# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 2 Passive and Active Structures of CPE Approximants  
So far, from the viewpoint of practical applications in lumped analog signal processing, the passive RC ladder structure of CPE approximation appears to be the most promising concept. The advantages are evident: passivity (CPEs without a supply voltage), straightforward understanding, simple frequency and impedance rescaling, easily calculated numerical values of the internal circuit components, non-laborious practical implementation, and verification. The authors of [15; 16] provide systematic design approaches and complete calculation routines. There are four input parameters: required phase shift (the order of CPEs to be approximated multiplied by \(90^{\circ}\)), phase error (definesthe width of the canal where the phase frequency response will unfold), frequency band (where the predefined phase error will be kept), and basic immittance constant (changes the pseudo-capacitance if the FO capacitor is designed). The order of CPE approximation circuits is a compromise between the first three input parameters; this statement will be clarified in upcoming text. First, and the most popular, two-terminal realization of CPEs is visualized by means of Figure 1a and described by the following function in Laplace transform  
\[Y(s)=\frac{I(s)}{V(s)}=sC_{p}+\frac{1}{R_{p}}+\sum\
olimits_{k=1}^{7}\frac{sC _{k}}{sC_{k}R_{k}+1}=sC_{p}+\frac{1}{R_{p}}+\sum\
olimits_{k=1}^{7}\frac{1}{R _{k}}\cdot\frac{s}{s+\omega_{k}}\quad\omega_{k}=\frac{1}{C_{k}R_{k}}, \tag{1}\]  
where \(s\) is complex frequency. The sum indicated in (1) de facto represents the complex admittance function decomposed into the partial fractions. Hence, CPEs with a math order less than one will be substituted by a circuit with total order eight. Because of the topology, this RC circuit will be denoted in further text as a series-parallel structure. An alternative circuit suitable for CPE approximation is provided in Figure 1b, and can be described by the following network functions  
\[Y(s)=\frac{I(s)}{V(s)}=\frac{1}{R_{s}+\frac{1}{sC_{s}}+\sum\
olimits_{k=1}^{7} \frac{R_{k}}{sC_{k}R_{k}+1}}\quad Z(s)=\frac{V(s)}{I(s)}=R_{s}+sL_{s}+\sum \
olimits_{k=1}^{7}\frac{sL_{k}R_{k}}{sL_{k}+R_{k}}. \tag{2}\]  
Again, the sum in (2) can be interpreted as the decomposition of complex impedance functions into partial fractions. Because of the internal topology, this RC circuit will be referred in this paper as a parallel-series structure. Both network structures presented above can be used in voltage-mode as well as current-mode applications. Furthermore, both admittances can be transformed into impedances (admittance function has the same form as impedance function) by following the duality principle. Corresponding schematics are given in Figure 1c,d. This paper provides only realizations of FO capacitors. To obtain FO inductors, capacitors should be substituted for inductors (with the same normalized value), resistors interchanged with conductors (that is, resistors having inverse normalized values), and the parallel connection of components connected in series and vice versa. The behavior of the FO inductor provided in Figure 1c in the frequency domain can be approximated by the function  
\[Z(s)=\frac{V(s)}{I(s)}=sL_{s}+R_{s}+\sum\
olimits_{k=1}^{7}\frac{sL_{k}}{sL_{ k}/R_{k}+1}=sL_{s}+R_{s}+\sum\
olimits_{k=1}^{7}R_{k}\cdot\frac{s}{s+\omega_{k} }\quad\omega_{k}=\frac{R_{k}}{L_{k}}, \tag{3}\]  
and the FO inductor depicted in Figure 1d is described by the function  
\[Z(s)=\frac{1}{Y(s)}=\frac{V(s)}{I(s)}=\frac{1}{1/sL_{p}+1/R_{p}+\sum\
olimits_ {k=1}^{7}\frac{1}{sL_{k}+R_{k}}}. \tag{4}\]  
Now, let us discuss general properties of the network functions (1)-(4).