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 [111] including electronics [103, 112, 11 3],

electromagne tic [114], ﬂuidic-dynamics [115], material technology [116], quan-

tum mechanics [117], etc.

Fractio nal calculus was initiated by a question of half-order derivative by L’

Hopital in 1695 and was generalized by Euler in [118]. In fractional calculus,

the integration and diﬀerentiation can be gener alized by the operator

a

D

α

t

[119],

a

D

α

t

=

d

α

dt

α

, α > 0

R

t

a

(dτ)

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)

α−n+1

dτ, (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 [103] 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 [112] 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

0.5πjα

C

(3.7)

Z

′

(ω) = ω

α

L

L

′

e

0.5πjα

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

0.5πjα

L

G

0

+ ω

α

C

C

′

e

0.5πjα

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

0.5πjα

L

and G

0

<< ω

α

C

C

′

e

0.5πjα

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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