}\) (mV) & \(I_{\text{LOAD}}\) (\(\mu\)A) \\ \hline
4 & 87.5 & 1 & 50 & 1.75 \\
25 & 206 & 1 & 50 & 4.13 \\
48 & 280 & 1 & 50 & 5.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Maximum allowed voltages and currents in the gyrator structure. The delay between input current and voltage found by the simulation is 4.53 msec; the corresponding \(\alpha\) is 0.453 which is close to the required value of 0.4687.