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No credit card required 50 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
phase function deﬁned by the following condition [127, 128]:
lim
ω→∞
γ(ω)
jω
0. (3.35)
For T-line, the minimum phase function can be calculated by substituting
(3.13) into (3.35) as
lim
ω→∞
γ(ω)
jω
= lim
ω→∞
ω
α
L
+α
C
2
·
L
C
·
sin
(α
L
+α
C
)π
4
j cos
(α
L
+α
C
)π
4
.
(3.36)
We can observe that (3.3 6) shows very diﬀerent responses for fractional-
order and integer-order T-line models. For the fractional-order model, (3.36)
equals zero when α
L
+ α
C
< 2. So the causality is always ensur ed as the
minimum phase function condition when (3.35) is satisﬁed. On the o ther hand,
α
L
and α
C
are both equal to one for the integer-order T-line model, and (3.36)
results in a constant value of
L
C
, where L
and C
become the normal
inductance and capacitance, respectively. Thus the minimum phase function
condition in (3.3 5) is violated and the model be comes non-causal.
Note that the major reason for the non-causal issue in the traditional inte-
ger T-line model is due to the linear frequency dependence of the propagation
constant γ(ω) when α
L
+ α
C
= 2 in (3.13). This cannot model the disper-
sion loss and non-quasi-static eﬀects in the high-frequency application like
THz. The reality of the integer-order T-line model is lost in the THz re gion,
and so is the causality. In contrast, the non-ideal eﬀects are considered in the
proposed fractional-order T-line model by fractional-order dis persion terms,
which c an largely improve the model r eality. As s uch, both the model accu-
racy and causality are improved. The causality of the fractio nal-order model
can also be veriﬁed in numerical calculation by computing the er ror terms in
(3.33), which will be discussed in the following section.
3.4 Prototypin g and Measurement
3.4.1 T-Line Fractional-Order Model Veriﬁcation
As shown in Fig. 3.5(a), a c oplanar wave guide transmission line (CP W- T L )
testing structure w ith RF-PADs is fabricated with Global Foundry 1P8M
65nm CMOS process, of which the dimensions are given in Fig. 3.5(b). The
CPW-TL is implemented on the top metal layer with thickness of 3.3 µm.
It is measur ed on a CASCADE Microtech Elite-300 probe station by Agilent
PNA-X (N5247A) with frequency sweep up to 110 GHz. The measurement
setup of S- parameters up to 110 GHz is illustrated in Fig. 3.6. The reference
plane of PNA is calibrated to the ends of GSG probes by SOLT method. Note
that both the probes and the impedance standard substrate are provided CMOS THz Modeling 51
5µm
35µm
5µm
5µm
160µm
(a) (b)
Figure 3.5: T-li ne testing structure: (a) die photo, and (b) detailed
dimensions .
measurement results with the “open-short” method. Table 3.1 summa rizes
extracted model parameters of both integer-order and fractional-or de r models
based on measurement results. The parameters of the traditional integer-order
model are ex tracted according to the pro cedure by . The parameters of
fractional-order model are extracted according to Sectio n 3.3.1.
The resulting S-parameters and characteristic impedance (Z
0
) of integer-
order and fractional-order RLGC models are compared in Fig. 3.8. We can
observe that bo th the traditional integer-order mo de l and the proposed
fractional-order model can ﬁt the measurement results in magnitude in Fig.
3.7. Here a relatively large de viation is observed in magnitude of S11 between
the simulation and measurement results. This deviation comes fr om the equip-
ment noise and calibration error, which is unavoidable as the absolute magni-
tude o f S11 is small (15 50 dB). Moreover, the phase delay o f both the
Figure 3.6: Measurement setup of on-wafer S-parameter testing up
to 110 GHz.

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