3376 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 11, NOVEMBER 2012
A High Slow-Wave Factor Microstrip Structure
With Simple Design Formulas and Its
Application to Microwave Circuit Design
Wei-Shin Chang and Chi-Yang Chang, Member, IEEE
Abstract—This paper proposes a new m icrostrip slow-wave
structure. The unit cell comprises a Schiffman section of me-
ander line and a shunt open-circuited stub. No via-holes an d
ground-plane patterns are required. Simple design formulas can
be used to obtain line parameters, such as the characteristic
impedance and phase velocity. According to the analysis, the
characteristic impedance and slow-wave factor of the proposed
slow-wave lin e can be independently controlled by merely two
layout parameters. The p roposed uniplanar structure only re-
quires a single-layer substrate and is simply constructe d u sing
the conventional printed circuit board manufacturing process.
A branch-line and a rat-race c oupler were designed and fabri-
cated using the proposed structure to demonstrate its feasibility.
Their sizes are only 8.49% and 4.87% of the conventional ones,
respectively. This n ovel slow-wave structure should find wide
applications in compact microwave circuits.
Index Terms—Branch-line coupler, microstrip line, periodic
structure, rat-race coupler, slow-wave structure.
I. INTRODUCTION
A
MICROSTRIP line plays an important role in microwave
circuits since it can be fabricated by photolithographic
processes and is easily integrated with passive and active mi-
crowave devices. The length of the con ventional microstrip line
is dominated by the dielectric constant
[1] so that the circuit
constructed by the conventional microstrip line cannot reduce
the phase velocity less than
of the free-space light ve-
locity. Therefore, the circuit may occupy a large area, which
results in a ser ious problem for m ini atur izati on. To reduce the
circuit size, the high dielectric-constant substrate may be em -
ployed.
Slow-wave guid ing structures have been extensively studied
to reduce the circuit size [2]–[20]. The m echanism behind the
slow-wave propagation is to separately store the electric and
magnetic energies a s much as possible in the guided-wave
media. Among these structures, this paper focuses on the
microstrip slow-wave structures where the conductor-back ed
ground plane is required. For the microstrip line, slow-wave
Manuscript received March 25, 2012; revised August 14, 2012; accepted Au-
gust 20, 2012. Date of publication September 20, 2012; date of current version
October 29, 2012. This work was supported in part by the National Science
Council (NSC), Taiwan, under Grant NSC 9 9-2221-E-009-050-MY3.
The authors are with the Institute of Communic ations Engineering, National
Chiao Tung University, Hsinchu 300, Taiwan (e-mail: aa494412338@hotmail.
com; mhchang@cc.nctu.edu.tw).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Dig
ital Object Identi fier 10.1109/TMTT.2012.2216282
TABLE I
C
OMPARISON OF MICROSTRIP SLOW-WAV E STRUCTURES
structures can be construct
ed in multilayer substrates [ 6 ] –[ 8].
However, to simplify the fa
brication process and to maintain
the low cost, the slow-wav
e structures with only a sin gle- layer
substrate are more prefer
able [9]–[20]. They can be realized
on a single-layer subs
tratewithaperiodicdielectricconstant
[9], or they have the pe
riodic perturbations on the signal and
ground planes [10]–[
20]. Nonetheless, they m ay not have a
simple and efficient
synthesis method so that the try and error
procedure wou ld be n
ecessary for the prescribed characteristic
impedance an d slow
-wave factor. In add it ion, the substrate
of some structure
s may be required to be suspended due to
the d efected gro
und plane. In [19], we pro pose a slow-wave
transmission l
ine with the signal strips and the inserted ground
strips period
ically loaded in the internal part of the conven-
tional micros
trip line. It has a sim ple structure and a higher
slow-wave fa
ctor compared to the previous studies. However,
the a djustm
ent of the dimension al parameters in this structure
influences
the characteristic impedance and the slow-wave
factor sim
ultaneously so that the co ntrol of these tw o param-
eters wou
ld not be straightforward. In addition, it may be
difficul
t or impossible to drill many via-holes in a small region
due to fa
brication tolerances and substrate intensity. Table I
compar
es seve ral microstrip slow-wave structures.
In this p
aper, we propose a novel slow-wave microstrip
struct
ure that comprises a Schiffman section of m eander line
and a s
hunt open-circuited stub. Its dimensions are easily
synth
esized for the prescribed characteristic impedance and
slow
-wave f actor. In other words, the characteristic impedance
and t
he slow-wave factor can be controlled individually, which
sol
ves the problem mentioned above. The proposed slow-wave
0018-9480/$31.00 © 2012 IEEE
CHANG AND CHANG: HIGH SLOW-WAVE FACTOR MICROSTRIP STRUCTURE 3377
Fig. 1. (a) Proposed slow-wave microstrip structur e. ( b) Meander line portion
and its equivalent lum pe d elements.
: characteristic impedance of the m i-
crostrip line with a wid th
. (c) Op en -cir cuited stub an d its equivalent lumped
elements.
and : characteristic impedances of th e microstri p line with
widths
and , respectively.
line has an extremely low fabrication cost due to the conven-
tional single-layer printed circuit board (PCB) process without
via-holes and ground-plane patterns. The design equations and
characteristics of the proposed slow-wave structure are exam-
ined in de tail . Finally, we design a branch-line and a rat-race
coupler usi ng the proposed transmission lin e to demonstrate its
applications.
II. P
ROPOSED SLOW-WAV E STRUCTURE
Fig. 1 d epicts the schematic of the proposed slow-w ave mi-
crostrip structure. Each unit cell consists of a meander lin e and
a shu nt open -circuited stub. The lengths of the unit cell corre-
sponding to the m eand e r line portion and the shunt stub por-
tion are
and , respectively, and their total transverse widths
are both
. T hereby, the length and width of the unit cell are
and , respectively. The spacings between adjacent
lines are all
. The line width of a meander line is .Thus,
. The characteristic impedances of the me-
ander line with a width
and the shunt stub with a width
are and , respectively. The slow-wave transmission line is
characterized by the characteristic impedance
and the prop-
agation constant
as follows [21]:
(1)
(2)
where
and are the total inductance and capacitance o f the
unit cell, respectively. The idea behind the pro posed slow-wave
structure is that the in ductance
is mainly attributed to the
high-impedance meander line (i.e.,
), and the capacitance
is primarily controlled by the shunt open-circuited stub (i.e.,
). As a result, we can choose as high as possible first and
then determine
according to . Furthermore, in practical
calculations, all inductances and capacitances associated with
the meander line portion and the shunt stub portion should be
taken into account in each unit cell. In applications,
is usually
long to obtain large capacitance, and the coupling is sm all for
wide coupled lines. Thus, in the followin g discussio n, for sim-
plicity, we ignore the coup lin g between the meander line a n d
the shunt open-circuited stub and the influ ence o f the T-jun c-
tion effect.
First, consider the hig h- impedance meander l ine in the unit
cell, as shown in Fig. 1(b). The configuration of the meander line
can be regarded as the parallel coupled lines where one end is
connected, which forms a Schiffman section. For fixed
and
, the image impedance and phase constant of a Schiffman
sectionaregivenby[22]
(3)
(4)
where
and are th
e even- and odd-mode impedances of
the parallel cou
pled lines, respectively.
is the electrical length
of the transmiss
ion lin e. In the proposed unit cell ,
,
where
is t h e pr
opagation constant of the microstrip line w ith
a width
.Sin
ce in the real situation
and
,(4)isfu
rther reduced to
(5)
Moreover, on the basis of the Taylor-series expansion of cosine
function, (4) can also be written as
(6)
The pro pagation constant
of a Schiffman section is
(7)
Comparing (5) and (6),
is proportional to , and consequently
from (7),
is p roportio nal to as well as to the frequency. The
inductance
and the capacitance due to the meander line
in the unit cell can be calcu lated as follows:
(8)
3378 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 11, NOVEMBER 2012
(9)
The second part of the above two equations corresponds to the
three short lines with a length
in Fig. 1(b).
Now, consider the sh unt open-circuited stub in the unit cell,
as show n in Fig. 1(c). The inductance
and the capacitance
due to this portion are derived as
(10)
(11)
where
and are the characteristic impedance and propa-
gation constant, respectively, of the mi crostrip line with a width
. and are the characteristic impedance and propaga-
tion constant, respectively, of the micro strip line with a width
. S um marizing from (8)–(11), the per-unit-length inductance
and capacitance in the propo sed unit cell are
(12)
(13)
It is worthwhile to discuss the proposed unit cell and the
above equations in more detail. Note that the longer the length
is, the smaller the characteristic impedance is. Thereby,
increases based on (11). Under this condition, since is mai nly
determined by
, will increase. is prim arily controlled
by
,and is attributed to the meander line portion so that
is almost not influenced by belonging to the shunt open-cir-
cuited stub. In summ ary, as
becomes longer, increases and
remains almost constan t. Consequ e ntly, from (1), the char-
acteristic impedance
of the unit cell becomes smaller. This
indicates that
can be easily changed by adjusting for fixed
and .
Since usually
, ,and in prac-
tical applications, from (8)–(11),
and can be simplified as
follows:
(14)
(15)
Accordingly,
and are both proportional to so that from
(1),
remains constant as changes. Moreover, according
to (2),
is proportional to . These two properties are very
important in the d e sign of th e proposed s lo w-wave line since
we can control the propagation constant
and the slow-wave
factor by adjusting
without changing .
To demonstrate the p roperty of the proposed stru cture, the
substrate with a dielectric constant
and a th ickness
mm is taken a s an exam ple. The electrical param-
eters of the microstrip line in the design equations (i.e.,
,
, , , , , ,and ) are quickly obtained by
using the circuit simulator AWR Microwave Office [23]. The
commercial full-wave electromagnetic (EM) sim ulation soft-
ware Sonnet [24] is used to compare the calculated and simu-
Fig. 2. (a) Slow-wave factor, (b) characteristic impedance ,and(c)per-
unit-guided wavelength loss versus total transverse width
at 0.9 GHz for
mm (35.36- case), mm ( 50 - case), and mm
(70.7-
case).
lated results. In the following discussion, we fix mm
(i.e.,
) based on the allowable fabrication process
and
mm (i.e., mm).
Taking the frequency at 0.9 GHz as an example, Fig. 2(a)
plots the slow-wave factor defined by
versus total trans-
verse width
,where is the free-space wavelength and
is the guided wavelength of the proposed slow-wave line. It
is seen that the proposed structure has a very high slow-wave
CHANG AND CHANG: HIGH SLOW-WAVE FACTOR MICROSTRIP STRUCTURE 3379
Fig. 3. Characteristic impedance versus length for mm.
factor. Furthermore, the slow-wave factor increases as the width
of the unit cell increases, and these two parameters are linearly
proportional to each other. The r esults from the equations and
the EM simulations are in good agreement with each other. The
small discrepancy between the calculated and sim ulated results
is m a in ly due to the coupling between the shunt open-circuited
stub and the meander line. This is especially obvious for the
mm case since the coupling is stronger for narrow
coupled lines. Fig. 2(b) shows the characteristic impedance
of the proposed slow-wave line versus . Apparently, as
varies from 2.5 to 4 mm, the slow -w ave factor increases signif-
icantly, whereas
remains almost constant. This is con sistent
with the theoretical results. For the substrate with a loss tangent
of 0.0021 in the simulatio n, Fig. 2(c) gives the per-unit-guided
wavelength lo ss ver sus
for the three sl ow- wave lines with
different
. The simulated result indicates that the loss has a
small variation as
changes.
Fig. 3 shows th e characteristic impedance
versus len gth
for mm. Apparently, the long er th e len gth is, the
smaller the characteristic im pedance
is, which corresponds
with the above d iscussion . The calculated and simulated results
are consistent with each other.
To observ e the dispersive property of the proposed structure,
take the dimensions in Fig. 2 as an example. For
mm,
Fig. 4(a) plots the slow-wave factor versus frequency from 0.5
to 1.3 GHz, which covers more than the operating frequency
range in the following circuit examples. Fig. 4(b) shows the
simulated characteristic impedance
versus frequency for dif-
ferent
(i.e., different s low -w ave factors). The calculated and
simulated results indicate that both the slow-wave factor and
the c haracter istic impedance remain a lm ost constant w it h re-
spect to th e frequency. Again, for the s ubstrate with a loss tan-
gent of 0 .0021 in the simulation, Fig. 4(c) g ives the per-unit-
guided wavelength loss of t he propo sed slow-wave structure.
At 0.9 GHz, the loss of the proposed unit cell with
is approximately 0.934 3 dB and decreases as the frequency
increases. The proposed structure has a larger loss compared to
the 50-
conventional microstrip line, which is 0.441 8 dB ,
where
is the guided wavelength of the 50- conventional
microstrip line on the substrate at 0.9 GHz.
Fig. 4. (a) Slow-wave factor, (b) simulated characteristic impedance ,and
(c) per-unit-guided wa velength loss versus frequency for
mm,
mm (35.36- case), mm (50- case), and mm (70.7-
case).
For mm as an example, Fig. 5 shows t he simu-
lated
-parameters of the prop osed slow-wave line consisting
of five unit cells. The frequency where
drops quickly cor-
responds to the cutoff frequency. Since the same
in the three
cases implies almost the same
based on (14), the larger
is, the smaller the characteristic impedance and the cutoff fre-
quency are. Therefore, the structure with
mm (i.e.,
) has the smallest cutoff frequency among the
three cases.
3380 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 11, NOVEMBER 2012
Fig. 5. Simulated -parameters of the proposed structur e with five unit cells
for
mm, mm (35.36- case), mm (50- case),
and
mm (70.7- case).
III. APPLICATION OF THE PROPOSED SLOW-WAV E STRUCTURE
To demonstrate t he proposed structure, one branch-line and
one rat-race coupler were designed and implemented. Both cir-
cuits w ere fabricated on the Rogers RO4003 substrate w ith a
dielectric constant of 3.38, a loss tangent of 0.0021, and a thick-
ness of 0.203 mm. The circuit sim ulator AWR M icro wave Of-
fice [23] and t he full-wave E M simulation software Sonnet [24]
were used to obtain the electrical param eter s in th e desig n equ a-
tions and to perform the simulation, respectively. The measure-
ments were carried out using an Agilent 8720ES network ana-
lyzer. The de sign steps of t he proposed unit cell are summ arized
as follows.
Step 1) Identify the characteristic im pedance
and slow-
wave factor of the slow-wave line. Set the transverse
width
of the unit cell arbitrarily since this width
will be adjusted for the specific slow-wave factor in
Step 4). The procedure of choosing
arbitrarily
here has almost no effect on the calculation of
in
Step 3). This feature has been illustrated in Fig. 2(b).
Step 2) Determine th e width
and spacing of the
Schiffman section o f meander line. U sually, for
large
and small area, the values of and are
very small and should be lim ited by the allowable
fabrication process. Once
and are given, the
even- and odd-mode impedances
and of
a Schiffman section can be obtained. According to
(3), (4), and (7), we calculate
and .
Step 3) From (1) and (8)–(13), the length
is available for
the specific
.
Step 4) As mentioned earlier, the adjustment of
has al-
most no effect on
. Hence, calculate from (2)
and (8)–(13) for the specific slow-wave factor. As
a result, the unit cell with the prescribed
and
slow-wave factor is achieved.
Step 5) Rep lace the conventional microstrip line with the
designed structure. Finally, t he circuit is simulat e d
with a full-wave EM simulator to take the effects
of couplings, discontinuities, an d ju nctions into ac-
count.
Fig. 6. Configuration of the proposed branch-line coupler. Circuit dimensions:
mm, mm, mm, mm,
mm, mm, mm, mm,
mm, mm, mm, and mm.
Fig. 7. Photograph of the fabricated branch-line coupler.
A. Branch -Line Coupler
The branch-line coupler comprises four
line sections,
two of which have the characteristic impedance of 35.36
,
and two of which have the characteristic impedance of 50
.
The proposed slow-wave structure is used to replace these four
transmission lines. The slow-wave factor is chosen as 5.6 for
the 35.36-
and 9.2 for the 50- conventional microstrip line.
First, we set
mm arbitrarily. H ere, mm
and
mm are fixed so that mm,
,and . Thus, and
from (3) and (4), and (7). Applying (1) an d (8)–(13),
the lengths of the unit cell for the shunt stub portion are ob-
tained as
mm for and mm
for
. From ( 2) and (8)–(13) with the prescrib ed
slow-wave factor, we calculate
mm fo r the 35.36-
line and mm for the 50- line. The proposed
branch-line coupler was designed at the center frequency of
940 MHz. O n t he b asis of the calculated valu es and after slightly
fine tunin g the whole cir c uit using the full-wave EM simulator,
