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Comparison of network analysis methods for computing complex propagation
coefficients of dispersive transmission lines
Article in Microwave and Optical Technology Letters · March 2014
DOI: 10.1002/mop.28183
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Remillard, Microwave and Optical Technology Letters, Submitted July 1, 2013, Accepted for publication
Comparison of network analysis methods for computing complex
propagation coefficients of dispersive transmission lines
S.K. Remillard, Caitlin Ploch, Kyle McLellan, V. Andrew Bunnell
Physics Department, Hope College, Holland, MI 49423, USA
Abstract — Transmission lines of various degrees of dispersion are often designed and fabricated for use
in high speed circuits and in metamaterials. The quick generation of dispersion diagrams and attenuation
coefficient curves can be made from S-parameter measurements, with S21 alone being sufficient for
generating dispersion diagrams. S21 is also sufficient to accurately measure the attenuation coefficient,
group velocity, phase velocity and effective index of refraction of highly dispersive transmission lines.
Index Terms — Dispersion, Scattering Parameters, Transmission Lines, Microwave Measurement
I. INTRODUCTION
Dispersion, which can result from resonance in a device’s transmission characteristics, can be harnessed
in the case of metamaterials [1], and needs to be modeled, measured and controlled in the case of high speed
signal transmission.
Highly dispersive transmission lines have been shown to effectively control
simultaneous switching noise in high clock speed circuits [2]. Transmission lines with discontinuities are
highly dispersive with complex dispersion diagrams that need to be quickly measured with a vector network
analyzer (VNA), or modeled with an electromagnetic field simulator. Measurements of the dispersion
characteristics of periodic structures have followed two approaches with differing degrees of computational
complexity.
The purpose of this short paper is to present and compare these two methods and to
demonstrate the simple realization of a dispersion diagram using data from a single calibrated S21 sweep
with a VNA.
Periodic variations in geometry produce a photonic bandgap at a wavelength commensurate with the
lattice dimension even at microwaves [3]. With a periodic variation in wave number, a periodic
transmission line is, in fact, a photonic crystal described by a Bloch wave function. Even transmission lines
without designed features are dispersive at high frequencies [4] due to the distributed reactance, causing a
Remillard, Microwave and Optical Technology Letters, Submitted July 1, 2013, Accepted for publication
distinction between the group and phase velocities. The complex propagation constant can be de-embedded
from the vector S-parameters [5] in order to highlight the dispersion features, and this has been done in
recent work on periodic transmission lines [6,7]. This short paper will describe two methods to find
dispersion features in the S-parameters. One of these two methods results from simplification of the other
method, and is justified by the analogy between the wave number of electromagnetic fields and the crystal
momentum of solid state physics.
Electromagnetic wave dispersion occurs in numerous other physical systems, most notably optical
multilayer thin films. These optical band periodic variations in wave impedance are used as optical filters
[8], producing forbidden transmission bands. The optical response to periodic impedance variation is also
well known to exhibit a crystalline dispersion [9]. Similar to optical photonic crystals, the variation of
index of refraction through electrical permittivity [10] and magnetic permeability [11] has also been used to
realize photonic crystals at microwave frequencies. The use of the complex S21 to easily find the index of
refraction of the transmission line will be shown.
II. COMPUTING TRANSMISSION LINE DISPERSION DIAGRAMS
The complex propagation constant, γ=α+jβ was found two ways. The first approach uses a direct deembedding of the propagation constant from measured S-parameters that was developed for characterization
of IC interconnects [5]. The second approach uses the measured complex insertion loss. Benchmarking
these two methods validates that group delay dispersion and crystal momentum dispersion are equivalent
quantities.
Method 1. The wave number β in meter-1 and the attenuation coefficient α in nepers/meter of a real
device are related to the reflection and transmission S-parameters through [5]
Remillard, Microwave and Optical Technology Letters, Submitted July 1, 2013, Accepted for publication
2 2
1 − S112 + S 212 + (1 + S 112 − S 21
) − ( 2 S11 ) 2
e α L e jβ L =
2 S 21
(1)
which is derived from the ABCD matrix of the transmission line. S11 and S21 are the complex S-parameters
S11 = S11 e jφ11 and S 21 = S 21 e jφ21
(2)
and φ11 and φ21 are the reflection phase and the insertion phase, in radians, measured directly from the
vector network analyzer with full 2-port calibration. L is the physical length between the ports of the
transmission line.
Method 2. With the phase delay, τp, calculated from τp=φ21/ω, the phase velocity is vp=L/τp. The
crystal momentum, which is also the imaginary part of γ, can be computed from the phase velocity using
β=
ω
vp
=
φ21
(3)
L
giving two ways, (1) or (3), to find the wave number from the measured S-parameters where (1) also
includes the loss. The wave’s attenuation coefficient α is found in nepers per meter directly from the S21
measured using a calibrated Agilent 8753E VNA using


α ≈
−
(
1
ln ( S 21
L
)
| S 21 |>>| S11 |
(4)
)
 1 ln 1 − 2 S 2 cos ( 2θ ) + S 4 − 1 ln ( S ) | S |>>| S |
11
11
11
21
11
21
 2 L
L
where the second line is a limiting case of Eq. (1), ignoring S21 in comparison to S11, and is useful outside a
pass band. Eq. (4) does not cover all frequencies. However, in the limits shown, Eq. (1) reduces to these
expressions. Eq. (1) for β reduces to Eq. (3) in all cases.
In both methods, the measured insertion phase φ21 will need to be unwrapped since the VNA returns
phase values confined between ±180o.
determined.
To accomplish this, the frequency slope of φ21 needs to be
Assuming linear dispersion at frequencies below measurement, the value of φ21 at the
Remillard, Microwave and Optical Technology Letters, Submitted July 1, 2013, Accepted for publication
minimum measurement frequency can be found by forcing the phase to zero at zero frequency. Low
frequency phase linearity is quite a fair assumption especially if the VNA measurement is started at a low
frequency (such as 30 MHz). Besides the offset at the lowest measurement frequency, 360o is added to the
phase each time it jumps by 360o.
III. RESULTS
The equivalence of Methods 1 and 2 in calculating the wave number β and attenuation coefficient α was
first established for a 50Ω transmission line of uniform width fabricated on high loss 1.5 mm thick FR4
using photolithography. The wave number β was computed at each frequency, f, and plotted using Eq. (1) of
Method 1 and subsequently Eq. (3) of Method 2. The result shown in Figure 1a confirms the equivalence of
the two methods in computing wave number, despite the added mathematical complexity of Method 1. Eq.
(1) of Method 1 was first used, followed by the simpler |S21|>>|S11| part of Eq. (4). Since the transmission
line of Figure 1 was fabricated without periodicity, |S11|<<|S21| at all frequencies. The result shown in Figure
1b confirms the equivalence of the two methods for the FR4 transmission line.
A comparison of Methods 1 and 2 was applied to a highly dispersive transmission line similar to those
reported in Ref [7] fabricated on low loss R3003 Duroid [12] and synthesized with periodic variations in
impedance to have a rejection band between 2 GHz and 4 GHz, with measured S-parameters shown in
Figure 2. Figure 3a shows the result of computing the propagation coefficient β using both Eq. (1) of
Method 1 and Eq. (3) of Method 2. The similarity in the curves demonstrates the equivalence of the two
methods and the ability of Eq. (3) to accurately account for the effects of periodicity. Through this
similarity, the equivalence of crystal momentum, Eq. (3), and wave number, Eq. (1), is bolstered.
The attenuation coefficient α was first computed using Eq. (1) of Method 1. Because of the band gap in
the highly dispersive structure, both parts of Eq. (4) had to be used. The frequency points for the
Remillard, Microwave and Optical Technology Letters, Submitted July 1, 2013, Accepted for publication
transmission line were divided into “out-of-band” and “in-band” regions according to Figure 2. For regions
outside the band gap, using the first part of Eq. (4) produced nearly identical attenuation coefficients found
using Eq. (1), including the Fabry-Perot features evident outside the bandgap. Inside the bandgap, where
reflection dominates over transmission, the second part of Eq. (4) was used to calculateα. This resulting
section of the curve also matched the corresponding Method 1 section, confirming that the two methods are
nearly equivalent in their calculations of the attenuation coefficient α.
The group and phase velocities, as well as the group and phase indices, are also directly computed from
the insertion magnitude and phase. Using vp=β/ω for the phase velocity, the group velocity is then
vg =
vp
1−
ω dv p .
v p dω
(5)
Within the context of Method 2, the group index of refraction, ng=c/vg, is related to the phase index
through the well-known inhomogeneous differential equation [13],
ng = n p + ω
dn p
dω
(6)
which is of the form
F ( x) = y + xy'
(7)
and can be solved numerically using the Euler method with ng coming from a data array generated by Eq.
(5). All of this comes from the measured insertion magnitude and phase alone. Phase index
can also be solved within the context of Method 1 by multiplying β/f from the dispersion diagram, Figure
3(a), by c/2π. These two calculations of the phase index of the highly dispersive line are compared in
Figure 4. Above the highly dispersive region of 2 to 4 GHz, α and β from the two methods do not agree as
well. Likewise and possibly related to that, in the high frequency linear dispersion regime np does not return
to its low frequency, linear dispersion value.
Remillard, Microwave and Optical Technology Letters, Submitted July 1, 2013, Accepted for publication
The group index, ng, computed from the measured group delay τg as well as from the inverse slope of
the dispersion diagram in Figure 3 is shown in Figure 5. Note that far from the bandgap, the group and
phase indices have similar values, which should in principle be the same due to the linear dispersion.
Fabry-Perot resonances caused by impedance mismatch between the transmission line and the 50Ω cables
are responsible for the oscillatory behavior outside the bandgap [7]. The agreement found using the
measured group delay and the dispersion diagram shows the equivalence between group delay dispersion
and crystal momentum dispersion of periodic transmission lines.
The group velocity L/τg and the phase velocity ω/β were computed at each frequency point. The group
velocity becomes very small, and in principle approaches zero, at the band edge. Inside the bandgap, the
group velocity oscillates between the value that the phase velocity has at the band edge and a much larger
value. This differs from the phase velocity which is smoothly varying outside the bandgap and of constant
value throughout the bandgap. It may seem strange to contemplate a phase velocity in the bandgap since
the wave is evanescent and does not propagate. However, being of finite length, the transmission line still
couples energy between its ports, and the speed of this energy is governed by the phase velocity and is
ultimately limited by special relativity. As the number of periods is increased, the group velocity is
expected to increase without bound, similar to the quantum mechanical Hartman effect [14].
IV. CONCLUSION
Dispersion diagrams can be produced for transmission line devices ranging from barely dispersive to
strongly dispersive directly from the insertion phase, yielding results consistent with the more rigorous full
S-Parameter analysis. The propagation coefficient, β, found from Eq. (1) is corresponds to the insertion
phase divided by the length of the device, regardless of whether or not the device is dispersive, and
regardless of whether or not the device has significant attenuation. In addition, the attenuation coefficient α
is accurately computed from the simplified form of Eq. (1) that appears in the second line of Eq. (4).
Remillard, Microwave and Optical Technology Letters, Submitted July 1, 2013, Accepted for publication
ACKNOWLEDGEMENTS
This material is based upon work supported by the National Science Foundation under Research
Experiences for Undergraduates Grant No. PHY/DMR 1004811.
FIGURE CAPTIONS
Figure 1
50Ω transmission line on lossy FR4. a. (left) The wave number β determined using (1) and (3). b. (right) The
attenuation coefficient α determined using Eq. (1) and the “out-of-band” part of Eq. (4).
Figure 2.
S11 and S21 parameters of a highly dispersive transmission line measured with an Agilent 8753E VNA.
Figure 3
both parts of (4).
a. The dispersion diagram for β determined using (1) and (3). b. The attenuation coefficient α determined using (1) and
Figure 4
Phase index, np, computed with Methods 1 and 2, compared to np in the linear dispersion regime.
Figure 5
Group index for the periodic transmission line determined from group delay as well as from the inverse slope of the
dispersion diagram.
Remillard, Microwave and Optical Technology Letters, Submitted July 1, 2013, Accepted for publication
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Remillard, Microwave and Optical Technology Letters, Submitted July 1, 2013, Accepted for publication
.
Figure 1a. 50 Ω transmission line on lossy FR4. The wave number β determined using (1) and (3).
Figure 1b. 50 Ω transmission line on lossy FR4. The attenuation coefficient α determined using Eq. (1) and
the “out-of-band” part of Eq. (4).
Remillard, Microwave and Optical Technology Letters, Submitted July 1, 2013, Accepted for publication
Figure 2. S11 and S21 parameters of a highly dispersive transmission line measured with an Agilent 8753E VNA.
Remillard, Microwave and Optical Technology Letters, Submitted July 1, 2013, Accepted for publication
Figure 3 a. The dispersion diagram for β determined using (1) and (3).
Figure 3 b. The attenuation coefficient α determined using (1) and both parts of (4).
Remillard, Microwave and Optical Technology Letters, Submitted July 1, 2013, Accepted for publication
Figure 4. Phase index, np, computed with Methods 1 and 2, compared to np in the linear dispersion regime.
Figure 5. Group index for the periodic transmission line determined from group delay as well as from the inverse
slope of the dispersion diagram.
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