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

by Cascade Microtech. RF-PADs on both sides are de-embedded from the

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 [121]. 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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