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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