# Active and Passive Realization of Fractance Device of Order 1/2

Authors: B. T. Krishna; K. V. V. S. Reddy
Journal: Active and Passive Electronic Components
 (2008)

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## Abstract

Active and passive realization of Fractance device of order 1/2 is presented. The crucial point in the realization of fractance device is finding the rational approximation of its
impedance function. In this paper, rational approximation is obtained by using continued fraction expansion. The rational approximation thus obtained is synthesized as a ladder network. The results obtained have shown considerable improvement over the previous techniques.


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## Body

## 1. Introduction

A system which is defined by fractional order differential equations is
called as Fractional order System [1]. The significant advantage of Fractional
order systems compared to integer order systems are that they are characterized
by memory. Fractional order systems is characterized by infinite memory,
whereas they are finite for integer order systems. Fractance Device, semi-infinite
lossy Transmission Line, diffusion of heat into the semi-infinite solid, PIλDμ controllers, and so forth are some of the
examples of fractional order systems [1, 2]. For a semi infinite lossy
transmission line current is related to applied voltage as, I(s)=sV(s). In
the case of diffusion of heat into the semi-infinite solid, the temperature at
the boundary of the surface is related to half integral of heat. However, in
this paper, focus is made only on fractance device and its realization.Fractance device is an electrical element which
exhibits fractional order impedance properties. The Impedance of the fractance
device is defined as(1)Z(jω)=(jω)α, where ω is the angular frequency and α takes the values as −1, 0, 1 for capacitance,
resistance, and the inductance,
respectively [1]. Fractance device finds applications in Robotics, Hard Disk
drives, signal processing circuits, fractional order control, and so forth
[2–7]. The following are some of the important points about Fractance
device.(i)
The phase angle is constant with frequency but
depends only on the value of fractional order,α. Hence
this device is also called as constant phase angle device or simply fractor [3].(ii)
Moderate characteristics between inductor,
resistor, and capacitor can be obtained using fractance device.(iii)
By making use of an operational amplifier, a
fractional order differentiation and integration can be accomplished easily [2].Fractance device can be realized as either tree,
chain, or a net grid type networks. Different recursive structure realizations
were presented in [3, 4]. But, the disadvantage is hardware complexity [3, 4].
The second way of realizing fractance device is synthesizing network from the
rational approximation describing its fractional order behavior. So, the key
point in the realization of fractance device is finding the rational
approximation of the fractional order operator. A rational approximation
transfer function is characterized only by poles, whereas irrationality of
fractional transfer functions gives a cut on the complex s-plane. Due to the irrationality, the
fractional linear oscillations have a finite number of zeros [8]. The continued
fractions approximation ignores this feature. So, in this paper, rational
approximation for s is obtained using continued fraction
expansion. Fractance device of order 1/2 is defined by the following volt-ampere
characteristic: (2)Z(s)≈RsC≈k0s, where k0=R/C. By
making use of well known Regular Newton Process, Carlson and Halijak [6] have obtained
rational approximation of 1/s as (3)H(s)=s4+36s3+126s2+84s+99s4+84s3+126s2+36s+1.In [7], by approximating an irrational function with
rational one, and fitting the original function in a set of logarithmically
spaced points, Mastuda has obtained rational approximation of 1/s as, (4)H(s)=0⋅08549s4+4⋅877s3+20⋅84s2+12⋅995s+1s4+13s3+20⋅84s2+4⋅876s+0⋅08551.Oustaloup [2] has approximated the fractional
differentiator operator sα by a rational function and derived the
following approximations: (5)1s=s5+74⋅97s4+768⋅5s3+1218s2+298⋅5s+1010s5+298⋅5s4+1218s3+768⋅5s2+74⋅97s+1,s=10s5+298⋅5s4+1218s3+768⋅5s2+74⋅97s+1s5+74⋅97s4+768⋅5s3+1218s2+298⋅5s+10.In this paper, a rational approximation fors is obtained using continued fraction
expansion. The rational approximation thus obtained is synthesized. In
Section 2, realization of fractance device is presented. Numerical Simulations
are presented in Section 3. Finally, Conclusions were drawn in section 4.


## 2. Realization

We have the continued fraction expansion for(1+x)α as [9] (6)(1+x)α=11−αx1+(1+α)x2+(1−α)x3+(2+α)x2+(2−α)x5+−−−−−. The above continued fraction
expansion converges in the finite complex s-plane, along the negative real axis
from x=−∞ to x=−1. Substituting x=s−1 and taking number of terms of (6), the
calculated rational approximations for s are presented in Table 1. In order to get the
rational approximation of 1/s, the expressions has to be simply reversed.Table 1
Rational approximations fors.
S. NONo. of termsRational approximation123s+1s+3245s2+10s+1s2+10s+5367s3+35s2+21s+1s3+21s2+35s+7489s4+84s3+126s2+36s+1s4+36s3+126s2+84s+951011s5+165s4+462s3+330s2+55s+1s5+55s4+330s3+462s2+165s+11Figures1(a) and 1(b) compare the magnitude and phase
responses of the rational approximations with the ideal one. It is observed from Figures 1
and 2 that fifth-order rational approximation is best fit to
ideal response up to certain range of frequencies. So, (7)s=11s5+165s4+462s3+330s2+55s+1s5+55s4+330s3+462s2+165s+11,1s=s5+55s4+330s3+462s2+165s+1111s5+165s4+462s3+330s2+55s+1.Comparison of magnitude and phase responses
of rational approximation functions with ideals.
(a)(b)Figure 2
Pole-zero plot ofs.In order to check for the stability of the obtained rational
approximations, pole-zero plot is drawn and are shown in Figures2
and 3.Figure 3
Pole-zero plot of1/s.From Figures2
and 3, it is evident that pole and
zeros interlace on negative real axis making the system as stable one. So, the
obtained rational approximation can be synthesized using RC or RL elements [10].
The realized active and networks are shown in Figures 5
and 4, respectively.Figure 4
Passive realization of the fractance device.Figure 5
Active realization of the fractance
device.


## 3. Results

The
following plots from Figures6–11 compare the magnitude and phase responses for s1/2 and s−1/2 obtained using Oustaloup method and the
proposed method.Figure 6
Magnitude response ofs.Figure 7
Phase response ofs.Figure 8
Error plot ofs.Figure 9
Magnitude response of1/s.Figure 10
Phase response of1/s.Figure 11
Error plot of1/s.


## 4. Conclusions

Realization of fractance device of order1/2 using continued fraction expansion is
presented. From the results, it can be observed that the magnitude and phase
responses have shown considerable improvement than compared to Oustaloup
method. The percent relative error is almost zero for larger range of
frequencies using proposed method. So, the proposed method can be used effectively
for the realization like fractance device, fractional order differentiators,
fractional order integrators, and so forth.



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*Source: 369421-2008-05-18.xml*