## 4 Circuit Simulations  
To validate the proposed fractional-order filters with approximated elliptic characteristics, two cases were simulated and experimentally verified against the theoretical expectations. For these validations, two circuit topologies were selected and are given in Figure 11 where the component \(\mathrm{C_{1}}\) is a fractional-order capacitor with impedance \(Z_{\mathrm{C_{1}}}=1/s^{\mathrm{*}}\mathrm{C_{1}}\) and \(\mathrm{C_{2}}\) is a traditional integer-order capacitor. The topologies in Figure 11a,b realize Equation (7) when the coefficient \(a_{3}\) is positive and negative, respectively. It is necessary to present two topologies to realize the complete range of responses with the identified coefficients in this work, which take both positive and negative values depending on the value of \(\alpha\). Figure 10: Simulated magnitude responses of \((1+0.75)\) lowpass fractional order transfer function using coefficients from different asymmetrical frequency fitting ranges with: (**a**) lower limits of \(\omega=10^{-5}\) rad/s; and (**b**) upper limits of \(\omega=10^{5}\). The topology in Figure 11a realizes the transfer function given by:  
\[T_{1}^{1+\alpha}(s)=-\frac{R_{6}}{R_{1}}\frac{s^{1+\alpha}\frac{C_{1}C_{2}R_{1} R_{2}R_{4}}{R_{3}}+1}{s^{1+\alpha}\frac{C_{1}C_{2}R_{2}R_{4}R_{6}}{R_{5}}-s^{ \alpha}\frac{C_{1}R_{2}R_{6}}{R_{3}}+1}=-a_{4}\frac{\frac{s^{1+\alpha}}{\omega_ {0}^{1+\alpha}}a_{1}+1}{\omega_{0}^{1+\alpha}a_{2}+\frac{s^{\alpha}}{\omega_{0} ^{\alpha}}a_{3}+1} \tag{11}\]  
and the topology in Figure 11b realizes the transfer function given by:  
\[T_{2}^{1+\alpha}(s)=-\frac{R_{6}}{R_{1}}\frac{s^{1+\alpha}\frac{C_{1}C_{2}R_{4} ^{2}}{R_{5}}+1}{s^{1+\alpha}\frac{C_{1}C_{2}R_{4}^{2}R_{6}}{R_{5}}-s^{\alpha} \frac{C_{1}R_{2}R_{6}}{R_{3}}+1}=-a_{4}\frac{\frac{s^{1+\alpha}}{\omega_{0}^{ 1+\alpha}}a_{1}+1}{\omega_{0}^{1+\alpha}a_{2}+\frac{s^{\alpha}}{\omega_{0}^{ \alpha}}a_{3}+1} \tag{12}\]  
Note that in both cases the term \(\omega_{0}\) is the frequency scaling factor to shift the response from the normalized frequency of 1 rad/s. The topology in Figure 11a with transfer function given by Equation (11) was used to realize a \((1+\alpha)=1.8\) order filter. For this case, the utilized coefficients were \(a_{1}=0.01626\), \(a_{2}=1.6844\), \(a_{3}=0.3317\), and \(a_{4}=0.5622\), selected from coefficients returned using the \(10^{-5}\) rad/s to \(10^{5}\) fitting bandwidth given in Figure 2. Using these coefficients, the necessary resistances and capacitances for the topology were calculated using the system of equations built by equating the coefficients in Equation (11) (i.e., \(C_{1}C_{2}R_{1}R_{2}R_{4}/R_{3}=a_{1}/\omega_{0}^{1+\alpha}\); \(C_{1}C_{2}R_{2}R_{4}R_{6}/R_{5}=a_{2}/\omega_{0}^{1+\alpha}\); \(C_{1}R_{2}R_{6}/R_{5}=a_{3}/\omega_{0}^{\alpha}\); \(R_{6}/R_{1}=a_{4}\)). The specific resistors and capacitors are given in Table 1 when a frequency of \(\omega_{0}=(2\pi)10\) krad/s is used. At this time, fractional-order capacitors are not available for either simulations or physical  
Figure 11: Circuit topologies to realize fractional-order low-pass notch filter response given by Equation (7) when \(a_{3}\) is: (**a**) positive; and (**b**) negative. Note that in both topologies C\({}_{1}\) is a fractional-order capacitor with impedance \(Z_{C_{1}}=1/s^{\alpha}C_{1}\). realizations and require the use of approximations; however, it is important to note the recent progress in realizing devices with fractional-order impedances [26; 27; 28]. For the simulations and experimental circuits in this work, the fractional-capacitors were approximated using the 5 th order Foster-I topology given in Figure 12. The circuit components required to realize the \(\alpha=0.8\) device were calculated using the process detailed in [29], with these calculated values given in Table 2. The experimental results were collected using an Omicron Bode 100 network analyzer from a circuit realized using discrete components on a breadboard.