# 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 c} \hline
**R/Cs** & **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
**200 \(\Omega\)** & **150 k\(\Omega\)** & **54 k\(\Omega\)** & **20 k\(\Omega\)** & **7.1 k\(\Omega\)** & **2.6 k\(\Omega\)** & **938 \(\Omega\)** & **310 \(\Omega\)** \\
100 \(\Omega\) \(\angle\) & 150 k\(\Omega\) & 47 k\(\Omega\) \(\angle\) & 10 k\(\Omega\)\(\angle\) & 5.6 k\(\Omega\)\(\angle\) & 2.2 k\(\Omega\)\(\angle\) & 820 \(\Omega\)\(\angle\) & 270 \(\Omega\)\(\angle\) \\
100 \(\Omega\) & & 6.8 k\(\Omega\) & 10 k\(\Omega\) & 1.5 k\(\Omega\) & 390 \(\Omega\) & 120 \(\Omega\) & 39 \(\Omega\) \\ \hline
**69 nF** & **100 nF** & **60 nF** & **36.3 nF** & **22 nF** & **13 nF** & **7.9 nF** & **4.7 nF** \\
68 nF \(\parallel\) & 100 nF & 120 nF \(\angle\) & 33 nF \(\parallel\) & 22 nF & 12 nF \(\parallel\) & 4.7 nF \(\parallel\) & 4.7 nF \\
1 nF & & 120 nF & 3.3 nF & & 1 nF & 3.3 nF & \\ \hline \end{tabular}
\end{table} 
Table 27: CPE with math order \(\alpha=\) 2/3; fully passive RC parallel-series topology of fractal capacitor. \begin{table}
\begin{tabular}{c c c c c c c c} \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
**1.9 k\(\Omega\)** & **1 k\(\Omega\)** & **342 \(\Omega\)** & **118 \(\Omega\)** & **41 \(\Omega\)** & **14 \(\Omega\)** & **5 \(\Omega\)** & **1.8 \(\Omega\)** \\
1.8 k\(\Omega\)\(\angle\) & 1 k\(\Omega\) & 330 \(\Omega\)\(\angle\) & 100 \(\Omega\)\(\angle\) & 82 \(\Omega\)\(\parallel\) & 10 \(\Omega\)\(\angle\) & 10 \(\Omega\)\(\angle\) & 10 \(\Omega\)\(\parallel\) & 1.8 \(\Omega\)\(\Omega\) \\
100 \(\Omega\) & & 12 \(\Omega\) & 18 \(\Omega\) & 82 \(\Omega\) & 3.9 \(\Omega\)\(\Omega\) & 10 \(\Omega\) & \\ \hline
**3.15 μF** & **9 μF** & **6.4 μF** & **4 μF** & **2.6 μF** & **1.6 μF** & **1** μF** & **650 nF** \\
1 \(\mu\)F \(\parallel\) & 18 μ\(\angle\) & 18 μ\(\angle\) & 18 μ\(\angle\) & 3.9 μF \(\parallel\) & 2.2 μF \(\parallel\) & 1.5 μF \(\parallel\) & 1 μF & 470 nF \(\parallel\) \\
100 nF & 18 μF & 10 uF & 100 nF & 390 nF & 100 nF & & 150 nF \\ \hline \end{tabular}
\end{table} 
Table 28: CPE with math order \(\alpha=\) 7/10; fully passive RC series-parallel topology of fractal capacitor. \begin{table}
\begin{tabular}{c c c c c c c c} \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
**82 \(\Omega\)** & **92 k\(\Omega\)** & **34.5 k\(\Omega\)** & **12 k\(\Omega\)** & **4.1 k\(\Omega\)** & **1.4 k\(\Omega\)** & **485 \(\Omega\)** & **167 \(\Omega\)** \\
82 \(\Omega\)\(\Omega\) & 82 k\(\Omega\)\(\angle\) & 33 k\(\Omega\)\(\angle\) & 12 k\(\Omega\) & 3.9 k\(\Omega\)\(\angle\) & 1.2 k\(\Omega\)\(\angle\) & 470 \(\Omega\)\(\angle\) & 150 \(\Omega\)\(\angle\) \\  & 10 k\(\Omega\) & 1.5 k\(\Omega\)\(\Omega\) & & 220 \(\Omega\) & 220 \(\Omega\) & 15 \(\Omega\) & 18 \(\Omega\) \\ \hline
**120 Audio CPE for Mathematical Order \(\alpha\) = 7/9, Prescribed Phase Shift \(\varphi\) = \(\pm\)70\({}^{\circ}\)  
The fundamental frequency of this CPE is \(f_{0}\) = 1/(2\(\pi\)\(R_{0}\)\(C_{0}\)) = 1/(2\(\pi\)\(\cdot\)2\(\cdot\)10\({}^{5}\)\(\cdot\)10\({}^{-7}\)) = 8 Hz and the impedance constant is 10\({}^{5}\). Values important for realization can be found inside Tables 32 and 33, and pseudo-capacitance is 9.77 \(\mu\)F/s\({}^{2/9}\) (type I) and 73.8 nF/s\({}^{2/9}\) (type II). Audio CPE for Mathematical Order \(\alpha\) = 4/5, Prescribed Phase Shift \(\varphi\) = \(\pm\)72\({}^{\circ}\)  
This CPE has a starting frequency \(f_{0}\) = 1/(2\(\pi\)\(R_{0}\)\(C_{0}\)) = 1/(2\(\pi\)10\({}^{4}\)10\({}^{-6}\)) = 15.9 Hz and the impedance constant is 10\({}^{6}\).