# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 3 Approximation of CPE in Frequency Domain and Numerical Results
The values to build this CPE are given in Table 14 to Table 15, and pseudo-capacitance is 97.3 \(\mu\)F/s\({}^{2/3}\) (type I) and 4.37 \(\mu\)F/s\({}^{2/3}\) (type II). \begin{table}
\begin{tabular}{c c c c c c c} \hline
**R/Cs** & **R\({}_{1}\)/C\({}_{1}\)** & **R\({}_{2}\)/C\  
Audio CPE for Mathematical Order \(\alpha=2/5\), Prescribed Phase Shift \(\varphi=\pm 36^{\circ}\)  
This CPE begins with frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi\cdot 10^{4}\cdot 2\cdot 10^{-6})=8\) Hz, adopts an impedance constant of \(10^{4}\), and can be implemented using the values given in Tables 16 and 17. Pseudo-capacitance is about 88 \(\upmu\)F/s\({}^{3/5}\) (type I) and 9.4 \(\upmu\)F/s\({}^{3/5}\) (type II). Audio CPE for Mathematical Order \(\alpha=2/5\), Prescribed Phase Shift \(\varphi=\pm 36^{\circ}\)  
This CPE begins with frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi\cdot 10^{4}\cdot 2\cdot 10^{-6})=8\) Hz, adopts an impedance constant of \(10^{4}\), and can be implemented using the values given in Tables 16 and 17. Pseudo-capacitance is about 88 \(\upmu\)F/s\({}^{3/5}\) (type I) and 9.4 \(\upmu\)F/s\({}^{3/5}\) (type II). Audio CPE for Mathematical Order \(\alpha=\)1/2, Prescribed Phase Shift \(\varphi=\pm 45^{\circ}\)  
This CPE starts with frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi 10^{4}1.5\cdot 10^{-6})=10.6\) Hz and the impedance constant is adopted as \(10^{4}\). Numerical values for the design of this CPE are specified in Tables 20 and 21. Pseudo-capacitance is 85.25 \(\mu\)F/s\({}^{1/2}\) (type I) and 6 \(\mu\)F/s\({}^{1/2}\) (type II). Audio CPE for Mathematical Order \(\alpha=\)5/9, Prescribed Phase Shift \(\varphi=\pm 50^{\circ}\)  
This CPE uses start-up frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi 10^{5}10^{-7})=15.9\) Hz, an impedance constant of \(10^{4}\), and can be realized by using the values provided in Tables 22 and 23. Pseudo-capacitance is about 80.9  
Audio CPE for Mathematical Order \(\alpha=\) 7/10, Prescribed Phase Shift \(\varphi=\pm 67.5^{\circ}\)  
The starting frequency for this CPE is \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi\cdot 10^{5}10^{-7})=15.9\) Hz and the impedance constant is \(10^{4}\). Values calculated for CPE are given in Tables 28 and 29, and pseudo-capacitance is about 92.57 \(\mu\)F/s\({}^{3/10}\) (type I) and 254 nF/s\({}^{3/10}\) (type II). Audio CPE for Mathematical Order \(\alpha=\) 7/10, Prescribed Phase Shift \(\varphi=\pm 63^{\circ}\)  
This CPE starts with frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi\cdot 10^{5}10^{-7})=15.9\) Hz and the impedance constant is \(10^{5}\). Values required for design are in Tables 30 and 31. The estimated value of pseudo-capacitance is 10.38 \(\mu\)F/s\({}^{1/4}\) (type I) and 109 nF/s\({}^{1/4}\) (type II). \begin{table}
\begin{tabular}{c c c c c c c} \hline
**R/Cp** & **R\({}_{1}\)/C\({}_{1}\)** & **R\({}_{2}\)/C\({}_{2}\)** & **R\({}_{3}\)/C\({}_{3}\)** & **R\({}_{4}\)/C\({}_{4}\)** & **R\({}_{5}\)/C\({}_{5}\)** & **R\({}_{6}\)/C\({}_{6}\)** & **R\({}_{7}\)/C\({}_{7}\)** \\ \hline
**2.6 k\(\Omega\)** & **1.5 k\(\Omega\)** & **533 \(\Omega\)** & **193 \(\Omega\)** & **70 \(\Omega\)** & **25 \(\Omega\)** & **9.2 \(\Omega\)** & **3.3 \(\Omega\)** \\
2.2 k\(\Omega\)\(\angle\) & 1.5 k\(\Omega\) & 1.5 k\(\Omega\) \(\parallel\) & 180 \(\Omega\)\(\angle\) & 68 \(\Omega\)\(\angle\) & 15 \(\Omega\)\(\angle\) & 8.2 \(\Omega\)\(\angle\) & 3.3 \(\Omega\) \\
390 \(\Omega\) & & 820 \(\Omega\) & 12 \(\Omega\) & 2.2 \(\Omega\) & 10 \(\Omega\) & 1 \(\Omega\) & \\ \hline
**710 nF** & **10 μ**F & **6.1 μ**F** & **3.7 μ**F** & **2.2 μF** & **1.3 μF** & **806 nF** & **470 nF** \\
560 nF \(\parallel\) & 10 μF & 3.9 μF \(\parallel\) & 3.3 μF \(\parallel\) & 2.2 μF \(\parallel\) & 1.2 μF \(\parallel\) & 470 nF \(\parallel\) & 470 nF \\
150 nF & & 2.2 μF & 390 nF & & 100 nF & 330 nF & \\ \hline \end{tabular}
\end{table} 
Table 26: CPE with math order \(\alpha=\) 2/3; fully passive RC series-parallel topology of fractal capacitor.