ast- Dispersive Periodic Slow-Wave Structures >
Accepted for publication in IEEE Transaction on Microwave Theory and Techniques, to appear on May 2008


1

AbstractSlow-wave structures using distributed periodic
inductive and capacitive loadings have found many microwave
circuit applications as left-handed (band-pass) or right-handed
(low-pass) transmission lines. A large slow-wave factor could
result in a much smaller passive component, but also a much
lower band-gap (cut-off) frequency and a larger dispersion. This
paper addresses the issues and the design trade-off between slow-
wave factor, group delay (dispersion), and the cut-off frequency
of a right-handed (low-pass) quasi-lumped transmission line. A
new two-layer transmission line structure using three-
dimensional substrate metallization with a slow-wave factor of
5.8 is designed. A prototype of a 3GHz branch-line coupler with
a 70% size reduction using such a transmission line structure is
fabricated and tested.

Index TermsElectromagnetic band-gap, slow wave, periodic
structures, branch-line coupler, and dispersion.
I.
I
NTRODUCTION
he demands for device miniaturization together with the
advances of microelectronics manufacturing technology
lead to system on chip; this is a process by which an
entire communication system can be built on a small-size (less
than 1 cm by 1cm) semiconductor-wafer packaged as a chip.
As the radio and digital chip size becomes smaller and
smaller, RF passives remain a bottleneck for device
miniaturization on circuit boards and on chips. There has
been much less attention to the antennas and passive
components, as compared to the semiconductor process
technology. The size of RF passives (distributed circuits)
including antennas, baluns, and transmission-line components
is usually proportional to their electrical lengths (physical
length to wave length ratio). Size reduction can be achieved
by enhancing the transmission-line slow-wave factor, defined
as the wavelength ratio of the free-space wave to the guided
wave. The slow-wave factor can be enhanced by embedding a
transmission line in high dielectric-constant materials (smaller
wavelength), such as doped semiconductors and
ferromagnetic materials, in the form of a microstrip line or
coplanar wave guides [1-8].
The use of distributed inductive or capacitive periodic
loadings to a transmission line to form a slow-wave structure

Manuscript received May 7, 2007 revised August 28, 2007.
Chengzhi Zhou and H.Y. David Yang are with the Department of Electrical
and Computer Engineering, University of Illinois at Chicago, Chicago, IL
60607 USA (phone:312-996-2509; fax: 312-996-6465; e-mail:
czhou2@uic.edu
,
hyang@ece.uic.edu
).

has been known for a long time [9]. Depending on the
loadings, the slow-wave structure could be used for low-pass,
band-stop, or band-pass filters. In different contents, the
periodic slow-wave structures are called either right-handed
(low-pass) or left-handed (high-pass or band-pass)
metamaterial transmission lines or electromagnetic band-gap
(EBG) structures [9-16].
Tailoring the electromagnetic band-gap (EBG) and the
slope of the dispersion curves (double-negative metamaterials)
has found many useful applications. This paper discusses the
use of high-density three-dimensional substrate metallization
for passive component miniaturization. The periodic loadings
are wound into multiple layers to form distributed capacitors
and inductors (quasi-lumped circuits) within each unit cell.
The emphasis is on the right-handed slow-wave structures,
due to less dispersion and more stable slow-wave factor over a
broad-frequency range.
A goal of this paper is to provide the design criteria of
maximum slow-wave factor under the constraint of the
fundamental mode cut-off frequency, line impedance, group
delay, and the available space. Analytic formulas are
provided to illustrate the design trade-off. A new three-layer,
slow-wave structure is proposed to demonstrate the design
methodology. Finally, a branch-line coupler, evolved from
this slow-wave structure, is designed and tested. The results
show significant size reduction as compared to the use of a
normal microstrip line structure.
II. C
HARACTERISTICS OF PERIODIC SLOW
-
WAVE STRUCTURES

A basic left-handed transmission line structure (also called
a metamaterial transmission line) [17-23] is a periodically
loaded shunt inductance and series capacitance with high-pass
characteristics (dc isolation). In the first transmission band,
d
d /
(group velocity) is positive while
/
(phase
velocity) is negative. A left-handed transmission line is
usually dispersive (group velocity varies with frequency) for
guided-wave applications.
On the other hand, the cascade of series inductance and
shunt capacitance in a periodic transmission line is a right-
handed structure, where phase and group velocities are in the
same direction. As an example, the equivalent circuit of a RH
structure and its dispersion diagram are shown in Fig. 1.
Common transmission line structures, such as a microstrip line
and a co-planar waveguide, are right-handed transmission
lines with continuously distributed L-C components. The
fundamental mode is inherently low-pass and a slow-wave
mode. The transmission line dispersion curves can be
Design Considerations of Miniaturized Least-
Dispersive Periodic Slow-Wave Structures
Chengzhi Zhou, Student Member IEEE, and H.Y. David Yang, Fellow IEEE
T
>
Accepted for publication in IEEE Transaction on Microwave Theory and Techniques, to appear on May 2008


2
tailored through the periodic (distributed) loading of inductive
and capacitive elements. The distributed L and C are
frequency dependent. In the 2
nd
transmission band, the
equivalent circuit could become effectively a series capacitor
with a shunt inductor. In such a case, the right-handed line
becomes a left-handed line in the 2
nd
transmission band, as
shown in Fig. 1b.

f
First Brillouin
Zone Boundary
SWF
1
0
Band
Gap
Air line
f
c
Slow-wave
transmission zone
L/2
L/2
C
1
2
d
(a)
(b)

Fig. 1. Unit cell equivalent circuit of a periodic slow-wave
structure and its dispersion diagram.

Although a microstrip-line periodically loaded with shunt
capacitors or serial inductors can increase the slow-wave
factor, the wave length reduction is limited by its long
transmission line sections and is space inefficient. In this
paper, the investigation focuses on the use of distributed
inductance and capacitance to form the slow-wave
transmission lines as indicated in Fig. 1. This set up is also
referred to as quasi-lumped set up and is distinct from the
inductance and capacitance loading on a microstrip line. The
inductance and capacitance shown in Fig.1 are distributed
three-dimensionally over a unit cell. For circuit and antenna
applications, sections of such a slow-wave line are used in the
frequency below the first band-gap (below the cut-off
frequency f
c
). In the first pass band below the cut-off, the
slow-wave factor usually increases with frequency. A useful
slow-wave factor for evaluation is likely at half of the cut-off
frequency f
c
where it is larger than the dc case and is not very
dispersive yet.
For a linear transmission system, group delay is another
important factor for evaluating the transmission-line quality.
For a slow-wave unit cell (shown in Fig. 1), within the pass
band, a larger value of group delay corresponds to larger
dispersion. It is also referred to as envelope delay, which
indicates the delays experienced by the envelope of the
transmission packets [18].
For a matched transmission line system with the physical
length l and the propagation constant , the group delay is
given as
( )
l
g =
. (1)

Using Bloch-Floquet theorem, for a slow-wave structure
with a unit cell shown in Fig. 1, its propagation constant
(
)
j +
=
and characteristic impedance
c
Z
are the found as
(assuming lossless) [24]


2
2
1
cosh
LC
d =
(2)
and
4
2
1
LC
C
L
c
Z
=
(3)
The cut-off frequency
c
f
can be predicted by setting (2)
equals to -1.

LC
c
f 1
=
. (4)
The group delay of the unit cell below the cut-off frequency
can be derived as
4
2
1
LC
LC
g =
. (5)
The primary parameters for the periodic slow-wave line
design are the slow-wave factor, group delay, and
characteristic impedance (assuming lossless). They can be
expressed in terms of the cut-off frequency as






=
2
2
2
1
arccos
2
c
f
f
fd
c
SWF , (6)
2
1
1





=
c
c
g
f
f
f
, (7)
and
2
1
0





=
c
f
f
Z
Zc
, (8)

where
C
L
Z
=
0
.
Eqs. 6-8 reveal the design criteria of a periodic slow-
wave structure. The unit cell length should be as small as
possible; the inductance and capacitance (per unit cell) should
be as large as possible, but not to lower the cut-off (band-gap)
frequency too much. A smaller cut-off frequency corresponds
to larger frequency-sensitive group delay, a larger slow-wave
factor, and a less usable frequency range. A small unit cell
with large inductance and capacitance requires a transmission