# Accurate Constant Phase Elements Dedicated for Audio Signal Processing
## 2 Passive and Active Structures of CPE Approximants
Each immittance function of the CPE approximant can be expressed in form as a rational fraction  
\[F(s)=\frac{\sum\
olimits_{k=0}^{n}a_{k}s^{k}}{\sum\
olimits_{k=0}^{m}b_{k}s^ {k}}, \tag{5}\]  
where \(n\) and \(m\) are natural numbers (with unity maximal difference) and \(a_{k}\) and \(b_{k}\) are real positive constants, or equivalently as  
\[F(s)=\frac{a_{n}}{b_{m}}\cdot\prod\
olimits_{k=1}^{n}(s+\omega_{Zk})\cdot \prod\
olimits_{k=1}^{m}\frac{1}{s+\omega_{Pk}}, \tag{6}\]where \(\omega_{Zk}\) and \(\omega_{Pk}\) are the so-called zeroes and poles of the network function \(F(s)\). Note that zeroes and poles are supposed to be real numbers. Moreover, these zeroes and poles are singular, located on the horizontal axis of the open left half-plane of complex planes, and alternates with respect to frequency. The corresponding phase frequency response can be written as  
\[\varphi(\omega)=\sum\
olimits_{k=1}^{n}atan\frac{\omega}{\omega_{Zk}}-\sum \
olimits_{k=1}^{m}atan\frac{\omega}{\omega_{Pk}}. \tag{7}\]  
As is known from the theory of filters, each zero causes a smooth \(90^{\circ}\) positive contribution to phase shift, while each pole causes the same, but negative, contribution--a \(-90^{\circ}\) phase shift. Thus, the appropriate deployment of zeroes and poles (the values of angular frequencies \(\omega_{Zk}\) and \(\omega_{Pk}\)) can lead to a symmetrical phase ripple around any value in the range \(-90^{\circ}\) to \(90^{\circ}\). There are three important consequences coming from this kind of CPE approximation, and all should be respected within their intended applications. Firstly, CPE approximation is valid only in a finite frequency range. Secondly, the phase frequency response is no longer constant--it is rippled around a predefined value. The module frequency response is not a straight line--it has a stair-like shape. Furthermore, for small phase errors, zeroes and poles need to be located close to each other. Additional zeroes and poles always mean a more complicated final circuit structure of the CPE approximant. Finally, there is always a tendency to keep the order of CPE approximation circuits as low as possible because the mathematical order of a designed functional block (where CPE is incorporated) is increased by this number. Now, let us summarize the properties of CPEs that we need for audio applications. The frequency band begins with 20 Hz and stops at 20 kHz, and a predefined maximal phase deviation within this interval should be \(0.5^{\circ}\). Computer analysis and practical experiments supported by the formula provided in [17] returns the required number of seven RC sections regarding Figure 1a,b. Namely, the admittance function of the circuit in Figure 1a has eight zeroes (\(n=8\)) and seven poles (\(m=7\)), and approximation begins and ends with zero. At low frequencies, the phase shift starts to increase, and in the operational frequency range, zero-pole interactions take place, and approximation ends with a high frequency zero. Thus, phase shift asymptotically reaches \(90^{\circ}\) for high frequencies. Remember, for very high frequencies, the RC ladder network can no longer be considered as a circuit with lumped parameters. Straightforward analysis of the second network connection provided in Figure 1b gives the same number of zeroes and poles of complex impedance, \(m=n=8\). Series capacitor \(C_{s}\) leads to the first zero \(\omega_{k1}=0\). Consequently, phase shift asymptotically returns to zero for frequencies above the approximated range. This property can be important regarding the stability of the electronic system. General Expression (6) represents a cascade connection of \(m+n\) fundamental network functions.