# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 4 Applications of CPE
### Application of Audio CPEs in Subsonic Two-Port Phase Keeper  
This subsection describes design process toward an electronically tunable generator of harmonic waveforms. A two-port CPE is used to provide a constant angle difference between the two node voltages, over a frequency range dedicated to electronic tuning. The design flow can be divided into four steps. Firstly, for two-terminal passive CPEs picked from Section 3, the positions of zeroes and poles are calculated. Secondly, the approximation is transformed into the subsonic band via frequency rescaling. Next, the CPE is transformed into two-port as shown in Figure 11. Finally, the CPE is inserted into a closed loop together with only one additional two-port-- a first-order all-pass filter (APF) with amplification. This structure will work if [PERSON]'s conditions for oscillation are satisfied simultaneously. Assume that a \(20^{\circ}\) phase shift between two node voltages needs to be kept over the frequency range \(100~{}\mu\)Hz to \(1~{}\)Hz. Both passive ladder RC approximants of CPEs provided in Section 3.4 can be transformed into the desired frequency range by introducing the new frequency norm \(\gamma=10^{-5}\). This operation moves whole phase frequency responses horizontally into the desired band. To do this, values of all the capacitors need to be divided by \(\gamma\) while the resistors remain unchanged. Obviously, these significantly increased values capacitors can cause the CPEs to no longer be implementable by commercially available components. This problem can be still avoided by the application of impedance rescaling \(\psi\). This is real number that moves the module frequency response vertically without changing the phase frequency response. It can be applied by dividing the values of all capacitors by \(\psi\) while the values of all resistors are increased \(\psi\) times. For example, the application of \(\psi=100\) and \(\gamma=10^{-5}\) on the passive components given in Table 8 leads to set \(R_{p}=500~{}\)k\(\Omega\), \(R_{1}=1.25~{}\)M\(\Omega\), \(R_{2}=890~{}\)k\(\Omega\), \(R_{3}=640~{}\)k\(\Omega\), \(R_{4}=460~{}\)k\(\Omega\), \(R_{5}=320~{}\)k\(\Omega\), \(R_{6}=240~{}\)k\(\Omega\), \(R_{7}=160~{}\)k\(\Omega\), \(C_{p}=900~{}\)nF, \(C_{1}=2~{}\)mF, \(C_{2}=732~{}\mu\)F, \(C_{3}=224~{}\mu\)F,  
Figure 14: Module frequency responses in dB of proposed bass-corrector with (**a**) a standard capacitor, (**b**) a CPE with a math order \(\alpha=2/3\), (**c**) CPE with a math order \(\alpha=1/2\), (**d**) CPE with a math order \(\alpha=1/3\). \(C_{4}=69\)\(\mu\)F, \(C_{5}=21\)\(\mu\)F, \(C_{6}=6.4\)\(\mu\)F, and \(C_{7}=2\)\(\mu\)F. These values can still be implemented using a combination of existing discrete passive components. However, different realizations of subsonic oscillator will be suggested. Let us adopt the values of resistors and capacitors provided in Table 8. By substitution of these values into Equation (1), and by algebraic manipulations with resulting fraction, we obtain following coefficients of the numerator of (5)  
\[ \begin{array}{c}a_{8}=3.28\cdot 10^{12},a_{7}=9.04\cdot 10^{18},a_{6}=2.33\cdot 1 0^{24},a_{5}=1.14\cdot 10^{29},a_{4}=1.2\cdot 10^{33},\\ a_{3}=2.73\cdot 10^{36},\ a_{2}=1.36\cdot 10^{39},a_{1}=1.45\cdot 10^{41},\ a_{0}=2. 98\cdot 10^{42},\end{array}  \tag{23}\]  
together with the denominator of (5)  
\[ \begin{array}{c}b_{7}=3.65\cdot 10^{21},\ b_{6}=1.45\cdot 10^{26},\ b_{5}=1.01 \cdot 10^{28},\ b_{4}=1.49\cdot 10^{36},\\ b_{3}=4.74\cdot 10^{39},b_{2}=3.3\cdot 10^{42},\ b_{1}=5\cdot 10^{44},b_{0}=1. 49\cdot 10^{46}.\end{array}  \tag{24}\]  
Having admittance function as a fraction in a semi-symbolic form, we can easily calculate the roots of both polynomials. Hence, we obtain the locations of zeroes and poles that occur in Formula (6), namely  
\[