IEEE Circuits and Systems Magazine - Q2 2022 - 41

It is a direct generalization of expression (10) that gives
M
Gs
1 #a
() = N
/
/
k=0
k=0
where 01 . This is a fractional commensurate
LS.
3) Tempered CT-FARMA [24]
M
Gs
with m ! R .
4) Fractional non commensurate LS [64].
It generalizes (11)
M
Gs
/
/
() = N
k=0
k=0
5) Tempered Fractional non commensurate LS [24]
M
Gs
() = N
/
/
k=0
k=0
where mc = ,,
,,
kk k 01 f are real parameters.
6) Other generalizations can be found as ()
with v 2 0 .
Gs Hs=
v
()
It is important to remark that:
■ The complex variable functions introduced above
may represent, at least, two different TF, a causal
and an anti-causal, depending on the selected region
of convergence [56], [57], [65];
■ For stability reasons and without loosing generality,
we will assume
MN .1
C. The Differintegrator
Classically, the sequence
f ff f !
-- -
ss ss ss n
1
nn N
21 12
,
(15)
has a clear meaning due to the relation between integrals
(negative exponents) or derivatives (positive exponents)
and the LT. We note that:
■ As we said previously each term with negative exponent
represents two TF corresponding to two
disjoint regions of convergence, namely ()Re s
(causal system) and Re ()s 1 0 (anti-causal system),
with inverse LT given by
2 0
ZFDNR = K-2
()
SECOND QUARTER 2022
j~ 2
IEEE CIRCUITS AND SYSTEMS MAGAZINE
41
as
bs
kk
kk
+
+
m
c
()
()
bk
,
ak
(14)
for any real order (we can consider complex orders, but
the resulting systems are not Hermitian). The elemental
system with TF () ,,Gs s
tor. If () 2 0, it will be called forward, otherwise if
Re ()s 1 0 it will be denoted backward.
Re s
The impedance of a circuit with a differintegrator,
called constant phase element (CPE) [67], assumes
the form
Zj Kj a
a
() () .~~=
This impedance, called fractance, is complex for non
integer .a An ideal fractional coil ()02a
[68], [69]
has fractance
ZL ()j
L = a
~
a
with the inductance, L ,a with units 6Hs .a@ Similarly, an
$
ideal fractional capacitor has fractance [16]
()
ZC =
Cj
1
a ~ a
where the capacitance Ca has units 6Fs .
$
1-a@ With a 1=
we obtain the classic coil and capacitor reactances. An
interesting case is the frequency-dependent negative resistor
(FDNR) [67], [70], [71] that corresponds to
and also in (16) and (17) through
()
L sa
-1
=
!!
C a
t
-
--1
a
ut
(),
(20)
as
bs
k
k
bk
.
ak
(13)
() = N
/
/
k=0
k=0
as
bs
k
k
()
()
+
+
m
m
ka
,
ka
(12)
where we recall a relation derived by Gel'fand and
Shilov [66]:
d
()n
()
t =
()!
--t
n
1
n
1
(),.
ut n $ 0
gaps in sequence (15) yielding
ss ss ss ss
-- -r
22 11/21
ff ff ff ff f
r
2
(18)
The fractionalization of (18) allows us to fill in the
e
(19)
=
as
bs
k
,
ka
k
(11)
ka
gg ()t ut!! !! !! !!
()!12 1!
t
n 12 1
-
ut
n -
()
!
t ut ut() ()
(16)
■ The terms with positive or null exponents are
analytic on the whole complex plane. The corresponding
inverse LT are
()!-()
dd
d
12 n 1
ut
()
t
t
t
-- -t
()!
ut
ff
()!
()
ut
-n
()
12
1
()n
l () ff=
t
()
t
(17)
= a a ! R is called differintegra

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