# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 3 Approximation of CPE in Frequency Domain and Numerical Results
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**R/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
**72 \(\Omega\)** & **220 k\(\Omega\)** & **61 k\(\Omega\)** & **20 k\(\Omega\)** & **5.8**\(\Omega\)** & **1.8**\(\Omega\)** & **537 \(\Omega\)** & **165 \(\Omega\)** \\
68 \(\Omega\)\(\angle\) & 220 k\(\Omega\) & 33 k\(\Omega\)  
Audio CPE for Mathematical Order \(\alpha=\) 8/9, Prescribed Phase Shift \(\varphi=\pm 80^{\circ}\)  
This CPE has a start frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi 10^{5}10^{-7})=15.9\) Hz and the impedance constant is \(10^{5}\). Tables 36 and 37 provide complete numerical entries for CPE design, and pseudo-capacitance is about 18 \(\upmu\)F/s\({}^{1/9}\) (type I) and 51 nF/s\({}^{1/5}\) (type II). Audio CPE for Mathematical Order \(\alpha=\) 8/9, Prescribed Phase Shift \(\varphi=\pm 80^{\circ}\)  
This CPE has a start frequency \(f_{0}=1/(2\pi R_{0}C_{0})=1/(2\pi 10^{4}10^{-6})=15.9\) Hz and uses an impedance constant of \(10^{6}\). Optimal values for the design of this CPE can be found in Tables 38 and 39, pseudo-capacitance is about 2 \(\upmu\)F/s\({}^{1/9}\) (type I) and 232 nF/s\({}^{1/5}\) (type II). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Rp/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
**15.7 k\(\Omega\)** & **5 k\(\Omega\)** & **1.4 k\(\Omega\)** & **366 \(\Omega\)** & **94.6 \(\Omega\)** & **24.4 \(\Omega\)** & **6.8 \(\Omega\)** & **1.7 \(\Omega\)** \\
15 k\(\Omega\)\(\angle\) & 10 k\(\Omega\) \(\parallel\) & 1 k\(\Omega\)\(\angle\) & 5.6 k\(\Omega\)\(\angle\) & 82 \(\Omega\)\(\angle\) & 22 \(\Omega\)\(\angle\) & 6.8 \(\Omega\) & 1.8 \(\Omega\) \\
680 \(\Omega\) & 10 k\(\Omega\) & 390 \(\Omega\) & 390 \(\Omega\) & 12 \(\Omega\) & 2.2 \(\Omega\) & & \\ \hline
**3.6**\(\upmu\)F & **1.8**\(\upmu\)F & **1.5**\(\upmu\)F & **1.3**\(\upmu\)F & **1.1**\(\upmu\)F & **927 nF** & **783 nF** & **680 nF** \\
3.3 \(\upmu\)F \(\parallel\) & 1.8 \(\upmu\)F & 1.5 \(\upmu\)F & 1.2 \(\upmu\)F \(\parallel\) & 1 \(\upmu\)F \(\parallel\) & 820 nF \(\angle\) & 680 nF \(\parallel\) & 680 nF \\
330 nF & & & 100 nF & 100 nF & 100 nF & 100 nF & 100 nF & \\ \hline \hline \end{tabular}
\end{table} 
Table 36: CPE with math order \(\alpha=\) 8/9; fully passive RC series-parallel topology of fractal capacitor. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Rp/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
**170 k\(\Omega\)** & **71 k\(\Omega\)** & **21 k\(\Omega\)** & **6.2 k\(\Omega\)** & **1.84 k\(\Omega\)** & **543 \(\Omega\)** & **161 \(\Omega\)** & **48 \(\Omega\)** \\
3.3 M\(\Omega\)\(\parallel\) & 56 k\(\Omega\)\(\angle\) & 470 k\(\Omega\)\(\parallel\) & 5.6 k\(\Omega\)\(\angle\) & 1.5 k\(\Omega\)\(\angle\) & 390 \(\Omega\)\(\angle\) & 150 \(\Omega\)\(\angle\) & 33 \(\Omega\)\(\angle\) \\
180 k\(\Omega\) & 15 k\(\Omega\) & 22 k\(\Omega\) & 560 \(\Omega\) & 330 \(\Omega\) & 150 \(\Omega\) & 10 \(\Omega\) & 15 \(\Omega\) \\ \hline
**62 nF** & **141 nF** & **104 nF** & **77 nF** & **57 nF** & **42 nF** & **31 nF** & **25 nF** \\
47 nF \(\parallel\) & 120 nF \(\parallel\) & 100 nF \(\parallel\) & 330 nF \(\angle\) & 330 nF \(\angle\) & 390 nF \(\angle\) & 470 nF \(\angle\) & 100 nF \(\angle\) \\
15 nF & 22 nF & 3.9 nF & 100 nF & 68 nF & 47 nF & 33 nF \\ \hline \hline \end{tabular}
\end{table} 
Table 34: CPE with math order \(\alpha=\) 4/5; fully passive RC series-parallel topology of fractal capacitor.