CMOS THz Modeling 55
Figure 3.11: Measurement setup of on-wafer S-parameter testing from
220 GHz to 325 GHz.
Table 3.2: Modeling Parameters of Integer-Order and Fractional-
Order Models of CRLH T-Line Unit Cell
Integer-Order Model Fractio nal-Order Model
Parameter Value Unit Parameter Value Unit
L
S
15.6 pH α
L
S
/α
L
P
0.9847/0.9766 —
C
S
14.7 fF α
C
S
/α
C
P
0.9939/0.9973 —
G 1.3 mS L
′
S
14 V s
−α
L
S
A
−1
R 2.8 Ω C
′
P
1732.1 As
−α
C
P
V
−1
C
P
13.8 S/m L
′
P
39.41 V s
−α
L
P
A
−1
L
P
28.3 pH C
′
S
1408 As
−α
C
S
V
−1
— — — R
0
/G
0
0.3396/902 Ω/mS
a remarkable difference between integer-or de r and fractional-order res ults in
terms of characteristic impedance Z
0
. The measure ment Z
0
fit very well to
fractional-order model at zero -phase-shift region from 260 GHz, also a sma ller
error of Z
0
at low-frequency re gion compared to integer-order fitting result.
The average accuracy improvement of 78.8% is obtained by fra ctional-order
model c ompared to the integer-order counterpart with correlated measur ement
and simulation results of Z
0
. Moreover, the measurement res ults of CRLH T-
line agree well with the EM simulation results for the frequency range of 220
∼ 325 GHz.
3.4.3 Causality Verification and Comparison
The causality of the proposed fractional-order T-line model can be verified
by comparing imaginary parts o f S-para meters with the Hilbert transform of
56 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
Figure 3.12: Measurement, EM, integer-order and fractional-order
circuits simulation results: magnitude of S21 and S11 in dB.
real parts. T he n, the error term e
ij
(ω
n
) is calculated by (3.33) a s discussed in
Section 3.3. For the purpose of comparison, the causality of traditional integer -
order T-line model is also verified in the same way. The tabulated results
for both models are obtained by two-port S-parameter simulation in Ag ilent
Advanced Design System (ADS) based on the extracted model parameters
shown in Table 3.1. For a two-port network, four sets o f complex S-parameter
results can be obtained including S11, S22, S12 and S21. However, according
to the reciprocal property of the T-line structure (S11 = S22 and S12 = S21),
only S11 and S21 are considered in the causality analysis. In order to minimize
reconstruction and discretization er rors [125] introduced by finite spectrum,
the S-parameter simulation is conducted from 0Hz to 20THz with a step size
of 1GHz.
Firstly, the causality of return loss (S11) is verified for both integer and
fractional order T-line models. Figs. 3.15 and 3.16 show the comparison be-
tween ℑ(S11) and the value obtained by Hilbert tr ansformation from the re al
part HilbertReal(S11) for both integer-or de r and the proposed fractional- order
T-line models in the frequency rang e of 0.001 ∼ 1 THz, respectively. For the
traditional integer-order RLCG T-line model, the ℑ(S11) starts to deviate
CMOS THz Modeling 57
Figure 3.13: Measurement, EM, integer-order and fractional-order
circuits simulation results: phase of S 21 in degree and the absolute
value of extracted phase constant β.
from the causal response at 10 GHz and shows large deviation in 0.1 ∼ 1 T Hz
as depicted in Fig. 3.15. But for the proposed fractional-order T- line model
as shown in Fig. 3.16, we can observe that the ℑ(S11) closely fits the causal
response obtained from the Hilbert transformation Hilbert{ℜ(S11)}. The er-
ror magnitude of ℑ (S11) from both models are compared in Fig. 3.17, where
a dramatic erro r r eduction is observed by the application of fraction o rder
T-line model. Note that the error magnitude is calculated by
e
11
= |Hilbert{ℜ(S11)}− ℑ(S11)|. (3.37)
Secondly, the causality of return loss (S21) is verified for both integer
and fractional-order T-line model. The c omparison between Imag(S21) and
causal response for both models are illustrated in Figs. 3.18 and 3.19. For the
traditional integer-order RLCG T-line model, the ℑ(S21) obtained from the
integer-order RLCG T-line model deviates from the c ausal response (1∼10
GHz) as de pic ted in Fig. 3.18. But for the proposed fractional-order T-line
model as shown in Fig. 3.19, we can observe that the ℑ(S21) of the fractional-
order T-line mode l closely fits the causal respons e. A clear comparison by
58 Design of CMOS Millimeter-Wave and Terahertz Integrated Circuits
Figure 3.14: Measurement, EM, integer-order and fractional-order
circuits simulation resu lts: characteristic impedance of CRLH T-line
(Z
0
).
Figure 3.1 5: Causality verification by Hilbert transformation: integer-
order T-lin e model S11.
error magnitude of ℑ(S21) from both models is illustrated in Fig. 3.20, where
a dramatic error reduction is also observed by the proposed frac tional-order
T-line model. Note that the error magnitude is calculated by
e
21
= |Hilbert{ℜ(S21)}− ℑ(S21)|. (3.38)
CMOS THz Modeling 59
Figure 3.16: Causality verification by Hilbert transformation:
fractional-order T-line mode l S11.
Figure 3.17: Causality verification by Hilbert transformation: error
magnitude comp arison of S11.
Note that since both e
11
and e
21
for the fractional-order T-line model are
rather small, the causality enforcement by (3.31) and (3.32) is not required.
The resulting frequency model can be directly used to estimate the time-
domain model by the rational fitting. As such, the best a ccuracy could be
ensured in the time-domain simulation such as Transient Analysis or Periodic
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.