PIERS ONLINE, VOL. 3, NO. 7, 2007
1102
Extraction of Subterahertz Transmission-line Parameters of
Coplanar Waveguides
Jingjing Zhang and Thomas Y. Hsiang
Department of Electrical and Computer Engineering and Laboratory for Laser Energetics
University of Rochester, Rochester, NY 14627-0231, USA
Abstract— Building on our previous work that compared computed propagation characteristics of picosecond pulses on coplanar waveguides (CPW’s) with experiments, we extract
transmission-line parameters, including resistance, inductance, conductance, and capacitance,
from S-parameters up to subterahertz frequencies. The distributed-element circuit model is analyzed for applications in CPW-based circuit design.
DOI: 10.2529/PIERS060912144405
1. INTRODUCTION
The coplanar waveguide (CPW) is commonly used as passive components and interconnects in
high-speed circuits. The ideal CPW contains infinitely wide ground planes and can be exactly
analyzed with conformal mapping [1]. Recent research interests, however, have focused on the
practically important CPW’s that have finite ground-plane widths and can only be analyzed by
numerical simulations [2–6]. In our previous studies [5, 6], we used full-wave analysis to characterize the subterahertz propagation characteristics of both wide- and narrow-ground CPW’s and
compared the results with experimental measurements and, in the case of wide-ground lines, with
closed-form analysis. Effects of ground-plane width and lateral line dimensions on the attenuation
and dispersion characteristics were investigated up to subterahertz frequencies. The good agreement with experiments validates the numerical simulations. In this paper we extend the study to
further extract the frequency-dependent transmission-line parameters from the numerical analysis.
A distributedelement circuit model of CPW is described up to subterahertz frequencies.
2. TRANSMISSION-LINE PARAMETER EXTRACTION
A transmission line of length l, characteristic impedance Zc , and propagation constant γ can be
modeled as a two-port network, as shown in Fig. 1. The ABCD matrix for the network is given
as [6]
#
¸ "
cosh γl Zc sinh γl
A B
= sinh γl
.
C D
cosh γl
Zc
·
(1)
Due to symmetry of the transmission line, the scattering matrix can be simplified with S11 = S22
and S12 = S21 . Provided that both the input and output impedances equal the reference impedance
Zo , the relationship between the S-parameters and the ABCD matrix becomes
2 + S2
1 − S11
21
2S
21
¡
¢
2 − S2 Z
1 + 2S11 + S11
0
21
B=
2S21
,
2 − S2
1 − 2S11 + S11
21
C=
2S21 Z0
2 + S2
1 − S11
21
D=
2S21
A=
(2)
PIERS ONLINE, VOL. 3, NO. 7, 2007
Figure 1: Two-port network of a transmission line.
1103
Figure 2: Distributed-element circuit model of a
transmission line.
Substituting A, B, C, and D into Eq. (1), the propagation constant and characteristic impedance
are expressed in terms of S-parameters as below:
2S
q¡ 21
¢
2 + S2 ±
2 − S 2 2 − 4S 2
1 + S11
1 − S11
21
11
21
s
.
2
(1 + S11 )2 − S21
Zc = ±Z0
2
(1 − S11 )2 − S21
e−γl =
(3)
Experimentally, γ and Z are the measured quantities and are directly compared with the computed S parameters [5, 6].
In an alternative model, Fig. 2, which is more usefully for the circuit designer, the transmission
line can be described with a distributed-element circuit model:
p
γ = r(R + jωL) (G + jωC) = α + jβ
,
(4)
R + jωL
Zc =
G + jωC
where R, L, G, C are the unit-length resistance, inductance, conductance, and capacitance; α and
β are the attenuation and phase constants of signal propagation, respectively. Solving Eq. (4) yields
the circuit-element parameters of the transmission line:
R = < {γZc }
L = = {γZc } /ω
.
G = < {γ/Zc }
C = = {γ/Zc } ω
(5)
Since the S-parameters are readily obtained from our measurements or simulations, Eqs. (3-5) then
provide the algorithm from which the circuit-model parameters, shown in Fig. 2, can be extracted.
3. RESULTS AND DISCUSSION
In our previous work [5, 6], two classes of CPW lines (made of Au on GaAs substrates) were
studied experimentally and numerically. The first class contains ground planes at least 10 times
wider than the center conductor or the conductor spacing (the “wg lines”), closely approximating
an ideal CPW, while the second class uses narrow ground planes with the same width as the center
electrode (the “ng lines”). Simulations of signal propagation that make use of full-wave analysis
were compared with and verified by experimental data.
For computing the distributed-element circuit parameters, we chose to use the numerically simulated S-parameters as the starting point and solve Eqs. (3) and (5). Examples of these parameters
are shown in Fig. 3 that correspond to the CPW’s previously studied [5].
The strongest frequency-dependent parameter is the resistance term, R. At low frequencies, R
approaches a constant, corresponding to the quasi-static limit where the resistance per unit length
of the CPW can be calculated by
1
R=
,
(6)
σW t
where σ is conductivity of the conductor metal, W and t are the width and thickness of the center
conductor. For the 10-µm CPW, R is 7.63 Ω/mm and, for the 50-µm CPW, it is 1.53 Ω/mm; both
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(a)
(b)
(c)
(d)
Figure 3: Transmission line parameters R, G, C, L of CPW’s extracted from S-parameters. wg and ng refer
to wide ground and narrow ground, respectively. 10 µm and 50 µm indicate the lateral line dimensions of
the CPW’s.
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closely matching the values in Fig. 3(a). As frequency increases (up to 100 GHz), the skin depth
becomes smaller than the conductor thickness and surface resistance increases as the square-root
of frequency. In this region where conductor loss dominates, wider lines would have lower loss
compared with narrower lines, which can be seen from Fig. 3(a). At still higher frequencies, above
100 GHz, radiation loss becomes dominant. In this case, the narrower lines would have lower loss.
The crossover of the value of R’s for different line-widths as frequency is increased is prominently
shown in the data.
The width of the ground plane also has an effect on the value of R. Comparing the left and
right curves of Fig. 3(a), it can be seen that at higher frequencies, the ng lines have a lower value
of R, as a result of the reduced coupling of surface modes with the ground conductors [3, 5]. This
effect shows that the ng lines are preferred over the wg lines for high-frequency applications.
A secondary loss component for the CPW is its conductance term, G. In the quasi-static limit,
G arises when the dielectric contains a finite loss tangent. The value of G increases with decreasing
line separation. For each of the examples shown in Fig. 3(b), the line separation equals the line
width, thus giving rise to a higher conductance for the narrower lines. In the 100’s GHz range,
we observe an increase in the value of G, followed by a cross-over in frequency for the two line
widths shown, similar to that observed for R. A plausible explanation is that, in this region, part
of the radiation loss contributes to the coupling between the conductors and increases G; since
the loss due to the conductance is two orders of magnitude lower than that due to the resistance
for the samples considered, a small part of the radiation loss dominates the dielectric loss at high
frequencies.
The other two terms are the capacitive and inductive terms. In the low-frequency region and
for the wg lines, capacitance is determined by [7]
C = (εr + 1)ε0 2
K(k)
,
K 0 (k)
(7)
where εo is the vacuum permittivity, εr is relative permittivity of the substrate, and K(k)/K’(k) is a
geometry coefficient that depends only on the line separation-to-width ratio. Using this expression,
the capacitance of wg CPW’s is found to be 0.157 pF/mm and closely approximates the quasi-static
values in Fig. 3(c). We also observe that the ng lines have almost identical capacitance to that of
the wg lines, showing that Eq. (7) is applicable irrespective of the ground-plane widths. Further,
the capacitance for the 10-µm lines remains nearly constant to the highest frequency considered,
showing that the surface modes which dominate the high-frequency losses have only a small effect
on the overall field distribution. For the 50-µm lines, we observe a drop in C at higher frequencies;
this is attributed to a computational artifact.
Inductance is proportional to the magnetic flux contained in the CPW. Its dominant component
is the external inductance, associated with the magnetic flux between the electrodes, and can be
expressed as:
1
L= 2 ,
(8)
c Ca
where c is speed of light and Ca is the capacitance with the substrate replaced by air. Using Eq. (7),
we obtain L = 0.49 nH/mm, in agreement with the quasi-static values in Fig. 3(d). Secondary (and
frequency-dependent) components in L include the magnetic flux in the conductor and the surface
modes that modify the field distribution. These two effects appear to be negligible, as evidenced
by the nearly flat frequency dependence. The drop in L at higher frequencies for the 50 µm lines is
attributed to a computational artifact only.
4. CONCLUSIONS
In summary, we have extracted broadband transmission-line parameters R, L, G, C of CPW’s from
S-parameters computations, up to subterahertz frequencies. The origin, physics, and validation of
the circuit model are discussed. Examples are given that demonstrate how these model parameters
are affected by changing ground-plane widths and lateral dimensions.
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