On the Analysis and Design of Fractional-Order Chebyshev Complex Filter

Abstract

This paper introduces the concept of fractional-order complex Chebyshev filter. A fractional variation of Chebyshev differential equations is introduced based on Caputo fractional operator. The proposed equation is solved using fractional Taylor power series method. The condition for fractional polynomial solutions is obtained and the first four polynomials scaled using an appropriate scaling factor. The fractional-order complex Chebyshev low-pass filter based on the obtained fractional polynomials is developed. Two methods for obtaining the transfer functions of the complex filter are discussed. Circuit implementations are simulated using Advanced Design System (ADS) and compared with MATLAB numerical simulation of the obtained transfer functions to prove the validity of the two approaches.

Appendix: Valsa CPE

The calculation of the elements values used in the Valsa CPE approximation starts with the given data which are the start frequency $\omega _d=\frac{1}{R_1C_1}$, the phase angle $\phi _{av}$, the allowable phase variation $\varDelta \phi$, the number of sections m and the impedance value at $\omega =1$ ($D_p$). The end frequency is calculated as:

$$\begin{aligned} \omega _h=\frac{\omega _d}{(ab)^{m}}, \end{aligned}$$

(43)

where ab is calculated from the allowable ripple as:

$$\begin{aligned} ab\cong \frac{0.24}{1+\varDelta \phi }. \end{aligned}$$

(44)

a can be calculated as:

$$\begin{aligned} \alpha =\phi /90,\qquad a=\alpha \log (ab). \end{aligned}$$

(45)

The values of the resistors and capacitances in section are:

$$\begin{aligned} R_k=R_1a^{k-1}, \qquad k=1,2,\dots m, \end{aligned}$$

(46a)

$$\begin{aligned} C_k=C_1b^{k-1}, \qquad k=1,2,\dots m, \end{aligned}$$

(46b)

and the correction elements:

$$\begin{aligned} R_p=R_1\frac{1-a}{a},\qquad C_p=C-1\frac{b^m}{1-p}. \end{aligned}$$

(47)

The input admittance can be calculated as:

$$\begin{aligned} Y(j\omega _{av})=\frac{1}{R_p}+j\omega _{av}C_p+\sum \limits _{k=0}^{m}\frac{j\omega _{av}C_k}{1+j\omega _{av}R_kC_k}, \end{aligned}$$

(48)

where

$$\begin{aligned} \omega _{av}=\left( \frac{a}{b}\right) ^{0.25}z_k,\qquad z_k=\frac{\omega _d}{(ab)^{k-1}},\qquad k=\lceil m\rceil . \end{aligned}$$

(49)

The actual impedance value at $\omega =1$ is calculated as:

$$\begin{aligned} D=\frac{1}{|Y(j\omega _{av})|\omega _{av}^\alpha }. \end{aligned}$$

(50)

In order to obtain the required value $D_p$, all the resistances must be multiplied by $D_p/D$ and all the capacitances must be divided by the same ratio.

Throughout this work, the fractional capacitance is set to $C=10\,\hbox {nF}\cdot \hbox {s}^{\alpha -1}$, the number of branches is set to $m=7$ and the allowable phase variation is set to $\varDelta \phi =1^\circ$. The list of values used to implement fractional capacitors of order 0.9 and 0.7 are summarized in Table 2. The phase and capacitance response of the used capacitors are depicted in Fig. 15. It can be seen that the phase approximation is below $1^\circ$ of variation and the capacitance error is below $2\%$ within the frequency range of 100 Hz to 1 MHz.