# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 3 Approximation of CPE in Frequency Domain and Numerical Results  
Upcoming subsections show detailed numerical results associated with CPE approximants for individual mathematical orders. These subsections are sorted from the lowest value of order (1/10) and ends with the highest value (9/10). All designed CPEs for demanding audio applications have a very small phase error--lower than 0.5\({}^{\circ}\), and in some cases even lower than 0.2\({}^{\circ}\). This section continues with the design of CPEs that are provided in a sequence given in ascending order of CPE. Firstly, audio CPEs are provided in Sections 3.1-3.19. Graphical visualizations of numerical verification of designed CPEs are the subject of Section 3.20. To create as universal a range of CPEs as possible, quarters, and decimal mathematical orders of approximants, as well as elements with 10\({}^{\circ}\) phase shift steps, are considered. A complete list of designed CPEs is provided in Table 1. Figure 2 shows selected results for audio CPEs. Fundamental properties of two-terminal CPEs form headers of the subsections such that desired realization (phase shift) can be easily found. In upcoming tables, the symbols \(\angle\) and \(\parallel\) denote series and parallel connection of passive circuit components, respectively. Considering the E12 value series only, parallel combinations of circuit components are preferred due to the lower sensitivities. Of course, the denser fabrication series of the passive components, such as E24 and E48, can be also used. However, different values for the passive components do not provide much better results (more accurate CPE approximation). For CPEs with as low phase errors as announced in this paper, the precision of the final CPE product in practice depends on the diligence and manual skills of the engineer. The FO nature of ideal two-terminal admittance can be expressed in form \(Y(s)=Y_{0}\cdot s^{\alpha}\), where \(Y_{0}\) is the so-called pseudo-capacitance. It is, de facto, a module of admittance\(|Y(j\omega)|\) measured at the angular frequency 1 rad/s. Such a frequency is out of range for audio CPEs. Thus, the module frequency response was linearly extrapolated (in the log-scaled axis system) to obtain this roughly estimated measure. The so-called pseudo-inductance \(Z_{0}\) is calculated using an analogical approach, and is bounded to ideal expression for the FO impedance \(Z(s)=Z_{0}\cdot s^{\alpha}\). Besides numerical values of directly calculated resistors and capacitors (normal font), a suitable series-parallel interconnection is presented (bold letters). Of course, it is not unique, but it represents the best choice as thoroughly verified by AC analysis in Orcad Pspice circuit simulator. Neither values of coefficients of numerator and denominator of Fraction (5) nor Decomposition (6) are provided; these can be easily calculated using a suitable mathematical tool such as Matlab or Mathcad. Audio CPE for Mathematical Order \(\alpha=\) 1/10, Prescribed Phase Shift \(\varphi=\pm\)9\({}^{\circ}\)  
This CPE begins with frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi\):5\(\cdot\)10\({}^{4}\):10\({}^{-6})=3.2\) Hz and the impedance constant is 10\({}^{3}\). Numerical values are given in Tables 2 and 3, and pseudo-capacitance is 2.044 mF/s\({}^{9/10}\) (type I) and 1.985 mF/s\({}^{9/10}\) (type II). \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**CPE Order**} & \multirow{2}{*}{**Phase Shift [\({}^{\circ}\)]**} & \multicolumn{2}{c}{**Phase Error**} & \multicolumn{2}{c}{**Table with Values**} \\ \cline{3-6}  & & **1a** & **1b** & **1a** & **1b** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 1: Designed CPEs for audio frequency bands summarized in a comprehensive list. Figure 2: Polar plots of the complex admittance functions for individual RC approximants of CPE, fully passive ladder realizations based on Figure 1a dedicated for audio signal processing, infrasound frequencies (**red**), active audio range (**green**), and ultrasound frequencies (**blue**).