IEEE Circuits and Systems Magazine - Q2 2022 - 42

Remark II.2. The question of the unities in fractional
elements like coils and capacitors is an open subject
[72], [73].
There are interesting realizations of differintegrators
and applications to circuits [44], [48], [74]-[77]. An interesting
generalization of the Kramers-Krönig relations
was presented in [21].
Example II.1. A device with fractional characteristics is
the electrochemical capacitor for which several fractional
models were proposed for its impedance. This subject was
studied in [78], [79] and the following model was obtained
and validated experimentally
Zs R0
()
where R0
cc
12
,,
CC and C 3c
=+ ++c
c
1$$ $
c2
c2
cc
12
+
1
sC sC sC
11 1
,
c3
represents a series resistance, the symbols
stand for capacitances of the electrochemical
capacitor model, and 1c and 2c are the fractional orders.
It is interesting to remark that
1) the model is a series or a resistor and 3 ideal capacitors;
2) the order of one ideal capacitor is the sum of the
other two, assuming a value slightly greater than 1.
III. Study of Continuous-Time
FARMA Linear Systems
A. The Commensurate Case
The CT-FARMA LS are the most important class of fractional
systems, since they constitute the base for the
study of fractional electric circuits. This type of system
was introduced in (11) by means of its TF that we reproduce
here together with its partial fraction decomposition
M
Gs
()== = / () kn .
k
As
Bs
()
()
a
a
/
/
k=0
N
k=0
The parameters ,, ,,pk 12 N
kp
Az = R =0
k az
N
k
.
Bz
= f are called pseudopoles
and are the roots of the characteristic polynomial
()
k The roots of () = R =0
k bz
M
k
k are the
pseudo-zeroes and Rk stands for the residues of the partial
fraction decomposition of ()/( ).Bz Az
Remark III.1. Any system defined by (13), with rational
orders can be converted into the form (11), that is to
commensurate. We only may have to join additional null
coefficients. The same happens with (14) that gives (12).
Remark III.2. Let p be a complex number,
and the equation sp 0-=a
01 ,11a
that has infinite roots. However,
there is only one solution for this equation in the first
Riemann surface, if
0 = ;1aa
by
sp e/( )(/ )
j arg p
;
-ar 1 arg p # ar, and it is given
. For example, if () ,
()
arg p r=
then
the root of the equation is in the second Riemann surface.
Therefore, it does not lead to a pole or zero of a system.
42
IEEE CIRCUITS AND SYSTEMS MAGAZINE
as
bs
k
ak
k
k=1
sp
R
a
-
k
ak
Np
(22)
(21)
The case a 1= corresponds to the classic model
(10). As it is well-known [56], [57] the IR corresponding
to this TF has the form
Np
gt
() =!! (23)
k
/ Rk
t
k ,, ,,
ut
k=1 ()!
nk - 1
eu t
pt (),
where pk 12= f are the poles, nk denotes their
multiplicity, and Rk
!! is the causal/anti-causal Heaviside unit
represents the corresponding residues
in the partial fraction decomposition of (10). The
function ()
step used to make a clear distinction between the causal
(right) and anti-causal (left) solutions. This is important
if the variable t does not stand for time, but, for example,
it represents space.
For
01 ,11a
fraction /(
1 sp).n
a -
n
a
sp
-
-
()
()!( )
1
n =
dn-1
sp
a
dn-1
-
p
we only have to pay attention to the case:
()
Gs = a -
sp
1
.
(24)
1
nk-1
we need to invert a generic simple
Since
(25)
We will treat the causal case only. There are two different
ways of inverting (25).
1) Using the Mittag-Leffler Function
Using the geometric series, we can write successively
()
Fs = a
ps
sp 01
- nn
valid for ;;11a1
// (26)
1
==
33
-- -spsp
s
aa a
==
nn nn
,
. Choosing the ROC ()Re sps2;;awill
arrive at the causal inverse of F(s):
()
3
ft = / n-1
n=1
p
t
na-1
C()
na
f t
().
normally expressed in terms of the MLF as
()
ft tE pta
= a-1
aa, () ()f t
ft et
=
sp s
-
,
(27)
This function also called alpha-exponential [31] is
(28)
and gives the IR corresponding to partial fraction (26). If
a 1 ,= then we have the classic result ()
step response () of (26) can be obtained from
3
rtf
L rt @ = a
6 ()
f
leading easily to
()
11 nn1
1
$
= /ps
--a -1
(29)
ptf(). The
, we
rt tEaa+ () ()
t
f == -
a
p
a
,,1 Eptt111
$$
pta ff (30)
a
6 () (),
@
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