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Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits with Metamaterials by Yang Shang, Hao Yu

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46 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
Assuming the fractionality for both inductance a nd capacitance are all
constants that α
L
S
= α
L
P
and α
C
S
= α
C
P
, (3.11 ) can be simplified as
γ =
s
ω
α
L
S
+α
C
P
L
S
C
P
e
0.5πj(α
L
S
+α
C
P
)
+
1
ω
α
C
S
+α
L
P
C
S
L
P
e
0.5πj(α
C
S
+α
L
P
)
.
(3.22)
The zer o-phase-shift freq ue nc y (ω
0
) can be o btained from (3.22) with β =
0:
ω
0
= (L
S
C
P
C
S
L
P
)
1
α
L
S
+α
C
P
+α
C
S
+α
L
P
. (3.23)
Eq. (3.23) reveals an exponential relationship between the prefactors and
fractional-order terms, which can be used as a guideline in the fractional-orde r
modeling o f CRLH T-line network .
3.3 Model Extraction and Ca usality Analysis
T-line is a passive, linear and time-invar iant (LTI) network. The extracted
T-line model is there by needed to be causal. The extr action flow of fractional-
order T-line model is introduced with the additional causality checking and
enforcement followed by comparison with the traditional integer-order c oun-
terpart.
3.3.1 Fractional-Order Model Extraction
A fractional-order model para meters extraction flow for T-line at THz is illus-
trated in Fig. 3.3. The extraction begins with the measurement data obtained
from a Vector Network Analyzer. Firstly, the measurement data is converted
into transfer matrix (T matrix) for an easy operation, and the error terms
contributed by the testing pads are removed by de-embedding process. Sec-
ondly, characteristic impedance Z
0
and propagation constant γ are calculated
from de-embedded T-matrix according to [121]. Afterward, one can define the
modeling frequency interval [ω
1
, ω
2
] in the THz r egion based on his interests.
From (3.10), one can have
α
L
α
C
= 2log
ω
1
ω
2
Z
0
(ω
1
)
Z
0
(ω
2
)
(3.24)
where Z
0
(ω
1
) and Z
0
(ω
2
) are the characteristic impe dances at frequencies ω
1
and ω
2
in THz region, respectively.
From (3.13), one can have
α
L
+ α
C
= 2log
ω
1
ω
2
γ(ω
1
)
γ(ω
2
)
(3.25)
where γ(ω
1
) and γ(ω
2
) are the propagation constants at frequencie s ω
1
and ω
2
CMOS THz Modeling 47
Figure 3.3: Fractional-order T-line mo deling parameters e xtraction
flow.
in THz region, respectively. By c ombining (3.24) and (3.25), α
L
and α
C
can
be obtained in the fra ctional-order model; and by substituting α
L
and α
C
into
(3.24) and (3.25), fractional-order L
and C
can be obtained as well. Note that
L
and C
are the p.u.l. (per-unit-length) prefactors with corresponding units
of V s
α
L
A
1
/m and As
α
C
V
1
/m, respectively, but not p.u.l. inductance
and capacitance anymore.
Moreover, in order to apply the fractional-order T-line model in the time-
domain simulator to Cadence Spectre, the model needs to be converted from
frequency domain into time-domain by rational functional approximation
f(s)
N
X
j=1
c
j
s a
j
+ d + sh. (3.26)
Here, N is the rational order, a
j
and c
j
are the poles and residues in
complex conjugate pairs, d and h are real. The coefficients in (3.26) can be
obtained by vector fitting alg orithm as introduced in [122]. Note that the error
introduced by the frequency-to-time conversion is well c ontrolled by increasing
the rational-fitting order.
3.3.2 Causal LTI System and Causality Enforcement
To understand the causality for the extracted fractional-order T-line model,
the fundamentals of the c ausal LTI system are first rev iewed here. In a LTI

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