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No credit card required CMOS THz Modeling 41
3.2 Fractional-Order T-Line Model
3.2.1 Fractional Calculus
Generally, most o f dyna mic systems can be de scribed with fractional dynam-
ics, though the fractionality is rather low to be considered than the integer-
order behavior. Recently, the fractional-order models are re-examined when
considering loss terms in many ﬁelds  including electronics [103, 112, 11 3],
electromagne tic , ﬂuidic-dynamics , material technology , quan-
tum mechanics , etc.
Fractio nal calculus was initiated by a question of half-order derivative by L’
Hopital in 1695 and was generalized by Euler in . In fractional calculus,
the integration and diﬀerentiation can be gener alized by the operator
a
D
α
t
,
a
D
α
t
=
d
α
dt
α
, α > 0
R
t
a
()
α
, α < 0
(3.1)
where α is a real number, a and t are the lower and upper bounds. The
diﬀerentiation and integration can be treated as the special cases when α
equals 1 or -1, respectively. By Riemann–Liouv ille deﬁnition, the fractional
operation can be expressed by the following equation (n 1 < α < n),
a
D
α
t
f(t) =
1
Γ(n α)
d
n
dt
n
Z
t
a
f(τ)
(t τ)
αn+1
, (3.2)
where Γ(·) is the Euler’s gamma function.
Assuming the lower bound a = −∞, one can take the Fourie r transform
of 3.2 to o btain the generalized expression of fractional integral in frequenc y
domain for 0 < α < 1,
F{
−∞
D
α
t
f(t)} = (jω)
α
G(ω). (3.3)
Similarly, the generalized expression of fractiona l deriva tive in the fre-
quency domain is
F{D
α
f(t)} = (jω)
α
G(ω). (3.4)
As shown in 3.3 and 3.4, the fractional-operator in the frequency domain
can be treated as the product of a magnitude scaling factor ω
α
and a phase
rotation facto r j
α
. Theoretically, the physical behavior of any electronic de-
vice can be descr ibed by these two frac tional factors. More importantly, both
scaling factor and rotation factor are linked by a fractional-or de r α, w hich en-
sures the reality of one dynamic system with dissipation. T he refore, in order to
examine the loss terms for electronic devic es at T Hz, the aforementioned frac-
tional calculus can be applied with many interesting observa tions as explored
in the following s ections. 42 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
3.2.2 Fractional-Order Capacitance and Inductance
Fractio nal-order model for T-line can be built by introducing fr actional-order
terms in the conventional RLGC model a s shown in Fig. 3.1 (b). A fractional-
order capacitor model  with the I-V relation can be given by
I(t) = C
d
α
C
V (t)
dt
α
C
= C
0
D
t
αC
V (t) (3.5)
where C
is the fractional capacitance with order α
C
, and α
C
(0, 1] is the
fractional-order relating to the loss of capacitor.
Similarly, the I-V relation of fractional-order inductor  is
V (t) = L
d
α
L
I(t)
dt
α
L
= L
0
D
t
αL
I(t) (3.6)
where L
is the fractional inductance with order α
L
, and α
L
(0, 1] is
the fractional-order relating to the loss of inductor. The admittance and
impedance of fractional-order capacitor and inductor can be obtained from
(3.5) and (3.6) by
Y
(ω) = ω
α
C
C
e
0.5π
C
(3.7)
Z
(ω) = ω
α
L
L
e
0.5π
L
. (3.8)
When α
L
or α
C
6= 1, we can expect the existence of real-parts at the
right-hand sides of (3.7) and (3.8), which represent the frequency-dependent
loss. Physically in a particular device, the fractional-order operator indicates
the transfer of the energy storage to energ y loss . As such, the distributed
frequency-dependent terms are considered by L
and C
elements in fractional-
order terms.
3.2.3 Fractional-Order T-Line Model
Note that the fractional-order T-line can be analyzed in a similar fashion as the
traditional T -line. The characteristic impedance (Z
0
) of T-line can be found
by
p
Z/Y , where Z and Y ar e the series impedance and shunt admittance,
respectively. Based on (3.7) and (3.8) with consideration of resistance R
0
and
conductance G
0
, one can have
Z
0
=
s
R
0
+ ω
α
L
L
e
0.5π
L
G
0
+ ω
α
C
C
e
0.5π
C
. (3.9)
In THz frequency region, ω is in the order of 10
11
10
13
. At such a high
frequency, we have R
0
<< ω
α
L
L
e
0.5π
L
and G
0
<< ω
α
C
C
e
0.5π
C
, so (3.9)
can be approximated as
Z
0
=
r
L
C
· ω
α
L
α
C
2
·
cos
(α
L
α
C
)π
4
+ j sin
(α
L
α
C
)π
4
. (3.1 0)

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