## 3 Fractances with Tunable Factor's Order
### Tuning of the Fractor's Order  
In order to calculate a good approximation of the fractional impedance, every time the fractor's order \(\alpha\) changes, the whole representation should be re-calculated. However, representing the resistances in a logarithmic scale, all the values are on a line that rotates clockwise to the variation of \(\alpha\) around the \((n+3)/2\) value. For the values of \(n\), \(\omega_{0}\), \(Z_{0}\), and \(\delta\) the variation is show in Figure 5. So, once a set of resistances is found for a given value of \(\alpha_{0}\), is possible to suppose to find a rule for the variation of \(R_{i}\) as function of \(\alpha\).  
The family of straight lines can be expressed as:  
\[\log_{10}R_{i}-\log_{10}R_{(n+3)/2}=m(\alpha)(i-(n+3)/2) \tag{14}\]  
Once the equivalent network is calculated for a reference fractor's order \(\alpha_{0}\), it is possible to easily calculate \(m(\alpha_{0})\). The, it is found that applying the empirical law:  
\[m(\alpha)=m(\alpha_{0})\ e^{\alpha/\alpha_{0}-1} \tag{15}\]  
Additionally, practically it found that the resistances changes with \(\alpha\) much more than the capacitances. So, it is worth to consider a network where the resistances vary with \(\alpha\) according to (14) and (15) and capacitors are fixed. Such a network is a very good approximation of the fractional impedance for the value of \(\alpha_{0}\) adopted to compute the capacitances. However, it will be able to approximate fractional impedances for different values of \(\alpha\), in a bandwidth always as smaller as the fractor's order differs from the starting value of \(\alpha_{0}\).  
Such a kind of circuit can be easily implemented by means of the same configuration in Figure 2 and adopting traditional capacitors and variable resistors. For instance, by using digital potentiometers connected to a controller, it is possible to program the synchronous variation of all the potentiometers according to the programmed rules, in order to obtain the wished set of resistances and so in the end the wished fractional impedance.  
In order to provide a numerical example, we can consider once again the previously values of \(n\), \(\omega_{0}\), \(Z_{0}\), and \(\delta\). If we consider as reference fractor's order \(\alpha_{0}=0.5\), then we obtain \(m(\alpha_{0})=-0.402\). This allows to compute the coefficients \(m(\alpha)\) and then new values of \(R_{i}\) for different values of \(\alpha\).  
In Figures 6 and 7, we show the amplitude and the phase of the rational approximation \(G(\omega)\) for different values of \(\alpha\) computed with this procedure. The reference amplitudes and phases of \(Z(\omega)\) are not shown to make the pictures more readable. Anyway, the amplitude  
Figure 5: Variation of the resistances \(R_{i}\) as function of \(\alpha\) (variable between 0.3 and 0.7).  
comparison would have been neglectable, as for the special case of Figure 3, while the phase of \(G(\omega)\) has to be compared with a constant value of course.  
Again, the amplitude plot represents the fractional impedance in a wider bandwidth than the phase plot.  
Moreover, as expected, the quality of the approximation, especially in the phase plot, decreases for higher values of \(\alpha\).