# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 2 Passive and Active Structures of CPE Approximants
For any combination of two zeroes and two poles, a cascade of the bilinear network functions can be rewritten into the product of biquadratic network functions, i.e.,  
\[F_{1}(s)=\frac{a_{8}}{b_{7}}.\frac{s^{2}+\frac{\sqrt{\omega_{ZZ}\omega_{Z8}}}{ Q_{Z4}}s+\omega_{Z7}\omega_{Z8}}{s+\omega_{P7}}.\prod_{k=1}^{3}\frac{s^{2}+ \frac{\sqrt{\omega_{Z(2k-1)}\omega_{Z(2k)}}}{Q_{Zk}}s+\omega_{Z(2k-1)}\omega_ {Z(2k)}}{s^{2}+\frac{\sqrt{\omega_{P(2k-1)}\omega_{P(2k)}}}{Q_{Pk}}s+\omega_{P (2k-1)}\omega_{P(2k)}}, \tag{8}\]  
where the quality factor associated with a pair of zeroes, \(\omega_{1}\) and \(\omega_{2}\), (or poles) is \(Q=\sqrt{(}\omega_{1}\cdot\omega_{2})/(\omega_{1}+\omega_{2})\) For real zeroes and poles with significant distances, we reach very low-quality factors, much lower than \(Q=1/2\). Note that network function \(F_{1}(s)\) is related to the RC approximation given in Figure 1a. The second structure has a similar decomposition, namely  
\[F_{2}(s)=\frac{a_{8}}{b_{7}}.\frac{s(s+\omega_{Z7})}{s^{2}+\frac{\sqrt{\omega _{P7}\omega_{P8}}}{Q_{P4}}s+\omega_{P7}\omega_{P8}}.\prod_{k=1}^{3}\frac{s^{2} +\frac{\sqrt{\omega_{Z(2k-1)}\omega_{Z(2k)}}}{Q_{Zk}}s+\omega_{Z(2k-1)}\omega_ {Z(2k)}}{s^{2}+\frac{\sqrt{\omega_{P(2k-1)}\omega_{P(2k)}}}{Q_{Pk}}s+\omega_{P (2k-1)}\omega_{P(2k)}}. \tag{9}\]Significant frequencies and quality factors of zeroes and poles of CPEs depend on the choice of coupled zero pairs and pole pairs (this choice can be arbitrary). A low quality of zeroes and poles means that the individual cells of two-port approximation can be realized by the RC-only circuits, for example, by using the tree structure shown in Figure 1k, bridge-T network, etc. Next, the RC network structure in Figure 1e can be considered as a single-ended finite (but long enough for good approximation) transmission line divided into cells, each cell being described by two out of four primary parameters. The corresponding admittance function can be written as the continued fraction  
\[Y(s)=\frac{I(s)}{V(s)}=sC_{1}+\frac{1}{R_{1}+\frac{1}{sC_{2}+\frac{1}{R_{2}+ \frac{1}{sC_{3}+\frac{1}{sC_{4}+\frac{1}{sC_{5}+\frac{1}{sC_{6}+\frac{1}{sC_{7}+ \frac{1}{R_{7}+1/sC_{8}}}}}}}}}. \tag{10}\]  
Because of the long and very complex expression for individual coefficients of the polynomials of Ratio (5), Admittance (5) and (6) will not be provided in this form. However, Admittance (10) can be rewritten using the matrix method of unknown nodal voltages as  
\[Y(s)=\frac{I(s)}{V(s)}=\frac{det(\mathbf{Y})}{det(\mathbf{Y_{1,1}})}\ \mathbf{Y}=\left( \begin{array}{ccc}sC_{1}+G_{1}&-G_{1}&0\\ -G_{1}&sC_{k}+G_{k-1}+G_{k}&-G_{k}\\ 0&-G_{k}&\ddots\end{array} \right), \tag{11}\]  
where \(\mathbf{Y}\) is determinant of the admittance matrix, \(\mathbf{Y_{1,1}}\) is sub-determinant after remov its first row, column \(k=2\), \(3\),..., \(8\) is the index of independent nodes, and \(G_{k}=1/R_{k}\). Admittance matrix \(\mathbf{Y}\) is terminated by the element \(y_{88}(s)=sC_{8}+G_{7}\). By the principle of duality, we can construct a single-ended finite transmission line composed of the two remaining primary parameters as demonstrated in Figure 1f. The impedance function of this complex two-terminal element can be expressed as \[Z(s)=\frac{V(s)}{I(s)}=sL_{1}+\frac{1}{\frac{1}{R_{1}}+\frac{1}{sL_{2}+\frac{1}{ R_{2}}+\frac{1}{sL_{3}+\frac{1}{sL_{4}}+\frac{1}{sL_{5}+\frac{1}{sL_{6}}+\frac{1}{sL_{7}+ \frac{1}{sL_{8}}}}}}}. \tag{12}\]  
Similarly, as it is in the case of admittance, Complex Impedance Function (12) can be expressed as ratio of determinant \(\mathbf{Z}\) and sub-determinant \(\mathbf{Z_{1,1}}\) after removal of the first row and column  
\[Z(s)=\frac{V(s)}{I(s)}=\frac{det(\mathbf{Y_{1,1}})}{det(\mathbf{Y})}\ \mathbf{Y}=\left( \begin{array}{ccc}1/sL_{1}&-1/sL_{1}&0\\ -1/sL_{1}&1/sL_{k-1}+1/sL_{k}+G_{k-1}&-1/sL_{k}\\ 0&-1/sL_{k}&\ddots\end{array} \right), \tag{13}\]  
where \(k=2\), \(3\), \(\ldots\), \(8\) is the index of independent nodes and \(G_{k}=1/R_{k}\).