# Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems
## 9 Illustration of the Use of Basis Functions for Transient Analysis
### Commensurate Fractional-Order Sallen-Key Filter
This observation is consistent with the well-known fact that lowering the order \(2\alpha\) below 2 (i.e., \(\alpha=0.8<1\)) acts as additional damping. The filter in Figure 2 was also modeled in SPICE. Two different models were used:  
Model 1: Modeling transfer function (61) using the Laplace voltage-controlled voltage source. Model 2: Model of the circuit in \[K_{F}(s)=\frac{\omega_{0}^{2}}{s^{2\alpha}+s^{\alpha}\frac{\omega_{0}}{Q}+\omega_ {0}^{2}}, \tag{59}\]7  
In Model 2, for a conclusive comparison with the MATLAB results, the OpAmp follower was modeled with a simple controlled source (gain = 1). The factors were modeled with Laplace current-controlled voltage sources. Regarding the Micro-Cap outputs, Models 1 and 2 show a perfect match with the MATLAB results. However, the experiments revealed errors in the SPICE transient analysis if the fractional-order filter was set to an unstable mode due to an inappropriately chosen parameter \(\alpha\). In contrast, the method based on basis functions gives correct results. The stability analysis of transfer function (59) or (61), respectively, leads to the observation that the fractional-order Sallen-Key filter in Figure 2 is stable for  
\[K_{F}(s)=\frac{\omega_{0}^{2}}{s^{2\alpha}+s^{\alpha}\frac{\omega_{0}}{Q}+\omega_ {0}^{2}}, \tag{59}\]8  
For \(Q=5\), the threshold value of the parameter \(\alpha\) is 1.0638. The filter was re-designed for \(\alpha=1.15\), i.e., for the unstable mode. For \(Q=5\) and \(f_{0F}=1\) kHz, it now comes out as  
\[K_{F}(s)=\frac{\omega_{0}^{2}}{s^{2\alpha}+s^{\alpha}\frac{\omega_{0}}{Q}+\omega_ {0}^{2}}, \tag{59}\]9  
Figure 4 shows the results of the simulation of the step response in Micro-Cap. The simulation was performed with default simulation parameters and with a step ceiling of 10 us, which is one thousandth of the simulation time. For comparison, data from MATLAB were imported into the simulation program environment, showing the results obtained with the basis functions method. Figure 4 shows that Micro-Cap calculates completely different responses for Models 1 and 2, which should be interchangeable. The response for Model 2 is close to the response computed from the basis functions in MATLAB. The FFT check cannot be used to verify  
\[\omega_{0}=\frac{1}{\sqrt{R_{1}R_{2}C_{1}C_{2}}},Q=\frac{\sqrt{R_{1}R_{2}}}{R_ {1}+R_{2}}\sqrt{\frac{C_{1}}{C_{2}}}. \tag{60}\]0 Simulated step responses \(h_{F}\) of the commensurate Sallen–Key filter with transfer function (59) for \(\alpha=1.15\) and \(Q=5\) (unstable mode); S1 (SPICE-Micro-Cap, Method 1), S2 (SPICE-Micro-Cap, Method 2), M (MATLAB, basis functions). the results because the system is unstable. However, by lowering the step ceiling 100-fold to 0.1 us, the simulation program can be forced to perform a more accurate, but time-consuming, transient analysis. Then, the step response obtained using Method 2 is overlaid with the response computed using MATLAB. The above pitfalls of the transient analysis of FOS in a SPICE simulation environment certainly deserve attention. The unpleasant fact is that different SPICE-family programs may deal with the same simulation task differently depending on the properties of their internal algorithms, and that the particular simulator chosen may evaluate the response differently depending on how the model of a fractional-order circuit is constructed. The cause should be sought in the specifics of the method of a convolutional transient analysis that SPICE uses to solve circuits with Laplace sources. Anyway, the method utilizing basis functions does not deal with such problems in principle.