CC BY
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PA£IOOI3HKA
PA^iO®i3HKA PA^HO®H3HKA RADIOPHISICS
YAK 621.372.2
A.B. Gnilenko, O.V. Paliy
MICROSTRIP DISCONTINUITY ANALYSIS USINGFINITE-DIFFERENCE TIME-DOMAIN METHOD
The finite-difference time-domain method with Mur's firstorder absorbing boundary conditions is applied to the analysis of a microstrip open end and a microstrip gap. Based on Maxwell's equation discretization, computational algorithms are developed to simulate Gaussian pulse diffraction by the discontinuities. Qualitative time-domain data are obtained to illustrate scattering process evolution. Frequency-domain results in terms of -matrix elements are calculated and compared with other available data.
1 INTRODUCTION
Microstrip discontinuities serve as basic building elements of microwave printed circuits. Based on the knowledge of frequency-dependent parameters of discontinuities involved in a circuit, complex microwave devices can be easily analyzed from interconnects of microstrip discontinuities and microstrip line segments. That is why accurate modeling microstrip discontinuities in terms of S-matrix elements is a problem of great importance for modern microwave printed circuit design. To meet the requirements of microwave integrated circuit technology, one should apply sophisticated computer-aided design (CAD) tools for the accurate simulation of electromagnetic field scattering by circuit components.
Various techniques have been used for recent years to solve microstrip discontinuity problems. A full-wave spectral-domain approach has been applied to the analysis of arbitrary shaped open microstrip discontinuities in [1]. The full-wave analysis of shielded passive microstrip components has been provided in [2] using the method of moments. In [3], the mode matching technique has been presented for the analysis of a shielded microstrip step discontinuity. The method of integral equations for overlapping regions along with the planar dispersive waveguide model has been proposed in [4] for microstrip discontinuity analysis.
Presently, the finite-difference time-domain method (FDTD) is widely used for solving various problems of
electromagnetic theory including the analysis of printed circuit components. The method has been firstly presented in [5] to solve three-dimensional electromagnetic scattering problems and applied to microstrip discontinuity analysis by many authors [6], [7]. The main advantage of the FDTD method is its great flexibility in the analysis of a variety of microwave printed circuit configurations. Moreover, the ability of field simulation both in time and frequency domains allows us to get an insight into a time-dependent process of signal scattering by circuit discontinuities and to obtain the frequency-dependent characteristics of circuit elements in terms of S-matrices.
In this paper, the FDTD method with Mur's first-order absorbing boundary conditions (ABC) is applied to the analysis of a microstrip open end and a microstrip gap. Qualitative results are obtained in time domain to illustrate the electromagnetic field propagation and scattering. S-matrix elements are derived from the time-domain data using the Fourier transform. Frequency-dependent results are compared with other available data showing good agreement.
2 METHOD OF ANALYSIS
The finite-difference time-domain method is based on dividing a limited computational domain with an analyzed structure inside into unit cubic cells and calculating field values at nodal points of the space mesh at successive instants of time. An approximate solution of the electromagnetic problem can be obtained solving Maxwell's equations:
Y7 t dH
V x E = -u -=-, ^ dt
T7 dE
V x H = 8 "it-dt
6
ISSN 1607-3274 "Pa^ioe^eKTpoHiKa. Ii^opMaTHKa. YnpaB^iHHfl" № 2, 2003
A.B. Gnilenko, O.V. Paliy: MICROSTRIP DISCONTINUITY ANALYSIS USINGFINITE-DIFFERENCE TIMEDOMAIN METHOD
with appropriate initial, source and boundary conditions imposed on conductors, interface surfaces and the computational domain walls. The discretization of Maxwell's equations using the centered difference approximation leads to a set of algebraic equations providing an explicit computational algorithm for the simulation of electromag-
. . . netic field scattering. Each component of H field can be
. . . . derived from certain components of E field at four surrounding nodal points. Similarly, E field components are
calculated through H field values at four surrounding nodal points. To achieve the centered differences for time
derivatives, E and H field components should be alternatively calculated one through another at all nodal points at successive instants of time.
The ground plane and strip conductors are assumed to be perfect electric conductors of zero thickness and involved in the analysis by setting tangential electric field components on the conductors to zero. Field components on the dielectric-air interface are calculated using the average permittivity (Eg + EgEr)/2 . For tangential field components on the computational domain walls (except the ground plane) special treatment must be done. These field components cannot be calculated using field values at the nodal points outside the computational domain and should simulate outgoing waves with no reflection. To this end,
Mur's first-order approximate absorbing boundary condi-> .
tions [8] are imposed on tangential E field components on the domain walls.
The initial conditions force all field components to be zero at t = 0 throughout the computational domain. At t = 1 Gaussian pulse is excited on the front wall with the vertical electric field component
Ex( t) = exp
( t - tg ) 2 ■
T2
(2)
tion interval; k is the stability criterion constant (k = 0, 514) and c 1 is the velocity of light in air. Timedomain results shown in Fig.2 illustrate the field distribution calculated for Ex component on the plane just underneath the strip surface at the moment when Gaussian pulse has been reflected by the discontinuity.
Figure 1 - Computational domain for microstrip open end
The other electric field components on the source wall of the computational domain are forced to be zero. To avoid undesired return wave reflection by the front wall, the electric source wall conditions should be switched on the absorbing boundary conditions when Gaussian pulse reaches a discontinuity.
3 MICROSTRIP OPEN END
The microstrip open end is a microstrip discontinuity that is widely used in microwave printed circuits. The computational model of a microstrip open end is shown in Fig.1. The structural parameters used for calculation are Ax = Ay = Az = 0, 06mm, \t =kAz/c 1, a = c = 100, b = 50 , l = 50 , h = w = 10 , Er = 9, 6 , where Ax, Ay and Az are the space discretization intervals in x, y, and z directions, respectively; At is the time discretiza-
Figure 2 - Field distribution for microstrip open end
In the figure, one can also see a surface wave travelling from the discontinuity to the far end of the computational domain. It can be seen that the time-domain results explicitly illustrate Gaussian pulse scattering by the microstrip discontinuity. But for CAD purposes frequency-dependent data are required in terms of S -matrix elements. The scattering matrix of the microstrip open end consists of only one element that corresponds to the reflection coefficient. It can be obtained from the time-domain data as
S11 f) = Eref(f'Z = X 1 ) e2fl -11) 11 EXnc(f, z = X1)
(3)
where E^f is the Fourier-transformed reflected electric field component calculated underneath the strip at the ref-
PA£IOÔI3HKA
erence plane z = A^ Az, EXnc is the Fourier-transformed incident electric field component at the same point. The incident field is obtained as a result of the FDTD modeling of the regular microstrip line and the reflected field is calculated as the difference between the total scattered field and the incident field. The dispersion characteristic of the reflection coefficient is shown in Fig.3 together with the results provided in [6]. It can be seen that both results agree well with a discrepancy less then 4 %.
4 MICROSTRIP GAP
The computational model of a microstrip gap is shown
in Fig.4. The structural parameters are as follows: Ax = Ay = Az = 0, 06 mm, a = 60 c, c = 100, b = 50, L = 5, h = w = 10, 8r = 9, 6. Fig.5 demonstrates timedomain results for Ex component distribution on the plane just underneath the strip surface after Gaussian pulse scattering by the gap. It can be observed that the pulse has been partly reflected into incident port 1 and partly transmitted into port 2. Frequency-dependent characteristics of the microstrip gap obtained from the time-domain data are shown in Fig.6. Calculated results for the scattering matrix elements are compared with data of [6] revealing good agreement with maximum discrepancy of 3%.
!
---- [6]
- -
- -
- ■ -
4 s
Frequency (hiz)
7
.„its [ 1.0
Figure 3 - Reflection coefficient of microstrip open end Figure 4 - Computational domain for microstrip gap
- -FDTE'method ---- [6]
'■
- ----
- |s21|
1 1.5 2 2.5
3 3.5 4 Frequency (Hz)
5.5 6 x 10°
Figure 5 - Field distribution for microstrip gap Figure 6 - Scattering matrix elements of microstrip gap
8 ISSN 1607-3274 "Pa^ioe^eKTpoHiKa. iH^opMaTHKa. ynpaB.niHHH" № 2, 2003
М.В. Андреев, В.Ф. Борулько, О.О. Дробахт, Д.Ю. Салтыков: СПРОЩЕН1 П1ДХОДИ ДРОБОВО-РАЦЮНАЛЬНОГО ПРЕДСТАВЛЕННЯ В СПЕКТРАЛЬНО-СПОЛУЧЕН1Й ОБЛАСТ1 ДЛЯ ВИЗНАЧЕННЯ
5 CONCLUSION
The finite-difference time-domain method with Mur's first-order absorbing boundary conditions has been applied to the analysis of a microstrip open end and a microstrip gap. Both time-domain and frequency-domain numerical results have been obtained and discussed for the microstrip discontinuities under consideration. Frequency-dependent characteristics for scattering matrix elements have been compared with other available data showing good agreement.
REFERENCES
1. Horng T.S., Alexopulos N.G., Wu S.C., Yang H.Y. Full-wave spectral domain analysis for open microstrip discontinuities of arbitrary shape including radiation and surface-wave losses // Int. J. of MIMICAE, 1992, vol. 2, No. 4, pp. 24240.
2. Hill A., Tripathi V.K. An efficient algorithm for the three-dimensional analysis of passive microstrip components and discontinuities for millimeter-wave integrated circuits // IEEE Trans. Microwave Theory Tech., 1991, vol. MTT-39, No. 1, pp. 83-91.
3. Uzunoglu N.K., Capsalis C.N., Chronopoulos C.P. Frequency-dependent analysis of a shielded microstrip step discontinuity using an efficient mode-matching technique / / IEEE Trans. Microwave Theory Tech., 1988, vol. MTT-36, No. 6, pp. 976-984.
4. Yakovlev A.B., Gnilenko A.B. Analysis of microstrip discontinuities using the method of integral equations for overlapping regions // IEE Proc.-Microw. Antennas Propag., 1997, vol. 144, No. 6, pp. 449-457.
5. Yee K.S. Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media //
IEEE Trans. Antennas Propagat., 1966, vol. AP-14, No. 5, pp. 302-307.
6. Zhang X., Mei K.K. Time-domain finite difference approach to the calculation of the frequency-dependent characteristics of microstrip discontinuities // IEEE Trans. Microwave Theory Tech., 1988, vol. MTT-36, No. 12, pp.1775-1787.
7. Sheen D.M., Ali S. M., Abouzahra M.D., Kong J.A. Application of the three-dimensional finite-difference time-domain method to the analysis of planar microstrip circuits // IEEE Trans. Microwave Theory Tech., 1990, vol. MTT-38, No. 7, pp.849-857.
8. Mur G. Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equation // IEEE Trans. Electromagn. Compat., 1981, vol. EMC-23, No. 4, pp. 377-382.
Надшшла 16.10.2003
Представлен анализ открытого конца и разрыва микрополос ковой линии методом конечных разностей во временной области с поглощающими условиями Мура первого порядка. На основе дискретизации уравнений Максвелла построены вычислительные алгоритмы для моделирования дифракции гауссова импульса на неоднородностях. Получены результаты во временной области, качественно иллюстрирующие процессы дифракции. Проведено сравнение дисперсионных характеристик элементов матриц рассеяния с данными других авторов.
Проведено анал1з в1дкритого ктця та розриву мжро-смужково1 лгни методом скгнченних ргзниць у часовш областг з поглинаючими умовами Мура першого порядку. На тдстав1 дискретизацИ р1внянь Максвела побудовано обчислювальш алгоритми для моделювання дифракци гаусова гмпульсу на неодноргдностях. Одержано якгсш результати у часовш област1, що iлюструють процеси дифракци. Зроблено порiвняння дисперстних характеристик елементiв матриць розсiяння з iншими даними.
УДК 537.8:620.179:621.391:621.396
М.В. Андреев, В.Ф. Борулько, 0.0. Дробахш, Д.Ю. Салтиков
СПР0ЩЕН1 П1ДХ0ДИ ДРОБОВО-РАЩОНАЛЬНОГО ПРЕДСТАВЛЕНИЯ В СПЕКТРАЛЬНО-СПОЛУЧЕН1Й ОБЛАСТ1 ДЛЯ ВИЗНАЧЕННЯ ПАРАМЕТРА В1ДБИВАЛЬНИХ СТРУКТУР
Розглянута можливгсть застосування лшшних nidxodie для апроксимацп спектральных даних однополюсною функ-щею i використання дробово-рацюнальноЧ ттерполяцп спектральних даних з метою прискорення процедури визначення nараметрiв адитивно'г експоненщальноЧ моделi. Проаналiзованi точтст характеристики запропонованих методiв i проведене ïхне nорiвняння з аналогiчними характеристиками тших методiв даного класу.
ВСТУП
В багатьох задачах прикладно'' радюф1зики виникае проблема визначення параметр1в адитивно!' експонен-щально!' модель Як показуе досвщ практичного використання, застосування адитивно!' експоненщально!' модел1 дае змогу 1м1тувати р1зш властивосп вщбиваль-
них об'екпв [1-3]. Зокрема, така модель може бути використана для розгляду вщбиття в багато-шарових д1електричних структурах [4].
На попередньому етат дослщжень вщбивальних структур позитивш результати вим1рювань були досягнут! за рахунок використання метод1в цифрового спектрального анал1зу результапв вим1рювань в частотнш обласп [3]. Такий шдхщ потребуе розгляду моделей високого порядку, що веде до нестшкосп результапв. Необхщшсть використання моделей високого порядку обумовлено тим фактом, що в експериментальних даних, отриманих, наприклад, при дослщженш д1електричних структур, присутня шформащя про ви розсшвач1, яю були опромшеш зондом, а не т1льки про неоднорщносп структури, що безпосередньо дослщжуються. Альтер-
