# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 2 Passive and Active Structures of CPE Approximants
In this case, admittance matrix \(\mathbf{Y}\) is terminated by the element \(y_{88}(s)=1/sL_{8}+G_{7}\). The numerical values of \(k\)-th transmission line cell \(C_{k}\), \(G_{k}\), and \(L_{k}\) can be obtained by an optimization routine. Two-port circuitry realizations of CPEs based on the locations of zeroes and poles of the network function (voltage-mode, current-mode, trans-impedance-mode, or trans-admittance-mode) are given symbolically in Figure 1g. The required configuration of zeroes and poles of the final voltage transfer function can be constructed by using a cascade of the bilinear filters. A huge number of available topologies can be found via the internet. One promising active implementation of voltage-mode two-port using direct knowledge of the locations of zeroes and poles is provided in Figure 1i. The values of capacitors \(C_{Z}\) and \(C_{P}\) in one active cell predefine the frequency decade where the zero-pole pair is placed (high \(C\) means low \(f\) and vice versa). The advantage of this concept is obvious: these bilinear filters have a very simple relationship between circuit components and Voltage Transfer Function (6). As mentioned above, biquadratic filters can also be utilized to model CPEs if they are placed in a cascade. Both band-pass and band-reject filters can be used. The decomposition of voltage transfer function \(F_{1}(s)\), i.e., Formula (8), is visualized in Figure 1h, where \(K_{0}=a_{8}/b_{7}\). A slight modification of the branch impedances in Figure 1i leads to two-port realization where a significant frequency and quality factor are the key parameters (see Figure 1j). Alternative modifications with parallel resonant sub circuits also lead to suitable forms of voltage transfer function. Inside both cells, a commercially available integrated circuit AD844 (a second generation current conveyor followed by a voltage buffer)is utilized. The second circuitry can be considered as a single-band audio equalizer. The voltage transfer function of these cells can be expressed as follows  
\[K(s)=\frac{V_{out}}{V_{in}}=\frac{C_{Z}}{C_{P}}\frac{s+1/(R_{Z}C_{Z})}{s+1/(R_{P} C_{P})}\,K(s)=\frac{V_{out}}{V_{in}}=\frac{L_{Z}}{L_{P}}\frac{s^{2}+\frac{R_{Z}}{L_{Z}} \cdot s+\frac{1}{L_{Z}C_{Z}}}{s^{2}+\frac{R_{P}}{L_{P}}\cdot s+\frac{1}{L_{P}C_ {P}}}. \tag{14}\]  
Note that for both circuits given in Figure 1i,j, the locations of zeroes and poles are independently and uniquely determined by values of the passive circuit components. Quality factors associated with series resonant circuits are very low, such that a synthetic grounded inductor can be used. Synthetic lossy inductors are also supposed to be used in the case of the implementations provided in Figure 1c,d. There are many network topologies that can emulate very large values of self-inductance. For example, [PERSON]'s topology is simple (one or two operational amplifiers are needed for the simulation of grounded and floating inductors, respectively) and can be used for large self-inductance at the cost of large resistance in the series. [PERSON]'s structure utilizes two (grounded inductor) or four (floating inductor) operational amplifiers, and can model very large self-inductance with negligible resistance in series.