quasihybrid scheme of nonuniform rational b-spline–uniform geometrical theory of diffraction...
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frequency is above 11 KHz, the curve of h(f) is declined
abruptly. Figure 5(b) shows the amplitude-frequency response of
the FFP tunable filter. The amplitude-frequency of the filter is
kept under �2dB below 1.8 KHz, and the resonant frequency of
the filter is about 13 KHz for the minimum amplitude-frequency
response. For applications of this device to fast and accurate
scanning or locking, it is better to select the frequency of sinu-
soidal signal below 1.8 KHz.
4. CONCLUSIONS
In summary, a simple and an accurate method for measuring the
phase-frequency response of FFP tunable filters has been pro-
posed, which is based on a modification to the dynamic transfer
function of the filter. The method has been demonstrated suc-
cessfully with measurements of the phase-frequency response
and the amplitude- frequency response of a FFP tunable filter.
The phase-frequency characteristic is important for designs of
the device and surrounding control electronics, especially for the
wavelength locking application.
ACKNOWLEDGMENTS
This work was supported in part by the National Natural Science
Foundation of China under Grant No. 60677024 and the National
High Technology Research Development Program of China under
Grant No. 2009AA03Z418.
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VC 2011 Wiley Periodicals, Inc.
QUASIHYBRID SCHEME OFNONUNIFORM RATIONAL B-SPLINE–UNIFORM GEOMETRICAL THEORY OFDIFFRACTION METHOD FOR RADIATIONOF ANTENNAS
Nan Wang, Xiaojie Dang, Haobo Yuan, and Changhong Liang
Science and Technology on Antenna and Microwave Laboratory,Xidian University, Xi’an, Shaanxi 710071, China; Correspondingauthor: wangnan@mail.xidian.edu.cn
Received 1 March 2011
ABSTRACT: The nonuniform rational B-spline–uniform geometrical
theory of diffraction (NURBS-UTD) method is studied in this paper. Todeal with arbitrary wire antennas without analytical directional
functions, a quasihybrid scheme is presented to discuss the radiationpattern of wire antennas mounted around electrically large targets. TheNURBS technique is introduced to construct targets, the wire antennas,
and electrically large targets are analyzed by method of moment (MoM)and NURBS-UTD, respectively. Information from results of ray tracing
is chosen as interface between MoM and NURBS-UTD to formquasihybrid scheme. The practicability and validity can be seen fromexamples given. VC 2011 Wiley Periodicals, Inc. Microwave Opt Technol
Lett 53:2963–2967, 2011; View this article online at
wileyonlinelibrary.com. DOI 10.1002/mop.26374
Key words: NURBS; UTD; MoM; radiation pattern
1. INTRODUCTION
With the development of computer-aided geometric design, the
nonuniform rational B-spline (NURBS) technique becomes
another reliable way to construct targets for electromagnetic
simulation beside flat patches and analytical patches. As an uni-
versal computer-aided design technique, NURBS has the advan-
tages of high precision in modeling as illustrated in Figure 1
and great flexibility in forming algorithms.
In the area of electromagnetic simulation, the NURBS tech-
nique was first introduced into physical optics (PO) to calculate
radar cross section of electrically large targets [1, 2] and since
then, the NURBS-PO method was carefully studied in various
applications [3–5]. In the method of moment (MoM), lots of
studies have been made from basic thoughts to different basis
functions to try to absorb the NURBS technique in Refs. 6
and 7.
Uniform geometrical theory of diffraction (UTD) method is a
highly efficient method dealing with electrically large problems.
Theoretically, UTD can solve problems with arbitrary electrical
size without any extra need for computation resources. Due to
the successful introduction of NURBS technique into PO and
MoM, many works have been done on UTD method based on
NURBS (NURBS-UTD) to take advantage of the convenience
of NURBS in modeling [8–10]. In NURBS-UTD method, the
source can be point source, line source, and plane waves due to
their simplicity. Also, the source can be dipole or monopole,
because they have recognized analytical expression of direc-
tional functions. But, if the source is an antenna without analyti-
cal expression of directional function, the radiation of the
antenna will be hard to analyze only using NURBS-UTD.
To analyze complex wire antenna mounted around electri-
cally large targets while inheriting the advantages of NURBS-
UTD in dealing with electrically large problems, a quasihybrid
scheme is presented in this paper where MoM is used to analyze
wire antenna, and NURBS-UTD is used to analyze electrically
DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 12, December 2011 2963
large targets. The information (like information of reflected point
or diffracted point and so on) resulting from ray tracing is chosen
as interface between MoM and NURBS-UTD to form the quasihy-
brid scheme. Several examples are given in this paper to show the
validity of the quasihybrid scheme and the root-mean-square
(RMS) error is used to evaluate the results and show the precision.
2. QUASIHYBRID SCHEME
The mathematical expression of NURBS surface [11], which is
a kind of parametric surface, can be written as follows.
rðu; vÞ ¼Pn
i¼0
Pmj¼0 aijPijN
ipðuÞNj
qðvÞPni¼0
Pmj¼0 aijN
ipðuÞNj
qðvÞ���! u 2 ½0; 1�
v 2 ½0; 1��
(1)
where Pij is the control point, and aij is the corresponding weight
value, Nip(u) is the normalized B-spline basis function of degree p.
Any point on the surface can be derived from this expression. To-
gether with corresponding algorithm and the concept of differential
geometry, the geometrical information of any point on the surface
can also be derived easily, which becomes the joint between UTD
method and NURBS surfaces where basic aspects like ray tracing,
shadowing test, and calculating ray field are managed through nu-
merical measures, respectively [12, 13]. The NURBS-UTD method
which is based on the expression above can be applied directly to
arbitrary electrically large convex surfaces with highly accurate
approximation and can improve the precision of scattered field
especially when the surface has changed curvatures.
In NURBS-UTD method, all types of ray fields can be con-
cluded into the format given below
EðR0Þ ¼ EðQÞ � D¼ �AðsÞe�jks (2)
where R0 is the observation point; Q denotes the interactive
point, such as reflected point and diffracted point; D¼
is the
dyadic coefficient corresponding to different actions, such as the
dyadic reflected coefficient and the dyadic diffracted coefficient;
A(s) is the amplitude fading factor; and e�jks is the phase fading
factor. As can be seen in Figure 2, the first item, E(Q), in (2) is
the field traveling from the source point to the interactive point.
If a point source, line source is placed at the source point,
E(Q) is easily written as
EðQÞ ¼C e�jkr
r spherical wave
C e�jkrffiffir
p cylindrical wave
Ce�jkr plane wave
8><>: (3)
where C is a vector related to the strength of source.
Figure 1 NURBS surfaces. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com]
Figure 2 Illustration of ray field
Figure 3 Major procedure of the quasihybrid method
Figure 4 An arbitrary surface with 16 control points
2964 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 12, December 2011 DOI 10.1002/mop
If what is placed at the source point is simple wire antenna
like dipole or monopole which can be seen as a point source
without length, E(Q) is also easy to get by their recognized ana-
lytical directional function
EðQÞ ¼ Ccosðkl cos hÞ � coskl
sin h(4)
where C is a vector related to the strength of source, and l isthe half length of the dipole.
Problems appeared when the antenna placed at the source point
is complex, under this condition, the antenna may not have a ana-
lytical directional function, so that the radiation of antenna is really
hard to achieve only from NURBS-UTD. Aiming at arbitrary wire
antennas, a quasihybrid scheme is presented to combine the MoM
which is highly effective in analyzing wire antennas with NURBS-
UTD method. The main process of NURBS-UTD can be subdi-
vided into three parts, namely, ray tracing, shadowing test, and cal-
culating ray field. Given a source point and a observation point, the
ray-tracing process of NURBS-UTD is first taken, and all possible
kinds of rays are traced to find interactive point Q and the dyadic
coefficient D¼
and other useful information; second, the ray path
is tested to ensure that it does not shaded by any part of the tar-
get; third, valid information past the second step is imported to
the process of MoM to get E(Q) radiated from the antenna to Q;Finally, together with information like D
¼, E(Q) is substituted into
(3) to calculate total ray field received at the observation point.
Major procedure can be seen in Figure 3.
3. NUMERICAL RESULTS
In the examples given below, the working frequency is set as
300 MHz, the dimension unit of the targets is meter, and the
electrical size of the targets is set over square size of 150k2.To see the precision better, the RMS error is introduced here as
RMS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
N
XNi¼1
jrreference � rpresentedj2vuut (5)
where N is the number of data, rreference and rpresented are the
data corresponding to the referenced method and the presented
method, respectively.
The results are compared to FEKO, the commercial software
built based on MoM. Considering the computational constrain of
MoM, the examples given below are set to fulfill the extreme
computation of four common PC with parallel means.
It is quite necessary to point out that problems of arbitrary
large electrical size can be solved on one single PC by applying
the method presented in this paper.
Case 1. An arbitrary surface can be described by 16 control
points as shown in Figure 4, and a half-wave dipole is placed
above at (0.0, 0.0, 15.0) pointing to (1, 1, 1). The radiation pat-
tern of the antenna in zox-plane is calculated and shown in Fig-
ure 5 and compared to FEKO with 65,341 triangles.
The RMS error is only 0.82379 dB in this example, which
greatly shows the precision of the scheme presented. It is needed
Figure 5 Radiation pattern on zox-plane. [Color figure can be viewed
in the online issue, which is available at wileyonlinelibrary.com]
Figure 6 A wire helical antenna is put above the surface
Figure 7 Radiation pattern. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com]
DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 12, December 2011 2965
to point out that there are also direct ray field, reflected ray
field, and surface ray field contribute to the total ray field, and
obviously, the UTD method based on board, cylinder, and cone
cannot deal with this kind of surface.
In fact, for simple wire antennas like dipole and monopole,
radiation pattern from the quasihybrid scheme is much the same
as that from the analytical directional function defined in
Eq. (4), which shows the versatility of the presented scheme. If
the antenna is changed to antenna without any analytical direc-
tional function so that NURBS-UTD is limited, the quasihybrid
scheme will be more practical.
Case 2. A wire helical antenna is chosen with its radii set as
0.5, pitch set as 0.3, and height set as 0.3 to put above the sur-
face given at (0.0, 0.0, 15.0) and pointing to (0, 0, 1), as shown
in Figure 6. Radiation pattern on zox-plane and yoz-plane is cal-
culated and given in Figure 7.
The RMS error is 1.91644 dB for zox-plane and 1.71837 dB
for yoz-plane in this example, which shows that the presented
scheme can be applied to complex wire antennas without any
analytical directional function mounted around large scale sur-
face and achieve good precision.
Case 3. An aircraft constructed by 16 parametric patches is
chosen and shown in Figure 8. The helical antenna is put above
at (�45, 0.0, 9.0) and pointing to (�1, 0, 0).
The quasihybrid scheme is used to calculate radiation pattern
on yoz-plane and xoy-plane and illustrated in Figure 9. A three-
dimensional radiation pattern is also given in Figure 10. It is
needed to point out that the size of the craft is over 5000 square
wavelength, but the problem can be solved on one simple PC by
the quasihybrid scheme in NURBS-UTD method.
It can be concluded from the examples given above that
good precision can be achieved through the scheme presented,
and results are not higher than 2 dB compared to commercial
software judging by RMS error, which shows the validity; the
scheme can be applied to electrically large targets directly,
which shows the practicability.
4. CONCLUSIONS
The UTD method based on NURBS is studied in this paper. To
analyze complex wire antenna mounted around electrically large
targets while inheriting the advantages of NURBS-UTD in deal-
ing with electrically large problems, a quasihybrid scheme is
presented where MoM is used to analyze wire antenna, and
NURBS-UTD is used to analyze electrically large targets. The
information (information of reflected point, diffracted point,
etc.) resulting from ray tracing is chosen as interface between
MoM and NURBS-UTD to form the quasihybrid scheme. Sev-
eral examples are given in this paper to show the validity of the
quasihybrid scheme, and the RMS error is used to evaluate the
results and show the precision.
Figure 8 Helical antenna mounted around an aircraft
Figure 9 Radiation pattern. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com]
Figure 10 Three-dimensional pattern. [Color figure can be viewed in
the online issue, which is available at wileyonlinelibrary.com]
2966 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 12, December 2011 DOI 10.1002/mop
ACKNOWLEDGMENTS
This work was supported by the National Science Foundation of
China under Grant 60901030 and the Fundamental Research Funds
for the Central Universities JY10000902024.
REFERENCES
1. J. Perez and M.F. Catedra, Application of physical optics to the
RCS computation of bodies modeled with NURBS surfaces, IEEE
Trans Antennas Propag 42 (1994), 1404–1411.
2. C. Ming, Z. Yu, and L. ChangHong, Calculation of the field distri-
bution near electrically large NURBS surfaces with physical-optics
method, JEMWA 19 (2005), 1511–1524.
3. K. Huang, Z.-L. He, and C.H. Liang, Improved NURBS MOM-PO
method for analyzing antenna around electrically large platform,
Microwave Opt Technol Lett 52 (2010).
4. M. Chen, Y. Zhang, X.W. Zhao, and C.H. Liang, Analysis of
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IEEE Trans Antennas Propag 55 (2007), 407–413.
5. C. Ming, Z. XunWang, and L. ChangHong, Analysis of antenna
around NURBS surface with iterative mom-po technique, JEMWA
20 (2006), 1667–1680.
6. Y. Haobo, W. Nan, and C. H. Liang, Fast algorithm to extract the singu-
larity of higher order moment method, JEMWA 22 (2008), 1250–1257.
7. H.B. Yuan, N. Wang, and C.H. Liang, Application of higher order
method of moments to the RCS computation of bodies modeled with
NURBS surfaces, IEEE Trans Antennas Propag 57 (2009), 3558–3563.
8. J. Perez, J.A. Saiz, and O.M. Conde, Analysis of antennas on board
arbitrary structures modeled by NURBS surfaces, IEEE Trans
Antennas Propag 45 (1997), 1045–1053.
9. W. Nan, L. Changhong, and Y. Haobo, Calculation of pattern in
UTD method based on NURBS modeling with the source on sur-
face, Microwave Opt Technol Lett 49 (2007).
10. W. Nan, Z. Yu, and C. H. Liang, Creeping ray-tracing algorithm
of UTD method based on NURBS models with the source on sur-
face, JEMWA 20 (2006), 1981–1990.
11. L. Piegl, NURBS: A survey, IEEE Computer Graph Appl (South
Florida) (1991).
12. W. Nan, Z. Yu, and L. Chang-hong, Study on reflected ray tracing
algorithm of NURBS-UTD, Chin J Radio Sci 21 (2007), 834–837.
13. W. Nan, C. H. Liang, Z. Yu, and C. Ming, Study on the creeping ray-
tracing algorithm of NURBS-UTD, J Xidian Univ 34 (2007), 600–604.
VC 2011 Wiley Periodicals, Inc.
A LOW PHASE NOISE OSCILLATOR WITHA HIGH-Q SPLIT RING RESONATORUSING MNG METAMATERIAL
Ki-Cheol Yoon and Jong-Chul Lee
Department of Wireless and Communications Engineering,Kwangwoon University, Nowon-ku, Seoul 139-701, Korea;Corresponding author: kcyoon98@kw.ac.kr or jclee@kw.ac.kr
Received 3 March 2011
ABSTRACT: In this article, a novel high QL triple-split ring resonator(T-SRR) of mu-negative metamaterial is suggested. It is used in the
design of an I- band oscillator. A very low phase noise can be obtainedfor the oscillator due to the sharp band-rejection characteristic ofT-SRR. The oscillator is designed to operate at 10 GHz using the novel
T-SRR. Experiments show that the output power is 9.82 dBm, the secondharmonic suppression is �26.48 dBc, and the phase noise is �105.05
dBc/Hz at a 100-kHz offset. VC 2011 Wiley Periodicals, Inc. Microwave
Opt Technol Lett 53:2967–2971, 2011; View this article online at
wileyonlinelibrary.com. DOI 10.1002/mop.26404
Key words: T-SRR; MNG metamaterial; high loaded quality factor; low
phase noise; oscillator
1. INTRODUCTION
In general, all modern military satellite communication systems
use oscillators for the I-band (8–10 GHz), where it is transmitted
in piloting radar using the I-band satellite [1]. There have
increasingly been requests for low phase-noise in oscillator
design, because it determines the overall performance of the sys-
tems [2]. The double-split ring resonator (D-SRR) [3] with mu-
negative (MNG) metamaterials has been reported in the design
of resonators with band-rejected performance [3]. D-SRRs are
appropriate for the oscillators and other circuits required for a
high-loaded quality factor, (QL) and band-stop resonant per-
formance due to their sharp band-rejection characteristic [3].
MNG metamaterial is artificially engineered material exhibiting
features not readily available in natural materials. The split ring
resonator (SRR), which has as its theoretical basis Veselago’s
work on metamaterial in the late 1960s [3], has engendered
great research recent interest. After Pendry’s pioneering experi-
mental work on the SRR of negative permittivity and permeabil-
ity in 1992 [4], a great deal of research has been reported in the
literature. This took advantage of the compact size and high-QL
of the novel SRR structure. This article suggests a new micro-
strip SRR resonator with high-QL and an oscillator with low
phase noise using this resonator. The proposed triple-split ring
resonator (T-SRR) has three-sections of SRRs, where the high-
QL characteristic can be obtained due to the strong coupling
effect between multiple ring structures. In addition, the resonator
size can be adjusted without variation of the resonant frequency
due to the geometric characteristics.
2. DESIGN OF A NEW HIGH-QL T-SRR WITH MNGMETAMATERIAL
2.1. High-QL T-SRR StructureThe T-SRR is composed of a triple section of rings and small
center-pad (b) connected to the inner SRR by three-short lines,
lm, as shown in Figure 1(a). From the figure, d1 and d2 are gaps
between the rings and gi (g1� g3) are gaps of the open rings. In
addition, a is the side length of the outer ring and w is the width
of the rings; these are the same in this case. Figure 1(b) shows
the equivalent T-SRR circuit. Ls is an inductance corresponding
to the total electrical length at the T-SRR. Lm is inductance due
to three-short lines at the center-pad and Cm is mutual capaci-
tance in the center-pad, used to adjust the resonant frequency.
s indicates magnetic coupling between the transmission-line and
T-SRR structure, as shown in Figure 1(a). Inductance (Ls) and
capacitances (Ci ¼ C1//C2) determine the resonant frequency
(x0) at the equivalent circuit of the T-SRR. From Figure 1(a),
the length, ln of each ring in the T-SRR is defined by the fol-
lowing Eq. (1) [5],
ln ¼ N½a� 2wðn� 1Þ � 2diðn� 1Þ� (1)
where N is the number of the ring, 3 in this case. The resonant
frequency (x0) of the T-SRR is given by (2) [6],
xo ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
pr0LsCi
r(2)
where ro is the average radius of the T-SRR. The characteristic
impedances are given by (3) and (4) [7],
Z1;Zn ¼ � j
2½ðZoecotcelnÞ þ ZoocotðcolnÞ� (3)
DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 12, December 2011 2967
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