An order-reduced volume-integral equation approach ... - IEEE Xplore

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 3, MARCH 2005

799

An Order-Reduced Volume-Integral Equation Approach for Analysis of NRD-Guide and -Guide Millimeter-Wave Circuits

H

Duochuan Li, Ping Yang, and Ke Wu, Fellow, IEEE

Abstract—An order-reduced volume-integral equation approach is proposed for modeling and analyzing of nonradiative dielectric (NRD)-guide and -guide millimeter-wave circuits that involve arbitrarily shaped planar geometry and inhomogeneous dielectric. A half-sinusoidal vertical variation of fields is used so that the discretization of current for the volume-integral equation is made in the parallel plane. Combined basis functions of propagating and local modes are used in the Garlerkin’s method of moments on the basis of spectrum analysis of the NRD-guide and -guide. A vertical integration in the space domain is carried = 1 out analytically and a first-order Green’s function with is developed. The solution for the volume-integral equation in modeling NRD-guide and -guide circuits is then reduced to a two-dimensional planar problem. This technique can be applied for calculating the characteristics of various waveguide components and multiport circuits such as resonant frequencies and -parameters. The framework of this technique is demonstrated through its application to an NRD-guide open-end. In addition, an -guide open-end, three types of resonators, and three-pole gap-coupled NRD-guide filters are modeled and analyzed. The results are in good agreement with the measurements and results obtained by other methods. Index Terms—Green’s function, method of moments, millimeter wave, nonradiative dielectric (NRD)-guide and -guide circuits, order-reduced volume integral equation (ORVIE), vertical integration.

I. INTRODUCTION

M

ILLIMETER-WAVE techniques become increasingly important because of emerging wireless communication and noncommunication applications. Over this frequency range, the commonly used planar techniques, namely, microwave integrated circuit (MIC), miniaturized hybrid microwave integrated circuit (MHMIC), and monolithic microwave integrated circuit (MMIC), generally suffer from problems of prohibitive conductor loss and critical dimensional tolerance. Therefore, nonplanar technologies, which include metallic and dielectric waveguides, should be considered. Among those nonplanar schemes, the nonradiative dielectric (NRD) waveguide is promising for the making of passive components because it can effectively suppress radiation loss along circuit bends and discontinuities. Since its inception in 1981 [1], this technology has been used in the design and fabrication of a large class of Manuscript received February 9, 2004; revised May 24, 2004. The authors are with the Poly-Grames Research Center, École Polytechnique de Montréal, Montréal, QC, Canada H3V 1A2 (e-mail: liduo@ grmes.polymtl.ca; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2004.842511

integrated circuits and antennas that have demonstrated superior electrical performance at millimeter-wave frequencies [2]–[4]. Several numerical methods were used to analyze discontinuities and components in NRD circuits. The mode-coupling theory was applied in [5] to analyze the loss characteristics of the NRD-guide bends. A rigorous expression for coupling and modes (also referred to coefficient between as , or , in some publications) was derived and then used in the two-mode coupling equations to be solved for the bending loss analysis. This theory was reexamined in [6] to improve the design technique of NRD-guide bends. An efficient design technique of an NRD-guide filter based on a variational principle has been developed in [7], and both gap-coupled-type and alternating-width-type filters were successfully designed. However, those methods are geometrically specific and hard to extend for general modeling problems of NRD-guide discontinuity and components. The mode-matching technique has also been used to solve a large class of NRD-guide components and discontinuities, which include open ends, junctions, steps, gaps, T-junctions, and diplexer [8]–[10]. Nevertheless, it will fail once the planar section (the top view of dielectric circuit) of the guiding structure is of arbitrary shape and/or has inhomogeneous dielectric permittivity. In this paper, an order-reduced volume-integral equation (ORVIE) method is developed, which is able to solve not only NRD circuits of an arbitrarily shaped planar section and inhomogeneous dielectric permittivity in the planar section, but also -guide circuits in which radiation loss must be considered since the spacing between the two parallel plates is larger than a half-wavelength in free space [11]. An electric-field integral-equation (EFIE) method was used to obtain eigenmodes of dielectric guide in [12] and [13]. The EFIE in its standard, also referred to as the domain integral equation, treats dielectric strip domains as local perturbation of the configuration, replacing them with equivalent polarization currents. An electric dyadic Green’s function is then used for the integral representation of fields in the layer in which the strips are embedded. Due to its rigorous full-wave formulation, the EFIE technique is capable of handling both open and closed structures and describing physical effects such as wave leakage. However, when this method extends to three dimensions, referred to as the volume integral-equation method, only a few simple problems with finite dielectric regions can be solved because of large memory and CPU time requirements [14], [15]. This method has not been used in the modeling of NRD-guide

0018-9480/$20.00 © 2005 IEEE

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Fig. 1. Cross section of NRD-guide and

The mode spectrum in this structure can be divided into a discrete number of surface waves, which is classified into LSE and LSM modes, and a continuous set of radiation modes. As far as the eigenvalue problem is concerned, the presence of the parallel metal plates implies a discretization of the vertical wavenumber , with null or integer ), while the horizontal ( wavenumbers can be derived from transcendental equations related to the dielectric slab guide. The transcendental equations for the eigenvalues along are expressed as

H -guide.

and -guide circuits. In this study, this method is applied to the modeling of NRD-guide and -guide circuits by reducing the original three-dimensional (3-D) problems to two-dimensional (2-D) planar problems. In the NRD-guide and -guide structures, a half-sinusoidal vertical ( ) variation of fields remains unchanged all over the circuit with the - or -mode excitation. Therefore, the current discretization in the volume integral equation can be implemented in the – -plane just as in planar circuits. With such basis functions, the vertical integration in the space dofirst-order main can be obtained analytically and an Green’s function can be constructed. Thus, the element integrations are carried out in two dimensions in the spectral domain. The solution of NRD-guide and -guide circuits are reduced to 2-D planar problems. This largely reduces calculation effort and makes it possible to simulate NRD circuits of any shaped planar section and any varied dielectric permittivity in the planar section. In particular, it can accurately calculate radiation loss along bends and discontinuities of the -guide by extracting poles in the Green’s function. In what follows, a complete description of the mode spectrum for the NRD-guide and -guide is presented first, followed by the formulation of the volume integral-equation and spectral dyadic Green’s function. Afterwards, the ORVIE technique is outlined. Subsequently, techniques for the integration are briefly described. In Section III, the solution procedure of an NRD-guide open-end is presented in detail to demonstrate features of the ORVIE scheme. The -guide open-ends, resonators, notched square, and rectangular sections with inhomogeneous dielectric variation, and three-pole gap-coupled NRD filter are then modeled and discussed. II. FORMULATIONS A. Mode Spectrum of NRD-Guide and

-Guide

To understand the proposed ORVIE method, a detailed description of the mode spectrum of the NRD-guide and -guide is given in the following. The NRD-guide and -guide have the same structure as in Fig. 1, which can be viewed as a rectangular dielectric rod (width , height , and relative permittivity ) sandwiched between two parallel metal plates. The desired mode in this structure. The prinoperating mode is the cipal advantage of the -guide is known for its low transmission loss, which is achieved by keeping the plate spacing more than , but it suffers radiation loss at bends and discontinuities. On the other hand, the advantage of the NRD guide is its ability to suppress such radiation loss by keeping the plate spacing less . than

symmetric modes antisymmetric modes (1) for LSE modes and for LSM modes. The where , is the -directed wavenumber propagation factor is inside the dielectric, and is the decay constant in the air region. , , and have an analytical relationship as follows: (2) (3) Solving the transcendental equations gives rise to a system of eigenvalues with odd integers ( ) in symmetrical ) in antisymmetric modes and even integers ( modes. The propagation constants are then obtained by

Surface-wave modes occur at , where satisfies . Radiation modes appear while , where satisfies . In this case, the waves can propagate away at an angle from the dielectric strip on both sides. The space of the mode spectrum can be divided into surfacewave modes, continuous radiation modes, and prohibited regions, as depicted in Fig. 2 by simply defining two boundary and in the dialines gram where is a decay constant, is a propagation constant, and

LI et al.: ORVIE APPROACH FOR ANALYSIS OF NRD-GUIDE AND

-GUIDE MILLIMETER-WAVE CIRCUITS

Fig. 3.

Top view of the dielectric circuits of NRD-guide and

801

H -guide.

as an NRD-guide. The radiation mode is partly evanescent region where the and partly propagating in the structure works as an -guide. Only the propagating radiation mode can cause radiation loss and this is why an NRD-guide can suppress radiation loss at bends and discontinuities with . The case of is similar to the operating mode where the boundaries and shift to a higher that of frequency range and the radiation mode cuts off at , as shown in Fig. 2(c). B. Spectral Dyadic Green’s Function and Volume-Integral Equation Formulation

H m

m

Fig. 2. Mode spectrum of NRD-guide and -guide. (a) = 0 (spectral condition). (b) = 1 (spectral condition). (c) = 2 (spectral condition).

m

A full spectrum for was generated (with mm mm and , as shown in Fig. 2. , as described in Fig. 2(a), the LSE In the case of modes reduce to modes and no modes exist. mode, and The fundamental guided TE mode, i.e., the the propagating radiation mode exist over all frequency range. , , and The higher order TE modes, namely, modes, merge to a continuous radiation mode at a low-frequency range where . The boundaries and shift , as shown in to a high-frequency region in the case of Fig. 2(b). The continuous radiation mode becomes completely region where the structure works evanescent in the

Fig. 3 shows the top-view (planar section) of a multiport NRD-guide or -guide circuit that is sandwiched between two parallel infinitely extended metal plates. The core circuit relates to the external circuits through physical ports attached to an equal number of intrinsic standard NRD-guide or -guide feed lines. These feed lines are assumed to be semi-infinitely long with the same height , different width , and constant dielectric permittivity . The core circuit may be in the form of an NRD-guide or -guide as the dielectric permittivity varies . The whole circuit is surrounded by a in the – -plane medium with a constant dielectric permittivity . Suppose that the total field is , and according to the volume equivalent theorem, the resulting field will remain the same if we replace the inhomogeneous dielectric body with the surrounding medium, but assume there is a specific electric current density in its volume with the following condition (a time dependency is considered): of (4) The electric field generated by the polarized volume current is (5)

802


where is the space occupied by the dielectric strips. The in the spectral domain is dyadic Green’s function formulated as

(6) The explicit form of for the two parallel metal plates can be obtained by letting the reflection coefficient at the two plates [16], and can be expressed as (7), shown at the bottom of this page, where

and

In this Green’s function, the poles at which the denominator is equal to zero represent the parallel-plate waveguide modes, which form the radiation loss mechanism in discontinuities -guide circuits. This phenomenon is in NRD-guide and similar to the surface wave in microstrip circuits [17], [18]. , , where These poles occur at ( ). There is always at least continuous radiation mode, one pole that represents the as shown in Fig. 2(a). Another pole, representing the propagating radiation mode, as shown in Fig. 2(b), appears in the Green’s function at . With the increasing of frequency ( ) or increasing of the distance between the two ) will appear in this Green’s plates ( ), more poles ( function.

C. Current Discretization and System Matrix Structure The volume integral equation (4) can be solved with Garlerkin’s method of moments. Generally, the volume polarized current discretization should be made in three dimensions and this would make the matrix too large and make this method inefficient because of large memory and CPU time requirement. However, the NRD-guide and -guide circuits in Fig. 3 can be viewed as sections of dielectric waveguides that are short circuited by two infinite parallel metal plates placed perpendicularly to the longitudinal axis ( ) at a certain distance apart. In this is fixed by case, the propagating constant along , namely, , with null or integer ). the presence of the plates ( eigenmode excitation is considIn this paper, only the modes, including surface-wave modes ered. Thus, only and a continuous radiation wave mode, appears in this circuit modes. due to the orthogonal properties between different The eigenmode excitation with other values is similar to the case. With the eigenmode excitation, a half-siin nusoidal vertical ( ) variation of fields ( and components and in the component) remains unchanged over the entire circuits. The discretization can be made only in the – -plane just as planar circuits. In this case, the matrix is greatly reduced and becomes solvable for complicated circuits. In Fig. 3, the volume current of polarization can be represented by combined local and propagating modes basis functions. Local mode means that the field is restricted in a finite region of the circuit while the propagating mode means that the field will propagate out of the circuit or propagate into the circuit from outside through the feed lines. The local mode can be either entire mode or sub-sectional mode and the propagating mode, of course, is the entire mode. The field in the core circuit can be represented by the local modes that either can be entire modes or sub-sectional modes with rectangular or triangular meshes to fit different planar sections. The field along the feed lines consists of a discrete number of surface-wave modes and a continuous radiation mode, as shown in Fig. 2(b). The propagating surface-wave modes can be represented by propagating mode basis functions and the exponent terms in propagating modes are truncated after several cycles. The evanescent surface wave modes, if they appear, can be represented by local modes, and the entire-domain basis functions are more efficient for them since they decay much slower than the evanescent radiation mode. The radiation mode is totally evanescent in the frequency region of the NRD-guide, and becomes partly evanescent, and partly propagating in the operating frequency of the -guide. The evanescent radiation mode can be represented by local modes for which the sub-sectional basis functions are more efficient. The propagating part of the radiation mode that

(7)



will cause radiation loss in bends and discontinuities is reprepole in the Green’s function (7). sented by the With the introduced basis functions, the polarized volume current in the circuits and feed-lines can be written as

803

These entries can be obtained by a spectral-domain integration with the vertical part still in the space domain as follows:

(10)

(11) (8)

where ( ) are planar parts of the propagating modes in the th feed line including LSE or LSM modes, are planar which have exact analytical expressions. are planar parts of the inparts of the local modes. mode cidental propagating mode that can be either the mode. We call , , and or as planar basis functions. are the unknown transmission coefficients of propagating modes at port l. is the number of is the coefficients of the the unknown planar local modes. unknown planar local modes. , Only the local modes , and are used as testing functions in this Galerkin’s method of moments. The discretized integral-equation system is solved and a linear system of equations can be obtained as

Here, we merge the incidental terms with the transmission terms in (11) and use the subscript to represent and . and are Fourier transforms of the and . The functions planar basis functions and are of and . The power flow at ports is formulated by (12) Since the surface waves have exact analytic expressions, the power of the propagating modes at port l can be obtained analytically as

(13)

for the

mode and

(9) are impedance sub-matrices of self and mutual cou) and plings between the two local modes of ( components. and are impedance sub-matrices of mutual couplings between the local modes and propaand gating modes where . and are the coupling between the local mode and ’s component of the propagating mode. One of the crucial points of the method of moments lies in effective and accurate evaluation of the system matrix entries.

(14)

mode. is the propagating constant of the mode, is the transverse wavenumber inside the dielectric, and is the transverse wavenumber outare coefficients in the analytical side the dielectric. and and modes. expressions of the for the

804


Suppose that the circuit is excited by an or mode at port l, then the -parameters can be obtained by

Once again, results of the vertical integration of the ( , ) function in (7) can be obtained component dealing with the by (15)

where

can be obtained in (9).

D. Analytical Evaluation in Space-Domain Integrations and Green’s Function The space-domain integrals within (10) and (11) in connection with the variables and can be evaluated analytically. The vertical integration of ( , ), ( , ), ( , ) and ( , ) components that are concerned with the function in (7) is

(19)

and results of the vertical integration concerned with source in in (7) can be obtained by

(20) (16)

The results of the vertical integration of ( , ) and ( , ) components related to the function in (7) can be obtained by

The component in the square brackets in (7) can then be simplified by combining the source term (20) with the principle term (19) and using the separation condition as

(21) The vertical integration in connection with the second term in (10) and (11) can be obtained by (17)

Similarly, results of the vertical integration of ( , ) and ( , ) components related to the function in (7) can be obtained by

(22) With the help of (16)–(19), a new Green’s function can be constructed as

(18) (23)



805

where . With this Green’s function, the integration in (10) and (11) can be written as

Fig. 4. NRD-guide and

(24)

H -guide open ends.

can also be applied in the rectangular coordinate. The symmetric property and redundancy reduction techniques can also be used to reduce calculation effort in this method. Some of these properties will be shown in the following examples. III. NUMERICAL RESULTS AND DISCUSSION

(25) The Green’s function in (23) is simpler than (7) without triancomponent in the square bracket gular functions in it. The pole apis greatly simplified as a constant. Only the . pears in this Green’s function in the condition of Other poles, especially the ever-existing pole, are eliminated through the vertical integrations in (20)–(23). Since this case for NRD-guide Green’s function only governs the Green’s function and -guide circuits, we call it the and refer to (7) as the full-scale Green’s function. The integrations in (24) and (25) are 2-D in the spectral domain with respect and , as in planar circuits. With the 2-D discretization to technique, the solution of the volume integral equation for NRDguide and -guide circuits has been reduced to a completely 2-D planar problem. We call this method the ORVIE method. E. Some Technique for Numerical Integrations The integrations in (24) and (25) are similar to those used in planar circuits, but much simpler because of the simple Green’s functions. Many kinds of planar basis functions, such as piecewise sinusoidal functions, pulse basis functions with a rectangular or triangular mesh scheme, or the entire mode can be used for the local modes to fit different cross sections of circuits just as used in planar circuits. When poles appear, the numerical integration must be carand ried out in the polar coordinate with because the poles appear at . However, when there are no poles in the integration, they can be carried out either in the polar coordinate or in the rectangular coordinate. In the case of planar basis functions in an NRD-guide circuit, integrands and can be rearranged as and , and oscillate with a sinusoidal behavior. Thus, the integrations in the rectangular coordinate is more efficient than its polar counterpart. The integrations in the polar coordinate with poles were discussed in [17] and [19]. A general asymptotic subtraction technique was used in [19]. In this technique, the asymptotic behavior of integrands was searched prior to performing the numerical computations. Subtracting these representations leads to a fast decaying integrand, and the integral thus becomes well suited for a numerical integration; the complete analytical solutions can be determined for the asymptotic part. This technique

Several examples are discussed here to demonstrate properties and efficiency of the ORVIE method. The main emphasis focuses on NRD-guide circuits because of their important applications at millimeter-wave frequencies. The following solution procedure of the NRD-guide open end is presented to showcase the features of the ORVIE method. A. NRD-Guide Open Ends The most difficult part of the ORVIE approach lies in dealing with feed lines in which both propagating- and local-mode basis functions are used. Here, we will discuss the calculation procedure of an NRD-guide open-end whose field distribution is similar to that along the feed lines. The NRD-guide open end itself is also frequently encountered in the design of passive components and active devices. The effects of the open end are studied first in [8] with a mode-matching method. The planar section of an NRD-guide open end with width mm, thickness mm, and is mode and a continshown in Fig. 4. Only a reflected uous evanescent radiation mode are produced in this structure mode is used as the incident mode with the when an above parameters. Fields of the radiation mode can be represented by sub-sectional basis functions over a certain region of calculation. The length of region of calculation , as shown in Fig. 4, depends on the average decay constant of the radiation mode. By defining a weighting average decay constant , generally

is

enough. We divide the calculation region into by cells along the of equal area. The length of each cell is -direction, and along the -direction. The center , is denoted by ( , ) with coordinate of the ( , ) cell, and . Planar basis functions for the radiation mode can thus be represented by a set of pulse basis functions as follows: (26) where elsewhere (27) elsewhere and

.

806


The planar propagating mode basis functions are (28) where

(29) The upside sign in (28) and (29) represents the reflected wave and the downside sign represents the incident wave. Fourier transand are given in the Appendix . The exforms of ponent terms in (28) are truncated after several cycles. It is found that choosing a length of the exponent as an integral number of wavelength speeds the convergence of integrals. The length of depends on the length of the calculation region the exponent of the radiation mode. Typically, the solution is insensitive to the wave exponent length for greater than three or four times . We choose the same set of sub-sectional basis functions as the testing functions. The even and odd properties of the integrand can also be used to reduce the integration range to the first quadrant in the – coordinate, the integration of the element in (24) and (25) can be written as

Fig. 5. Frequency characteristics of the phase of open-end structure with different mesh schemes.

S

for the NRD-guide

, is equal to . To make The asymptotic term valid for all , should be much greater than . Since the integrand with large contributes little to the total integrais enough. Subtracting the asymptotic terms tion, and , which are given in the Appendix , in the first integral of (32) leads to a fast decaying integrand. The integral thus becomes well suited for numerical integration. For the second integral, called the asymptotic part, complete analytical solutions can be determined as (33) (34)

(30)

(31) The integrands , , and terms are given in the Appendix. The integrations in (30) and (31) can be carried out in the rectangular coordinate ( , ) since the integrands oscillate in coordinates and , respectively, as shown in the Appendix . The integrations are defined through infinite spectral integrals and these can be evaluated numerically by restricting the unto the finite region bounded integration interval and , where and are sufficiently large numbers. The integrands exhibit a rapidly oscillating behavior, and amand . For the plitude of the integrands decreases as integer number , it is found that , and the convergence of the integrals is thus ensured. The weakest convergence characterizes the second terms of and . At this point, it would be beneficial to search for asymptotic behavior of the integrands prior to performing the numerical computation. The can be written as integration term relating to

(32)

have a similar solution for Integration terms concerning that is changed to . In the case of a uniform mesh on a rectangular structure with basis functions, there are shifting terms and , where and in (46), (47), (52), and (53). Thus, only different system matrix must be calculated. matrix elements of the In this case, only several seconds of CPU time is needed for one frequency point. The NRD open-end is simulated with a frequency sweep between 26–30 GHz. To estimate the error introduced by the discretization of the structure, the simulation was repeated on the basis of four different meshes with increasing cell density , 6 3, 9 5, 12 7. The calculated phases of at the reference plane are plotted in Fig. 5 as a function of frequency, while the magnitudes of remains almost unit constant in all the cases. The convergence is very satisfactory at low frequencies. However, more meshes should be used at high frequency since more fields due to the radiation mode are produced and decay slowly in the high frequencies’ region. In Fig. 6, our theoretical results are verified by comparing the amwith calculated and measured results plitude and phase of in [8], showing a very good agreement. Measurement errors observed over the frequency range of interest were explained in [8], i.e., the 26.5-GHz cutoff effect of the mode launcher from the rectangular waveguide, to a large extent, causes the phase deviation and a difference in results. Around the upper part of the frequency range, the errors may come from the fact that is



Fig. 8.

Fig. 6. Calculated and measured results of S -parameters for the NRD-guide open-end discontinuity. (a) Amplitude of S . (b) Phase of S .

Calculated amplitude of

807

S

of

H -guide open end.

appears in the integrations, which must be carried out in the as a funcpolar coordinate. Fig. 8 shows the magnitude of tion of frequency. It can be seen that the magnitude decreases very rapidly just after the radiation loss appears, then slowly as the frequency continues to increase. This phenomenon can continuous radiation mode be well explained from the configuration in Fig. 2(b). The level of radiation loss depends on the continuous spectrum width of the propagating radiation . The wider the spectrum is, the higher the mode radiation loss becomes. The slope rate of is infinite at the start , and this point of the propagating radiation mode indicates that the increasing rate of the spectrum width is infinite. Therefore, the decreasing rate of radiation loss is infinite at GHz, where , and the slope the critical point curve is also infinite at this critical point. As frerate of the becomes smaller and the quency increases, the slope rate of increasing of the continuous spectrum width becomes slower. Therefore, the decreasing of the radiation loss becomes slower. C. NRD-Guide Resonators The ORVIE approach can easily be used to solve the NRD-guide resonators. A homogeneous linear system of equations can be obtained as follows for the core circuit in Fig. 3:

(35)

Fig. 7. Field distribution of the end.

m = 1 radiation mode in the NRD-guide open

very close to the spacing limitation governed by the nonradiative condition of the guide, which is a half-wavelength in free space. Fig. 7 shows the field distribution of the radiation mode in the – -plane inside the dielectric medium. It can be seen that the field varies sinusoidally in the -direction and exponentially in the -direction. B.

-Guide Open-End

As the frequency increases, the structure in Fig. 4 will become an -guide. In this case, a pole at

Resonance frequencies can be obtained by letting the determinant of the coefficient matrix be zero as follows: (36) Four rectangular planar section resonators with different dimensions are calculated for comparison with measured results to validate the proposed method. Table I summarizes the measured and simulated resonant frequencies of the four different resonators. Theoretical results obtained with a numerical approach in [20] are also presented in Table I for comparison, and this approach effectively combines the method of lines with the mode-matching method. It is found that those results coincide well with each other for both LSE and LSM modes.

808


COMPARISON

OF

TABLE I THEORETICAL AND EXPERIMENTAL RESULTS FOR THE RESONANT FREQUENCY (IN GIGAHERTZ) OF DIFFERENT NRD RESONATORS (" = 2:58)

Fig. 10. Variation of the quasi-dual resonant frequencies f and f for a square-section NRD-guide resonator [illustrated in Fig. 9(b)] as the corner’s cut d varies with parameters " = 2:53, a = 12:3 mm, and b = 10 mm.

Fig. 9. NRD-guide resonators. (a) Rectangular-section resonator. (b) Notched square-section resonator.

A notched square-section resonator is then modeled, as shown in Fig. 9(b), which was first calculated precisely with a boundary-element method (BEM) in [21]. A symmetric cut is considered along a corner of the square section (height , un). In this perturbed sides , and notched sides so that case, a pair of resonance is related to perfect electric and perfect magnetic symmetry walls placed along the notched-square minor diagonal [dashed line in Fig. 9(b)]. Fig. 10 shows a and as a variation of the quasi-dual resonant frequencies function of in the notched square-section NRD resonator for the lowest pair of resonance. It is shown that results from BEM and ORVIE methods agree very well. The maximum difference between the two methods is less than 0.01 GHz and it mainly comes from the finite discretization in these two methods. The BEM can solve resonators of arbitrary shape, but with a constant dielectric-permittivity profile. The method in [20] can model an inhomogeneous dielectric-resonator problem, but the permittivity can vary in one direction only. Our proposed ORVIE method has no such limitation, and it can handle a resonator with any shape of planar section and with dielectric permittivity varying in both the - and -direction. Fig. 11 depicts resonant characteristics of an inhomogeneous resonator

Fig. 11. Resonant frequency characteristics of an inhomogeneous dielectric resonator against the exponential decaying factor for both LSM and LSE fundamental modes (a = 2:25 mm, b = 0:6 mm, c = 1:25 mm).

with a profile function of the dielectric permittivity in the following: (37) (in this in which is an exponentially decaying factor and example, ) the maximum permittivity of the dielectric resonator. It indicates that both the LSE and LSM modes have a quasi-linear increase with an exponential factor. This has also verified the analysis in [20] in which only the LSM mode has



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on the analysis of the mode spectrum in the NRD-guide and -guide. Vertical integration in the space domain can be carGreen’s function can be ried out analytically, and an constructed. The solution of the volume integral equation for NRD-guide and -guide circuits has then been reduced to a 2-D planar modeling problem. This method can be applied to the calculation of both the resonant frequency and scattering parameter for various dielectric circuits. The framework of the proposed technique was demonstrated through its application to an NRD-guide open-end structure. A number of examples include -guide open-end, resonators, and an air gap-coupled NRD-guide filter have been successfully modeled, and theoretical results have also been compared with other available simulations and measurements. APPENDIX The Fourier transforms of

and

are

Fig. 12. (a) Top view of a three-pole gap-coupled NRD-guide filter (a = 3:5 mm b = 2:7 mm d = d = d = 2:72 mm. L = l = 1:60 mm, l = l = 3:5 mm " = 2:04). (b) Calculated and measured transmission loss and return loss of the air gap-coupled filter. Theory (——). Measured (, – –*– –), Theory in [7] (–.–.–.).

(38) this behavior, while the LSE mode is not sensitive to a high decaying factor of the inhomogeneous dielectric. This phenomenon was explained in [20] because the dielectric permittivity only varies in the -direction, but remains constant in the -direction in that case.

(39) (40)

D. NRD-Guide Filter The last example is an air gap-coupled type three-pole 0.1-dB Chebyshev ripple bandpass filter with a 2% bandwidth at a center frequency of 49.5 GHz, which was designed in [7] based on a variational technique and fabricated with a Teflon dielectric. The ORVIE, of course, can be used to design this kind of NRD-guide filter. In this study, it was used to model the whole filter and to find its frequency characteristics. Fig. 12(a) shows the configuration of the gap-coupled NRD-guide filter with gemm, mm, ometrical dimensions mm, and mm. Simulated filter response is shown in Fig. 12(b), together with measured and simulated results [7] for comparison. Agreement between modeling and measurements of insertion loss is quite satisfactory. The excess insertion loss was measured and found to be 0.3 dB, i.e., 0.06 dB in our method, and it is more close to the designed value of 0.1 dB. The return loss in our method is a little shift to the low-frequency region.

(41)

(42) The integrands

and

are

IV. CONCLUSION In this paper, an ORVIE approach has been proposed and presented for modeling and analysis of NRD-guide and -guide circuits having any kind of section shape and inhomogeneous dielectric media. By introducing a half-sinusoidal vertical variation of field, the discretization can be made only in the parallel plane. Combined propagating mode and local mode basis functions are used in the Garlerkin’s method of moments based

(43) and (44)–(53), shown on the following page.

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(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)



has the following expressions:

(54)

(55)

The

asymptotic are

equations

of

and

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Duochuan Li was born in Huainan, Anhui Province, China. He received the B.Sc. degree in physics from Peking University, Beijing, China, in 1990, the M.Sc. and Ph.D. degrees in controlled nuclear fusion and plasma physics from the Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, Anhui, China, in 1993 and 1998 respectively, and is currently working toward the Ph.D. degree in electrical engineering at the École Polytechnique de Montréal, Montréal, QC, Canada. His research interests include computational electromagnetics, NRD waveguides, 3-D hybrid planar/nonplanar integration techniques, and substrate integrated waveguides.

Ping Yang was born in Hunan Province, China. He received the B.Eng. and M.Eng degrees in radio engineering from the Nanjing Institute of Technology (now Southeast University), Nanjing, Jangsu, China, in 1986 and 1989, respectively, and is currently working toward the Ph.D. degree in electronic engineering at the École Polytechnique de Montréal, Montréal, QC, Canada. From 1989 to 2000, he was with the Department of Information and Control Engineering, Shanghai Jiao Tong University, Shanghai, China, where he was formerly an Assistant Professor and then an Associate Professor. His current research interests involve computational electromagnetic and modeling of multilayered integrated circuits.

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Ke Wu (M’87–SM’92–F’01) was born in Liyang, Jiangsu Province, China. He received the B.Sc. degree (with distinction) in radio engineering from the Nanjing Institute of Technology (now Southeast University), Nanjing, China, in 1982, and the D.E.A. and Ph.D. degrees in optics, optoelectronics, and microwave engineering (with distinction) from the Institut National Polytechnique de Grenoble (INPG), Grenoble, France, in 1984 and 1987, respectively. He conducted research with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada, prior to joining the École Polytechnique de Montréal (Engineering School affiliated with the University of Montreal), Montréal, QC, Canada, as an Assistant Professor. He is currently a Professor of Electrical Engineering and Canada Research Chair in Radio-Frequency and Millimeter-Wave Engineering. He has been a Visiting or Guest Professor with the Telecom-Paris, Paris, France, INPG, the City University of Hong Kong, Hong Kong, the Swiss Federal Institute of Technology (ETH-Zurich), Zurich, Switzerland, the National University of Singapore, Singapore, the University of Ulm, Ulm, Germany, and the Technical University Munich, Munich, Germany, as well as many short-term visiting professorships with other universities. He also holds an honorary visiting professorship and a Cheung Kong endowed chair professorship (visiting) with Southeast University, Nanjing, China, and an honorary professorship with the Nanjing University of Science and Technology, Nanjing, China. He has been the Director of the Poly-Grames Research Center, as well as the Founding Director of the Canadian Facility for Advanced Millimeter-Wave Engineering (FAME). He has authored or coauthored over 390 referred papers and also several books/book chapters. His current research interests involve hybrid/monolithic planar and nonplanar integration techniques, active and passive circuits, antenna arrays, advanced field-theory-based computer-aided design (CAD) and modeling techniques, and development of low-cost RF and millimeter-wave transceivers. He is also interested in the modeling and design of microwave photonic circuits and systems. He serves on the Editorial Board of Microwave and Optical Technology Letters, Wiley’s Encyclopedia of RF and Microwave Engineering, and Microwave Journal. He is also an Associate Editor of the International Journal of RF and Microwave Computer-Aided Engineering (RFMiCAE). Dr. Wu is a member of the Electromagnetics Academy, the Sigma Xi Honorary Society, and the URSI. He is a Fellow of the Canadian Academy of Engineering (CAE). He has held numerous positions in and has served on various international committees, including the vice-chairperson of the Technical Program Committee (TPC) for the 1997 Asia–Pacific Microwave Conference, the general cochair of the 1999 and 2000 SPIE International Symposium on Terahertz and Gigahertz Electronics and Photonics, the general chair of the 8th International Microwave and Optical Technology (ISMOT’2001), the TPC chair of the 2003 IEEE Radio and Wireless Conference (RAWCON’2003), and the general co-chair of the 2004 IEEE Radio and Wireless Conference (RAWCON’2004). He has served on the Editorial or Review Boards of various technical journals, including the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, and the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He served on the 1996 IEEE Admission and Advancement Committee and the Steering Committee for the 1997 joint IEEE Antennas and Propagation Society (AP-S)/URSI International Symposium. He has also served as a TPC member for the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS). He was elected into the Board of Directors of the Canadian Institute for Telecommunication Research (CITR). He served on the Technical Advisory Board of Lumenon Lightwave Technology Inc. He is currently the chair of the joint chapters of the IEEE MTT-S/AP-S/LEOS in Montreal, QC, Canada, and the vice-chair of the IEEE MTT-S Transnational Committee. He was the recipient of a URSI Young Scientist Award, the Oliver Lodge Premium Award of the Institute of Electrical Engineer (IEE), U.K., the Asia–Pacific Microwave Prize, the University Research Award “Prix Poly 1873 pour l’Excellence en Recherche” presented by the École Polytechnique de Montréal on the occasion of its 125th anniversary, and the Urgel-Archambault Prize (the highest honor) in the field of physical sciences, mathematics, and engineering from the French–Canadian Association for the Advancement of Science (ACFAS). In 2002, he was the first recipient of the IEEE MTT-S Outstanding Young Engineer Award.