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A generalized CAD model for printed antennas and arrays with arbitrary multilayer geometries
Computer Physics Communications 68 (1991) 393—440 North-Holland
Computer Physics Communications
A generalized CAD model for printed antennas and arrays with arbitrary multilayer geometries Nirod K. Das Weber Research Institute, Polytechnic University, Route 110, Farmingdale, NY 11735, USA
and David M. Pozar Electrical and Computer Engineering Department, University of Massachusetts, Amherst, !vL4 01003, USA Received 16 August 1990; in revised form 12 February 1991
A general full-wave solution to arbitrary multilayer geometries of printed antennas and arrays coupled to multiple printed feed lines has been developed. It uses spectral-domain Green’s functions for multilayer substrates to account for the substrate layering of the printed antenna, and a newly developed generalized multiport reciprocity analysis to treat the coupling of the transmission feed lines to the antenna elements. The analysis is completely rigorous, including all surface wave or other non-radiating modal effects. The solution has been developed for both isolated elements as well as infinite arrays. The theoretical formulation of the generalized multiport reciprocity analysis is described in details and results for several practical examples are presented to demonstrate the accuracy and versatility of the analysis used. Considering the rigor and versatility of the analysis, it should find useful for full-wave simulation and computer aided design of a variety of multilayered printed phased-array configurations.
1. Introduction Printed antennas monolithically integrated with active circuits and a feed network is a cost-effective way of realizing a phased array for electronic beam steering [1,2]. In its most basic form, a printed antenna is a microstrip patch on a single-grounded dielectric substrate, fed by a coaxial probe or a microstrip line. This configuration has long been studied and theoretically characterized in detail. The most direct approach of using this type of element for a monolithic phased array is to edge-feed the patch with a microstrip line and integrate the feed network and active circuits on the same substrate in the space available between the elements. But, besides its notoriously small bandwidth there are three other immediate problems with this architecture: (a) There will be serious topological problems in laying out the feed network, and/or the space available between radiating elements (limited by grating lobe considerations) may not be sufficient to accommodate the feed network and active circuits. (b) There may be detrimental coupling between the radiating elements and active circuits. Also, the spurious radiation from feed network and active circuits could degrade the cross-polarization level and radiation pattern. (c) In a monolithic phased-array application, where a high-dielectric constant substrate is preferred for the integration of active devices, scan blindness is a potential problem. The underlying cause of this 0010-4655/91/$03.50 © 1991
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Elsevier Science Publishers B.V. All rights reserved
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problem is the fact that a single substrate is being used for the two quite distinct electrical functions of radiation and circuitry. (A thick low-dielectric-constant substrate is preferable for the antenna elements, while a thin high-dielectric-constant substrate is optimum for circuitry and active devices.) In order to solve these problems it is necessary to electrically isolate the antenna and active circuits in separate parallel substrates (some possible configurations are discussed in ref. [3]). The available space for integration is thus increased by reusing the physical surface area in separate electrically isolated planes. The RF interconnections between these separate layers can be realized by electrical connections through via-holes, or by electromagnetic coupling [4,5]. From manufacturing considerations, making a via-hole in a large-scale monolithic fabrication is often difficult, and can be avoided by use of the electromagnetic coupling, where electrical connection is established not by a physical connection, but by close “proximity” to a feed line or aperture. Besides isolating the antenna and feed network, there are other reasons to use multilayer integrated geometries with more layers of dielectrics and ground planes. Use of a dielectric cover as a protective radome, and use of stacked layers of microstrip patches [6—91for bandwidth enhancement, are two clear examples of antenna considerations. Also, a radome layer properly configured on the face of the antenna has potentials for achieving wide-angle impedance matching (WAIM) [10] of a phased array. From feed-network considerations, the surface area available exclusively for active circuit integration may be insufficient, especially when topologically complex feeds are required, such as dual feeds for circular or dual polarization. In this case, it may be desired to use a “sub-array” architecture as discussed in ref. [1]. This would require two levels of RF feed network. The primary feed network for the elements in the sub-array module may be required to be isolated by a ground plane from the active circuitry and secondary feed network in a different layer, and electromagnetic coupling between them can be established via a proximity-transition similar to that of ref. [11]. A number of techniques have been developed over the last several years for the analysis of a single-layered microstrip antenna printed on a dielectric substrate, with the most rigorous of these probably being the full-wave solutions such as refs. [12—161.Similar techniques have also been applied to infinite arrays of microstrip antennas on a dielectric substrate [17—19].More recently, some specific cases of the multilayer geometries mentioned above in an isolated or infinite-array environment have begun to be treated using full-wave analysis techniques. The analyses for the covered dipoles [20,211, the proximity-coupled dipoles [22—24],the proximity-coupled microstrip patches [25],and the aperture-coupled microstrip patches [26,27] are the representative cases. More complicated multilayered geometries that have been studied only experimentally include parasitic stacked patches [281, a dual-polarized aperture-coupled stacked-patch configuration [29], and an array of aperture-coupled patches with a radome [30]. Up until this time, the analysis of a new type of printed-antenna geometry has usually required the development of a new solution and computer program. In some cases, this might simply involve using a new Green’s function (for example, to add a radome layer to a geometry that has already been analyzed). In other cases, considerably more effort may be needed to adequately model the geometry of interest. Therefore, in order to avoid a large number of individual solutions and software for specific printed-antenna geometries, we have developed a generalized full-wave solution that can handle a large class of problems involving (a) multiple dielectric or magnetic layers and ground planes (each with or without loss), (b) various printed feed lines such as microstrip lines, slotlines, striplines, coplanar waveguides, coupled lines, etc., and (c) radiating slot, patch, or printed dipole elements. This technique can be used to model practically any geometry that can be constructed using rectangular planar elements. Layers can be coupled by proximity coupling to a transmission line, or by aperture coupling; idealized delta-gap excitation can also be considered, but not probe coupling. The solution uses the extended generalized Green’s functions of refs. [31,44] for the multilayer geometry, and a newly developed generalized
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N.K Das, D.M Pozar
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395
AN INTEGRATED ANTENNA ARCHITECTURE IN MULTIPLE LAYERS
////7 ~
Radome ParasiticAntenna Primary Layer
~-
Slotsj~~
~‘
Ground Plane
__________________
/
/~
Electro-Magnetic Coupling
~If//~—
Ground Plane
2(
Primary Feed Network
Coup
via
/~.“~7
/
~_~“
~
~e~y
Feed network
Fig. 1. An integrated multilayer architecture of printed antenas and transmission feed lines.
multiport reciprocity analysis for treating the coupling of transmission lines to coupling apertures, or to radiating patches, dipoles, or slots. It is completely rigorous, including all surface-wave or other non-radiating modal effects. The solution has been developed for both infinite arrays and isolated elements. Figure 1 depicts a possible geometry of a multiple-layer architecture, suitable for a monolithic phased array that can be treated using the proposed solution. It shows a sub-array module consisting of two distinct layers of primary and secondary feed networks, the primary feed network for circular polarization, stacked configuration for bandwidth enhancement, and a protective radome layer on top. The following section will describe the theoretical formulations of the spectral-domain momentmethod/ generalized multiport scattering analysis. For convenience, most of the complex mathematical formulas for direct numerical implementation are included in the appendices. In section 3, several practical examples are presented to illustrate the versatility of the method, and theoretical results for several of these cases are compared with measurements to verify the accuracy of the solution. It is hoped that this work will reduce the need for the development of a large number of solutions and computed programs to treat specific multilayer printed-antenna problems, and support further developments on multilayer antenna technology.
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2. Theory There are three important parts of this general analysis: the multilayer substrate configuration or the “substrate topology”, the transmission feed-line configuration or the “transmission-line topology”, and the radiating-elements configuration or the “source topology”. Any variation of the layered medium is rigorously accounted for by the use of a rigorous set of exact Green’s functions. To account for a general set of planar transmission-line configurations to feed the antenna system, the eigensolution to these transmission lines should be separately performed. The propagation constant, ke, characteristic impedance, Z~,and the eigenfields and eigencurrents of these transmission lines are required in the general analysis to completely describe the feed system. Finally, an arbitrary configuration of antenna (or parasitic) elements is incorporated into the analysis by replacing them with a set of current (electric or equivalent magnetic) expansion functions with unknown amplitudes. These unknown current amplitudes are solved using a moment-method procedure. The excitation to the general antenna is provided by means of incident waves at the feed-line ports, which are scattered due to the above currents induced on the antenna elements. These scattered waves at the feed-line ports are the other set of unknowns that are solved by relating them to the various induced currents on the antenna elements by use of a generalized multiport scattering formulation. This general analysis is schematically described in fig. 2, and the three important steps of the analysis, the multilayer Green’s function, the general transmission-line analysis, and the multiport-scattering / method of moments (MOM) analysis, will be separately discussed in the following sections. Emphasis will be given on detailed description of the multiport scattering modeling and the moment-method analysis. Spectral-domain expressions of all Green’s function components will be given, with only outlines of their derivation. Also, detailed discussions on the analysis of the transmission lines is beyond the scope of this presentation. Related literature is cited for reference on spectral-domain transmission-line analyses. 2.1. The multilayer Green function ~‘
The Green’s functions are vital to the numerical implementation of the general analysis. A method of derivation of spectral-domain Green’s functions for a multilayer geometry has been described in refs. [31,44], which uses equivalent transmission-line sections to account for several layers. This method has
GREEN’S FUNCTION SUBSTRATE TOPOLOGY
EVALUATE SPECTRAL
TOPOLOGY
MOM
EXCITATION FROM TRANSMISSION FEED LINES MODELED BY A RECIPROCITY ANALYSIS Fig. 2. General analysis of multilayered printed antennas and arrays fed by transmission-feed lines.
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Fig. 3. Geometry of a generalized configuration of multiple layers used for the analysis.
been extended for a more general configuration of multilayer structures to include several additional features. The details of the derivations are not described, since the basic steps are the same as in refs. [31,44]. A general configuration of a multilayer geometry of dielectric or magnetic substrates, and perfect electric or perfect magnetic conductors, is shown in fig. 3. The multilayer medium (“the substrate topology”) can include losses in dielectrics and conductors, with electric or equivalent magnetic sources between 11 and 21 layers directed along any coordinate axis, parallel or perpendicular to the substrate layers. A complete set of Green’s functions for this general structure needs to take into account all components of sources and fields, electric or magnetic, defined at any plane of the layered structure. Understanding the number of possible components of such a general set of Green’s functions, use of systematic notations to conveniently refer to specific Green’s function components is deemed essential. A set of notations recommended for general use is given in appendix A. In appendix B necessary algorithmic steps for obtaining the Green’s functions using an iterative scheme are described. Definition of various intermediate terms are the same as in refs. [31,44], and are consistently followed through. The flow-chart of fig. 4 highlights the basic steps discussed in appendix B. It should be noted that the implementation of the Green’s function algorithm ensures optimal computer time that is comparable with various closed-form expressions available in the literature for simple substrate topologies [14,32]. Appendix B treats sources in x- and z-directions. To obtain Green’s functions for y-directed sources, an easy-to-use look-up table is given in appendix C. Also, a table describing the symmetry properties of various Green’s functions in different quadrants of the spectral parameters, k~and k5, are presented in appendix D, and are often useful for transforming spectral integrals over the four quadrants of the k~—k~-plane into only one quadrant [14]. Other useful properties of the spectral-domain Green’s functions, such as the 4i (azimuth) variation on the k~—k~-plane and the asymptotic orders for large spectral values are also tabulated in appendices E and, respectively F. In appendix E the parameters a
398
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SUBSTRATE ‘TOPOLOGY’
Calculate the Green’s function correspondmg to the required field = 11 or 21
+4 ___
(~REE~ FIELDS AT INTERFACE
~ 11 or 21 Calculate the Green’s function corresponding to the required field
___
+
+
(~RE~) with Separate
Z-Dependent Part
BETWEEN TWO LAYERS Fig. 4. Flow chart of a four-step procedure used for the computation of Green’s functions.
and f are the vector magnetic and electric potentials, respectively. While considering power radiated into free space, corresponding to the values of a less than or equal to k0, the En-component of radiation
is directly proportional to a, whereas the E,~-componentof radiation is directly proportional to f. Hence, the çb-dependence of the far-field E9- and E~-componentsdue to a spectral source sheet can be obtained from the table in appendix E corresponding to respective a’s and f’s. For example, the çb-dependence of the far-field E6 pattern for a J~source sheet is a cos 4 pattern. With the above detailed algorithmic expressions of the Green’s functions, and various tables to describe their useful properties, one can numerically implement these Green’s functions, and compute various spectral integrals that use them in an efficient and straight-forward way. That these Green’s functions have poles on the k5—k~(or a) spectral plane is vital to correct evaluation of spectral integrals. Locations of these poles correspond to the excitation of various surface wave, parallel plate, or other trapped waveguide modes of the multilayer structure, and the spectral integrals of the Green’s functions
N.K Das, D.M Pozar
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need to include the contribution of these poles in the residue sense. Residues can be numerically computed using the following limiting definition of the residue of a function G(a) at the pole, a~. res G(a) lim G(a)(a —ar). (1) =
a=a~
This limiting expression can be approximated in two possible ways, res G(a)
G(a~+ 6)6,
(2)
a
or, as res G(a)
~6[G(a~
+
6)
—
G(a~ 6)],
(3)
—
a
where 6 is an arbitrary small value, optimally chosen to insure numerical stability [44]. Residues can also be computed analytically, but considering the complexity of the Green’s function expressions for a general multilayer geometry, this is practical for every simple layered structures only (for example, the case of a single-grounded dielectric slab, or a parallel-plate structure with a single dielectric substrate in between). 2.2. Generalized planar transmission-line analysis in multiple layers Planar transmission lines in multiple layers of dielectrics and ground planes are useful for many microwave- and millimeter-wave applications, and are of special interest for the characterization of various feed structures for the multilayered integrated phased arrays. A variety of geometries are possible with variations in their configurations of dielectric and conducting layers (the “substrate topology”), as well as the configurations of the guiding structures (the “source topology”). The substrate topology can consist of a number of dielectric substrates with different dielectric constants covered by ground planes on both sides (as in a stripline) or only one side (as in a microstrip line), or completely uncovered on both sides (as in a slotline). On the other hand, the source topology can consist of multiple (typically one or two) strips of conducting lines, and/or slots on ground planes, in the same or in different layers. The general multilayer configuration of fig. 3 can account for the possible substrate topologies of a generalized planar transmission line with different source configurations placed at one or
Mkrostrip Line
Coplanar waveguide
Slotline
Edge Coupled Stripline
Broadskie Coupled Striphne
Dual Sidtline
Coplanar Striplirie
Coupled Microstrip Line
StriPline
Fig. 5. Transmission-line configurations of common interest with variation in their source topologies.
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A generalized CAD model for printed antennas
more of its interfaces. Figure 5 shows various possible source topologies of interest that can be placed in one or more layers to completely define the source configuration of any type of planar transmission line. In ref. [31] a preliminary attempt was made for the spectral-domain analysis of simple configurations of planar transmission lines in multiple layers. The general analysis framework for the unified full-wave analysis of a large class of planar transmission lines has been presented in ref. [33] with special emphasis on computation of different loss components. User-oriented software on mainframe and personal computers have been implemented to perform such full-wave spectral moment-method analysis of general transmission lines [34,35,44]. An integrated analysis package for mainframe computers, “UNIfied Computer Aided Analysis of Multilayer Transmission lines”, UNICAAMT [34,44], has been developed at the University of Massachusetts, the general procedure of which has been schematically shown in fig, 6. Besides computing the various eigensolution of the transmission lines (ke, Z~or P~etc.), it also graphically displays various forms of field patterns and contours for a better qualitative understanding. A comprehensive personal-computer version of UNICAAMT, called “Personal Computer Aided Analysis of Multilayer Transmission lines” (PCAAMT), has also been made available recently from the authors [35]. The results of such general printed transmission-line analyses (the propagation constant, ke, the characteristic impedance, Z~,or the characteristic power, P~,and the current basis-function amplitudes) are directly used in the general multiport antenna analysis discussed below to completely describe the various feedlines. Further related details on spectral-domain moment-method analysis of printed transmission lines can be obtained from various references [36—39]. 2.3. Spectral-domain moment-method
/ generalized multiport scattering analysis
2.3.1. Generalized formulation of multiport scattering Before describing the general analysis of the multilayer antenna, it is useful to develop a general formulation for scattering in a multiport system. Any general multiport system with scattering sources inside can be completely described by its scattering matrix, [S,~].The scattering matrix elements can be characterized by exciting only one port at a time using a matched source, and quantifying the outgoing waves at all match-terminated ports. To solve this particular problem with one port excitation, it can be decomposed into two separate parts: In the first part of the problem (part A), all the sources are physically removed from inside the system and power at all ports determined due to the incident port excitation (say the jth port). Also, let the scattering matrix of this source-free multiport system be [SA~J]• Assume that this source-free system is such that it is exactly characterizable with known eigenfields. These electric and magnetic eigenfields are completely describable by two matrices of field vectors, [EAJ] and [HA)], respectively, corresponding to the jth port excitation. EAJ and HA] denote the fields due to the jth port excitation in general anywhere inside the system. Assume that the individual feed lines at different ports are also characterizable with known eigencurrents and fields, propagation constant (kej) and characteristic impedance (Z~1).Let the electric and magnetic eigenfields for the outgoing wave of the ith port be e[ and hI, respectively, and the corresponding incoming eigenfields are e~=eI, h~ —hI, such that =
ff
[eixhl]
ds=1.
(4)
ith port
Thus, if EA~Jand HA~Jare the eigenfields of the source-free system at the ith port when the jth port is excited, HAJ=HAtJ=SAtJhi, EAJ
=
EALJ
=
i~j,
(5a)
#j,
(5b)
SA~Je1, i
N.K Das, D.M Pozar
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A. Substrate topology. B. Source and field identifiers, C. Output Options
INPUT CALL ~
~
——+—— Set up Eigen value
GREEN1
RFTURN
Matrix elements for
-
Ke
I
Numerical
INPUT]
trial
_______
. 4~..
Solve the Eigen value forKe
) Output
• The total current
LII}~~
I
JPOWER4d
I R E T U R N
r~l
I I
I
N-_-~\. T R V
• unknown current co-effs
• The current-density transform
MODULE _____
C1
1
Find
GRAPHICS
Yc
or Z~
Dielectric
Conductor
Loss
LOSS
I
/~
/4~ f
.
Field Pattern CALL
GREEN 1 & GREEN 2 —~ Green’s function module
Fig. 6. Algorithm describing the basic steps of the generalized transmission-line analysis.
or HAJ
=
HAI]
=
(SAl]
—
1)I~~ (1 =
—
SAjJ)h7, i =j,
(5c)
i=j.
(5d)
EAJ=EAlJ=(SAlJ+1)eI=(1+SAlJ)e,~,
Figure 7a shows such a 2 IV~port system without the inside sources. In the second part (part B) of the problem, the incident port excitation is dropped, but the inside sources are replaced by their respective actual/ equivalent currents induced when the jth port is excited by a matched source, with other ports match-terminated (see fig. 7b). The volume sources are a set of .17 and M7, and the scattered fields at the ith port is SBI). For this second part of the problem let us
402
/
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A generalized CAD model for printed antennas
/
2/ •
.
7 ISAI
1I (CAll, [HAIl
•
7 [Ml
=~—
S
S
No Source
S -
-
EBII, H61j
5AIj, EAII,
All
SB
V ii
S
S
11
•
/ A
B
Fig. 7. Part A and B of the multiport scattering problem.
assume, however, that the corresponding electric and magnetic fields, EBJ and HBJ, are not known everywhere inside the system. But at the feed-line ports they can be described in terms of SBI, as HBJ
=
HBIJ
EB)
=
EBIJ
=
=
(6a)
SBjJhi,
SBIJeI,
for all i.
(6b)
Now the actual problem can be treated as a superposition of the two parts: S11
(7) at the port p due to the volume sources .17 and M, to It possiblebetween to relatethe theeigenfields scattered field, theisreaction EAP, HAP and .17, M~’1.A reciprocity equation of the following form can be used to establish the above relationship: =
SAl] + SBI].
5Bpj’
V’(Ea xHb—EbXHQ)
=
~
~Eb+Ha ‘Mb,
(8)
where Ea, Ha, and Eb, Hb are solutions of Maxwell’s equations in a closed volume, V, with the corresponding volume sources as Ja’ Ma, and .1,,, Mb, respectively. For the present case let Ea and Ha correspond to EAP and HAP, respectively, with Ja Ma 0, and Eb and Hb correspond to EB] and HBJ with Jb ,JJ”, and Mb MJ”, respectively. Thus, the above reciprocity equation can be rewritten as =
=
=
=
V.(EAPxHBJ—EBJxHAp)=—J~EAp+M~HAp.
(9)
Consider the closed surface, 5, in fig. 7b on which the port references are defined. Applying the divergence theorem to eq. (9) on this closed surface S, and using the expressions (5) and (6) on S for
N.K Das, D.M Pozar EAIP, HAIP, EBI]
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A generalized CAD model for printed antennas
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and H8~1,it can be shown after some straight-forward mathematical simplifications that
Jj(EAPxHBJ—EBJxHAP)ds=fff(—.17.EAP+MJ”.HAP)dv,
(10)
SBP]~Jff(_Jj’EAP+Mj~HAp)dV~fJ
(11)
~‘
V
(—JJ.EAp+Mi,.~HAp)ds, sources
1, have been replaced by the corresponding surface currents, J~and
where volume sources, J and M,’the source currents are always represented by such surface currents M~.Forthe printed-antenna applications due to their planar characteristics. Equation (11) is the multiport general scattering condition that states: When various currents are placed as sources in a multiport system with no external sources of excitation, and all ports are matched, the amplitude of the scattered wave propagating out of the pth port with proper port-phase reference can be expressed as one half the sum of positive or negative reactions on all source currents due to the eigenfields of the source-free multiport system. The above reacting eigenfields should correspond to the pth incident port, the same port where the outgoing scattered wave needs to be evaluated (note that in fig. 7a the incident port for the source-free system is the jth port). Positive reaction is considered if the source is a magnetic current, whereas negative reaction is used if the source is an electric current. 2.3.2. Simplified model for a multiport antenna system Figure 8 describes the topology of a general geomerty of a multiport antenna system with Nf’ different antenna alements fed by N~feed lines (per unit cell, if an infinite array). Certain useful conventions are consistently followed, and various practical assumptions are made to simplify the general analysis: • As shown in fig. 8, all the transmission feed lines run through without being terminated inside the multiport boundary. Thus the IV~feed lines result in 2N~ports. Any external termination can, however, be accounted for via the scattering parameters of the circuit, as long as the terminations can be assumed not to directly interact with the internal fields of the multiport circuit. For certain practical geometries, for example a dipole electromagnetically coupled to the open-end of a microstrip line [22,40], such an assumption would not be valid. Suitable modifications discussed in section 2.3.5 need to be incorporated using an additional non-Galerkin testing procedure to account for such practical geometries. • The reference incident electric-current directions on the two opposite ports (the ith and the N~+ ith ports, as a convention) of the same (ith) strip-type transmission feed line are assumed to be opposite, whereas the reference incident equivalent magnetic-current directions on the two opposite ports of the same slot-type transmission feed line are assumed the same (see fig. 8). This assumption enforces the same reference transverse electric-field directions for the two opposite ports. This is just to establish a convenient conventional consistency. • The scattered port electric current on a strip-type transmission feed line is negative of the corresponding scattering parameter, whereas the scattered port equivalent magnetic current (or slot field) on a slottype transmission line is the same as the corresponding scattering parameter. This is consistent with the standard electric field referenced definations of scattering parameters. • The individual transmission line fields do not directly couple with each other in the absence of the radiating elements. However, indirect coupling between them can always result via mutual coupling between associated radiating (finite-length) elements that are directly coupled to the transmission lines. This restriction does not exclude the use of coupled lines, such as coupled microstrip lines or coplanar waveguides, as long as they are treated in groups as a single transmission-line element. As per this restriction, the transmission feed lines are assumed to be far apart, or printed on different layers electrically isolated by conducting planes in between (fig. 8 does not show the layering, and the cross-overs are assumed to be on isolated layers).
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I t+’
~4—--\\~r----
~7
/~~\\\\ ~
/
\~N~-1
I/I
UNIT CELL) BOUNDARY
(for infinite array)
a
~
ARRAY
~
SLOT
N
1
i L ~1 CONDUCTOR Fig. 8. Generalized multiport model of printed antennas integrated with planar transmission feed lines. Various antenna elements and transmission lines are in general placed on different surfaces of a multilayer configuration.
• For infinite arrays the lengths of the feed lines are strictly restricted by the unit cell dimensions. For practical isolated geometries the feed lines are always of finite extent. These finite-length transmission lines are, however, approximately modeled by hypothetically extending them to infinity in both directions [27]. Any criss-crossings due to this hypothetical infinite-length extensions are eventually ignored. To summarize, the above restrictions are not very severe for most practical applications, but are necessary for analytical simplification purposes. The other fundamental idea behind the above assumptions is to model the dominant electromagnetic interaction in the system due to mutual coupling between the finite-length radiating elements, and not between feed structures. For off-broadside scanning of infinite arrays, particularly near blindness resonances, such assumptions may not be strictly valid, however. With the above assumptions for the multiport model of fig. 8, excite the mth port by an incident wave of unit amplitude. The source-free part (part A) containing transmission lines without direct coupling between them can be described as: 5Aim~’ i=P~+m, mN~, =1,
i=m—N~, m>N~,
=0,
otherwise,
(12)
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with EArn equal to the travelling eigenfields of the mth transmission line alone. Such simplification results in completely solving the first part of the multiport antenna problem if solutions for individual transmission lines are available. As mentioned before, general numerical solutions for multilayer transmission lines are available [34,35,44], and need to be used to support the present scattering formulation model. Now, denote the scattered waves, SBjrn, of part B as R1, dropping the second index, m, for convenience, with the understanding that R,’s are due to the sources induced by the incident wave at the mth port. Thus, using eqs. (7) and (12), the R1’s can be trivially related to the actual scattering parameters, ~ of the multiport antenna: Rj=Sim~1, i=N1+m, mN5, 51m1’ i=m—N~, m>1V, 5im’
otherwise.
(13)
2.3.3. Galerkin testing on the antenna elements The unknown currents on various finite-length strip elements, or equivalent magnetic currents due to electric fields on various finite-length slot elements (of only one unit cell, for an infinite array), can be expanded as a sum of a set of basis functions with unknown coefficients, N 1
Jm>JI,.J~,
(14)
i= 1
where J is used both for electric currents as well as equivalent magnetic currents for the sake of generality, with the longitudinal and the transverse variations suitably chosen from the set of basis functions of appendix H. Nf is the total number of finite-length modes of the system, distributed over Nf’ finite-length physical antenna or parasitic elements (see fig. 8). From the equivalence principle, it should be noted that the electric field in a slot on a ground plane is equivalent to two magnetic currents of equal magnitude and opposite directions placed above and below the slot with the ground plane closed. These two magnetic currents are described by only one unknown coefficient in eq. (14). Consider a perfectly conducting element of the general antenna configuration, for example an electric dipole or a patch. In order to obtain the solution to the entire system, a part of the boundary condition to be satisfied requires the tangential components of electric field to vanish at all points on the conducting element, E~~’ Escat + E~~c0, =
=
(15)
where ESC~~ is the component of the tangential electric field produced due to currents on all finite-length elements of the system, and EInC is that due to all travelling-wave elements of the system. It may be noted, using the two-part solution approach of section 2.3.1 and 2.3.2, that ~ includes the fields of the first part, and a part of the fields of the second part produced due to only the travelling-wave currents on the feed lines. The above boundary condition can be rewritten as N1
N,’ —
1=1
LE,’°~,
(16)
i=1
where E1 is the scattered tangential field due to the ith finite-length mode with coefficient I,, Nf is the total number of finite-length modes of the entire system, EnC is the total incident tangential field due to all travelling-wave currents on the ith transmission line, and N’ is the total number of transmission lines coupled to the particular conducting element. This exact boundary condition can, however, be enforced
406
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using a variational approach by means of a Galerkin testing procedure [14,32] resulting in a set of linear equations involving the unknown coefficients, 11’s; N~
N,
~I1Z~,=
~i’ç~,
i=1
1=1
(17)
for all jth current modes of the system used to expand the currents on the particular conductor on which the boundary condition is being tested, where Z~~,—J
E1’J, ds,
(18a)
jth mode
V,=
—Jjth modeE”~J,.ds,
(18b)
and are respectively refered to as elements of reaction and excitation vectors of the Galerkin moment method. On the other hand, if the element considered is a slot, or an equivalent magnetic conductor, the required boundary condition to be satisfied is given as (19)
t2 + Hm0~~ — HWC2,
0 ~ H~’~’ where the superscripts 1 and 2 refer to the top and bottom surfaces of the slot, respectively. Using a similar testing procedure as described before for conducting elements, the resulting set of linear equations for all j corresponding to the modes of the particular slot can be written as in eq. (18), where Htan
2
—
z~
=
=
—
1—H,2)’J
1=J
(H1
1ds,
(20)
jth mode
—fjth mode (H1~’~
J7~,=
2) .j
ds.
(21)
HiflC
Depending on the location of the ith mode (above, below, or on the same conducting ground plane as the jth mode), one of the two magnetic-field terms of eqs. (20) and (21) may be zero due to the ground-plane isolation. Note that the dipoles and patches are assumed to be made of perfect conductors in the above equations, and no impedance boundary condition is implemented for these antenna elements. However, such impedance boundary conditions can easily be implemented by suitable modification of the expressions for V, and ~ which would only involve computation of more Green’s function components. This may be essential for high-frequency applications to include conductor losses due to the radiating elements. However, losses due to dielectrics or various ground planes of the substrate topology are automatically included via the Green’s functions. Components of reaction and excitation vectors for various cases of expansion and testing modes as defined by eqs. (18), (20) and (21) are computed using spectral-domain Green’s functions. Specific details of the spectral-domain implementation of the reaction and excitation elements will be separately discussed. Infinite arrays as well as isolated-element cases will be considered. 2.3.3.1. Spectral domain expressions of reaction elements. Using the spectral-domain Green’s functions of appendix B, and extending them for an infinite-array environment [17], the expressions of (18) and (20) can be combinedly written as Z~ 1=
~ n
~ m
V~
~(kxrnn,
kyrnn) ~1j~
Uj(kxmn, kyrnn)L~*(kxrnn, kym~)~
Ak~,
(22)
N.K Das, D.M Pozar
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Ye Va
•
•
•
S
407
S
Fig. 9. Reference axes of various coordinate systems used in the generalized multilayer antenna analysis.
where G
G~,. if the jth mode is an electric current, or
G~/21~
if the jth mode is an equivalent magnetic current, with the source, J~,electric or magnetic as the ith expansion mode. The Fourier transforms of the ith and jth modes, respectively L~and U1, as described in appendix H are assumed to be with respect to the system coordinates, the expansion test 2e 1, and and are I~ respectively £~.The superscript, k,, of and the the Green’s modes with the respective current directions in functions refer to the particular substrate configuration with the ith mode between the 21 and 11 layers of the substrate topology. The indices, m and n, refer to various Floquet modes of the infinite-array configuration and depend on the periodicity of the array, the grid structure of the array, and the phasing of the array tQ obtain a particular scan angle. The scan angle of the array is given by the angles (0, 4,) of a spherical coordinate system with the z-axis of the multilayer substrate configuration coinciding with the 0 0 and the x—y-plane of the layered structure coincide with the 0 900 plane, with 4, = 00 along the array x-axis, Xa~The geometry of the array configuration with various reference directions is shown in fig. 9. With reference to fig. 9, the expressions for kxmn, kyrnn, ~ ~ in eq. (22) can be written as =
—
=
=
0,
=
2m’iT kxm
(23a)
0=~k05~flOCO54,+~_~__,
2n’rr
kyrnn
~k and,
a
X
—k0 sin 0 cos(4, Oa) + —i--— /sin Oa 2 4’Tr2 z~k =—, 4ir ~ =z~k=absinOa z.IA =
—
and b are the grid spacings along
Xa
—
kxmn/tan Oa,
(23b) (23e)
and y~axes, respectively. To handle the most general array
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A generalized CAD model for printed antennas
configuration, four local coordinate systems are used. (Xa, ya) are chosen here as the reference system coordinates, with Xa the array x-axis, and y~perpendicular to it. The array grid periodicity is along Xa ~fl one direction, but along y~where y~is not perpendicular to Xa in general, and is referred to as the array y-axis. Other than the system coordinates, (xa, ya), the array coordinates, (Xa, y~),the coordinate system (Xe, ye) is the expansion coordinate system and (x 1, y~)is the test-mode coordinate system, with various relative angles as shown in fig. 9. Clearly, the reaction vector of the moment analysis that accounts for the dominant coupling mechanism in the infinite array must be computed differently when extended to isolated-element analysis. Analytically it is equivalent to computing the Z,1’s of eq. (22) by using an infinite integral instead of an infinite series. Thus
~ffV•
=
~
k~).x~jjUj(kx,k~)U~*(k5, k~)dk~dk~,
(24)
with the terms defined as in eq. (22). Unlike the infinite-array analysis, the reaction matrix can now be shown to be symmetric with Z13 Z31 based on the reciprocity theorem. Also, unlike the infinite array, where the infinite sum must be computed over the entire spectral plane, the infinite integrals of eq. (24) can often be reduced to integrals over only one quadrant of the spectral plane, relating the contribution from other quadrants by symmetry considerations [14] of appendix D. The other major analytical considerations when replacing the infinite-sum expressions of (22) by the infinte integrals of (24) are due to the numerical problem associated with the computation of the spectral integrals. Efficient numerical methods have to be used to optimally account for the sometimes fast oscillating integrals of (24). As discussed earlier, the poles of the integrals of (24) associated with the poles of the Green’s function due to the characteristic waves of the multilayer structure impose the additional complexity for the numerical implementation of the infinite spectral integrals of (24). Contributions of the residues at these poles has to be added to the final principal values of Z11 to account for coupling due to the excitation of characteristic waves. Positions of the characteristic wave poles of the Green’s functions on the k~—k~-plane can be searched using a Newton—Rhapson iterative technique. Using the values of these pole locations the residues of the integrals of (24) can be extracted numerically in a way similar to that described in section 2.1. =
Spectral-domain expressions of excitation elements. Unlike the computation of reaction elements in section 2.3.3.1, the computation of excitation-vector elements is the same for infinite arrays as for isolated elements. This is based on similar assumptions as in section 2.3.2 that there can be no direct coupling between transmission lines of the same unit cell (for an infinite-array case), or of two different neighbouring unit cells. Also, due to a relatively more localized nature of the transmission-line fields, they usually do not couple to antenna elements of the neighbouring cells. In other words, the coupling between an antenna element and an infinite array of transmission lines is electrically equivalent to the same between the antenna element and a single transmission line. Such an assumption is valid for near-broadside scanning of the array, but fails near blindness angles or for near-endfire scanning. As discussed in section 2.3.2, when port m is excited, the outgoing wave for the second part of the problem at the ith matched port is equal to R,. The first part of the problem only consists of the eigenfields of the mth transmission line with unit amplitude. Under this condition of one port excitation, before precisely defining the expressions for various excitation-vector elements, V~’s,of the moment method system, it may be useful to define the following intermediate expressions: 2.3.3.2.
=
—
=
—
~I~ffE,”
‘J3. ds = ~
L~f~ [1~~(_k~,
~l’~
ky)1~*(_ke,k~)F~(k~) dk~,
(25a)
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409
if the jth element is an electric current, or,
=
—
~
~f~(i~
[~i~ij(_ke,
k~)— O~ij(~ke, k~)j)U
1*(_ke, k~)F,,(k~)dk~,
(25b) if the jth element is an equivalent magnetic current. The system coordinates are chosen with the direction of forward propagation of the ith transmission line, x~,,the same as the system x-axis. L~V~ is the excitation of the ith infinite transmission-line element to the jth finite-length mode associated to its discontinuity, when the current on the ith transmission-line element, described by the coefficients of its expansion modes (I,,’s), is such that unity power is transmitted across its cross-section in the forward direction. This means, for example, in the case of a microstrip line, a total longitudinal current along the conducting strip equal to 1/ ~ units, where Z,~is the characteristic impedance of the microstrip line. For a single slot line to transmit unit power across its cross-section, the required equivalent magnetic current along the slot line needs to be equal to 1/ ~ where )‘~ is the characteristic admittance. The source components associated with the Green’s functions of the equations (25) are decided by the nth mode of the transmission line, J,,, and can be J~,M1, .1,,, or M~as required. The Green’s functions are evaluated at the plane of the jth mode where k,, denotes the substrate topology with respect to the nth source currents placed between the 11 and 21 layers, as usual. is the Fourier transforms of the jth test mode directed in I~ 1,, with respect to the system coordinates defined (see appendix H), and ~, is the Fourier transform of the transverse variation of the nth mode of the ith transmission-line element. Finally, ke is the propagation constant of the ith transmission line element. A similar term, z~V~called the forward-discontinuity mode excitation, can be defined where instead of the ith transmission-line mode with infinite length in the forward direction, the ith transmission-line forward-discontinuity mode (see Appendix H) is considered as the source of excitation. =
_
~V~=
ff~[~..o~,(k
ky)IUJ*(kx, k~)U,,(k5,k~)dk~dk~,
(26a)
if the jth element is an electric current, or, — —
—
V’
~
~L
~
U I~11 . JJ
~k,, (~. ; \ ‘-‘H21J,,Y~x’‘by)
xL~.*(k~,k~)U~(k~, k~)dk~dk~,
—
,-‘k ii “JtilJ,,V’x’
n.y (26b)
if the jth mode is an equivalent magnetic current. The notations have similar interpretation as those of eq. (25). It should be noted that double spectral integrals in eq. (26) are required in place of single-dimensional spectral integrals of (25) to account for the discontinuity modes. U,, is the Fourier transform of the forward-discontinuity mode as described in appendix H, with the same transverse variation on the transmission line as before. Two additional terms equivalent to LW~and ~J’~ can be obtained by rotating the ith transmission line, and thus rotating the system coordinate axes by 1800, and recomputing ~ and ‘5d with reference to the new system coordinates. In addition, a negative sign would be added to or for a slot transmission-line mode, to take care of the port-reference current directions described earlier. These two new terms are called the backward excitation, z~J’~’, and the backward-discontinuity excitation, z~V~respectively, and correspond to the backward-propagating wave on the ith transmission line.
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A generalized CAD model for printed antennas
With these terms defined, it is easier to formulate the V.1’s of eqs. (18) and (21) of the moment-method system as follows: +
=
~Rl+N,(~l’~
+
if i ~ m or i
+~ #
+
~i’~fl,
(27a)
m, or (27b)
if i
=
m
N1 or
>
IV~,respectively.
Similar treatment of the transmission-line travelling-wave currents are presented in ref. [24] in the context of a specific simplified geometry with a microstrip feed line coupled to a perpendicularly placed dipole. The transmission-line discontinuity modes of appendix H take care of the semi-infiniteness of different R.’s. Various semi-infinite-length travelling-wave currents extending in one direction can be decomposed as the sum of an infinite mode amplitude, and a discontinuity mode of the type described in appendix H, each with half the amplitude of the semi-infinite mode. This is essentially the basis on which eqs. (27) are derived, with ~W~5 and zWJ’ resulting from the infinite mode parts, and z~VJ”and L~VE~~d resulting from the corresponding discontinuity mode parts. While performing numerical implementation, computations of ~j~3I~d or are further simplified by decomposing into odd or even discontinuity modes, and also incorporating a modified transform to smooth the poles due to the discontinuity mode transforms (see appendix H). It can be noted that or LW~’in the expression of ~ when i m is due to the transmission-line fields of the first part of the multiport problem when no discontinuities are present. As discussed before, when only the mth port is excited, this trivial case results in fields only in the transmission line corresponding to the mth port, travelling forward or backward depending on whether m is less than or equal to, or greater than, N~. =
2.3.4. Spectral-domain implementation of multiport scattering model The number of linear equations obtained from the Galerkin testing of the boundary conditions on the finite-length elements is equal to the total number of finite-length expansion modes of the system. But this set of equations is not sufficient to solve for the additional 2N~unknown values of R1’s. However, another set of 2N~equations can be obtained by using the generalized multiport scattering model. 2N5 + Nf unknowns of the moment-method system can thus be solved by solving the complete set of linear equations. This generalized scattering model is a rigorous generalization of the reciprocity arguments used in refs. [26,27] for a perpendicular slot-type discontinuity, and in ref. [24] for a perpendicular electric-dipole-type discontinuity. In both refs. [24] and [26,27] the transmission line is a single microstrip line. In a general environment where various types of transmission lines are coupled to dipoles or slots, or a combination of both at the same discontinuity plane, or when more than one transmission line is coupled to the same radiating elements, the reciprocity arguments of refs. [24] and [26,27] cannot be directly extended. The present multiport-scattering formulation, however, accounts for all such cases. Using the generalized scattering condition (11), the eigensolution of the ith transmission line described by its current expansion mode amplitudes (I,,’s), and the the expression (25) for coupling between the ith infinite transmission line and the jth antenna mode, gives the following result: R1= ~ =
~
±1,,~l’~J”, if iN~,
(28a)
±1,,L~V,~” if
(28b)
i>N~,
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/ A generalized CAD
modelfor printed antennas
411
where the summation over j is for all finite-length modes coupled to the ith transmission line, the negative sign is chosen if the jth mode is a magnetic current below the nth transmission-line mode, and the positive sign is chosen if the jth mode is a magnetic current above, or is an electric current above or below the nth transmission-line mode. Different signs for the expressions in eq. (28) for the cases of a magnetic-mode current above or below the transmission line are used to take care of the reversed signs of the equivalent magnetic currents for two sides of the ground plane. It should be noted that the different signs for the R1 expressions in (28) do not depend on the type of transmission lines, but on the type of the current mode coupled to it. For example, if one transmission line is used, with one dipole or slot perpendicular to it, with an electric or equivalent magnetic current, I, and the transmission line is excited from port 1, R1= +~Ii~V1~,
(29a)
R2= +~IAV1~,
(29b)
if i is due to a dipole, or R1= —~Ii~V1~,
(30a)
R2= ~
(30b)
if I is due to a slot below the transmission line, and thus S11 R1, ~21 1 + R2 1 + R1 for a dipole, and ~ R1, S21 1 + R2 1 R1 for a slot. This is due to further simplification based on the fact that the dipole or slot is perpendicular to the transmission line, and so R2 R1 for a dipole case, and R2 —R1 for the case of a slot. This can be easily verified using eqs. (29,30) and (25), and phase relationships between forward- and backward-travelling-wave-mode fields in a transmission line. The transmission lines feeding the slot or dipole can be of any type, a slot line or a stripline, as long as the 4~V expressions are properly computed as described under (25)—(27). These results agree with those of refs. [24,26,27] for the case of a microstrip line feed. =
=
=
=
=
=
—
=
=
2.3.5. General considerations Using the analysis described, a general-purpose computer routine, UNIFY, has been developed for unified simulation of multilayer multifeed planar antennas in an infinite array as well as an isolated environment. Various practical approximations discussed earlier are assumed, however, to simplify the most general formulation, and other special considerations not explicitly discussed in the analysis are discussed here to handle special cases: The analysis explicitly discusses the computation of scattering-port parameters. Various current expansion functions with the unknown coefficients solved from the moment method can, however, be used to evaluate the far-field components in a relatively straight-forward way via a stationary phase method with proper choice of spectral Green’s function components [45]. The two orthogonal far-field components can be treated as two additional ports, the scattering parameters of which can be used for radiation characterization. s The currents on the transmission lines can be completely described as a superposition of infinite transmission-line modes, transmission-line discontinuity modes of appendix H, and the additional finite-length non-travelling-wave modes in the vicinity of the radiating (or parasitic) elements coupled to the transmission lines. Inclusion of the additional finite-length modes on the transmission lines are not explicitly discussed in the analysis. These modes can be virtually treated as other finite-length modes on antenna elements, however. As discussed in appendix H, half PWS modes may be required to insure current continuity compensating for the inherent discontinuity of the transmission-line discontinuity modes. s
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A generalized CAD model for printed antennas
• As discussed in section 2.3.2, planar structures such as a dipole coupled to the open-end of a parallel microstrip feed line [24] could not be accounted for in the multiport-scattering formulation. This is different from the stub-tuned aperture-coupled microstrip antenna of refs. [26,27],where the open end of the microstrip feed line can be accounted for as a circuit element via the scattering parameters of the two-port antenna, because here the open end does not directly interact with the slot fields. For cases similar to ref. [24] the currents in the open-ended feed line can still be expressed as a superposition of finite length and transmission-line modes, and can be tested for appropriate boundary conditions in the same way discussed in the analysis. Half PWS modes would be required to compensate for current discontinuity of the transmission-line discontinuity modes. An additional equation instead of the multiport-scattering condition (not valid here) is required here, and can be obtained by use of an additional non-Galerkin testing function on the transmission line [46]. • Some of the numerical considerations for implementing the spectral integrals for an isolated element case have been discussed earlier. Convergence of results with respect to expansion modes of the moment method is equally critical, and carefully selecting the expansion current modes to describe the actual current distribution is useful. In order to get a first-order idea of the current distributions, the solutions for another closely related problem, the exact solution for which can be easily obtained, can be used as a guideline. For example, a closed cavity with magnetic side walls excited by an equivalent magnetic source at the slot is very similar to that of a slot-coupled microstrip antenna, so far as the currents on the microstrip patch are concerned. The additional strongly radiating current component on the microstrip patch is, however, excluded from the non-radiating magnetic-walled cavity solution. Figure 10 shows the coefficients of various cavity eigenmodes of the currents in the resonant direction corresponding to a specific slot-coupled microstrip antenna. As it clearly shows, the fundamental eigenmode is the most dominant, and a few other eigenmodes around the dominant eigenmode is enough to very closely describe the patch currents. Also, because the slot is placed at the center, odd mode variations (even m) about the patch center in the direction of the patch current, as well as even mode variations (odd n) in the transverse directions are absent. The Corresponding real-space variations of the patch currents are shown in fig. 11, clearly showing the dominant mode, along with other higher-order modes constituting the rapid variations at the center directly above the slot. This is similar to the rapidly varying currents on a probe-fed antenna [19], but unlike the probe-fed case, is produced by the proximity effect of the slot not in direct physical contact with the patch. Similar results for the patch currents in two orthogonal directions for the general case of an inclined slot are presented in fig. 12a and fig. 12b for below and above the respective dominant cavity-mode resonant frequencies. The dominant resonant mode is still clearly prominent, and changes its relative phase below and above resonance, as expected.
3. Applications 3.1. Aperture-coupled microstrip antenna with radome Figures 13 and 14 show a comparison between the theoretical and experimental values of equivalent impedances (as seen by the transmission line [26]) of an aperture-coupled antenna, for an infinite array and for an isolated-element environment, respectively. The experimental data for the infinite-array case was obtained from a waveguide simulator [27]. For both cases the results of the generalized analysis are clearly in good agreement with experiment. The results of the analysis for the uncovered geometries of figs. 13 and 14 are also found to be in similar agreement with experimental results (not presented here). As expected, for moderate values of cover-substrate thickness and dielectric constant the impedance
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A generalized CAD model for printed antennas
413
I
IJymni
13
0 .~
m
ylmi
H
Xlnl
Fig. 10. Spectral distribution of cavity eigenmodes for a slot-coupled microstrip patch with magnetic side walls. Patch: length in resonant dimension = 4.0 cm, width 3.0 cm; slot: at the center of the patch, 1.0 x 0.11 cm; substrate: 2.2, 0.16 cm, frequency = 2.375 GHz. Mode indices, m and n, correspond to integral multiples of half-wavelength variations in the respective directions.
locus of the uncovered case is found to be very close to that for the covered case, with the resonant frequency for the later less than that for the former. 3.2. Aperture-coupled microstrip antenna with a parasitic patch
Figure 15 compares the theoretical and experimental results for the equivalent series impedances of an infinite array of a two-layer stacked microstrip patch aperture coupled to a microtrip feed line. The agreement is excellent, and similar agreement has also been obtained for other sets of physical parameters. As can be seen, in contrast to the impedance loci of the single layered microstrip antennas of figs. 13 and 14, the impedance locus of the stacked geometry has a characteristic double-tuned loop corresponding to the resonances of the two patches. Detailed studies of the bandwidth characteristics have been performed, with the conclusion that bandwidths of the order of 15—20% can be obtained from
414
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A generalized CAD modelfor printed antennas
Im J~cv)
4cm 3cm
op Fig. 11. Dominant current, J~(x,y), as a function of x, y for the slot-coupled magnetic walled microstrip cavity of fig. 10.
a two-layer stacked geometry [41]. For electrically thin substrates it is observed that it may be preferable to use a slightly larger cover patch to obtain the maximum bandwidth. However, a significantly larger cover patch may result in dual-frequency operation of the antenna, instead of an enhanced-bandwidth mode. On the other hand, a significantly smaller cover patch may not couple to the feeding slot, since it tends to be isolated by the primary patch. 3.3. Dual aperture-coupled microstrip antenna for circular polarization More complex geometries with multiple feed lines and coupling apertures can also be rigorously analyzed using the present analysis. A dual slot-coupled configuration with two microstrip feed lines, as shown in fig. 16a, has been analyzed for impedance as well as polarization characteristics. By suitably exciting the two ports, a perfect circular polarization can be obtained. Due to the asymmetric mutual coupling effects between the slots, perfect circular polarization can not be obtained by exciting the two ports with equal amplitudes and a 90 degree phase difference [42]. Hence, the excitations must be compensated to account for such undesirable asymmetric mutual coupling effects. Figure 16b presents the input return loss of an infinite array of dual slot-coupled circularly polarized elements with a parallel reactive power divider for the two ports, with different line lengths to provide the 90 degree phase shift. As fig. 16b shows, compared with the return loss from a single aperture-coupled linearly polarized geometry, the circularly polarized case has a broader bandwidth due to mutual cancellation of reflections from the two ports. This is due to the 180 degree phase difference between the reflected waves from the two coupling apertures. However, the polarization bandwidth is less than the return loss bandwidth.
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A generalized CAD model for printed antennas
415
Im J~(x,v)
ImJ~(x,v)
a
b
Ci~~jx~~
Fig. 12 (a) Dominant patch current, J~(x,y), for the magnetic walled cavity of fig. 10 with the slot inclined at 45 degrees, at a frequency, 2.75 GHz, above resonance in y-dimension. (b) Orthogonal patch current, J~(x,y), for the magnetic walled cavity of (a) at 2.75 GHz, below resonance in x-dimension.
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A generalized CAD modelfor printed antennas
3.25 GHz
=
COVER LAYER Ec d~
PRIMARY
AN~~I a~b
ANT
50MHz
3.70
ENNA
SUBSTRATE Ea
3.45
•S •ThEORY
S S
X—)E—X EXPT.
ND PLANE
.l58cmdc~.152cm EtEa2.2 Ec~2.55 a- b~2.5cm FEED LINE: sLoT: 1.1 x.lScm w .50cm, 500 ARRAY UNIT CELL: ~607cm: H dfda
~~FEED SLOT
SUBSTRATE
__
df FEEDLINE
3.404cm: E-PLANE
Fig. 13. Waveguide-simulator measurements and calculations for a covered aperture-coupled microstrip antenna. 1.0
2.3
GHz 1~f=25MHz •..
+
EXPT.
+ + +THEORY
.475
2.35
2~
~
+ WITH A COVER
2.55, O~l6cm 0
0
Fig. 14. Comparison of theoretical and experimental results for an isolated offset aperture-coupled microstrip antenna covered by a dielectric sheet. Patch: 3.9x3.9 cm; slot: 1.2X0.17 cm; patch offset with respect to the slot (center to center): 0.7 cm in the resonant direction, 0.6 cm in the non-resonant direction; feed line: r = 10.2, w = 0.12 cm, Z~= 50 ~1, ~eff = 6.84; antenna substrate: Er 2.2, 0.16 cm; cover substrate: ~r = 2.55, 0.16 cm.
N.K Das, D.M. Pozar
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A generalized CAD model for printed antennas
:~~
//‘~ PARASITIC PATCH
bxb LAYER
__________
ANTENNA SUBSTRATE
_____
3.65 •
417
.5
GHz.
85
PRIMARY ANTEN~~J~
77 da
/ SLOT._,L_~. /‘\
,./ //
/ ~~
_____________
,~-
GROUND PLANE
X-X--K EXPT.
A/i
//
V,z~ d~
•SSTHEORY da~d~d~.158cm FEED SUBSTRATE E~
Ea=EfEc~’2.2
a= b~2.5cm FEED LINE: SLOT:1.lx.l5cm wr.5Ocm,500 ARRAY UNIT CELL: 3.607cm: H 3.404cm E-PLANE
FEEDLINE
Fig. 15. Waveguide-simulator measurements and calculations for a stacked microstrip antenna configuration with a
3.4 (a)
3.5
3.6
3.7
=
2.5
3.8
cm
=
b.
3.9
Freq.(GHz.) (b)
Fig. 16. (a) The two-port circuit configuration used for the analysis of the dual-slot-coupled structure. Antenna substrate: Er = 2.2, 0.16 cm; feed substrate: = 2.2, 0.16 cm; feed lines: width = 0.5 cm, 50 fi, ~eff = 1.9; antenna: 2.5 >< 2.5 cm; slots: 1.1 x 0.15 cm; slot offset: 0.7 cm from patch (center to center) in corresponding non-resonant dimensions; array unit cell: 4.11 x 4.11 cm; scan: broadside. (b) Return loss of the dual-slot circularly polarized configuration compared with that of the corresponding single-slot-fed linearly polarized antenna.
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N.K Das, D.M. Pozar
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A generalized CAD model for printed antennas
9.0
SLOTS ON GROUND PLANE
_____________
N
.
• ~I EX~. X X K THEORY
/
m 6.0
~~R~AT~OP
LH\
—FEED NETWORK ON BOTTOM SUBSTRATE
~.Q .~
>(
..
.
500
(a)
______________-
U-93
1.00
/
1.02
to
(b) Fig. 17. (a) Geometry of a dual-slot-coupled circularly polarized microstrip antenna designed, built and tested. Antenna substrate: Er = 2.2, 0.16 cm; feed substrate: ~ 10.2, 0.127 cm; patch: 3.9x3.9 cm; slots: 1.2x0.17 cm; patch offset with respect to slots (center to center): 0.7 cm in the non-resonant dimension; 75 £1 feed lines: 0.4 mm width; 90 degrees phase difference between the two lines; 50 Cl feed line: width 0.12 cm with a 37.5 (1—50 Cl tapered transition. (b) Comparison of theoretical and experimental results of normalized frequency performance of axial ratio of the isolated dual-slot-fed circularly polarized antenna: f 0 (theory) = 2.35 GHz, f0 (experiment) = 2.3875 GHz.
The infinite-array geometry of circularly polarized elements can not be simulated in a waveguide simulator. However, an isolated element version of the above case was designed, built and tested using the general analysis routine. Figure 17 compares the theoretical and experimental axial ratio performances of this geometry, and demonstrates the validity of the general analysis. However, the experimental resonant frequency was found to be slightly different from that of the theory, probably because of tolerances in the dielectric constant. 3.4. Microstrip dipole-coupled to an inclined covered microstrip feed line The analysis is now applied to an infinite array of printed dipoles coupled to inclined covered microstrip lines. The S11 and S22 of the two-port circuit are plotted in fig. 18 as a function of the angle of inclination, 0. Unlike the perpendicular dipole case [241, for a general inclined dipole S11 ~ S22, due to the physical asymmetry of the dipole with respect to the two ports. Also, unlike the perpendicular dipole case, we now have S21 1 + S11, which implies that for the general inclined dipole a shunt-equivalent impedance model is not valid. In order to obtain an equivalent circuit model for such inclined dipoles, it is required to use a T or m- network with three independent impedance parameters. With the general multiport analysis described here, these inclined dipole cases are completely modeled by the generalized scattering conditions. The multiport scattering parameters can be used to construct an equivalent circuit, the topology of which is not specifically assumed a priori. 3.5. A stripline aperture-coupled microstrip antenna All the structures discussed up to this point used a microstrip feed line. However, to avoid back radiation from the coupling slot, and to isolate the feed network, a stripline feed to the antenna element
N.K Das, D.M. Pozar
/ /~
txe=lo° /
/
/ /
/
~
/
A generalized CAD model for printed antennas
419
N\
d=O.l6cm w=O.5cm dipole= 3.5~j~L~
~
I~————~ \ /
S29f~
Uao°
\\\~
/ .~\
~
—.-------.I
s11~
Fig. 18. Locus of S
11 and S22 of an infinite array of printed dipoles inclined-coupled to covered microstrip feed lines, as a function of the inclination angle, 9. Substrate: ~r = 2.2, 0.16 cm; superstrate: Er = 2.2, 0.16 cm; feed line: 0.5 cm width, 50 (1, Eeff = 2.096; dipole: 3.5 xO.1 cm; array unit cell: 5.0x5.0 cm; offset: ~ = 1.0 cm, frequency = 3.143 GHz; scan angle: broadside.
can be useful. But a stripline type of feed has potential problems, caused by guided modes that need to be carefully considered [43]. The geometry of a representative stripline aperture-coupled antenna structure with a randome layer is shown in fig. 19 along with its scan performance for a specific set of physical parameters. For these physical parameters the scan performance exhibits a prominent scan blindness at 43.6 that corresponds to the excitation of a parallel-plate mode of the feed structure forced by the non-radiating slots. The possible solutions for avoiding such dangers of blindness due to parallel-plate feed structures are to use a low-dielectric-constant substrate for the feed and/or reduce the size of the array unit cell. 3.6. A proximity-coupled covered microstrip antenna This geometry is an example of a microstrip antenna proximity-coupled to an open-ended covered microstrip line. Unlike the previous examples, where the radiating elements were coupled to the dominant transverse fields of a through transmission line, in this case the antenna is coupled to the fringing fields of the open end. As discussed before, in this case the simplified reciprocity analysis cannot be used, and so the moment system is solved for the reflected traveling-wave amplitude by use of an additional testing function. Extra PWS current modes are used to model the non.traveling-wave current
420
N.K Das, D.M. Pozar
/
A generalized CAD model for printed antennas RADOME
COVER LAYER
PRIMARY ANNA ~7~T~ATE
~~Ed
~TRATE
FEED L(NE
43.6°
_____________________________________________ O.~ I I I~ I I I
ii
-
EH—•—.-D-PLANE
/
/
/
/
I
o-J
/
I I
-
z
I
/
•-‘
-
—
/. /
I
-
/
J.
/
I
U)
~.0
/
-
E~~IMENT
7/
-
p
//
—12.,
1/ /
-
4 4
~
p
2,40
/
-
/1
-
7 i8.C ~—1 0
If,//
S
7 I
.
I
30 SCAN ANGLE,
I
60
I
90
~f=50
2.25
2.
DEG.
e Fig. 19. Scan performance of an infinite array of stripline-fed aperture-coupled covered microstrip patches. Array unit cell: 0.4 X 0.4k 0, a b = 2.5 cm; slot: 1.1 X 0.15 cm; feed line: w = 0.1 cm, 50 ~ = 2.55, d~ = 0.158 cm, Ea 2.2, d, = 0.158 cm, E11 = 10.2, df1 = 0.127 cm, E~ = 2.2, d~= 0.158 cm, frequency = 3.45 GHz.
Fig. 20. Comparison of theoretical and experimental results for an isolated microstrip antenna covered by a dielectric sheet proximity-coupled to the open end of a covered microstrip line. Patch: 3.9 x 3.9 cm; feed line: w = 0.5 cm, Z~ = 50 Cl, E4f1 = 2.144; all substrates: Er = 2.2, 0.16 cm; open end at the patch center.
components near the open end, and play the dominant role in establishing effective coupling with the antenna. In contrast, for cases of through transmission line coupled antennas, the dominant coupling from the feed line to the antennas is due to the fields of the traveling-wave current components on the feed line, and extra non-traveling-wave current expansions on the feed line in the vicinity of the antenna only describe a refinement of the solution. In fact, for the through transmission-line case with an aperture-coupled microstrip antenna (section 3.1), for example, fairly accurate results can be obtained without including the non-traveling-wave current expansion modes on the microstrip feed line, whereas for the present open-end proximity-coupled geometry a sufficiently large number of finite-length current expansion functions are required to obtain reasonable results. Thus the solution of such proximity
N.K Das, D.M Pozar / A generalized CAD model for printed antennas
421
coupled geometries is computationally more involved, and also requires careful attention to the convergence of the solution with respect to the number and density of the non-traveling wave modes near the open end. Figure 20 shows the comparison of the experimental results with the results of the present analysis for a covered proximity-coupled patch antenna. The theoretical results were obtained with eight PWS modes of longitudinal currents on the feed line over a length of about 4.0 cm near the open end, and five fundamental entire-domain sinusoid modes (even and odd, with uniform transverse variation) on the patch with currents in the same direction as on the transmission line. Figure 20 shows the equivalent impedance as seen by the feed line with the phase reference at the open end. Clearly, the agreement between the results is good with a shift in resonant frequency on the order of approximately 1.0%, which probably can be attributed to tolerances in the value of the dielectric constant and other fabrication tolerances. Similar agreement was also obtained without the cover substrate as well as with other patch dimensions and feed positioning.
4. Conclusion Excellent agreement of the results of UNIFY with experiments for various configurations confirms the validity of the general analysis. Considering the rigor and versatility of the analysis, it should be useful for full-wave simulation of a wide variety of multilayered integrated array geometries.
Appendix A. Different notations for the generalized Green’s functions 1. (a) G~~
k~,z.1): the scalar spectral-domain Green’s function of the field component * (Er, E~,E~,H~,H~,H~)due to the source # (J~J~,~ M~,M~,Me), with source coordinate at (0, 0, 0). The field is computed in the dielectric layer (/]), at a plane given by z~,(see fig. 3). The field components of the Green’s functions can also be along any direction on the x—y-plane, and can be represented, for example as ~ where x’ is the axis along the field direction. As discussed in appendix B, Green’s functions with arbitrary orientation of field direction on the x—y-plane can be expressed as linear combination of Green’s functions with field components in x- and y-directions. Similarly, the source direction can also be arbitrarily oriented on the x—y-plane. The spectral domain Green’s function, G~11~(k~, k~,z11), is defined such that it is related to the corresponding Green’s function in the space domain, G~~1+(x,y, z11), as ~
=ff ~ ~Jf
k~,z1~)
G~1~~(x, y, z11)
=
y, z~1)~
G~~14(k~, k~,z~1)~
e_jkyY dx dy,
e~r~ dk~dk~.
(A.1)
(A.2)
(b) The superscript k of the Green’s function refers to a set of ~‘s(e’— je”), P~r’S and corresponding D’s identifying the topology of the particular multilayer structure. When a number of different multilayer structures are used, this superscript works as an identification to which multilayer structure the Green’s
N.K Das, D.M. Pozar
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/
A generalized CAD modelfor printed antennas
function corresponds to. For example, in the general multilayer structure of fig. 3 the Green’s function refers to the following 8 vectors: E~l ~i2 =
:
:
=
‘
(A.3)
62N
E1N E~hl
~i2
E 22
=
:
‘
[e~’]’
:
=
EIN
l.Lii 1
P~21
IL12 =
(A.4)
2N
[i.t~]
.
/~22
1
=
IL1N1
.
,
(A.5)
/~2N
D11 1=
D21 D22 :
(A.6)
[D2] DIN 1
D2N2
By convention, a perfect electric at of the1NI N1 or or 2N N2 layer is identified by making EiNi or ~2N2 01N, or conductor instead equal to 0, and specifying 2 corresponding values of D and p. are not necessary and are denoted by * If otherwise there is a perfect magnetic conductor at the N1 or N2 layer, it is identified by making ~iN1 or 2N2 equal to — 1, and corresponding values of e”, p. and D are redundant, and are denoted by *~ On the other hand, if EN1 or 2N2 is not equal to 0 or — 1, then the corresponding layer (N1 or N2) is assumed to extend to infinity and the corresponding D (i.e., DlN~or is not needed. Hence, DiN1 and D2N2 are always unnecessary and are represented as * .
2. The following are the recommended representative short forms of various spectral-domain Green’s functions and current transform expressions for use in spectral-domain moment analyses: (a) O~1~~(z,1): the other parameters are k~and k~.
(A.7)
(b) O~11~(k~, kr): the other parameter z~ 0. 2GEIJ#] and similarly for H in place of E. (c) ~ = ~ +Y~OEIJ# + (d) ~ iji = [O~, j~~I + O * ~ 9] and similarly for M in place of J.
(A.8) (A.9)
=
(e)
~ijJ
=
[O~~I+ OL. 9]
(A.10) (A. 11)
(~)GE
11) = [I~ + 9~~IJJ+ zGE,JJ] and similarly with other combinations of (E, H) and (J, M).
(A. 12)
N.K. Das, D.M. Pozar
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A generalized CAD model for printed antennas
(g) The scalar Green’s function with arbitrary field direction, E~,fLJJ
~,
423
and source direction, I~:
G~~JJ î~.
4= ~
(A.13)
.
(h) G~~: refers to any layer in general, and the parameters are (ks, k~,z).
(A.14)
(i) J~’J~,M~M~,L, M~:spectral-domain transforms of J~,J~,M~,M~,J~and M,, respectively, with k~and k~as parameters.
(A.15)
(i) O~#(z): refers to any layer and the other two parameters are
k~and k~.
(A.16)
(k) G~1~#(a,4)
4
=
G~,~ (a cos 4i, a sin
(I) O~11~(a, 4’, z~)= O~1~~(a cos
4’,
(m) O~ #(a,
4’,
z)
a sin
O~#(a,
4,)
=
(n)
=
O~,#(a cos 4’,
O~#(a
4,
0), where k~ a cos =
and k~= a sin
4,. Similarly,
a sin 4,, z~1).
4’,
(A.17) (A.18)
z).
(A.19)
cos 4,, a sin 4,, 0).
(A.20)
Appendix B. Systematic procedure for the computation of generalized Green’s functions Any component of the generalized Green’s functions at any plane and due to any source, electric or magnetic, x-, y-, or z-directed, can be obtained by following a four-step procedure. In the first step reflection coefficients at the source interface are computed using an iterative algorithm, and is independent of the type of source or field component of the Green’s function. The second step accounts for the particular source and computes the coefficients of excitation of TE and TM waves propagating above and below the source. The coefficients of TE and TM waves at the source interface computed in step two are transformed to any plane where the field is desired by use of another set of iterative procedure in step three. Finally, the Green’s function corresponding to any field component is obtained using a set of standard formulas relating different field components, and various excitation coefficients and the reflection coefficients obtained in earlier steps. Step 1 1A2l’ FF
The following equations and parameters are used to iteratively find TAII’ E~11, 2I: 2 = k~+ k~,where k~and k~are spectral domain variables, and are complex in general. (B.1) (a) e~’s,;ys a (b) and D 11’s are the real and imaginary parts of relative dielectric constants and thicknesses of the ijth layer, respectively. For a conducting layer on the top or bottom (~,‘N= 0), the respective conductivities, OIN~S are specified, instead of e~.(see fig. 3 and appendix A). 1~=;~—~~’3, jN_1 EjN,N/JE’~, iN1#O~ —1; tiN, = 1 —J(~N,/wEO), GIN. = 0. (B.2) 2, Im(f3,~) 0, where k (c) f3,~= i/k~e1~p.~~ —a 0 = free-space wave number. (B.3) (d) 1V = total number of layers in the ith side of the source, including a conductor, if present (see fig. 3). (B.4)
424
(e)
N.K. Das, D.M. Pozar ~
If
~
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A generalized CAD modelfor printed antennas
1, FAiN = 0 = 1, (i) FAIN= —1, FFIN= 1, (see appendix A), (ii) TAiN,—1 = —exp(—2j/3~N_lD~N_l) (iii) = exp(—2jf3~~. ~zN
—
E’~,
(B.5)
1D1~1).
~
‘-‘j’
ZAij
—
_______
(Oojj
Fij —
‘
________
ZA--—Z~--+l (g) FAjJ = exp( — 2jI3,1D,~)~ + T Aij T
~
FFIJ
Aij+l
exp( — 2jp1~D1~)1, ~
I +
1F1f+YF~f+1 T l—FFIJ+I 17FiJ+1’ ~“Fij+1 = 11+1 r Fij±1
l—FA~J+1
where ZAIJ+I = 1 ~ rAij+1 ZAIJ+l, for j <1V, if Ei’N ~ — 1, or for
=
j
—
1.
(B.7)
Step 2 A. Electric source (a) Tm=J[(13 1 FAI 1A21)l, 11/11X1 +FA2IX 1)+(1321/E2lXl +FA11X1 = ~[((3i 1/~iiX1 — 1F11X1 + FF21) + (/321 + /.L21X1 — F~1X1+ FFII)]. (b) f21 = k~wp.0(1+ TFIl)/Tea, 2, fa21 k~(1—FAI1XI311/ell)/Tma 2, +FF2l)/7~a a11 =k~wp.~(1 11 = —k1(1 — FA21X/32l/E21)/Tma.
(B.9) (B.10) (B.11) (B.12)
B. Magnetic source on a ground plane 2(1+F~ii), (a) 7~1=a = a(1 + FF
(B.14)
(b)
Tmi Tm
=
2
=
f21 =
(B.8)
(B.13)
21), 2(1 — FAll), (1311/e11)a2(1 — FA
(B.15) (B.16)
(/321/21)a —jk~/7~2,
(B.17) (B.18) (B.19) (B.20) (B.21)
2I).
a21 =jk~we0/ Tm2,
f11 =~k~/T~ a11 =jkyWo/ Tmi. C. Magnetic source not on a ground plane FA2IX11 — + FAll) F~ + (f321/e21X1 + FA1IX1 — F~1)], = j[(13u/p.nXl — FF11X 2, 1)+ (/321/p.21X1 — FF21X1 + FFI1)1. (b) fa21 = —k~o0(1+FA11)/Tma 2, a21 = k~(1— F~11Xf311/p.11)/2’a 2, 11 = —k~w0(1+ FA21)/Tma fii = —k~(1— F~ 2. 21)(/321/p.2l)/7~a D. z-directed electric source (a) Tm = j[(/311/11X1
+
(B.22) (B.23) (B.24) (B.25) (B.26) (B.27)
(a) Tm = j[(/3 =
11/e11X1 j[(/311/p.11X1
+ F~1X1— —
F~11)+ (f321/21X1 + FA11)(1 FA21)l, FF11)(l + F~1)+ (/321/p.21X1 — FF21X1 + FF1I)l. —
(B.28) (B.29)
N.K Das, D.M. Pozar
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A generalized CAD modelforprinted antennas
(1 + FA21)/ETm, (1 + FAll)/ETm, =f21 = 0.
(B.30) (B.31)
= =
E. z-directed magnetic source FA21X1 — FAll) + (/321/21X1 + FA11X1 — F~1)], (B.32) = j[(131i/p.iiX1 — 7~, FF11X1 + FF21) + (j321/p.21)(1 — F~1X1+ FFI1)]. (B.33) (B.34) (b) f11 f = (1 + F~21)/p. 21 = (1 + F~11/p.l~, (B.35) a11 = a21 = 0. It should be noted that for z-directed sources the substrate topology is such that the layers 21 and 11 are identical with a dielectric constant e, and a magnetic constant p.. In other words, the source is buried inside a uniform medium. (a) Tm
=
j[(1311/11X1
+
Step 3 — 2a,~exp( —jI3IJD1J)ZA~J ajj~1—(1+FAIJ+l)(ZA~J+((1—FAIJ+l)/(1+FAIJ+l))ZAlJ+l)’
—
(1
(
2f~~ ((1 exp(—j/3I)D~J)YF~J FF~J+l)/(1+ FFlf+1))YF~J±1)
+ FFIJ+l)(YF~J+
—
B 36
) B 37 ) .
.
—
Step 4a In the following equations the source # is J~,M 1, J~or M~,and correspond to step 2A, (2B or 2C), 2D, or 2E, respectively. ka-xli I (a) GEIJ#(z~J)= (—j/31~X 1) (exp( —jf3,~z11) FAll exp~j/3,1z1~)) .
—
—
0) C0
—jk~f,1(exp(—j/3~~z11) + FF~Jexp(j$~1z,~)). (b)
OEIJ#(zlJ) =
(c) GE,J#(zIJ)
=
TAIJ °‘
WEo Eli
(B.38) exp(j/3,~z
a,3( —jf311X — 1Y(exp( —j/31~z1~) —
11))
+jk~f,~(exp( 2a —j31~z,~) + FFIJ exp(j/31~z,~)). [a 1~(exp(—j/311z,~) + FAll exp(j/311z1~))].
(B.39) (B.40)
.
JO) E~~
(For .J~source a term ~(z) should be added to the expression inside the (d) GH~J#(zlJ)=jk~a13(exp(—j/3,~z11) + FAll exp(jf3,~z11)) + —
(e) GHIJ#(zIf)
= —
°‘
f~(—jf311X—1)’(exp(—jf31~z~~)FFIJ —
exp(j/3,~z~1)).
(B.41)
exp(j/31~z11)).
(B.42)
2f,
1
—
FFIJ
jk~a1~(exp( —jf3,~z1~) + FAll exp(jf3,~z~1))
+
(f) G~,1~(z11) =
f~(—jf3~X--1Y(exp(—jf31~z11)
—
wp.op.j)
[1 bracket.)
.
jwp.0p.11
[a
1(exp(
—j/31~z1~) +
FFIJ exp(jf3,1z1~))].
(For M~source a term ~(z) should be added to the expression inside the
(B.43)
[1 bracket)
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/ A generalized CAD
N.K Das, D.M. Pozar
model forprinted antennas
Step 4b For Green’s function components with y-directed sources see appendix C.
Step 4c Corresponding to a particular source, #, the Green’s functions with field-directions arbitrarily oriented on the x—y-plane, say in x ‘-direction at an angle 0 with the x-axis, can be expressed as =
GH,jj# =
GE
j~
cos 0
+
GE
0
+
G5~11~ sin 0
GHIJ# C05
~
sin 0 =i’ ~
~
(B.44)
~
(B.45)
.
Appendix C. Look-up table for obtaining a Green’s function with a y-directed source from another Green’s function with a x-directed source In appendix B Green’s function components with x-directed sources (~x or Mi), have been derived. By simple coordinate transformation and using the “quadrant symmetry” of the Green’s functions, different Green’s function components with y-directed sources (J~or M~)can be related to a Green’s function component with a x-directed source. These results have been tabulated in table 1. A bold arrow implies that the Green’s function component, which the arrow points at, is obtained from the Green’s function component at the tail of the
Table 1 Table to find a Green’s function component with a y-directed source in terms of another Green’s function component with a x-directed source
Source
Field
E~
M~
Ey
E~
H~
Hy
GExJx
GEJ
°EzJx
GHJ
GHJ
GEM
GEM
GEM
GHxMx
GHM
H~
GHzJx
GHM
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A generalized CAD modelfor printed antennas
427
Table 2 Table of quadrant symmetry of various Green’s function components
Source
J x
Field E~
Ey
±~ -:i:-Fi:
~± +1—
~J±
~± +1—
TANGENT
COSINE
TANGENT
UNIFORM
~
TANGENT
UNIFORM
-J±
~±
1~
TANGENT
UNIFORM
z
±~ +1+ UPW’ORM
±~ —I— SINE
.::I± —1+
—I—
+1+
+1—
SINE
UNIFORM
TANGENT
COSIE
=1± +1— TANGENT
~± —1+ COS1~
~j±
±~
~F SiNE
u,’,o~,i
+1+
~F
~l± ~
.Lt +1—
+1+
—I—
UNIFORM
TANGENT
COSINE
TANGENT
UNEORM
SEE
—1+
+k-
+1+
±[±
~1± ~W
COSINE
SINE
UNIFORM
SINE
COSINE
SINE
COSINE
COSINE
SEE
~
Jz
H~
~i±
+3±
~W
M
Hy
~±I±~ ±1±. ±1±. -=1±
~—1+
MX
H~
~F
±1±. ±1±.
~F
—
ISIFORM
arrow by interchanging kx and k~and adding a negative sign before it, whereas a thin arrow implies the same thing as a doubled-lined arrow except that it does not add a negative sign before it. For example, k~)=
~
GE~,J0(kX,k~)=
~GEYMr(kY~
ks),
(C.1)
GEJ(kY, ks).
(C.2)
Corresponding to a particular field component, *, the Green’s functions with source directions arbitrarily oriented on the x—y-plane, say in x ‘-direction at an angle 0 with the x-axis, can now be expressed, for example, as =
G~~11 cos 0
+
~
sin 0
=
G~112 1’.
(C.3)
Appendix D. Quadrant symmetry of different Green’s function components The symmetries of Green’s function components in different quadrants on the kx—ky-plane are tabulated in table 2. The symmetry of a function f(kx, k~)is described as follows: uniform:f(kx, k~)=f(—k~,ky)=f(~kx, —k~)——f(k~, —kr); sine:
f(kx, k~)=f(—k~,k~)= —f(-—k~,—kr)
=
—f(k~,—kr);
N.K. Das, D.M. Pozar
428
/ A generalized CAD
cosine: f(kx, k~)= —f(—k~, k~)= ~f(~kx, tangent:f(k~, k~)=~f(~kx,
modelfor printed antennas
—k),) =f(k~, —kr);
k,) =f(—k~,—k~)=~(kx,
—ky).
The + and — signs in the different quadrants are used to describe the relative signs of the Green’s function components in the different quadrants of the k~—k~-plane.
Appendix E. Patterns of TE and TM excitations for Green’s function components The patterns of excitation of various field and potential components due to various source configurations as a function of the angle, 4,, with respect to the x-direction are tabulated in table 3. In table 3 the blank spaces imply that the respective Green’s function components are either not excited or not applicable for the specific source. For example, the relative magnitudes of excitation of the E~field component due to an x-directed magnetic source varies as sin 4’. This is only due to the contribution from the TM part of the excited wave; the Er-component can never be produced by a TE to z-wave. Such patterns are applicable to the relative magnitudes of excitation of the TE or TM source-free modes, as well as the radiation patterns of, respectively, E,1, and E0 for a given value of 6.
Appendix F. Asymptotic orders of various Green’s function components The asymptotic orders of various Green’s function components, in terms of the powers of a, are given in table 4 for three different limiting cases. In one case kx approaches infinity which k~remaining finite, and in the other case k~approaches infinity with kx finite, whereas in the third case kx and k~ simultaneously approach infinity. The separate asymptotic contributions from TE and TM parts are also given. The real or imaginary nature of the asymptotic functions of table 4 are given in table 5, and are independent of the particular direction along which one approaches the limit of large a on the k~—k~ spectral plane.
Appendix G. Basis functions for transverse variation of source distribution used in generalized transmission-line analysis G. 1. Single conductor or slot case In the case of a transmission line with one conductor or slot, the electric or equivalent magnetic surface current along the direction of propagation can be expanded in the transverse direction using the following basis functions (see fig. 21): f~m(Y)e(Y1v)c05
kmY,
m=1,2,3,...,
(G.la)
with km
=
2(m
—
1)’rr/w,
(G.lb)
N.K Das, D.M Pozar
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Table 3 Table of patterns of TE and TM excitations for Green’s function components Source J~, TE TM TE TM J~ TE TM Mr TE TM M~TE TM M~TE TM
Field a
f
—
sin 4,
cos 4,
—
E~ 24, sin cos24,
E~ sin 24, sin 24, cos2çb sin24,
E~
Hr
—
sin 24,
H~ sin 4,
sin 24,
H~ sin24, cos24,
cos 4,
cos24, sin24,
sin 24, sin 24,
cos 4,
sin 4,
—
—
cos 4,
sin 4,
—
sin 24, sin 24,
-
-
-
-
-
-
-
-
1
—
cos 4,
sin 4,
1
sin 4,
cos 4,
—
sin 24, sin 24,
cos24, sin24,
cos24, sin24,
sin 24, sin 24,
cos 4,
sin 4,
sin 24, sin 24,
sin24, cos24,
sin 4,
cos 4,
—
—
—
cos 4,
sin 4,
—
—
sin 4,
cos 4,
—
sin2cb cos24,
sin 24, sin 24,
—
I
sin 4,
cos 4,
—
cos 4,
sin 4,
1
-
-
-
-
-
-
-
-
—
—
—
—
Table 4 Table of asymptotic orders of various Green’s function components Source J~ K~ TM TE K,, TM TE a TM TE
Field Er
E~
1 —3
0 —2
E~ 1 —
Hr
0
—
—1 —1 —1 0 0
—2 —2 0 0
—1
0
0
—1
—
0
0
—1
—2
—
1 —1
1
1
—1
—
—1 —1 1
—
—2 0
1
0
—2 1 —1
—3
—
1
1
—1
MrKrTM TE K,, TM TE a TM TE
—1
—2
—1
—1 —1 0 0
0 0 —2 0 0
MyKr
0 —2 —2 0 0 0
—1 —1 —1 —1 0 0
K~ TM TE K,, TM TE a TM TE
TM TE K~ TM TE a TM TE
0
—2 0
0
—
—2 0 0
—1 —1 —1 0 0
—1
—3
—2
—
1
0
0
—1
—2
—
—1 —1 1
—1 1
0
—2
—1
—
0
—1
—2 0 —1
—3 1 —1
1
1
—
0
—1 —
0 —
H~
—1
—1
J,,
H,,
0
—
0 —
0 —
—1 —
0 —
-
1 —
0 —
1 —
0 —
1 —
1
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Table 5 Table of asymptotic nature (real or imaginary) of various Green’s function components Source
Field
TM TE J~ TM TE ~r
Er
E,,
E~
j —j j j
j j j —j
—1
—1 -
1
-j
TE
—1
1
-
M,,TM TE
-1
-1 1
—1
H,,
1
—1 —1 —1 I
-1 1 1
-
1
MXTM
Hr
—j j i j
i -
Hr —
—j —
j j
-
-1
-j
-
j
—1
where
e(y; w)
=
1/w,
without edge condition,
I 2 2 1/’rry (w/2) y , with edge condition, y ~ w/2, and = 0, elsewhere. =
for
I
—
(G.lc)
The Fourier transform of these basis functions as defined by (G.2a)
~m(ky;1~)ffsm(y;H)exp(jkyy)dY,
fsm(Y
al
w)=
(G.2b)
~fFsm(ky;w)exp(ikyY)dky,
Dominant modes, no edge Condition
IbI
2.0
Dominant modes, with edge condition
5.0 —ru~41 mr2 — -n,4,3
—m=1 — —
/
00
,//
-~
~
/
‘
/
f
-~
, /
I
/~.
—2.0 —0.5
00
-m3 m 4
—5.0 0.0 y/W
0.5
—0.5
0.0
0.5
y/w
Fig. 21. Transverse variation of dominant electric or equivalent magnetic-current modes on single strip/slot transmission line (a) without, (b) with edge condition.
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431
can be expressed as F~,,,(k,,;w)
=
~(Fe(ky
—
km) +Fe(k~+ km)),
(G.3a)
where Fe(ky)
=
sin(k~w/2)/(k~w/2)= 1.0,
k~—* 0,
Ik~I—s-oc,
a/k~,
(G.3b)
for the case without any edge condition, and Fe(ky)
=J0(k~w/2)= 1,
k~-+0,
1/v1ii~T,
Ik~I
(G.3c)
—~,
for the case with an edge condition. In the above expressions the width of the source, w, is the parameter, and y is the transverse coordinate on the transmission line. On the other hand, for the above case of a single-source transmission line, the surface currents in the transverse direction can be expanded using the following basis functions (see fig. 22):
f~(y;w)=sinkmy, IyI
(G.4a)
with
2ii~m km=
,
m=1,2,3,...,
(G.4b)
and its Fourier transform set can be expressed as: F~,(k,,;w)
=
(1/2j)(F~(k~ km) —F~(k~ + km)),
(G.5a)
—
2.11
Orthogonal modes
m~2
y/w
Fig. 22. Transverse variation of orthogonal electric- or equivalent magnetic-current modes on a single strip/slot transmission line.
N.K. Das, D.M. Pozar
432
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A generalized CAD model for printed antennas
where F~(k~) = sin(k~w/2)/(k~w/2).
(G.5b)
It can be noted that the current components along the direction of the transmission line are the dominant components, and are even with respect to the center of the source, whereas the current components in the orthogonal direction are not dominant, and are excited only in the cases of non-TEM structures, with odd distribution with respect to the center of the source. Also, the former has an edge singularity at y = ±w/2of order 1/2, whereas the later goes to zero at the two edges. —
G. 2. Double conductors or slots on the same plane When there are two conductors or slots on the same plane of the multilayer structure, with center to center distance, d, as in the cases of edge coupled microstrip lines or co-planar waveguides, they can be treated as a single source using the odd or even symmetry of the currents with respect to the center point between two lines. In these cases, for the dominant current components, the basis functions are (see fig. 23a, b): fxoe( y; w, d) =f~m(y d/2) ±(— 1)mf~m(y + d/2), (G.6a) —
where
m=1,3,5,..., or
f~’me(Y)~0~mY,
=e(y;w)sinkmy, km=’TT(m—l)/w,
(G.6b) (G.6c)
m=2,4,6,...,and
(G.6d)
-
with their Fourier transforms are given as
w, d) =F,~(k,,;w)[exp(_Jk,,d/2) ±(_1)m exp(jk~d/2)],
(G.7a)
where F~m(ky)= ~(Fe(ky =
—
km)
(1/2j)(Fe(ky
—
+ Fe(ky +
km)
—
km)),
F~(k~ + km)),
m = 1, 3, 5,...,
(G.7b)
m = 2, 4, 6
(G.7c)
On the other hand, the basis functions for the orthogonal components are given as (see fig. 23c) mfc~m(Y+d/2), (G.8a) fr~e(y; w, d) =fg~(y—d/2) ±(1) where f~m5~kmY,
m=1,3,5,...,
=coskmy,
m=2,4,6,...,
(G.8b) (G.8c)
with ‘Tr(m + 1) km= =
w ~(m—1)
,
m=1,3,5,...,
(G.8d)
m=2,4,6,...,
(G.8e)
N.K Das, D.M. Pozar IaI
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A generalized CAD model for printed antennas
Dominant modes, no edge condition
Dominant modes, with edge condition
Ib)
2.0
433
5.0 —rrl=t
_~
rn—S
~
0.0
,~// /
rn2
~\~\\~ .
~
-
E 0.0 .
.
-~ ~.
:‘
\\\~
—2.0
—5.0
—0.5
0.0
0.5
—0.5
0.0
y/w
0.5
y/w Orthogonal modes
CI
2.0 —
0.0
=1 rnZ
~
-2.0 —0.5
0.0
0.5
y/w Fig. 23. Current distribution functions associated with double- or multi-conductor/slot lines.
and the corresponding Fourier transform expressions are F~~e(k~ w, d) =1~’m(ky;w, d)[exp(_jk~d/2) ±(_1)m exp(jk~d/2)],
(G.9a)
where F,j~~(ky) = (1/2j)(F~(k~ km) —
=
~(F~(k~
—
km)
—
F~(k~ + km)),
+ F~(k~ + km)),
m
=
1, 3, 5,...,
(G.9b)
m
=
2, 4, 6
(G.9c)
In these expressions “o” and “e” respectively refer to coupled sources with odd and even symmetry with respect to each other, and correspond respectively to the top and bottom signs of various double signs appearing in the expressions.
434
N.K. Das, D.M. Pozar / A generalized CAD modelfor printed antennas
G. 3. Multiple conductors and / or slots This is the most general case of which cases 1 and 2 are derived. In this case each conductor or slot is expanded with a set of basis functions with respect to the center of the conductor or slot. The basis functions for the dominant components are f,~(y;w) =f~m(y;w),
(G.10)
with its Fourier transform F,~(k,,;w) =F~m(ky;w).
(G.11)
And similarly, the basis functions for the orthogonal components are f,~(y;w) =fa~m(y;w),
(G.12)
and its Fourier transforms are F,~(k~: w) =F,~m(ky:w).
(G.13)
Appendix H. Basis functions used for expansion of various unknown currents of a generalized multilayered printed antenna configuration Various unknown electric currents, or equivalent magnetic currents (or slot fields) of a multilayered configuration of printed antennas are expanded using three major types of basis functions, i.e. piece-wise sinusoids (PWS), entire basis sinusoids (EBS) and transmission-line discontinuity modes. The first two types of modes are useful for expansion of currents on finite-length radiating structures, such as patches, dipoles or slots, whereas the third type of basis functions are useful for taking into account reflected and transmitted currents on a transmission feed line at various discontinuities. Unlike eigenmodes of an infinite transmission line, these transmission-line discontinuity modes extend to infinity only in one direction [24], and can be referred to as semi-infinite modes. In the first part of this appendix these basis functions are expressed with respect to a coordinate system at the center point of the mode. By convention, the x-axis of this coordinate system, referred to as the source coordinate system, coincides with the direction of current flow of the mode, with the y-axis as the transverse direction. A number of such modes with suitable orientation and offset with respect to the system coordinate system can define a complete system. When using them in a spectral-domain moment method, any orientations and/or offset with respect to the system coordinates can be easily taken care of by suitable transformation of the spectral plane, and are separately discussed in the second part of this appendix. Also, for consistency of notations, and to distinguish from the basis functions of appendix G used for expansion of transverse variation of transmission line currents, the basis functions for finite structures will be denoted as u( )‘s with the corresponding Fourier transforms as U( )‘s, with suitable subscripts, superscripts and indices to specifically denote their identity. The basis functions can be expressed as a product of two separate functions, one for the x-dependence and the other for the y-dependence, where x and y refer to the source coordinate system, and correspond to the directions along or transverse to the direction of the current flow, respectively. The
]
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two-dimensional Fourier transform of a basis function is thus the product of two separate one-dimensional transforms in kx and k,,, defined, for, say, k~,as U
1(k~)= fui(x)
u1(x)
=
exp(—jkxx) dx,
(H.la)
U1(k~)exp(jkxx) dk~.
(H.lb)
~—J
Different types of basis functions will be separately discussed. Their values at various critical points, as well as their asymptotic orders for large spectral arguments will be given in each case, and are useful for various convergence considerations while performing spectral integrals. H. 1. Basis functions in source coordinate system H. 1.1. Finite-length modes
(a) For x-variation: • Piece-wise sinusoid (PWS) modes (full): -
u’pw(x;
L, k~,N)
=
sin[k (z~L/2— I(x—~-)I)j e sin(k~~L/2) ,
=0, elsewhere, where z~L= 2L/(N + 1), zJ~, = —L/2
(z~L/2)i;
(H.2)
2ke[cos(ke z~L/2)—cos(k~z~L/2)] -
U~,w(kx;L, ke)
+
x—411
(k~—k~)Slfl(ke ~1~/2)
=
exp(—jk~~1)
=(~L/2) exp(—jkx~ij), k~— ±ke (H.3) where a is a real number, positive or negative. Full PWS modes are always considered as a group of N subdomain modes over a region of length L, all with respect to the same source coordinate system, and are indexed by i = 1,. , N (see fig. 24). • Piece-wise sinusoid (PWS) mode (half): Unlike the full PWS modes, these half PWS modes are used individually, and are discontinuous at one end. Such discontinuous half modes are sometimes useful to satisfy current continuity in a mixed-mode environment with travelling-wave and finite-length modes [24]. 4w(X’ 2L, ke, 1), x<0, uHP(x; L, ke) 1 =0, x>0. (H.4) k [cos k L cos k L] j [ k sin k L k sin k L UHP(k.Lk)=e e + x (kx2k~) e x e (k~—k~) sin keLx sine keL x . .
—
L 2
.
—
[i’~eL cos keL
sin keL] 2ke sin keL —
k-+-i-k X
—
kx-+OO,
where a is an arbitrary real number, positive or negative.
e’
(H.5)
N.K. Das, D.M. Pozar / A generalized CAD model forprinted antennas
436
5.0
PWS modes
2.0 —
—In
InS
—-1=3
A /
...
InN
-—
\,/
...
2.5
/
0.0 —0.5
—2.0 —0.5
0.0
0.0
0.5
0.5
x/L
Fig. 24. Layout of PWS modes over a length, L.
Fig. 25. Uniform and edge-condition modes used for transverse variation offinite-length current modes.
• Entire-basis sinusoids (EBS):
u~x(x)=Lfj,±1(x;L), =Lf~1(x; L),
i= 1,3,5,...,
(H.6a)
i=2,4,6,...,
(H.6b)
U~~(k~) =LF~±1(k~ L)
a/k~,
(H.7)
k~—*oc,
where f,~is the same as defined in appendix G for transmission-line transverse variation. (b) For y-variation • Uniform mode (see fig. 25):
u~(y;w)=1/w, IyIw/2, =0, U~(k~ w)
=
elsewhere;
(H.8)
sin(k,,w/2)/(k~w/2)= 1,
k~—+0,
a ~,
Ik~I—~.
(H.9)
• Edge-condition mode (see fig. 25): u~(y:w)
=
1/~V(w/2)2_y2,
I~Iw/2,
=
0,
elsewhere;
Ue(ky; w) =J0(k,,w/2)
=
1,
k,,
—~
Ik,,I
(H.10)
0, -+~.
(H.11)
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• Transverse entire-basis modes (EBS): u~y(y; w)
=f~(y;w),
(H.12)
U~y(k~W)=F~,(k~w)a/k~,
i=1, k~—+-oo,
a/k~,
i#1, k~—*co.
(H.13)
H. 1.2. Transmission-line discontinuity modes
(a) For x-variation • Forward-discontinuity mode: ufd(x, ke) =exp(—jkex),
—exp(—jkex),
=
Ufd(kX; ke)
x>0, x<0;
(H.14)
~2J/(kx+ke)
=
a/kx,
-
IkxI-+~
(H.15)
It has a pole at k~= —k~. • Backward-discontinuity mode: e~~P(jl’~ex), x>0,
ubd(x, ke)
=—exp(jkex), Ubd(kX; ke)
=
x<0;
(H.16)
—2j/(kx—k~)
a/kx,
IkxI-+oc~
(H.17)
It has a pole at k~= ke. For numerical efficiency considerations, these forward- and backward-discontinuity modes can be expressed as a combination of two other types of modes called as “even” and “odd” discontinuity modes: uOd(x,
ke)
=
~[ufd(x; ke)
+
(x; ke)1
= =
L’~d(kx;ke)
=
cos k~x, C05 keX,
x>0, x<0;
_2jkx/(k~ k~);
(H.19)
—
ued(x;ke)=(1/2j)[ufd(x;ke)ubd(x;kd)]—sinkex, =
~d(kx;
(H.18)
ke) =2ke/(k~~k~).
—sin
x>0, kex,
x<0;
(H.20) (H.21)
These even and odd functions are real functions, and so have even and odd symmetries, respectively, in the spectral plane. Such symmetries are useful in order to simplify various spectral-domain integrals used for moment-method solution that also exploit similar symmetries of Green’s function components as described in appendix D.
N.K. Das, D.M. Pozar / A generalized CAD model for printed antennas
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Forward- and backward-discontinuity modes are described as linear combinations of the even and odd functions: ufd(x, ke) ubd(x,
ke)
Ufd(x,
k~)=
Ubd(x;
ke)
1~od(~ ke)
+ju~~(x; ke),
(H.22a)
ke)
ju~d(~ ke),
(H.22b)
= uOd(x,
=
Uod(kx; ke) +jU~~(k~ ke),
(H.23a)
UOd(kX; ke) —jUed(kx; ke).
(H.23b)
At any discontinuity along an infinite transmissions line, the travelling-wave currents due to scattering at the discontinuity can always be represented by superposition of the discontinuity modes and infinite transmission line eigenmodes. It is important to note that these discontinuity mode transforms, unlike other finite-mode transforms, have poles at kx = ±ke.These poles would always create problems while performing various numerical integrations. Also, these poles, unlike the poles of Green’s functions, do not have any physical significance of exciting any additional modes of the structure. These poles are purely mathematical, and arise due to the choice of the expansion functions used, and so also the residue about these poles do not have any physical meaning. In fact, instead of having these modes of infinite extend, if we truncate them at very large lengths, the corresponding transform will result in smoothening these poles. For numerical convenience, it is suggested that instead of using the discontinuity-mode transforms the way presented before, they should be used, for example, as UOd(kX; ke, a) =Im UOd(kX; k~—ja),
(H.24)
where the superscript “s” stands for the respective smoother versions, and a is an arbitrary positive number. a is equivalent to making the transmission line slightly lossy, and should carefully be chosen so as to properly take care of the numerical problems due to the pole, while not severely affecting the real physical situation.
(b) For y-variation The y-variation of transmission-line discontinuity modes are chosen to be the same as that of the eigensolution of the respective infinite transmission line, f(y). Thus f(y) can be expressed as a sum of basis functions from appendix G with known coefficients. Strictly speaking, the transverse variation of current on a transmission line at the discontinuity plane is not the same as that of the corresponding infinite transmission line. Hence, even though sufficiently larger number of transverse modes might be required for the solution of the infinite transmission line, in order to model the traveling-wave currents at the discontinuity one can express the transverse variation of the discontinuity mode using a fewer number of modes, without significantly affecting the result. However, the propagation constant and the characteristic impedance of the transmission-line discontinuity mode remain the same as that of the infinite transmission line and are rigorously obtained with sufficient number of expansion modes. H.2. Spectral-domain transformation of basis functions for different orientations and offsets While computing various coupling terms for the spectral-domain moment-method solution of a general antenna geometry, the Fourier transforms of the basis functions (expansion or test modes) need to be expressed with respect to the system coordinates. The necessary transformations on the Fourier transform expressions are exp(—jk~z~~), (H.25) Ut(k~,k~)= U(k~,k~)exp(—jk~.~1~)
N.K. Das, D.M. Pozar / A generalized CAD modelfor printed antennas
439
where U( , ) is the Fourier transform in the source coordinate system, Ut(kx, k~)is- the transformed Fourier transform in system coordinates, ‘tx and 4~,respectively are the x and y offsets of the center of the source mode with respect to the system coordinates, k~= kx cos O~+ k~sin O~
(H.26a)
kt,,
(H.26b)
=
—k~sin O~+ k~cos O~,
and O~is the angle of the x-axis of the source coordinates with respect to the system x-axis. For notational convenience the superscript “t” of the Fourier transform, U, is dropped while using in computing various reactions, and should be understood to include proper transformations.
References [1] R.J. Mailloux, Phased array architecture for millimeter wave active arrays, IEEE Antennas Propag. Soc. Newslett. 28 (February 1986) 5. [2] J.A. Kinzel, GaAs technology for millimeter wave phased arrays, IEEE Antennas Propag. Soc. Newslett. 29 (February 1987) 12.
[3] D.M. Pozar and D.H. Schaubert, Comparision of architectures for monolithic phased array antennas, Microwave J. (March 1986) 93. [4] D.M. Pozar, Five model feeding techniques for microstrip antennas, in: IEEE Antennas and Propagation Society Symposium Digest, Virginia, 1987, p. 251. [51AC. Buck and D.M. Pozar, Aperture coupled microstrip antenna with a perpendicular feed, Electron. Lett. 22 (1986) 125. [61S.A. Long and M.D. Walton, A dual frequency stacked circular disc antenna, IEEE Trans. Antennas Propag. (1979) 270. [7] K. Araki, H. Ueda and M. Takahashi, Hankel transform domain analysis of complex resonant frequencies of double-tuned circular disc microstrip resonators/radiators, Electron. Lett. 21(1985) 277. [8] C.H. Chen, A. Tulinteseff and R.M. Sorbello, Broadband two-layer microstrip antenna, in: IEEE Antennas and Propagation Society Symposium Digest, Boston, 1984, p. 251. [9] F. Croq, A. Papiernik and P. Brachat, Wideband aperture coupled microstrip subarray, in: IEEE Antennas and Propagation Society Symposium Digest, Dallas, 1990, p. 1128. [10] E.G. Magill and H.A. Wheeler, Wide angle impedance matching of a planar array antenna by a dielectric sheet, IEEE Trans. Antennas Propag. 14 (1966) 49. [11] J.B. Knorr, Slot-line transitions, IEEE Trans. Microwave Theory Tech. 22 (1974) 548. [12] J.R. Mosig and F.E. Gardiol, General integral equation formulation for microstrip antennas and scatterers, lEE Proc. Pt. H 132 (1985) 424.
[13] M.D. Deshpande and M.C. Bailey, Input impedance of microstrip antennas, IEEE Trans. Antennas Propag. 30 (1982) 645. [14] D.M. Pozart, Input impedance and mutual coupling of rectangular microstrip antennas, IEEE Trans. Antennas Propag. 30 (1982) 1191.
[15] J.T. Aberle and D.M. Pozar, An improved probe feed model for printed antennas and arrays, in: IEEE Antennas and Propagation Symposium Digest, Syracuse, 1988, pp. 438—441. [16] D.M. Pozar and S.M. Voda, A rigorous analysis of a microstripline fed patch antenna, IEEE Trans. Antennas Propag. 35 (1987) 1343. [17] D.M. Pozar and D.H. Schaubert, Scan blindness in infinite phased arrays of printed dipoles, IEEE Trans. Antennas Propag. 32 (1984) 602.
[181 C.C. Liu, A. Hessel and J. Shmoys, Performance of probe-fed microstrip-patch element phased arrays, IEEE Trans. Antennas Propag. 36 (1988) 1501. [191 J.T. Aberle and D.M. Pozar, Analysis of infinite arrays of probe-fed rectangular microstrip patches using a rigorous feed model, Proc. lEE Pt. H 136 (1989) 110. [201N.G. Alexopoulos and D.R. Jackson, Fundamental superstrate (cover) effects on printed circuit antennas, IEEE Trans. Antennas Propag. 32 (1984) 807. [211J. Castaneda and N.G. Alexopoulos, Infinite arrays of microstrip dipoles with a superstrate (cover) layer, in: IEEE Antennas and Propagation Symposium Digest, 1985, p. 713. [22] PB. Katehi and N.G. Alexopoulos, On the modeling of electromagnetically coupled microstrip antennas — the printed strip dipole, IEEE Trans. Antennas Propag. 32 (1984) 1179.
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[23] M. Gaouyer, J.M. Floch and J. Citerne, Radome effects on an electromagneticallt coupled dipole, in: IEEE Antennas and Propagation Symposium Digest, 1989, pp. 612—615. [24] N.K. Das and D.M. Pozar, Analysis and design of series-fed arrays of printed-dipoles proximity-coupled to a perpendicular microstripline, IEEE Trans. Antennas Propag. 37 (1989) 435. [25] J. Herd, Scanning impedance of electromagnetically coupled rectangular microstrip patch arrays, in: IEEE Antennas and Propagation Symposium Digest, 1989, p. 1150. [26] D.M. Pozar, A reciprocity method of analysis for printed slot and slot-coupled microstrip antennas, IEEE Trans. Antennas Propag. 34 (1986) 1439. [27] D.M. Pozar, Analysis of an infinite array of aperture coupled microstrip patches, IEEE Trans. Antennas Propag. 37 (1989) 418. [28] R.Q. Lee and K.F. Lee, Effects of parasitic patch sizes on multi-layer electromagnetically coupled patch antenna, in: IEEE Antennas and Propagation Symposium Digest, 1989, p. 624. [29] C.H. Tsao, Y.M. Hwang, F. Kilburg and F. Dietrich, Aperture coupled patch antennas with wide-bandwidth and dual polarization capabilities, in: IEEE Antennas and Propagation Symposium Digest, 1988, p. 936. [30] J.F. Zurcher, The SSFIP: a global concept for high-performance broadband planar antennas, Electron. Lett. 24 (1988) 1433. [31] N.K. Das and D.M. Pozar, A generalized spectral-domain Green’s function for multilayer dielectric substrates with applications to multilayer transmissionlines, IEEE Trans. Microwave Theory Tech. 35 (1987) 326. [32] M.C. Bailey and M.D. Deshpande, Integral equation formulation of microstrip antennas, IEEE Trans. Antennas Propag. 30 (1982) 651. [33] N.K. Das and D.M. Pozar, Full-wave spectral-domain computation of material, radiation and guided wave losses in infinite multilayered printed transmission lines, IEEE Trans. Microwave Theory Tech. 39 (1991) 54. [34] N.K. Das and D.M. Pozar, UNIfied Computer Aided Analysis of Multilayer Transmissionline, UNICAAMT, a computer code for generalized planar transmissionline analysis, in: Proc. Int. Symp. Electronic Circuits, Devices and Systems, ISELDECS ‘87, vol. I, Kharagpur, India, December 1987, p. 85. [35] N.K. Das and D.M. Pozar, PCAAMT, Personal Computer Aided Analysis of Multilayer Transmission Lines, Version 1.0, User’s Manual (June 1990). [36] T. Itoh, Spectral domain immittance approach for dispersion characteristics of generalized printed transmission lines, IEEE Trans. Microwave Theory Tech. 28 (1980) 733. [37] A. Farrar and A.T. Adams, Multilayer microstrip transmission lines, IEEE Trans. Microwave Theory Tech. 22 (1974) 889. [38] E.J. Denlinger, A frequency dependent solution for microstrip transmission lines, IEEE Trans. Microwave Theory Tech. 19 (1971) 30. [391J.B. Davies and D. Mirshekar-Syahkal, Spectral domain solution of arbitrary coplanar transmission line with multilayer substrates, IEEE Trans. Microwave Theory Tech. 25 (1977) 143. [40] H.G. OIlman and D.A. Huebner, Electromagnetically coupled microstrip dipoles, IEEE Trans. Antenas Propag. 29(1981)151. [411 N.K. Das and D.M. Pozar, Infinite array analysis of an aperture coupled stacked microstrip antenna: considerations for bandwidth enhancement, in: Proc. Symp. Progress in Electromagnetic Research, MIT, Cambridge, July 1989, p. 510. [42] N.K. Das, J.T. Aberle and D.M. Pozar, Scan and frequency performance of circularly polarized microstrip antenna arrays, Proc. Symp. Progress in Electromagnetic Research MIT, Cambridge, July 1989, p. 512. [43] N.K. Das and D.M. Pozar, Printed antennas in multiple layers: general considerations and infinite array analysis using a unified method, in: Proc. lEE Int. Conf. Antennas and Propagation, ICAP, Univ. Warwick, UK, Part 1, April 1989, p. 364. [44] N.K. Das, A study of multilayered printed antenna structures, Ph.D. Dissertation, Dept. Electrical and Computer Engineering, Univ. Massachusetts, Amherst, September 1989 (Microfilm International, Michigan, 1989). [45] D.M. Pozar, Radiation and scattering from a microstrip patch on a uniaxial substrate, IEEE Trans. Antennas Propag. 35 (1987) 613. [46] P.L. Sullivan and D.H. Schaubert, Analysis of an aperture coupled microstrip antenna, IEEE Trans. Antennas Propag. 34 (1986) 977.
Computer Physics Communications
A generalized CAD model for printed antennas and arrays with arbitrary multilayer geometries Nirod K. Das Weber Research Institute, Polytechnic University, Route 110, Farmingdale, NY 11735, USA
and David M. Pozar Electrical and Computer Engineering Department, University of Massachusetts, Amherst, !vL4 01003, USA Received 16 August 1990; in revised form 12 February 1991
A general full-wave solution to arbitrary multilayer geometries of printed antennas and arrays coupled to multiple printed feed lines has been developed. It uses spectral-domain Green’s functions for multilayer substrates to account for the substrate layering of the printed antenna, and a newly developed generalized multiport reciprocity analysis to treat the coupling of the transmission feed lines to the antenna elements. The analysis is completely rigorous, including all surface wave or other non-radiating modal effects. The solution has been developed for both isolated elements as well as infinite arrays. The theoretical formulation of the generalized multiport reciprocity analysis is described in details and results for several practical examples are presented to demonstrate the accuracy and versatility of the analysis used. Considering the rigor and versatility of the analysis, it should find useful for full-wave simulation and computer aided design of a variety of multilayered printed phased-array configurations.
1. Introduction Printed antennas monolithically integrated with active circuits and a feed network is a cost-effective way of realizing a phased array for electronic beam steering [1,2]. In its most basic form, a printed antenna is a microstrip patch on a single-grounded dielectric substrate, fed by a coaxial probe or a microstrip line. This configuration has long been studied and theoretically characterized in detail. The most direct approach of using this type of element for a monolithic phased array is to edge-feed the patch with a microstrip line and integrate the feed network and active circuits on the same substrate in the space available between the elements. But, besides its notoriously small bandwidth there are three other immediate problems with this architecture: (a) There will be serious topological problems in laying out the feed network, and/or the space available between radiating elements (limited by grating lobe considerations) may not be sufficient to accommodate the feed network and active circuits. (b) There may be detrimental coupling between the radiating elements and active circuits. Also, the spurious radiation from feed network and active circuits could degrade the cross-polarization level and radiation pattern. (c) In a monolithic phased-array application, where a high-dielectric constant substrate is preferred for the integration of active devices, scan blindness is a potential problem. The underlying cause of this 0010-4655/91/$03.50 © 1991
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Elsevier Science Publishers B.V. All rights reserved
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N.K Das, D.M. Pozar
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A generalized CAD model for printed antennas
problem is the fact that a single substrate is being used for the two quite distinct electrical functions of radiation and circuitry. (A thick low-dielectric-constant substrate is preferable for the antenna elements, while a thin high-dielectric-constant substrate is optimum for circuitry and active devices.) In order to solve these problems it is necessary to electrically isolate the antenna and active circuits in separate parallel substrates (some possible configurations are discussed in ref. [3]). The available space for integration is thus increased by reusing the physical surface area in separate electrically isolated planes. The RF interconnections between these separate layers can be realized by electrical connections through via-holes, or by electromagnetic coupling [4,5]. From manufacturing considerations, making a via-hole in a large-scale monolithic fabrication is often difficult, and can be avoided by use of the electromagnetic coupling, where electrical connection is established not by a physical connection, but by close “proximity” to a feed line or aperture. Besides isolating the antenna and feed network, there are other reasons to use multilayer integrated geometries with more layers of dielectrics and ground planes. Use of a dielectric cover as a protective radome, and use of stacked layers of microstrip patches [6—91for bandwidth enhancement, are two clear examples of antenna considerations. Also, a radome layer properly configured on the face of the antenna has potentials for achieving wide-angle impedance matching (WAIM) [10] of a phased array. From feed-network considerations, the surface area available exclusively for active circuit integration may be insufficient, especially when topologically complex feeds are required, such as dual feeds for circular or dual polarization. In this case, it may be desired to use a “sub-array” architecture as discussed in ref. [1]. This would require two levels of RF feed network. The primary feed network for the elements in the sub-array module may be required to be isolated by a ground plane from the active circuitry and secondary feed network in a different layer, and electromagnetic coupling between them can be established via a proximity-transition similar to that of ref. [11]. A number of techniques have been developed over the last several years for the analysis of a single-layered microstrip antenna printed on a dielectric substrate, with the most rigorous of these probably being the full-wave solutions such as refs. [12—161.Similar techniques have also been applied to infinite arrays of microstrip antennas on a dielectric substrate [17—19].More recently, some specific cases of the multilayer geometries mentioned above in an isolated or infinite-array environment have begun to be treated using full-wave analysis techniques. The analyses for the covered dipoles [20,211, the proximity-coupled dipoles [22—24],the proximity-coupled microstrip patches [25],and the aperture-coupled microstrip patches [26,27] are the representative cases. More complicated multilayered geometries that have been studied only experimentally include parasitic stacked patches [281, a dual-polarized aperture-coupled stacked-patch configuration [29], and an array of aperture-coupled patches with a radome [30]. Up until this time, the analysis of a new type of printed-antenna geometry has usually required the development of a new solution and computer program. In some cases, this might simply involve using a new Green’s function (for example, to add a radome layer to a geometry that has already been analyzed). In other cases, considerably more effort may be needed to adequately model the geometry of interest. Therefore, in order to avoid a large number of individual solutions and software for specific printed-antenna geometries, we have developed a generalized full-wave solution that can handle a large class of problems involving (a) multiple dielectric or magnetic layers and ground planes (each with or without loss), (b) various printed feed lines such as microstrip lines, slotlines, striplines, coplanar waveguides, coupled lines, etc., and (c) radiating slot, patch, or printed dipole elements. This technique can be used to model practically any geometry that can be constructed using rectangular planar elements. Layers can be coupled by proximity coupling to a transmission line, or by aperture coupling; idealized delta-gap excitation can also be considered, but not probe coupling. The solution uses the extended generalized Green’s functions of refs. [31,44] for the multilayer geometry, and a newly developed generalized
/ A generalized CAD
N.K Das, D.M Pozar
model for printed antennas
395
AN INTEGRATED ANTENNA ARCHITECTURE IN MULTIPLE LAYERS
////7 ~
Radome ParasiticAntenna Primary Layer
~-
Slotsj~~
~‘
Ground Plane
__________________
/
/~
Electro-Magnetic Coupling
~If//~—
Ground Plane
2(
Primary Feed Network
Coup
via
/~.“~7
/
~_~“
~
~e~y
Feed network
Fig. 1. An integrated multilayer architecture of printed antenas and transmission feed lines.
multiport reciprocity analysis for treating the coupling of transmission lines to coupling apertures, or to radiating patches, dipoles, or slots. It is completely rigorous, including all surface-wave or other non-radiating modal effects. The solution has been developed for both infinite arrays and isolated elements. Figure 1 depicts a possible geometry of a multiple-layer architecture, suitable for a monolithic phased array that can be treated using the proposed solution. It shows a sub-array module consisting of two distinct layers of primary and secondary feed networks, the primary feed network for circular polarization, stacked configuration for bandwidth enhancement, and a protective radome layer on top. The following section will describe the theoretical formulations of the spectral-domain momentmethod/ generalized multiport scattering analysis. For convenience, most of the complex mathematical formulas for direct numerical implementation are included in the appendices. In section 3, several practical examples are presented to illustrate the versatility of the method, and theoretical results for several of these cases are compared with measurements to verify the accuracy of the solution. It is hoped that this work will reduce the need for the development of a large number of solutions and computed programs to treat specific multilayer printed-antenna problems, and support further developments on multilayer antenna technology.
396
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2. Theory There are three important parts of this general analysis: the multilayer substrate configuration or the “substrate topology”, the transmission feed-line configuration or the “transmission-line topology”, and the radiating-elements configuration or the “source topology”. Any variation of the layered medium is rigorously accounted for by the use of a rigorous set of exact Green’s functions. To account for a general set of planar transmission-line configurations to feed the antenna system, the eigensolution to these transmission lines should be separately performed. The propagation constant, ke, characteristic impedance, Z~,and the eigenfields and eigencurrents of these transmission lines are required in the general analysis to completely describe the feed system. Finally, an arbitrary configuration of antenna (or parasitic) elements is incorporated into the analysis by replacing them with a set of current (electric or equivalent magnetic) expansion functions with unknown amplitudes. These unknown current amplitudes are solved using a moment-method procedure. The excitation to the general antenna is provided by means of incident waves at the feed-line ports, which are scattered due to the above currents induced on the antenna elements. These scattered waves at the feed-line ports are the other set of unknowns that are solved by relating them to the various induced currents on the antenna elements by use of a generalized multiport scattering formulation. This general analysis is schematically described in fig. 2, and the three important steps of the analysis, the multilayer Green’s function, the general transmission-line analysis, and the multiport-scattering / method of moments (MOM) analysis, will be separately discussed in the following sections. Emphasis will be given on detailed description of the multiport scattering modeling and the moment-method analysis. Spectral-domain expressions of all Green’s function components will be given, with only outlines of their derivation. Also, detailed discussions on the analysis of the transmission lines is beyond the scope of this presentation. Related literature is cited for reference on spectral-domain transmission-line analyses. 2.1. The multilayer Green function ~‘
The Green’s functions are vital to the numerical implementation of the general analysis. A method of derivation of spectral-domain Green’s functions for a multilayer geometry has been described in refs. [31,44], which uses equivalent transmission-line sections to account for several layers. This method has
GREEN’S FUNCTION SUBSTRATE TOPOLOGY
EVALUATE SPECTRAL
TOPOLOGY
MOM
EXCITATION FROM TRANSMISSION FEED LINES MODELED BY A RECIPROCITY ANALYSIS Fig. 2. General analysis of multilayered printed antennas and arrays fed by transmission-feed lines.
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A generalized CAD model for printed antennas
397
MAGNETIC
GROUND PLANE 62N2= —1 ELECTRIC
Z
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~
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,
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,
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Fig. 3. Geometry of a generalized configuration of multiple layers used for the analysis.
been extended for a more general configuration of multilayer structures to include several additional features. The details of the derivations are not described, since the basic steps are the same as in refs. [31,44]. A general configuration of a multilayer geometry of dielectric or magnetic substrates, and perfect electric or perfect magnetic conductors, is shown in fig. 3. The multilayer medium (“the substrate topology”) can include losses in dielectrics and conductors, with electric or equivalent magnetic sources between 11 and 21 layers directed along any coordinate axis, parallel or perpendicular to the substrate layers. A complete set of Green’s functions for this general structure needs to take into account all components of sources and fields, electric or magnetic, defined at any plane of the layered structure. Understanding the number of possible components of such a general set of Green’s functions, use of systematic notations to conveniently refer to specific Green’s function components is deemed essential. A set of notations recommended for general use is given in appendix A. In appendix B necessary algorithmic steps for obtaining the Green’s functions using an iterative scheme are described. Definition of various intermediate terms are the same as in refs. [31,44], and are consistently followed through. The flow-chart of fig. 4 highlights the basic steps discussed in appendix B. It should be noted that the implementation of the Green’s function algorithm ensures optimal computer time that is comparable with various closed-form expressions available in the literature for simple substrate topologies [14,32]. Appendix B treats sources in x- and z-directions. To obtain Green’s functions for y-directed sources, an easy-to-use look-up table is given in appendix C. Also, a table describing the symmetry properties of various Green’s functions in different quadrants of the spectral parameters, k~and k5, are presented in appendix D, and are often useful for transforming spectral integrals over the four quadrants of the k~—k~-plane into only one quadrant [14]. Other useful properties of the spectral-domain Green’s functions, such as the 4i (azimuth) variation on the k~—k~-plane and the asymptotic orders for large spectral values are also tabulated in appendices E and, respectively F. In appendix E the parameters a
398
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SUBSTRATE ‘TOPOLOGY’
Calculate the Green’s function correspondmg to the required field = 11 or 21
+4 ___
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~ 11 or 21 Calculate the Green’s function corresponding to the required field
___
+
+
(~RE~) with Separate
Z-Dependent Part
BETWEEN TWO LAYERS Fig. 4. Flow chart of a four-step procedure used for the computation of Green’s functions.
and f are the vector magnetic and electric potentials, respectively. While considering power radiated into free space, corresponding to the values of a less than or equal to k0, the En-component of radiation
is directly proportional to a, whereas the E,~-componentof radiation is directly proportional to f. Hence, the çb-dependence of the far-field E9- and E~-componentsdue to a spectral source sheet can be obtained from the table in appendix E corresponding to respective a’s and f’s. For example, the çb-dependence of the far-field E6 pattern for a J~source sheet is a cos 4 pattern. With the above detailed algorithmic expressions of the Green’s functions, and various tables to describe their useful properties, one can numerically implement these Green’s functions, and compute various spectral integrals that use them in an efficient and straight-forward way. That these Green’s functions have poles on the k5—k~(or a) spectral plane is vital to correct evaluation of spectral integrals. Locations of these poles correspond to the excitation of various surface wave, parallel plate, or other trapped waveguide modes of the multilayer structure, and the spectral integrals of the Green’s functions
N.K Das, D.M Pozar
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399
need to include the contribution of these poles in the residue sense. Residues can be numerically computed using the following limiting definition of the residue of a function G(a) at the pole, a~. res G(a) lim G(a)(a —ar). (1) =
a=a~
This limiting expression can be approximated in two possible ways, res G(a)
G(a~+ 6)6,
(2)
a
or, as res G(a)
~6[G(a~
+
6)
—
G(a~ 6)],
(3)
—
a
where 6 is an arbitrary small value, optimally chosen to insure numerical stability [44]. Residues can also be computed analytically, but considering the complexity of the Green’s function expressions for a general multilayer geometry, this is practical for every simple layered structures only (for example, the case of a single-grounded dielectric slab, or a parallel-plate structure with a single dielectric substrate in between). 2.2. Generalized planar transmission-line analysis in multiple layers Planar transmission lines in multiple layers of dielectrics and ground planes are useful for many microwave- and millimeter-wave applications, and are of special interest for the characterization of various feed structures for the multilayered integrated phased arrays. A variety of geometries are possible with variations in their configurations of dielectric and conducting layers (the “substrate topology”), as well as the configurations of the guiding structures (the “source topology”). The substrate topology can consist of a number of dielectric substrates with different dielectric constants covered by ground planes on both sides (as in a stripline) or only one side (as in a microstrip line), or completely uncovered on both sides (as in a slotline). On the other hand, the source topology can consist of multiple (typically one or two) strips of conducting lines, and/or slots on ground planes, in the same or in different layers. The general multilayer configuration of fig. 3 can account for the possible substrate topologies of a generalized planar transmission line with different source configurations placed at one or
Mkrostrip Line
Coplanar waveguide
Slotline
Edge Coupled Stripline
Broadskie Coupled Striphne
Dual Sidtline
Coplanar Striplirie
Coupled Microstrip Line
StriPline
Fig. 5. Transmission-line configurations of common interest with variation in their source topologies.
400
N.K Das, D.M Pozar
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A generalized CAD model for printed antennas
more of its interfaces. Figure 5 shows various possible source topologies of interest that can be placed in one or more layers to completely define the source configuration of any type of planar transmission line. In ref. [31] a preliminary attempt was made for the spectral-domain analysis of simple configurations of planar transmission lines in multiple layers. The general analysis framework for the unified full-wave analysis of a large class of planar transmission lines has been presented in ref. [33] with special emphasis on computation of different loss components. User-oriented software on mainframe and personal computers have been implemented to perform such full-wave spectral moment-method analysis of general transmission lines [34,35,44]. An integrated analysis package for mainframe computers, “UNIfied Computer Aided Analysis of Multilayer Transmission lines”, UNICAAMT [34,44], has been developed at the University of Massachusetts, the general procedure of which has been schematically shown in fig, 6. Besides computing the various eigensolution of the transmission lines (ke, Z~or P~etc.), it also graphically displays various forms of field patterns and contours for a better qualitative understanding. A comprehensive personal-computer version of UNICAAMT, called “Personal Computer Aided Analysis of Multilayer Transmission lines” (PCAAMT), has also been made available recently from the authors [35]. The results of such general printed transmission-line analyses (the propagation constant, ke, the characteristic impedance, Z~,or the characteristic power, P~,and the current basis-function amplitudes) are directly used in the general multiport antenna analysis discussed below to completely describe the various feedlines. Further related details on spectral-domain moment-method analysis of printed transmission lines can be obtained from various references [36—39]. 2.3. Spectral-domain moment-method
/ generalized multiport scattering analysis
2.3.1. Generalized formulation of multiport scattering Before describing the general analysis of the multilayer antenna, it is useful to develop a general formulation for scattering in a multiport system. Any general multiport system with scattering sources inside can be completely described by its scattering matrix, [S,~].The scattering matrix elements can be characterized by exciting only one port at a time using a matched source, and quantifying the outgoing waves at all match-terminated ports. To solve this particular problem with one port excitation, it can be decomposed into two separate parts: In the first part of the problem (part A), all the sources are physically removed from inside the system and power at all ports determined due to the incident port excitation (say the jth port). Also, let the scattering matrix of this source-free multiport system be [SA~J]• Assume that this source-free system is such that it is exactly characterizable with known eigenfields. These electric and magnetic eigenfields are completely describable by two matrices of field vectors, [EAJ] and [HA)], respectively, corresponding to the jth port excitation. EAJ and HA] denote the fields due to the jth port excitation in general anywhere inside the system. Assume that the individual feed lines at different ports are also characterizable with known eigencurrents and fields, propagation constant (kej) and characteristic impedance (Z~1).Let the electric and magnetic eigenfields for the outgoing wave of the ith port be e[ and hI, respectively, and the corresponding incoming eigenfields are e~=eI, h~ —hI, such that =
ff
[eixhl]
ds=1.
(4)
ith port
Thus, if EA~Jand HA~Jare the eigenfields of the source-free system at the ith port when the jth port is excited, HAJ=HAtJ=SAtJhi, EAJ
=
EALJ
=
i~j,
(5a)
#j,
(5b)
SA~Je1, i
N.K Das, D.M Pozar
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A generalized CAD modelfor printed antennas
401
A. Substrate topology. B. Source and field identifiers, C. Output Options
INPUT CALL ~
~
——+—— Set up Eigen value
GREEN1
RFTURN
Matrix elements for
-
Ke
I
Numerical
INPUT]
trial
_______
. 4~..
Solve the Eigen value forKe
) Output
• The total current
LII}~~
I
JPOWER4d
I R E T U R N
r~l
I I
I
N-_-~\. T R V
• unknown current co-effs
• The current-density transform
MODULE _____
C1
1
Find
GRAPHICS
Yc
or Z~
Dielectric
Conductor
Loss
LOSS
I
/~
/4~ f
.
Field Pattern CALL
GREEN 1 & GREEN 2 —~ Green’s function module
Fig. 6. Algorithm describing the basic steps of the generalized transmission-line analysis.
or HAJ
=
HAI]
=
(SAl]
—
1)I~~ (1 =
—
SAjJ)h7, i =j,
(5c)
i=j.
(5d)
EAJ=EAlJ=(SAlJ+1)eI=(1+SAlJ)e,~,
Figure 7a shows such a 2 IV~port system without the inside sources. In the second part (part B) of the problem, the incident port excitation is dropped, but the inside sources are replaced by their respective actual/ equivalent currents induced when the jth port is excited by a matched source, with other ports match-terminated (see fig. 7b). The volume sources are a set of .17 and M7, and the scattered fields at the ith port is SBI). For this second part of the problem let us
402
/
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A generalized CAD model for printed antennas
/
2/ •
.
7 ISAI
1I (CAll, [HAIl
•
7 [Ml
=~—
S
S
No Source
S -
-
EBII, H61j
5AIj, EAII,
All
SB
V ii
S
S
11
•
/ A
B
Fig. 7. Part A and B of the multiport scattering problem.
assume, however, that the corresponding electric and magnetic fields, EBJ and HBJ, are not known everywhere inside the system. But at the feed-line ports they can be described in terms of SBI, as HBJ
=
HBIJ
EB)
=
EBIJ
=
=
(6a)
SBjJhi,
SBIJeI,
for all i.
(6b)
Now the actual problem can be treated as a superposition of the two parts: S11
(7) at the port p due to the volume sources .17 and M, to It possiblebetween to relatethe theeigenfields scattered field, theisreaction EAP, HAP and .17, M~’1.A reciprocity equation of the following form can be used to establish the above relationship: =
SAl] + SBI].
5Bpj’
V’(Ea xHb—EbXHQ)
=
~
~Eb+Ha ‘Mb,
(8)
where Ea, Ha, and Eb, Hb are solutions of Maxwell’s equations in a closed volume, V, with the corresponding volume sources as Ja’ Ma, and .1,,, Mb, respectively. For the present case let Ea and Ha correspond to EAP and HAP, respectively, with Ja Ma 0, and Eb and Hb correspond to EB] and HBJ with Jb ,JJ”, and Mb MJ”, respectively. Thus, the above reciprocity equation can be rewritten as =
=
=
=
V.(EAPxHBJ—EBJxHAp)=—J~EAp+M~HAp.
(9)
Consider the closed surface, 5, in fig. 7b on which the port references are defined. Applying the divergence theorem to eq. (9) on this closed surface S, and using the expressions (5) and (6) on S for
N.K Das, D.M Pozar EAIP, HAIP, EBI]
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A generalized CAD model for printed antennas
403
and H8~1,it can be shown after some straight-forward mathematical simplifications that
Jj(EAPxHBJ—EBJxHAP)ds=fff(—.17.EAP+MJ”.HAP)dv,
(10)
SBP]~Jff(_Jj’EAP+Mj~HAp)dV~fJ
(11)
~‘
V
(—JJ.EAp+Mi,.~HAp)ds, sources
1, have been replaced by the corresponding surface currents, J~and
where volume sources, J and M,’the source currents are always represented by such surface currents M~.Forthe printed-antenna applications due to their planar characteristics. Equation (11) is the multiport general scattering condition that states: When various currents are placed as sources in a multiport system with no external sources of excitation, and all ports are matched, the amplitude of the scattered wave propagating out of the pth port with proper port-phase reference can be expressed as one half the sum of positive or negative reactions on all source currents due to the eigenfields of the source-free multiport system. The above reacting eigenfields should correspond to the pth incident port, the same port where the outgoing scattered wave needs to be evaluated (note that in fig. 7a the incident port for the source-free system is the jth port). Positive reaction is considered if the source is a magnetic current, whereas negative reaction is used if the source is an electric current. 2.3.2. Simplified model for a multiport antenna system Figure 8 describes the topology of a general geomerty of a multiport antenna system with Nf’ different antenna alements fed by N~feed lines (per unit cell, if an infinite array). Certain useful conventions are consistently followed, and various practical assumptions are made to simplify the general analysis: • As shown in fig. 8, all the transmission feed lines run through without being terminated inside the multiport boundary. Thus the IV~feed lines result in 2N~ports. Any external termination can, however, be accounted for via the scattering parameters of the circuit, as long as the terminations can be assumed not to directly interact with the internal fields of the multiport circuit. For certain practical geometries, for example a dipole electromagnetically coupled to the open-end of a microstrip line [22,40], such an assumption would not be valid. Suitable modifications discussed in section 2.3.5 need to be incorporated using an additional non-Galerkin testing procedure to account for such practical geometries. • The reference incident electric-current directions on the two opposite ports (the ith and the N~+ ith ports, as a convention) of the same (ith) strip-type transmission feed line are assumed to be opposite, whereas the reference incident equivalent magnetic-current directions on the two opposite ports of the same slot-type transmission feed line are assumed the same (see fig. 8). This assumption enforces the same reference transverse electric-field directions for the two opposite ports. This is just to establish a convenient conventional consistency. • The scattered port electric current on a strip-type transmission feed line is negative of the corresponding scattering parameter, whereas the scattered port equivalent magnetic current (or slot field) on a slottype transmission line is the same as the corresponding scattering parameter. This is consistent with the standard electric field referenced definations of scattering parameters. • The individual transmission line fields do not directly couple with each other in the absence of the radiating elements. However, indirect coupling between them can always result via mutual coupling between associated radiating (finite-length) elements that are directly coupled to the transmission lines. This restriction does not exclude the use of coupled lines, such as coupled microstrip lines or coplanar waveguides, as long as they are treated in groups as a single transmission-line element. As per this restriction, the transmission feed lines are assumed to be far apart, or printed on different layers electrically isolated by conducting planes in between (fig. 8 does not show the layering, and the cross-overs are assumed to be on isolated layers).
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I t+’
~4—--\\~r----
~7
/~~\\\\ ~
/
\~N~-1
I/I
UNIT CELL) BOUNDARY
(for infinite array)
a
~
ARRAY
~
SLOT
N
1
i L ~1 CONDUCTOR Fig. 8. Generalized multiport model of printed antennas integrated with planar transmission feed lines. Various antenna elements and transmission lines are in general placed on different surfaces of a multilayer configuration.
• For infinite arrays the lengths of the feed lines are strictly restricted by the unit cell dimensions. For practical isolated geometries the feed lines are always of finite extent. These finite-length transmission lines are, however, approximately modeled by hypothetically extending them to infinity in both directions [27]. Any criss-crossings due to this hypothetical infinite-length extensions are eventually ignored. To summarize, the above restrictions are not very severe for most practical applications, but are necessary for analytical simplification purposes. The other fundamental idea behind the above assumptions is to model the dominant electromagnetic interaction in the system due to mutual coupling between the finite-length radiating elements, and not between feed structures. For off-broadside scanning of infinite arrays, particularly near blindness resonances, such assumptions may not be strictly valid, however. With the above assumptions for the multiport model of fig. 8, excite the mth port by an incident wave of unit amplitude. The source-free part (part A) containing transmission lines without direct coupling between them can be described as: 5Aim~’ i=P~+m, mN~, =1,
i=m—N~, m>N~,
=0,
otherwise,
(12)
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with EArn equal to the travelling eigenfields of the mth transmission line alone. Such simplification results in completely solving the first part of the multiport antenna problem if solutions for individual transmission lines are available. As mentioned before, general numerical solutions for multilayer transmission lines are available [34,35,44], and need to be used to support the present scattering formulation model. Now, denote the scattered waves, SBjrn, of part B as R1, dropping the second index, m, for convenience, with the understanding that R,’s are due to the sources induced by the incident wave at the mth port. Thus, using eqs. (7) and (12), the R1’s can be trivially related to the actual scattering parameters, ~ of the multiport antenna: Rj=Sim~1, i=N1+m, mN5, 51m1’ i=m—N~, m>1V, 5im’
otherwise.
(13)
2.3.3. Galerkin testing on the antenna elements The unknown currents on various finite-length strip elements, or equivalent magnetic currents due to electric fields on various finite-length slot elements (of only one unit cell, for an infinite array), can be expanded as a sum of a set of basis functions with unknown coefficients, N 1
Jm>JI,.J~,
(14)
i= 1
where J is used both for electric currents as well as equivalent magnetic currents for the sake of generality, with the longitudinal and the transverse variations suitably chosen from the set of basis functions of appendix H. Nf is the total number of finite-length modes of the system, distributed over Nf’ finite-length physical antenna or parasitic elements (see fig. 8). From the equivalence principle, it should be noted that the electric field in a slot on a ground plane is equivalent to two magnetic currents of equal magnitude and opposite directions placed above and below the slot with the ground plane closed. These two magnetic currents are described by only one unknown coefficient in eq. (14). Consider a perfectly conducting element of the general antenna configuration, for example an electric dipole or a patch. In order to obtain the solution to the entire system, a part of the boundary condition to be satisfied requires the tangential components of electric field to vanish at all points on the conducting element, E~~’ Escat + E~~c0, =
=
(15)
where ESC~~ is the component of the tangential electric field produced due to currents on all finite-length elements of the system, and EInC is that due to all travelling-wave elements of the system. It may be noted, using the two-part solution approach of section 2.3.1 and 2.3.2, that ~ includes the fields of the first part, and a part of the fields of the second part produced due to only the travelling-wave currents on the feed lines. The above boundary condition can be rewritten as N1
N,’ —
1=1
LE,’°~,
(16)
i=1
where E1 is the scattered tangential field due to the ith finite-length mode with coefficient I,, Nf is the total number of finite-length modes of the entire system, EnC is the total incident tangential field due to all travelling-wave currents on the ith transmission line, and N’ is the total number of transmission lines coupled to the particular conducting element. This exact boundary condition can, however, be enforced
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using a variational approach by means of a Galerkin testing procedure [14,32] resulting in a set of linear equations involving the unknown coefficients, 11’s; N~
N,
~I1Z~,=
~i’ç~,
i=1
1=1
(17)
for all jth current modes of the system used to expand the currents on the particular conductor on which the boundary condition is being tested, where Z~~,—J
E1’J, ds,
(18a)
jth mode
V,=
—Jjth modeE”~J,.ds,
(18b)
and are respectively refered to as elements of reaction and excitation vectors of the Galerkin moment method. On the other hand, if the element considered is a slot, or an equivalent magnetic conductor, the required boundary condition to be satisfied is given as (19)
t2 + Hm0~~ — HWC2,
0 ~ H~’~’ where the superscripts 1 and 2 refer to the top and bottom surfaces of the slot, respectively. Using a similar testing procedure as described before for conducting elements, the resulting set of linear equations for all j corresponding to the modes of the particular slot can be written as in eq. (18), where Htan
2
—
z~
=
=
—
1—H,2)’J
1=J
(H1
1ds,
(20)
jth mode
—fjth mode (H1~’~
J7~,=
2) .j
ds.
(21)
HiflC
Depending on the location of the ith mode (above, below, or on the same conducting ground plane as the jth mode), one of the two magnetic-field terms of eqs. (20) and (21) may be zero due to the ground-plane isolation. Note that the dipoles and patches are assumed to be made of perfect conductors in the above equations, and no impedance boundary condition is implemented for these antenna elements. However, such impedance boundary conditions can easily be implemented by suitable modification of the expressions for V, and ~ which would only involve computation of more Green’s function components. This may be essential for high-frequency applications to include conductor losses due to the radiating elements. However, losses due to dielectrics or various ground planes of the substrate topology are automatically included via the Green’s functions. Components of reaction and excitation vectors for various cases of expansion and testing modes as defined by eqs. (18), (20) and (21) are computed using spectral-domain Green’s functions. Specific details of the spectral-domain implementation of the reaction and excitation elements will be separately discussed. Infinite arrays as well as isolated-element cases will be considered. 2.3.3.1. Spectral domain expressions of reaction elements. Using the spectral-domain Green’s functions of appendix B, and extending them for an infinite-array environment [17], the expressions of (18) and (20) can be combinedly written as Z~ 1=
~ n
~ m
V~
~(kxrnn,
kyrnn) ~1j~
Uj(kxmn, kyrnn)L~*(kxrnn, kym~)~
Ak~,
(22)
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Ye Va
•
•
•
S
407
S
Fig. 9. Reference axes of various coordinate systems used in the generalized multilayer antenna analysis.
where G
G~,. if the jth mode is an electric current, or
G~/21~
if the jth mode is an equivalent magnetic current, with the source, J~,electric or magnetic as the ith expansion mode. The Fourier transforms of the ith and jth modes, respectively L~and U1, as described in appendix H are assumed to be with respect to the system coordinates, the expansion test 2e 1, and and are I~ respectively £~.The superscript, k,, of and the the Green’s modes with the respective current directions in functions refer to the particular substrate configuration with the ith mode between the 21 and 11 layers of the substrate topology. The indices, m and n, refer to various Floquet modes of the infinite-array configuration and depend on the periodicity of the array, the grid structure of the array, and the phasing of the array tQ obtain a particular scan angle. The scan angle of the array is given by the angles (0, 4,) of a spherical coordinate system with the z-axis of the multilayer substrate configuration coinciding with the 0 0 and the x—y-plane of the layered structure coincide with the 0 900 plane, with 4, = 00 along the array x-axis, Xa~The geometry of the array configuration with various reference directions is shown in fig. 9. With reference to fig. 9, the expressions for kxmn, kyrnn, ~ ~ in eq. (22) can be written as =
—
=
=
0,
=
2m’iT kxm
(23a)
0=~k05~flOCO54,+~_~__,
2n’rr
kyrnn
~k and,
a
X
—k0 sin 0 cos(4, Oa) + —i--— /sin Oa 2 4’Tr2 z~k =—, 4ir ~ =z~k=absinOa z.IA =
—
and b are the grid spacings along
Xa
—
kxmn/tan Oa,
(23b) (23e)
and y~axes, respectively. To handle the most general array
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configuration, four local coordinate systems are used. (Xa, ya) are chosen here as the reference system coordinates, with Xa the array x-axis, and y~perpendicular to it. The array grid periodicity is along Xa ~fl one direction, but along y~where y~is not perpendicular to Xa in general, and is referred to as the array y-axis. Other than the system coordinates, (xa, ya), the array coordinates, (Xa, y~),the coordinate system (Xe, ye) is the expansion coordinate system and (x 1, y~)is the test-mode coordinate system, with various relative angles as shown in fig. 9. Clearly, the reaction vector of the moment analysis that accounts for the dominant coupling mechanism in the infinite array must be computed differently when extended to isolated-element analysis. Analytically it is equivalent to computing the Z,1’s of eq. (22) by using an infinite integral instead of an infinite series. Thus
~ffV•
=
~
k~).x~jjUj(kx,k~)U~*(k5, k~)dk~dk~,
(24)
with the terms defined as in eq. (22). Unlike the infinite-array analysis, the reaction matrix can now be shown to be symmetric with Z13 Z31 based on the reciprocity theorem. Also, unlike the infinite array, where the infinite sum must be computed over the entire spectral plane, the infinite integrals of eq. (24) can often be reduced to integrals over only one quadrant of the spectral plane, relating the contribution from other quadrants by symmetry considerations [14] of appendix D. The other major analytical considerations when replacing the infinite-sum expressions of (22) by the infinte integrals of (24) are due to the numerical problem associated with the computation of the spectral integrals. Efficient numerical methods have to be used to optimally account for the sometimes fast oscillating integrals of (24). As discussed earlier, the poles of the integrals of (24) associated with the poles of the Green’s function due to the characteristic waves of the multilayer structure impose the additional complexity for the numerical implementation of the infinite spectral integrals of (24). Contributions of the residues at these poles has to be added to the final principal values of Z11 to account for coupling due to the excitation of characteristic waves. Positions of the characteristic wave poles of the Green’s functions on the k~—k~-plane can be searched using a Newton—Rhapson iterative technique. Using the values of these pole locations the residues of the integrals of (24) can be extracted numerically in a way similar to that described in section 2.1. =
Spectral-domain expressions of excitation elements. Unlike the computation of reaction elements in section 2.3.3.1, the computation of excitation-vector elements is the same for infinite arrays as for isolated elements. This is based on similar assumptions as in section 2.3.2 that there can be no direct coupling between transmission lines of the same unit cell (for an infinite-array case), or of two different neighbouring unit cells. Also, due to a relatively more localized nature of the transmission-line fields, they usually do not couple to antenna elements of the neighbouring cells. In other words, the coupling between an antenna element and an infinite array of transmission lines is electrically equivalent to the same between the antenna element and a single transmission line. Such an assumption is valid for near-broadside scanning of the array, but fails near blindness angles or for near-endfire scanning. As discussed in section 2.3.2, when port m is excited, the outgoing wave for the second part of the problem at the ith matched port is equal to R,. The first part of the problem only consists of the eigenfields of the mth transmission line with unit amplitude. Under this condition of one port excitation, before precisely defining the expressions for various excitation-vector elements, V~’s,of the moment method system, it may be useful to define the following intermediate expressions: 2.3.3.2.
=
—
=
—
~I~ffE,”
‘J3. ds = ~
L~f~ [1~~(_k~,
~l’~
ky)1~*(_ke,k~)F~(k~) dk~,
(25a)
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if the jth element is an electric current, or,
=
—
~
~f~(i~
[~i~ij(_ke,
k~)— O~ij(~ke, k~)j)U
1*(_ke, k~)F,,(k~)dk~,
(25b) if the jth element is an equivalent magnetic current. The system coordinates are chosen with the direction of forward propagation of the ith transmission line, x~,,the same as the system x-axis. L~V~ is the excitation of the ith infinite transmission-line element to the jth finite-length mode associated to its discontinuity, when the current on the ith transmission-line element, described by the coefficients of its expansion modes (I,,’s), is such that unity power is transmitted across its cross-section in the forward direction. This means, for example, in the case of a microstrip line, a total longitudinal current along the conducting strip equal to 1/ ~ units, where Z,~is the characteristic impedance of the microstrip line. For a single slot line to transmit unit power across its cross-section, the required equivalent magnetic current along the slot line needs to be equal to 1/ ~ where )‘~ is the characteristic admittance. The source components associated with the Green’s functions of the equations (25) are decided by the nth mode of the transmission line, J,,, and can be J~,M1, .1,,, or M~as required. The Green’s functions are evaluated at the plane of the jth mode where k,, denotes the substrate topology with respect to the nth source currents placed between the 11 and 21 layers, as usual. is the Fourier transforms of the jth test mode directed in I~ 1,, with respect to the system coordinates defined (see appendix H), and ~, is the Fourier transform of the transverse variation of the nth mode of the ith transmission-line element. Finally, ke is the propagation constant of the ith transmission line element. A similar term, z~V~called the forward-discontinuity mode excitation, can be defined where instead of the ith transmission-line mode with infinite length in the forward direction, the ith transmission-line forward-discontinuity mode (see Appendix H) is considered as the source of excitation. =
_
~V~=
ff~[~..o~,(k
ky)IUJ*(kx, k~)U,,(k5,k~)dk~dk~,
(26a)
if the jth element is an electric current, or, — —
—
V’
~
~L
~
U I~11 . JJ
~k,, (~. ; \ ‘-‘H21J,,Y~x’‘by)
xL~.*(k~,k~)U~(k~, k~)dk~dk~,
—
,-‘k ii “JtilJ,,V’x’
n.y (26b)
if the jth mode is an equivalent magnetic current. The notations have similar interpretation as those of eq. (25). It should be noted that double spectral integrals in eq. (26) are required in place of single-dimensional spectral integrals of (25) to account for the discontinuity modes. U,, is the Fourier transform of the forward-discontinuity mode as described in appendix H, with the same transverse variation on the transmission line as before. Two additional terms equivalent to LW~and ~J’~ can be obtained by rotating the ith transmission line, and thus rotating the system coordinate axes by 1800, and recomputing ~ and ‘5d with reference to the new system coordinates. In addition, a negative sign would be added to or for a slot transmission-line mode, to take care of the port-reference current directions described earlier. These two new terms are called the backward excitation, z~J’~’, and the backward-discontinuity excitation, z~V~respectively, and correspond to the backward-propagating wave on the ith transmission line.
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With these terms defined, it is easier to formulate the V.1’s of eqs. (18) and (21) of the moment-method system as follows: +
=
~Rl+N,(~l’~
+
if i ~ m or i
+~ #
+
~i’~fl,
(27a)
m, or (27b)
if i
=
m
N1 or
>
IV~,respectively.
Similar treatment of the transmission-line travelling-wave currents are presented in ref. [24] in the context of a specific simplified geometry with a microstrip feed line coupled to a perpendicularly placed dipole. The transmission-line discontinuity modes of appendix H take care of the semi-infiniteness of different R.’s. Various semi-infinite-length travelling-wave currents extending in one direction can be decomposed as the sum of an infinite mode amplitude, and a discontinuity mode of the type described in appendix H, each with half the amplitude of the semi-infinite mode. This is essentially the basis on which eqs. (27) are derived, with ~W~5 and zWJ’ resulting from the infinite mode parts, and z~VJ”and L~VE~~d resulting from the corresponding discontinuity mode parts. While performing numerical implementation, computations of ~j~3I~d or are further simplified by decomposing into odd or even discontinuity modes, and also incorporating a modified transform to smooth the poles due to the discontinuity mode transforms (see appendix H). It can be noted that or LW~’in the expression of ~ when i m is due to the transmission-line fields of the first part of the multiport problem when no discontinuities are present. As discussed before, when only the mth port is excited, this trivial case results in fields only in the transmission line corresponding to the mth port, travelling forward or backward depending on whether m is less than or equal to, or greater than, N~. =
2.3.4. Spectral-domain implementation of multiport scattering model The number of linear equations obtained from the Galerkin testing of the boundary conditions on the finite-length elements is equal to the total number of finite-length expansion modes of the system. But this set of equations is not sufficient to solve for the additional 2N~unknown values of R1’s. However, another set of 2N~equations can be obtained by using the generalized multiport scattering model. 2N5 + Nf unknowns of the moment-method system can thus be solved by solving the complete set of linear equations. This generalized scattering model is a rigorous generalization of the reciprocity arguments used in refs. [26,27] for a perpendicular slot-type discontinuity, and in ref. [24] for a perpendicular electric-dipole-type discontinuity. In both refs. [24] and [26,27] the transmission line is a single microstrip line. In a general environment where various types of transmission lines are coupled to dipoles or slots, or a combination of both at the same discontinuity plane, or when more than one transmission line is coupled to the same radiating elements, the reciprocity arguments of refs. [24] and [26,27] cannot be directly extended. The present multiport-scattering formulation, however, accounts for all such cases. Using the generalized scattering condition (11), the eigensolution of the ith transmission line described by its current expansion mode amplitudes (I,,’s), and the the expression (25) for coupling between the ith infinite transmission line and the jth antenna mode, gives the following result: R1= ~ =
~
±1,,~l’~J”, if iN~,
(28a)
±1,,L~V,~” if
(28b)
i>N~,
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where the summation over j is for all finite-length modes coupled to the ith transmission line, the negative sign is chosen if the jth mode is a magnetic current below the nth transmission-line mode, and the positive sign is chosen if the jth mode is a magnetic current above, or is an electric current above or below the nth transmission-line mode. Different signs for the expressions in eq. (28) for the cases of a magnetic-mode current above or below the transmission line are used to take care of the reversed signs of the equivalent magnetic currents for two sides of the ground plane. It should be noted that the different signs for the R1 expressions in (28) do not depend on the type of transmission lines, but on the type of the current mode coupled to it. For example, if one transmission line is used, with one dipole or slot perpendicular to it, with an electric or equivalent magnetic current, I, and the transmission line is excited from port 1, R1= +~Ii~V1~,
(29a)
R2= +~IAV1~,
(29b)
if i is due to a dipole, or R1= —~Ii~V1~,
(30a)
R2= ~
(30b)
if I is due to a slot below the transmission line, and thus S11 R1, ~21 1 + R2 1 + R1 for a dipole, and ~ R1, S21 1 + R2 1 R1 for a slot. This is due to further simplification based on the fact that the dipole or slot is perpendicular to the transmission line, and so R2 R1 for a dipole case, and R2 —R1 for the case of a slot. This can be easily verified using eqs. (29,30) and (25), and phase relationships between forward- and backward-travelling-wave-mode fields in a transmission line. The transmission lines feeding the slot or dipole can be of any type, a slot line or a stripline, as long as the 4~V expressions are properly computed as described under (25)—(27). These results agree with those of refs. [24,26,27] for the case of a microstrip line feed. =
=
=
=
=
=
—
=
=
2.3.5. General considerations Using the analysis described, a general-purpose computer routine, UNIFY, has been developed for unified simulation of multilayer multifeed planar antennas in an infinite array as well as an isolated environment. Various practical approximations discussed earlier are assumed, however, to simplify the most general formulation, and other special considerations not explicitly discussed in the analysis are discussed here to handle special cases: The analysis explicitly discusses the computation of scattering-port parameters. Various current expansion functions with the unknown coefficients solved from the moment method can, however, be used to evaluate the far-field components in a relatively straight-forward way via a stationary phase method with proper choice of spectral Green’s function components [45]. The two orthogonal far-field components can be treated as two additional ports, the scattering parameters of which can be used for radiation characterization. s The currents on the transmission lines can be completely described as a superposition of infinite transmission-line modes, transmission-line discontinuity modes of appendix H, and the additional finite-length non-travelling-wave modes in the vicinity of the radiating (or parasitic) elements coupled to the transmission lines. Inclusion of the additional finite-length modes on the transmission lines are not explicitly discussed in the analysis. These modes can be virtually treated as other finite-length modes on antenna elements, however. As discussed in appendix H, half PWS modes may be required to insure current continuity compensating for the inherent discontinuity of the transmission-line discontinuity modes. s
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• As discussed in section 2.3.2, planar structures such as a dipole coupled to the open-end of a parallel microstrip feed line [24] could not be accounted for in the multiport-scattering formulation. This is different from the stub-tuned aperture-coupled microstrip antenna of refs. [26,27],where the open end of the microstrip feed line can be accounted for as a circuit element via the scattering parameters of the two-port antenna, because here the open end does not directly interact with the slot fields. For cases similar to ref. [24] the currents in the open-ended feed line can still be expressed as a superposition of finite length and transmission-line modes, and can be tested for appropriate boundary conditions in the same way discussed in the analysis. Half PWS modes would be required to compensate for current discontinuity of the transmission-line discontinuity modes. An additional equation instead of the multiport-scattering condition (not valid here) is required here, and can be obtained by use of an additional non-Galerkin testing function on the transmission line [46]. • Some of the numerical considerations for implementing the spectral integrals for an isolated element case have been discussed earlier. Convergence of results with respect to expansion modes of the moment method is equally critical, and carefully selecting the expansion current modes to describe the actual current distribution is useful. In order to get a first-order idea of the current distributions, the solutions for another closely related problem, the exact solution for which can be easily obtained, can be used as a guideline. For example, a closed cavity with magnetic side walls excited by an equivalent magnetic source at the slot is very similar to that of a slot-coupled microstrip antenna, so far as the currents on the microstrip patch are concerned. The additional strongly radiating current component on the microstrip patch is, however, excluded from the non-radiating magnetic-walled cavity solution. Figure 10 shows the coefficients of various cavity eigenmodes of the currents in the resonant direction corresponding to a specific slot-coupled microstrip antenna. As it clearly shows, the fundamental eigenmode is the most dominant, and a few other eigenmodes around the dominant eigenmode is enough to very closely describe the patch currents. Also, because the slot is placed at the center, odd mode variations (even m) about the patch center in the direction of the patch current, as well as even mode variations (odd n) in the transverse directions are absent. The Corresponding real-space variations of the patch currents are shown in fig. 11, clearly showing the dominant mode, along with other higher-order modes constituting the rapid variations at the center directly above the slot. This is similar to the rapidly varying currents on a probe-fed antenna [19], but unlike the probe-fed case, is produced by the proximity effect of the slot not in direct physical contact with the patch. Similar results for the patch currents in two orthogonal directions for the general case of an inclined slot are presented in fig. 12a and fig. 12b for below and above the respective dominant cavity-mode resonant frequencies. The dominant resonant mode is still clearly prominent, and changes its relative phase below and above resonance, as expected.
3. Applications 3.1. Aperture-coupled microstrip antenna with radome Figures 13 and 14 show a comparison between the theoretical and experimental values of equivalent impedances (as seen by the transmission line [26]) of an aperture-coupled antenna, for an infinite array and for an isolated-element environment, respectively. The experimental data for the infinite-array case was obtained from a waveguide simulator [27]. For both cases the results of the generalized analysis are clearly in good agreement with experiment. The results of the analysis for the uncovered geometries of figs. 13 and 14 are also found to be in similar agreement with experimental results (not presented here). As expected, for moderate values of cover-substrate thickness and dielectric constant the impedance
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I
IJymni
13
0 .~
m
ylmi
H
Xlnl
Fig. 10. Spectral distribution of cavity eigenmodes for a slot-coupled microstrip patch with magnetic side walls. Patch: length in resonant dimension = 4.0 cm, width 3.0 cm; slot: at the center of the patch, 1.0 x 0.11 cm; substrate: 2.2, 0.16 cm, frequency = 2.375 GHz. Mode indices, m and n, correspond to integral multiples of half-wavelength variations in the respective directions.
locus of the uncovered case is found to be very close to that for the covered case, with the resonant frequency for the later less than that for the former. 3.2. Aperture-coupled microstrip antenna with a parasitic patch
Figure 15 compares the theoretical and experimental results for the equivalent series impedances of an infinite array of a two-layer stacked microstrip patch aperture coupled to a microtrip feed line. The agreement is excellent, and similar agreement has also been obtained for other sets of physical parameters. As can be seen, in contrast to the impedance loci of the single layered microstrip antennas of figs. 13 and 14, the impedance locus of the stacked geometry has a characteristic double-tuned loop corresponding to the resonances of the two patches. Detailed studies of the bandwidth characteristics have been performed, with the conclusion that bandwidths of the order of 15—20% can be obtained from
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Im J~cv)
4cm 3cm
op Fig. 11. Dominant current, J~(x,y), as a function of x, y for the slot-coupled magnetic walled microstrip cavity of fig. 10.
a two-layer stacked geometry [41]. For electrically thin substrates it is observed that it may be preferable to use a slightly larger cover patch to obtain the maximum bandwidth. However, a significantly larger cover patch may result in dual-frequency operation of the antenna, instead of an enhanced-bandwidth mode. On the other hand, a significantly smaller cover patch may not couple to the feeding slot, since it tends to be isolated by the primary patch. 3.3. Dual aperture-coupled microstrip antenna for circular polarization More complex geometries with multiple feed lines and coupling apertures can also be rigorously analyzed using the present analysis. A dual slot-coupled configuration with two microstrip feed lines, as shown in fig. 16a, has been analyzed for impedance as well as polarization characteristics. By suitably exciting the two ports, a perfect circular polarization can be obtained. Due to the asymmetric mutual coupling effects between the slots, perfect circular polarization can not be obtained by exciting the two ports with equal amplitudes and a 90 degree phase difference [42]. Hence, the excitations must be compensated to account for such undesirable asymmetric mutual coupling effects. Figure 16b presents the input return loss of an infinite array of dual slot-coupled circularly polarized elements with a parallel reactive power divider for the two ports, with different line lengths to provide the 90 degree phase shift. As fig. 16b shows, compared with the return loss from a single aperture-coupled linearly polarized geometry, the circularly polarized case has a broader bandwidth due to mutual cancellation of reflections from the two ports. This is due to the 180 degree phase difference between the reflected waves from the two coupling apertures. However, the polarization bandwidth is less than the return loss bandwidth.
N.K Das, D.M. Pozar
/
A generalized CAD model for printed antennas
415
Im J~(x,v)
ImJ~(x,v)
a
b
Ci~~jx~~
Fig. 12 (a) Dominant patch current, J~(x,y), for the magnetic walled cavity of fig. 10 with the slot inclined at 45 degrees, at a frequency, 2.75 GHz, above resonance in y-dimension. (b) Orthogonal patch current, J~(x,y), for the magnetic walled cavity of (a) at 2.75 GHz, below resonance in x-dimension.
416
N.K Das, D.M Pozar
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A generalized CAD modelfor printed antennas
3.25 GHz
=
COVER LAYER Ec d~
PRIMARY
AN~~I a~b
ANT
50MHz
3.70
ENNA
SUBSTRATE Ea
3.45
•S •ThEORY
S S
X—)E—X EXPT.
ND PLANE
.l58cmdc~.152cm EtEa2.2 Ec~2.55 a- b~2.5cm FEED LINE: sLoT: 1.1 x.lScm w .50cm, 500 ARRAY UNIT CELL: ~607cm: H dfda
~~FEED SLOT
SUBSTRATE
__
df FEEDLINE
3.404cm: E-PLANE
Fig. 13. Waveguide-simulator measurements and calculations for a covered aperture-coupled microstrip antenna. 1.0
2.3
GHz 1~f=25MHz •..
+
EXPT.
+ + +THEORY
.475
2.35
2~
~
+ WITH A COVER
2.55, O~l6cm 0
0
Fig. 14. Comparison of theoretical and experimental results for an isolated offset aperture-coupled microstrip antenna covered by a dielectric sheet. Patch: 3.9x3.9 cm; slot: 1.2X0.17 cm; patch offset with respect to the slot (center to center): 0.7 cm in the resonant direction, 0.6 cm in the non-resonant direction; feed line: r = 10.2, w = 0.12 cm, Z~= 50 ~1, ~eff = 6.84; antenna substrate: Er 2.2, 0.16 cm; cover substrate: ~r = 2.55, 0.16 cm.
N.K Das, D.M. Pozar
/
A generalized CAD model for printed antennas
:~~
//‘~ PARASITIC PATCH
bxb LAYER
__________
ANTENNA SUBSTRATE
_____
3.65 •
417
.5
GHz.
85
PRIMARY ANTEN~~J~
77 da
/ SLOT._,L_~. /‘\
,./ //
/ ~~
_____________
,~-
GROUND PLANE
X-X--K EXPT.
A/i
//
V,z~ d~
•SSTHEORY da~d~d~.158cm FEED SUBSTRATE E~
Ea=EfEc~’2.2
a= b~2.5cm FEED LINE: SLOT:1.lx.l5cm wr.5Ocm,500 ARRAY UNIT CELL: 3.607cm: H 3.404cm E-PLANE
FEEDLINE
Fig. 15. Waveguide-simulator measurements and calculations for a stacked microstrip antenna configuration with a
3.4 (a)
3.5
3.6
3.7
=
2.5
3.8
cm
=
b.
3.9
Freq.(GHz.) (b)
Fig. 16. (a) The two-port circuit configuration used for the analysis of the dual-slot-coupled structure. Antenna substrate: Er = 2.2, 0.16 cm; feed substrate: = 2.2, 0.16 cm; feed lines: width = 0.5 cm, 50 fi, ~eff = 1.9; antenna: 2.5 >< 2.5 cm; slots: 1.1 x 0.15 cm; slot offset: 0.7 cm from patch (center to center) in corresponding non-resonant dimensions; array unit cell: 4.11 x 4.11 cm; scan: broadside. (b) Return loss of the dual-slot circularly polarized configuration compared with that of the corresponding single-slot-fed linearly polarized antenna.
418
N.K Das, D.M. Pozar
/
A generalized CAD model for printed antennas
9.0
SLOTS ON GROUND PLANE
_____________
N
.
• ~I EX~. X X K THEORY
/
m 6.0
~~R~AT~OP
LH\
—FEED NETWORK ON BOTTOM SUBSTRATE
~.Q .~
>(
..
.
500
(a)
______________-
U-93
1.00
/
1.02
to
(b) Fig. 17. (a) Geometry of a dual-slot-coupled circularly polarized microstrip antenna designed, built and tested. Antenna substrate: Er = 2.2, 0.16 cm; feed substrate: ~ 10.2, 0.127 cm; patch: 3.9x3.9 cm; slots: 1.2x0.17 cm; patch offset with respect to slots (center to center): 0.7 cm in the non-resonant dimension; 75 £1 feed lines: 0.4 mm width; 90 degrees phase difference between the two lines; 50 Cl feed line: width 0.12 cm with a 37.5 (1—50 Cl tapered transition. (b) Comparison of theoretical and experimental results of normalized frequency performance of axial ratio of the isolated dual-slot-fed circularly polarized antenna: f 0 (theory) = 2.35 GHz, f0 (experiment) = 2.3875 GHz.
The infinite-array geometry of circularly polarized elements can not be simulated in a waveguide simulator. However, an isolated element version of the above case was designed, built and tested using the general analysis routine. Figure 17 compares the theoretical and experimental axial ratio performances of this geometry, and demonstrates the validity of the general analysis. However, the experimental resonant frequency was found to be slightly different from that of the theory, probably because of tolerances in the dielectric constant. 3.4. Microstrip dipole-coupled to an inclined covered microstrip feed line The analysis is now applied to an infinite array of printed dipoles coupled to inclined covered microstrip lines. The S11 and S22 of the two-port circuit are plotted in fig. 18 as a function of the angle of inclination, 0. Unlike the perpendicular dipole case [241, for a general inclined dipole S11 ~ S22, due to the physical asymmetry of the dipole with respect to the two ports. Also, unlike the perpendicular dipole case, we now have S21 1 + S11, which implies that for the general inclined dipole a shunt-equivalent impedance model is not valid. In order to obtain an equivalent circuit model for such inclined dipoles, it is required to use a T or m- network with three independent impedance parameters. With the general multiport analysis described here, these inclined dipole cases are completely modeled by the generalized scattering conditions. The multiport scattering parameters can be used to construct an equivalent circuit, the topology of which is not specifically assumed a priori. 3.5. A stripline aperture-coupled microstrip antenna All the structures discussed up to this point used a microstrip feed line. However, to avoid back radiation from the coupling slot, and to isolate the feed network, a stripline feed to the antenna element
N.K Das, D.M. Pozar
/ /~
txe=lo° /
/
/ /
/
~
/
A generalized CAD model for printed antennas
419
N\
d=O.l6cm w=O.5cm dipole= 3.5~j~L~
~
I~————~ \ /
S29f~
Uao°
\\\~
/ .~\
~
—.-------.I
s11~
Fig. 18. Locus of S
11 and S22 of an infinite array of printed dipoles inclined-coupled to covered microstrip feed lines, as a function of the inclination angle, 9. Substrate: ~r = 2.2, 0.16 cm; superstrate: Er = 2.2, 0.16 cm; feed line: 0.5 cm width, 50 (1, Eeff = 2.096; dipole: 3.5 xO.1 cm; array unit cell: 5.0x5.0 cm; offset: ~ = 1.0 cm, frequency = 3.143 GHz; scan angle: broadside.
can be useful. But a stripline type of feed has potential problems, caused by guided modes that need to be carefully considered [43]. The geometry of a representative stripline aperture-coupled antenna structure with a randome layer is shown in fig. 19 along with its scan performance for a specific set of physical parameters. For these physical parameters the scan performance exhibits a prominent scan blindness at 43.6 that corresponds to the excitation of a parallel-plate mode of the feed structure forced by the non-radiating slots. The possible solutions for avoiding such dangers of blindness due to parallel-plate feed structures are to use a low-dielectric-constant substrate for the feed and/or reduce the size of the array unit cell. 3.6. A proximity-coupled covered microstrip antenna This geometry is an example of a microstrip antenna proximity-coupled to an open-ended covered microstrip line. Unlike the previous examples, where the radiating elements were coupled to the dominant transverse fields of a through transmission line, in this case the antenna is coupled to the fringing fields of the open end. As discussed before, in this case the simplified reciprocity analysis cannot be used, and so the moment system is solved for the reflected traveling-wave amplitude by use of an additional testing function. Extra PWS current modes are used to model the non.traveling-wave current
420
N.K Das, D.M. Pozar
/
A generalized CAD model for printed antennas RADOME
COVER LAYER
PRIMARY ANNA ~7~T~ATE
~~Ed
~TRATE
FEED L(NE
43.6°
_____________________________________________ O.~ I I I~ I I I
ii
-
EH—•—.-D-PLANE
/
/
/
/
I
o-J
/
I I
-
z
I
/
•-‘
-
—
/. /
I
-
/
J.
/
I
U)
~.0
/
-
E~~IMENT
7/
-
p
//
—12.,
1/ /
-
4 4
~
p
2,40
/
-
/1
-
7 i8.C ~—1 0
If,//
S
7 I
.
I
30 SCAN ANGLE,
I
60
I
90
~f=50
2.25
2.
DEG.
e Fig. 19. Scan performance of an infinite array of stripline-fed aperture-coupled covered microstrip patches. Array unit cell: 0.4 X 0.4k 0, a b = 2.5 cm; slot: 1.1 X 0.15 cm; feed line: w = 0.1 cm, 50 ~ = 2.55, d~ = 0.158 cm, Ea 2.2, d, = 0.158 cm, E11 = 10.2, df1 = 0.127 cm, E~ = 2.2, d~= 0.158 cm, frequency = 3.45 GHz.
Fig. 20. Comparison of theoretical and experimental results for an isolated microstrip antenna covered by a dielectric sheet proximity-coupled to the open end of a covered microstrip line. Patch: 3.9 x 3.9 cm; feed line: w = 0.5 cm, Z~ = 50 Cl, E4f1 = 2.144; all substrates: Er = 2.2, 0.16 cm; open end at the patch center.
components near the open end, and play the dominant role in establishing effective coupling with the antenna. In contrast, for cases of through transmission line coupled antennas, the dominant coupling from the feed line to the antennas is due to the fields of the traveling-wave current components on the feed line, and extra non-traveling-wave current expansions on the feed line in the vicinity of the antenna only describe a refinement of the solution. In fact, for the through transmission-line case with an aperture-coupled microstrip antenna (section 3.1), for example, fairly accurate results can be obtained without including the non-traveling-wave current expansion modes on the microstrip feed line, whereas for the present open-end proximity-coupled geometry a sufficiently large number of finite-length current expansion functions are required to obtain reasonable results. Thus the solution of such proximity
N.K Das, D.M Pozar / A generalized CAD model for printed antennas
421
coupled geometries is computationally more involved, and also requires careful attention to the convergence of the solution with respect to the number and density of the non-traveling wave modes near the open end. Figure 20 shows the comparison of the experimental results with the results of the present analysis for a covered proximity-coupled patch antenna. The theoretical results were obtained with eight PWS modes of longitudinal currents on the feed line over a length of about 4.0 cm near the open end, and five fundamental entire-domain sinusoid modes (even and odd, with uniform transverse variation) on the patch with currents in the same direction as on the transmission line. Figure 20 shows the equivalent impedance as seen by the feed line with the phase reference at the open end. Clearly, the agreement between the results is good with a shift in resonant frequency on the order of approximately 1.0%, which probably can be attributed to tolerances in the value of the dielectric constant and other fabrication tolerances. Similar agreement was also obtained without the cover substrate as well as with other patch dimensions and feed positioning.
4. Conclusion Excellent agreement of the results of UNIFY with experiments for various configurations confirms the validity of the general analysis. Considering the rigor and versatility of the analysis, it should be useful for full-wave simulation of a wide variety of multilayered integrated array geometries.
Appendix A. Different notations for the generalized Green’s functions 1. (a) G~~
k~,z.1): the scalar spectral-domain Green’s function of the field component * (Er, E~,E~,H~,H~,H~)due to the source # (J~J~,~ M~,M~,Me), with source coordinate at (0, 0, 0). The field is computed in the dielectric layer (/]), at a plane given by z~,(see fig. 3). The field components of the Green’s functions can also be along any direction on the x—y-plane, and can be represented, for example as ~ where x’ is the axis along the field direction. As discussed in appendix B, Green’s functions with arbitrary orientation of field direction on the x—y-plane can be expressed as linear combination of Green’s functions with field components in x- and y-directions. Similarly, the source direction can also be arbitrarily oriented on the x—y-plane. The spectral domain Green’s function, G~11~(k~, k~,z11), is defined such that it is related to the corresponding Green’s function in the space domain, G~~1+(x,y, z11), as ~
=ff ~ ~Jf
k~,z1~)
G~1~~(x, y, z11)
=
y, z~1)~
G~~14(k~, k~,z~1)~
e_jkyY dx dy,
e~r~ dk~dk~.
(A.1)
(A.2)
(b) The superscript k of the Green’s function refers to a set of ~‘s(e’— je”), P~r’S and corresponding D’s identifying the topology of the particular multilayer structure. When a number of different multilayer structures are used, this superscript works as an identification to which multilayer structure the Green’s
N.K Das, D.M. Pozar
422
/
A generalized CAD modelfor printed antennas
function corresponds to. For example, in the general multilayer structure of fig. 3 the Green’s function refers to the following 8 vectors: E~l ~i2 =
:
:
=
‘
(A.3)
62N
E1N E~hl
~i2
E 22
=
:
‘
[e~’]’
:
=
EIN
l.Lii 1
P~21
IL12 =
(A.4)
2N
[i.t~]
.
/~22
1
=
IL1N1
.
,
(A.5)
/~2N
D11 1=
D21 D22 :
(A.6)
[D2] DIN 1
D2N2
By convention, a perfect electric at of the1NI N1 or or 2N N2 layer is identified by making EiNi or ~2N2 01N, or conductor instead equal to 0, and specifying 2 corresponding values of D and p. are not necessary and are denoted by * If otherwise there is a perfect magnetic conductor at the N1 or N2 layer, it is identified by making ~iN1 or 2N2 equal to — 1, and corresponding values of e”, p. and D are redundant, and are denoted by *~ On the other hand, if EN1 or 2N2 is not equal to 0 or — 1, then the corresponding layer (N1 or N2) is assumed to extend to infinity and the corresponding D (i.e., DlN~or is not needed. Hence, DiN1 and D2N2 are always unnecessary and are represented as * .
2. The following are the recommended representative short forms of various spectral-domain Green’s functions and current transform expressions for use in spectral-domain moment analyses: (a) O~1~~(z,1): the other parameters are k~and k~.
(A.7)
(b) O~11~(k~, kr): the other parameter z~ 0. 2GEIJ#] and similarly for H in place of E. (c) ~ = ~ +Y~OEIJ# + (d) ~ iji = [O~, j~~I + O * ~ 9] and similarly for M in place of J.
(A.8) (A.9)
=
(e)
~ijJ
=
[O~~I+ OL. 9]
(A.10) (A. 11)
(~)GE
11) = [I~ + 9~~IJJ+ zGE,JJ] and similarly with other combinations of (E, H) and (J, M).
(A. 12)
N.K. Das, D.M. Pozar
/
A generalized CAD model for printed antennas
(g) The scalar Green’s function with arbitrary field direction, E~,fLJJ
~,
423
and source direction, I~:
G~~JJ î~.
4= ~
(A.13)
.
(h) G~~: refers to any layer in general, and the parameters are (ks, k~,z).
(A.14)
(i) J~’J~,M~M~,L, M~:spectral-domain transforms of J~,J~,M~,M~,J~and M,, respectively, with k~and k~as parameters.
(A.15)
(i) O~#(z): refers to any layer and the other two parameters are
k~and k~.
(A.16)
(k) G~1~#(a,4)
4
=
G~,~ (a cos 4i, a sin
(I) O~11~(a, 4’, z~)= O~1~~(a cos
4’,
(m) O~ #(a,
4’,
z)
a sin
O~#(a,
4,)
=
(n)
=
O~,#(a cos 4’,
O~#(a
4,
0), where k~ a cos =
and k~= a sin
4,. Similarly,
a sin 4,, z~1).
4’,
(A.17) (A.18)
z).
(A.19)
cos 4,, a sin 4,, 0).
(A.20)
Appendix B. Systematic procedure for the computation of generalized Green’s functions Any component of the generalized Green’s functions at any plane and due to any source, electric or magnetic, x-, y-, or z-directed, can be obtained by following a four-step procedure. In the first step reflection coefficients at the source interface are computed using an iterative algorithm, and is independent of the type of source or field component of the Green’s function. The second step accounts for the particular source and computes the coefficients of excitation of TE and TM waves propagating above and below the source. The coefficients of TE and TM waves at the source interface computed in step two are transformed to any plane where the field is desired by use of another set of iterative procedure in step three. Finally, the Green’s function corresponding to any field component is obtained using a set of standard formulas relating different field components, and various excitation coefficients and the reflection coefficients obtained in earlier steps. Step 1 1A2l’ FF
The following equations and parameters are used to iteratively find TAII’ E~11, 2I: 2 = k~+ k~,where k~and k~are spectral domain variables, and are complex in general. (B.1) (a) e~’s,;ys a (b) and D 11’s are the real and imaginary parts of relative dielectric constants and thicknesses of the ijth layer, respectively. For a conducting layer on the top or bottom (~,‘N= 0), the respective conductivities, OIN~S are specified, instead of e~.(see fig. 3 and appendix A). 1~=;~—~~’3, jN_1 EjN,N/JE’~, iN1#O~ —1; tiN, = 1 —J(~N,/wEO), GIN. = 0. (B.2) 2, Im(f3,~) 0, where k (c) f3,~= i/k~e1~p.~~ —a 0 = free-space wave number. (B.3) (d) 1V = total number of layers in the ith side of the source, including a conductor, if present (see fig. 3). (B.4)
424
(e)
N.K. Das, D.M. Pozar ~
If
~
/
A generalized CAD modelfor printed antennas
1, FAiN = 0 = 1, (i) FAIN= —1, FFIN= 1, (see appendix A), (ii) TAiN,—1 = —exp(—2j/3~N_lD~N_l) (iii) = exp(—2jf3~~. ~zN
—
E’~,
(B.5)
1D1~1).
~
‘-‘j’
ZAij
—
_______
(Oojj
Fij —
‘
________
ZA--—Z~--+l (g) FAjJ = exp( — 2jI3,1D,~)~ + T Aij T
~
FFIJ
Aij+l
exp( — 2jp1~D1~)1, ~
I +
1F1f+YF~f+1 T l—FFIJ+I 17FiJ+1’ ~“Fij+1 = 11+1 r Fij±1
l—FA~J+1
where ZAIJ+I = 1 ~ rAij+1 ZAIJ+l, for j <1V, if Ei’N ~ — 1, or for
=
j
—
1.
(B.7)
Step 2 A. Electric source (a) Tm=J[(13 1 FAI 1A21)l, 11/11X1 +FA2IX 1)+(1321/E2lXl +FA11X1 = ~[((3i 1/~iiX1 — 1F11X1 + FF21) + (/321 + /.L21X1 — F~1X1+ FFII)]. (b) f21 = k~wp.0(1+ TFIl)/Tea, 2, fa21 k~(1—FAI1XI311/ell)/Tma 2, +FF2l)/7~a a11 =k~wp.~(1 11 = —k1(1 — FA21X/32l/E21)/Tma.
(B.9) (B.10) (B.11) (B.12)
B. Magnetic source on a ground plane 2(1+F~ii), (a) 7~1=a = a(1 + FF
(B.14)
(b)
Tmi Tm
=
2
=
f21 =
(B.8)
(B.13)
21), 2(1 — FAll), (1311/e11)a2(1 — FA
(B.15) (B.16)
(/321/21)a —jk~/7~2,
(B.17) (B.18) (B.19) (B.20) (B.21)
2I).
a21 =jk~we0/ Tm2,
f11 =~k~/T~ a11 =jkyWo/ Tmi. C. Magnetic source not on a ground plane FA2IX11 — + FAll) F~ + (f321/e21X1 + FA1IX1 — F~1)], = j[(13u/p.nXl — FF11X 2, 1)+ (/321/p.21X1 — FF21X1 + FFI1)1. (b) fa21 = —k~o0(1+FA11)/Tma 2, a21 = k~(1— F~11Xf311/p.11)/2’a 2, 11 = —k~w0(1+ FA21)/Tma fii = —k~(1— F~ 2. 21)(/321/p.2l)/7~a D. z-directed electric source (a) Tm = j[(/311/11X1
+
(B.22) (B.23) (B.24) (B.25) (B.26) (B.27)
(a) Tm = j[(/3 =
11/e11X1 j[(/311/p.11X1
+ F~1X1— —
F~11)+ (f321/21X1 + FA11)(1 FA21)l, FF11)(l + F~1)+ (/321/p.21X1 — FF21X1 + FF1I)l. —
(B.28) (B.29)
N.K Das, D.M. Pozar
(b) a11 a21
f11
/
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A generalized CAD modelforprinted antennas
(1 + FA21)/ETm, (1 + FAll)/ETm, =f21 = 0.
(B.30) (B.31)
= =
E. z-directed magnetic source FA21X1 — FAll) + (/321/21X1 + FA11X1 — F~1)], (B.32) = j[(131i/p.iiX1 — 7~, FF11X1 + FF21) + (j321/p.21)(1 — F~1X1+ FFI1)]. (B.33) (B.34) (b) f11 f = (1 + F~21)/p. 21 = (1 + F~11/p.l~, (B.35) a11 = a21 = 0. It should be noted that for z-directed sources the substrate topology is such that the layers 21 and 11 are identical with a dielectric constant e, and a magnetic constant p.. In other words, the source is buried inside a uniform medium. (a) Tm
=
j[(1311/11X1
+
Step 3 — 2a,~exp( —jI3IJD1J)ZA~J ajj~1—(1+FAIJ+l)(ZA~J+((1—FAIJ+l)/(1+FAIJ+l))ZAlJ+l)’
—
(1
(
2f~~ ((1 exp(—j/3I)D~J)YF~J FF~J+l)/(1+ FFlf+1))YF~J±1)
+ FFIJ+l)(YF~J+
—
B 36
) B 37 ) .
.
—
Step 4a In the following equations the source # is J~,M 1, J~or M~,and correspond to step 2A, (2B or 2C), 2D, or 2E, respectively. ka-xli I (a) GEIJ#(z~J)= (—j/31~X 1) (exp( —jf3,~z11) FAll exp~j/3,1z1~)) .
—
—
0) C0
—jk~f,1(exp(—j/3~~z11) + FF~Jexp(j$~1z,~)). (b)
OEIJ#(zlJ) =
(c) GE,J#(zIJ)
=
TAIJ °‘
WEo Eli
(B.38) exp(j/3,~z
a,3( —jf311X — 1Y(exp( —j/31~z1~) —
11))
+jk~f,~(exp( 2a —j31~z,~) + FFIJ exp(j/31~z,~)). [a 1~(exp(—j/311z,~) + FAll exp(j/311z1~))].
(B.39) (B.40)
.
JO) E~~
(For .J~source a term ~(z) should be added to the expression inside the (d) GH~J#(zlJ)=jk~a13(exp(—j/3,~z11) + FAll exp(jf3,~z11)) + —
(e) GHIJ#(zIf)
= —
°‘
f~(—jf311X—1)’(exp(—jf31~z~~)FFIJ —
exp(j/3,~z~1)).
(B.41)
exp(j/31~z11)).
(B.42)
2f,
1
—
FFIJ
jk~a1~(exp( —jf3,~z1~) + FAll exp(jf3,~z~1))
+
(f) G~,1~(z11) =
f~(—jf3~X--1Y(exp(—jf31~z11)
—
wp.op.j)
[1 bracket.)
.
jwp.0p.11
[a
1(exp(
—j/31~z1~) +
FFIJ exp(jf3,1z1~))].
(For M~source a term ~(z) should be added to the expression inside the
(B.43)
[1 bracket)
426
/ A generalized CAD
N.K Das, D.M. Pozar
model forprinted antennas
Step 4b For Green’s function components with y-directed sources see appendix C.
Step 4c Corresponding to a particular source, #, the Green’s functions with field-directions arbitrarily oriented on the x—y-plane, say in x ‘-direction at an angle 0 with the x-axis, can be expressed as =
GH,jj# =
GE
j~
cos 0
+
GE
0
+
G5~11~ sin 0
GHIJ# C05
~
sin 0 =i’ ~
~
(B.44)
~
(B.45)
.
Appendix C. Look-up table for obtaining a Green’s function with a y-directed source from another Green’s function with a x-directed source In appendix B Green’s function components with x-directed sources (~x or Mi), have been derived. By simple coordinate transformation and using the “quadrant symmetry” of the Green’s functions, different Green’s function components with y-directed sources (J~or M~)can be related to a Green’s function component with a x-directed source. These results have been tabulated in table 1. A bold arrow implies that the Green’s function component, which the arrow points at, is obtained from the Green’s function component at the tail of the
Table 1 Table to find a Green’s function component with a y-directed source in terms of another Green’s function component with a x-directed source
Source
Field
E~
M~
Ey
E~
H~
Hy
GExJx
GEJ
°EzJx
GHJ
GHJ
GEM
GEM
GEM
GHxMx
GHM
H~
GHzJx
GHM
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Table 2 Table of quadrant symmetry of various Green’s function components
Source
J x
Field E~
Ey
±~ -:i:-Fi:
~± +1—
~J±
~± +1—
TANGENT
COSINE
TANGENT
UNIFORM
~
TANGENT
UNIFORM
-J±
~±
1~
TANGENT
UNIFORM
z
±~ +1+ UPW’ORM
±~ —I— SINE
.::I± —1+
—I—
+1+
+1—
SINE
UNIFORM
TANGENT
COSIE
=1± +1— TANGENT
~± —1+ COS1~
~j±
±~
~F SiNE
u,’,o~,i
+1+
~F
~l± ~
.Lt +1—
+1+
—I—
UNIFORM
TANGENT
COSINE
TANGENT
UNEORM
SEE
—1+
+k-
+1+
±[±
~1± ~W
COSINE
SINE
UNIFORM
SINE
COSINE
SINE
COSINE
COSINE
SEE
~
Jz
H~
~i±
+3±
~W
M
Hy
~±I±~ ±1±. ±1±. -=1±
~—1+
MX
H~
~F
±1±. ±1±.
~F
—
ISIFORM
arrow by interchanging kx and k~and adding a negative sign before it, whereas a thin arrow implies the same thing as a doubled-lined arrow except that it does not add a negative sign before it. For example, k~)=
~
GE~,J0(kX,k~)=
~GEYMr(kY~
ks),
(C.1)
GEJ(kY, ks).
(C.2)
Corresponding to a particular field component, *, the Green’s functions with source directions arbitrarily oriented on the x—y-plane, say in x ‘-direction at an angle 0 with the x-axis, can now be expressed, for example, as =
G~~11 cos 0
+
~
sin 0
=
G~112 1’.
(C.3)
Appendix D. Quadrant symmetry of different Green’s function components The symmetries of Green’s function components in different quadrants on the kx—ky-plane are tabulated in table 2. The symmetry of a function f(kx, k~)is described as follows: uniform:f(kx, k~)=f(—k~,ky)=f(~kx, —k~)——f(k~, —kr); sine:
f(kx, k~)=f(—k~,k~)= —f(-—k~,—kr)
=
—f(k~,—kr);
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cosine: f(kx, k~)= —f(—k~, k~)= ~f(~kx, tangent:f(k~, k~)=~f(~kx,
modelfor printed antennas
—k),) =f(k~, —kr);
k,) =f(—k~,—k~)=~(kx,
—ky).
The + and — signs in the different quadrants are used to describe the relative signs of the Green’s function components in the different quadrants of the k~—k~-plane.
Appendix E. Patterns of TE and TM excitations for Green’s function components The patterns of excitation of various field and potential components due to various source configurations as a function of the angle, 4,, with respect to the x-direction are tabulated in table 3. In table 3 the blank spaces imply that the respective Green’s function components are either not excited or not applicable for the specific source. For example, the relative magnitudes of excitation of the E~field component due to an x-directed magnetic source varies as sin 4’. This is only due to the contribution from the TM part of the excited wave; the Er-component can never be produced by a TE to z-wave. Such patterns are applicable to the relative magnitudes of excitation of the TE or TM source-free modes, as well as the radiation patterns of, respectively, E,1, and E0 for a given value of 6.
Appendix F. Asymptotic orders of various Green’s function components The asymptotic orders of various Green’s function components, in terms of the powers of a, are given in table 4 for three different limiting cases. In one case kx approaches infinity which k~remaining finite, and in the other case k~approaches infinity with kx finite, whereas in the third case kx and k~ simultaneously approach infinity. The separate asymptotic contributions from TE and TM parts are also given. The real or imaginary nature of the asymptotic functions of table 4 are given in table 5, and are independent of the particular direction along which one approaches the limit of large a on the k~—k~ spectral plane.
Appendix G. Basis functions for transverse variation of source distribution used in generalized transmission-line analysis G. 1. Single conductor or slot case In the case of a transmission line with one conductor or slot, the electric or equivalent magnetic surface current along the direction of propagation can be expanded in the transverse direction using the following basis functions (see fig. 21): f~m(Y)e(Y1v)c05
kmY,
m=1,2,3,...,
(G.la)
with km
=
2(m
—
1)’rr/w,
(G.lb)
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Table 3 Table of patterns of TE and TM excitations for Green’s function components Source J~, TE TM TE TM J~ TE TM Mr TE TM M~TE TM M~TE TM
Field a
f
—
sin 4,
cos 4,
—
E~ 24, sin cos24,
E~ sin 24, sin 24, cos2çb sin24,
E~
Hr
—
sin 24,
H~ sin 4,
sin 24,
H~ sin24, cos24,
cos 4,
cos24, sin24,
sin 24, sin 24,
cos 4,
sin 4,
—
—
cos 4,
sin 4,
—
sin 24, sin 24,
-
-
-
-
-
-
-
-
1
—
cos 4,
sin 4,
1
sin 4,
cos 4,
—
sin 24, sin 24,
cos24, sin24,
cos24, sin24,
sin 24, sin 24,
cos 4,
sin 4,
sin 24, sin 24,
sin24, cos24,
sin 4,
cos 4,
—
—
—
cos 4,
sin 4,
—
—
sin 4,
cos 4,
—
sin2cb cos24,
sin 24, sin 24,
—
I
sin 4,
cos 4,
—
cos 4,
sin 4,
1
-
-
-
-
-
-
-
-
—
—
—
—
Table 4 Table of asymptotic orders of various Green’s function components Source J~ K~ TM TE K,, TM TE a TM TE
Field Er
E~
1 —3
0 —2
E~ 1 —
Hr
0
—
—1 —1 —1 0 0
—2 —2 0 0
—1
0
0
—1
—
0
0
—1
—2
—
1 —1
1
1
—1
—
—1 —1 1
—
—2 0
1
0
—2 1 —1
—3
—
1
1
—1
MrKrTM TE K,, TM TE a TM TE
—1
—2
—1
—1 —1 0 0
0 0 —2 0 0
MyKr
0 —2 —2 0 0 0
—1 —1 —1 —1 0 0
K~ TM TE K,, TM TE a TM TE
TM TE K~ TM TE a TM TE
0
—2 0
0
—
—2 0 0
—1 —1 —1 0 0
—1
—3
—2
—
1
0
0
—1
—2
—
—1 —1 1
—1 1
0
—2
—1
—
0
—1
—2 0 —1
—3 1 —1
1
1
—
0
—1 —
0 —
H~
—1
—1
J,,
H,,
0
—
0 —
0 —
—1 —
0 —
-
1 —
0 —
1 —
0 —
1 —
1
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Table 5 Table of asymptotic nature (real or imaginary) of various Green’s function components Source
Field
TM TE J~ TM TE ~r
Er
E,,
E~
j —j j j
j j j —j
—1
—1 -
1
-j
TE
—1
1
-
M,,TM TE
-1
-1 1
—1
H,,
1
—1 —1 —1 I
-1 1 1
-
1
MXTM
Hr
—j j i j
i -
Hr —
—j —
j j
-
-1
-j
-
j
—1
where
e(y; w)
=
1/w,
without edge condition,
I 2 2 1/’rry (w/2) y , with edge condition, y ~ w/2, and = 0, elsewhere. =
for
I
—
(G.lc)
The Fourier transform of these basis functions as defined by (G.2a)
~m(ky;1~)ffsm(y;H)exp(jkyy)dY,
fsm(Y
al
w)=
(G.2b)
~fFsm(ky;w)exp(ikyY)dky,
Dominant modes, no edge Condition
IbI
2.0
Dominant modes, with edge condition
5.0 —ru~41 mr2 — -n,4,3
—m=1 — —
/
00
,//
-~
~
/
‘
/
f
-~
, /
I
/~.
—2.0 —0.5
00
-m3 m 4
—5.0 0.0 y/W
0.5
—0.5
0.0
0.5
y/w
Fig. 21. Transverse variation of dominant electric or equivalent magnetic-current modes on single strip/slot transmission line (a) without, (b) with edge condition.
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can be expressed as F~,,,(k,,;w)
=
~(Fe(ky
—
km) +Fe(k~+ km)),
(G.3a)
where Fe(ky)
=
sin(k~w/2)/(k~w/2)= 1.0,
k~—* 0,
Ik~I—s-oc,
a/k~,
(G.3b)
for the case without any edge condition, and Fe(ky)
=J0(k~w/2)= 1,
k~-+0,
1/v1ii~T,
Ik~I
(G.3c)
—~,
for the case with an edge condition. In the above expressions the width of the source, w, is the parameter, and y is the transverse coordinate on the transmission line. On the other hand, for the above case of a single-source transmission line, the surface currents in the transverse direction can be expanded using the following basis functions (see fig. 22):
f~(y;w)=sinkmy, IyI
(G.4a)
with
2ii~m km=
,
m=1,2,3,...,
(G.4b)
and its Fourier transform set can be expressed as: F~,(k,,;w)
=
(1/2j)(F~(k~ km) —F~(k~ + km)),
(G.5a)
—
2.11
Orthogonal modes
m~2
y/w
Fig. 22. Transverse variation of orthogonal electric- or equivalent magnetic-current modes on a single strip/slot transmission line.
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A generalized CAD model for printed antennas
where F~(k~) = sin(k~w/2)/(k~w/2).
(G.5b)
It can be noted that the current components along the direction of the transmission line are the dominant components, and are even with respect to the center of the source, whereas the current components in the orthogonal direction are not dominant, and are excited only in the cases of non-TEM structures, with odd distribution with respect to the center of the source. Also, the former has an edge singularity at y = ±w/2of order 1/2, whereas the later goes to zero at the two edges. —
G. 2. Double conductors or slots on the same plane When there are two conductors or slots on the same plane of the multilayer structure, with center to center distance, d, as in the cases of edge coupled microstrip lines or co-planar waveguides, they can be treated as a single source using the odd or even symmetry of the currents with respect to the center point between two lines. In these cases, for the dominant current components, the basis functions are (see fig. 23a, b): fxoe( y; w, d) =f~m(y d/2) ±(— 1)mf~m(y + d/2), (G.6a) —
where
m=1,3,5,..., or
f~’me(Y)~0~mY,
=e(y;w)sinkmy, km=’TT(m—l)/w,
(G.6b) (G.6c)
m=2,4,6,...,and
(G.6d)
-
with their Fourier transforms are given as
w, d) =F,~(k,,;w)[exp(_Jk,,d/2) ±(_1)m exp(jk~d/2)],
(G.7a)
where F~m(ky)= ~(Fe(ky =
—
km)
(1/2j)(Fe(ky
—
+ Fe(ky +
km)
—
km)),
F~(k~ + km)),
m = 1, 3, 5,...,
(G.7b)
m = 2, 4, 6
(G.7c)
On the other hand, the basis functions for the orthogonal components are given as (see fig. 23c) mfc~m(Y+d/2), (G.8a) fr~e(y; w, d) =fg~(y—d/2) ±(1) where f~m5~kmY,
m=1,3,5,...,
=coskmy,
m=2,4,6,...,
(G.8b) (G.8c)
with ‘Tr(m + 1) km= =
w ~(m—1)
,
m=1,3,5,...,
(G.8d)
m=2,4,6,...,
(G.8e)
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Dominant modes, no edge condition
Dominant modes, with edge condition
Ib)
2.0
433
5.0 —rrl=t
_~
rn—S
~
0.0
,~// /
rn2
~\~\\~ .
~
-
E 0.0 .
.
-~ ~.
:‘
\\\~
—2.0
—5.0
—0.5
0.0
0.5
—0.5
0.0
y/w
0.5
y/w Orthogonal modes
CI
2.0 —
0.0
=1 rnZ
~
-2.0 —0.5
0.0
0.5
y/w Fig. 23. Current distribution functions associated with double- or multi-conductor/slot lines.
and the corresponding Fourier transform expressions are F~~e(k~ w, d) =1~’m(ky;w, d)[exp(_jk~d/2) ±(_1)m exp(jk~d/2)],
(G.9a)
where F,j~~(ky) = (1/2j)(F~(k~ km) —
=
~(F~(k~
—
km)
—
F~(k~ + km)),
+ F~(k~ + km)),
m
=
1, 3, 5,...,
(G.9b)
m
=
2, 4, 6
(G.9c)
In these expressions “o” and “e” respectively refer to coupled sources with odd and even symmetry with respect to each other, and correspond respectively to the top and bottom signs of various double signs appearing in the expressions.
434
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G. 3. Multiple conductors and / or slots This is the most general case of which cases 1 and 2 are derived. In this case each conductor or slot is expanded with a set of basis functions with respect to the center of the conductor or slot. The basis functions for the dominant components are f,~(y;w) =f~m(y;w),
(G.10)
with its Fourier transform F,~(k,,;w) =F~m(ky;w).
(G.11)
And similarly, the basis functions for the orthogonal components are f,~(y;w) =fa~m(y;w),
(G.12)
and its Fourier transforms are F,~(k~: w) =F,~m(ky:w).
(G.13)
Appendix H. Basis functions used for expansion of various unknown currents of a generalized multilayered printed antenna configuration Various unknown electric currents, or equivalent magnetic currents (or slot fields) of a multilayered configuration of printed antennas are expanded using three major types of basis functions, i.e. piece-wise sinusoids (PWS), entire basis sinusoids (EBS) and transmission-line discontinuity modes. The first two types of modes are useful for expansion of currents on finite-length radiating structures, such as patches, dipoles or slots, whereas the third type of basis functions are useful for taking into account reflected and transmitted currents on a transmission feed line at various discontinuities. Unlike eigenmodes of an infinite transmission line, these transmission-line discontinuity modes extend to infinity only in one direction [24], and can be referred to as semi-infinite modes. In the first part of this appendix these basis functions are expressed with respect to a coordinate system at the center point of the mode. By convention, the x-axis of this coordinate system, referred to as the source coordinate system, coincides with the direction of current flow of the mode, with the y-axis as the transverse direction. A number of such modes with suitable orientation and offset with respect to the system coordinate system can define a complete system. When using them in a spectral-domain moment method, any orientations and/or offset with respect to the system coordinates can be easily taken care of by suitable transformation of the spectral plane, and are separately discussed in the second part of this appendix. Also, for consistency of notations, and to distinguish from the basis functions of appendix G used for expansion of transverse variation of transmission line currents, the basis functions for finite structures will be denoted as u( )‘s with the corresponding Fourier transforms as U( )‘s, with suitable subscripts, superscripts and indices to specifically denote their identity. The basis functions can be expressed as a product of two separate functions, one for the x-dependence and the other for the y-dependence, where x and y refer to the source coordinate system, and correspond to the directions along or transverse to the direction of the current flow, respectively. The
]
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two-dimensional Fourier transform of a basis function is thus the product of two separate one-dimensional transforms in kx and k,,, defined, for, say, k~,as U
1(k~)= fui(x)
u1(x)
=
exp(—jkxx) dx,
(H.la)
U1(k~)exp(jkxx) dk~.
(H.lb)
~—J
Different types of basis functions will be separately discussed. Their values at various critical points, as well as their asymptotic orders for large spectral arguments will be given in each case, and are useful for various convergence considerations while performing spectral integrals. H. 1. Basis functions in source coordinate system H. 1.1. Finite-length modes
(a) For x-variation: • Piece-wise sinusoid (PWS) modes (full): -
u’pw(x;
L, k~,N)
=
sin[k (z~L/2— I(x—~-)I)j e sin(k~~L/2) ,
=0, elsewhere, where z~L= 2L/(N + 1), zJ~, = —L/2
(z~L/2)i;
(H.2)
2ke[cos(ke z~L/2)—cos(k~z~L/2)] -
U~,w(kx;L, ke)
+
x—411
(k~—k~)Slfl(ke ~1~/2)
=
exp(—jk~~1)
=(~L/2) exp(—jkx~ij), k~— ±ke (H.3) where a is a real number, positive or negative. Full PWS modes are always considered as a group of N subdomain modes over a region of length L, all with respect to the same source coordinate system, and are indexed by i = 1,. , N (see fig. 24). • Piece-wise sinusoid (PWS) mode (half): Unlike the full PWS modes, these half PWS modes are used individually, and are discontinuous at one end. Such discontinuous half modes are sometimes useful to satisfy current continuity in a mixed-mode environment with travelling-wave and finite-length modes [24]. 4w(X’ 2L, ke, 1), x<0, uHP(x; L, ke) 1 =0, x>0. (H.4) k [cos k L cos k L] j [ k sin k L k sin k L UHP(k.Lk)=e e + x (kx2k~) e x e (k~—k~) sin keLx sine keL x . .
—
L 2
.
—
[i’~eL cos keL
sin keL] 2ke sin keL —
k-+-i-k X
—
kx-+OO,
where a is an arbitrary real number, positive or negative.
e’
(H.5)
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436
5.0
PWS modes
2.0 —
—In
InS
—-1=3
A /
...
InN
-—
\,/
...
2.5
/
0.0 —0.5
—2.0 —0.5
0.0
0.0
0.5
0.5
x/L
Fig. 24. Layout of PWS modes over a length, L.
Fig. 25. Uniform and edge-condition modes used for transverse variation offinite-length current modes.
• Entire-basis sinusoids (EBS):
u~x(x)=Lfj,±1(x;L), =Lf~1(x; L),
i= 1,3,5,...,
(H.6a)
i=2,4,6,...,
(H.6b)
U~~(k~) =LF~±1(k~ L)
a/k~,
(H.7)
k~—*oc,
where f,~is the same as defined in appendix G for transmission-line transverse variation. (b) For y-variation • Uniform mode (see fig. 25):
u~(y;w)=1/w, IyIw/2, =0, U~(k~ w)
=
elsewhere;
(H.8)
sin(k,,w/2)/(k~w/2)= 1,
k~—+0,
a ~,
Ik~I—~.
(H.9)
• Edge-condition mode (see fig. 25): u~(y:w)
=
1/~V(w/2)2_y2,
I~Iw/2,
=
0,
elsewhere;
Ue(ky; w) =J0(k,,w/2)
=
1,
k,,
—~
Ik,,I
(H.10)
0, -+~.
(H.11)
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437
• Transverse entire-basis modes (EBS): u~y(y; w)
=f~(y;w),
(H.12)
U~y(k~W)=F~,(k~w)a/k~,
i=1, k~—+-oo,
a/k~,
i#1, k~—*co.
(H.13)
H. 1.2. Transmission-line discontinuity modes
(a) For x-variation • Forward-discontinuity mode: ufd(x, ke) =exp(—jkex),
—exp(—jkex),
=
Ufd(kX; ke)
x>0, x<0;
(H.14)
~2J/(kx+ke)
=
a/kx,
-
IkxI-+~
(H.15)
It has a pole at k~= —k~. • Backward-discontinuity mode: e~~P(jl’~ex), x>0,
ubd(x, ke)
=—exp(jkex), Ubd(kX; ke)
=
x<0;
(H.16)
—2j/(kx—k~)
a/kx,
IkxI-+oc~
(H.17)
It has a pole at k~= ke. For numerical efficiency considerations, these forward- and backward-discontinuity modes can be expressed as a combination of two other types of modes called as “even” and “odd” discontinuity modes: uOd(x,
ke)
=
~[ufd(x; ke)
+
(x; ke)1
= =
L’~d(kx;ke)
=
cos k~x, C05 keX,
x>0, x<0;
_2jkx/(k~ k~);
(H.19)
—
ued(x;ke)=(1/2j)[ufd(x;ke)ubd(x;kd)]—sinkex, =
~d(kx;
(H.18)
ke) =2ke/(k~~k~).
—sin
x>0, kex,
x<0;
(H.20) (H.21)
These even and odd functions are real functions, and so have even and odd symmetries, respectively, in the spectral plane. Such symmetries are useful in order to simplify various spectral-domain integrals used for moment-method solution that also exploit similar symmetries of Green’s function components as described in appendix D.
N.K. Das, D.M. Pozar / A generalized CAD model for printed antennas
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Forward- and backward-discontinuity modes are described as linear combinations of the even and odd functions: ufd(x, ke) ubd(x,
ke)
Ufd(x,
k~)=
Ubd(x;
ke)
1~od(~ ke)
+ju~~(x; ke),
(H.22a)
ke)
ju~d(~ ke),
(H.22b)
= uOd(x,
=
Uod(kx; ke) +jU~~(k~ ke),
(H.23a)
UOd(kX; ke) —jUed(kx; ke).
(H.23b)
At any discontinuity along an infinite transmissions line, the travelling-wave currents due to scattering at the discontinuity can always be represented by superposition of the discontinuity modes and infinite transmission line eigenmodes. It is important to note that these discontinuity mode transforms, unlike other finite-mode transforms, have poles at kx = ±ke.These poles would always create problems while performing various numerical integrations. Also, these poles, unlike the poles of Green’s functions, do not have any physical significance of exciting any additional modes of the structure. These poles are purely mathematical, and arise due to the choice of the expansion functions used, and so also the residue about these poles do not have any physical meaning. In fact, instead of having these modes of infinite extend, if we truncate them at very large lengths, the corresponding transform will result in smoothening these poles. For numerical convenience, it is suggested that instead of using the discontinuity-mode transforms the way presented before, they should be used, for example, as UOd(kX; ke, a) =Im UOd(kX; k~—ja),
(H.24)
where the superscript “s” stands for the respective smoother versions, and a is an arbitrary positive number. a is equivalent to making the transmission line slightly lossy, and should carefully be chosen so as to properly take care of the numerical problems due to the pole, while not severely affecting the real physical situation.
(b) For y-variation The y-variation of transmission-line discontinuity modes are chosen to be the same as that of the eigensolution of the respective infinite transmission line, f(y). Thus f(y) can be expressed as a sum of basis functions from appendix G with known coefficients. Strictly speaking, the transverse variation of current on a transmission line at the discontinuity plane is not the same as that of the corresponding infinite transmission line. Hence, even though sufficiently larger number of transverse modes might be required for the solution of the infinite transmission line, in order to model the traveling-wave currents at the discontinuity one can express the transverse variation of the discontinuity mode using a fewer number of modes, without significantly affecting the result. However, the propagation constant and the characteristic impedance of the transmission-line discontinuity mode remain the same as that of the infinite transmission line and are rigorously obtained with sufficient number of expansion modes. H.2. Spectral-domain transformation of basis functions for different orientations and offsets While computing various coupling terms for the spectral-domain moment-method solution of a general antenna geometry, the Fourier transforms of the basis functions (expansion or test modes) need to be expressed with respect to the system coordinates. The necessary transformations on the Fourier transform expressions are exp(—jk~z~~), (H.25) Ut(k~,k~)= U(k~,k~)exp(—jk~.~1~)
N.K. Das, D.M. Pozar / A generalized CAD modelfor printed antennas
439
where U( , ) is the Fourier transform in the source coordinate system, Ut(kx, k~)is- the transformed Fourier transform in system coordinates, ‘tx and 4~,respectively are the x and y offsets of the center of the source mode with respect to the system coordinates, k~= kx cos O~+ k~sin O~
(H.26a)
kt,,
(H.26b)
=
—k~sin O~+ k~cos O~,
and O~is the angle of the x-axis of the source coordinates with respect to the system x-axis. For notational convenience the superscript “t” of the Fourier transform, U, is dropped while using in computing various reactions, and should be understood to include proper transformations.
References [1] R.J. Mailloux, Phased array architecture for millimeter wave active arrays, IEEE Antennas Propag. Soc. Newslett. 28 (February 1986) 5. [2] J.A. Kinzel, GaAs technology for millimeter wave phased arrays, IEEE Antennas Propag. Soc. Newslett. 29 (February 1987) 12.
[3] D.M. Pozar and D.H. Schaubert, Comparision of architectures for monolithic phased array antennas, Microwave J. (March 1986) 93. [4] D.M. Pozar, Five model feeding techniques for microstrip antennas, in: IEEE Antennas and Propagation Society Symposium Digest, Virginia, 1987, p. 251. [51AC. Buck and D.M. Pozar, Aperture coupled microstrip antenna with a perpendicular feed, Electron. Lett. 22 (1986) 125. [61S.A. Long and M.D. Walton, A dual frequency stacked circular disc antenna, IEEE Trans. Antennas Propag. (1979) 270. [7] K. Araki, H. Ueda and M. Takahashi, Hankel transform domain analysis of complex resonant frequencies of double-tuned circular disc microstrip resonators/radiators, Electron. Lett. 21(1985) 277. [8] C.H. Chen, A. Tulinteseff and R.M. Sorbello, Broadband two-layer microstrip antenna, in: IEEE Antennas and Propagation Society Symposium Digest, Boston, 1984, p. 251. [9] F. Croq, A. Papiernik and P. Brachat, Wideband aperture coupled microstrip subarray, in: IEEE Antennas and Propagation Society Symposium Digest, Dallas, 1990, p. 1128. [10] E.G. Magill and H.A. Wheeler, Wide angle impedance matching of a planar array antenna by a dielectric sheet, IEEE Trans. Antennas Propag. 14 (1966) 49. [11] J.B. Knorr, Slot-line transitions, IEEE Trans. Microwave Theory Tech. 22 (1974) 548. [12] J.R. Mosig and F.E. Gardiol, General integral equation formulation for microstrip antennas and scatterers, lEE Proc. Pt. H 132 (1985) 424.
[13] M.D. Deshpande and M.C. Bailey, Input impedance of microstrip antennas, IEEE Trans. Antennas Propag. 30 (1982) 645. [14] D.M. Pozart, Input impedance and mutual coupling of rectangular microstrip antennas, IEEE Trans. Antennas Propag. 30 (1982) 1191.
[15] J.T. Aberle and D.M. Pozar, An improved probe feed model for printed antennas and arrays, in: IEEE Antennas and Propagation Symposium Digest, Syracuse, 1988, pp. 438—441. [16] D.M. Pozar and S.M. Voda, A rigorous analysis of a microstripline fed patch antenna, IEEE Trans. Antennas Propag. 35 (1987) 1343. [17] D.M. Pozar and D.H. Schaubert, Scan blindness in infinite phased arrays of printed dipoles, IEEE Trans. Antennas Propag. 32 (1984) 602.
[181 C.C. Liu, A. Hessel and J. Shmoys, Performance of probe-fed microstrip-patch element phased arrays, IEEE Trans. Antennas Propag. 36 (1988) 1501. [191 J.T. Aberle and D.M. Pozar, Analysis of infinite arrays of probe-fed rectangular microstrip patches using a rigorous feed model, Proc. lEE Pt. H 136 (1989) 110. [201N.G. Alexopoulos and D.R. Jackson, Fundamental superstrate (cover) effects on printed circuit antennas, IEEE Trans. Antennas Propag. 32 (1984) 807. [211J. Castaneda and N.G. Alexopoulos, Infinite arrays of microstrip dipoles with a superstrate (cover) layer, in: IEEE Antennas and Propagation Symposium Digest, 1985, p. 713. [22] PB. Katehi and N.G. Alexopoulos, On the modeling of electromagnetically coupled microstrip antennas — the printed strip dipole, IEEE Trans. Antennas Propag. 32 (1984) 1179.
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[23] M. Gaouyer, J.M. Floch and J. Citerne, Radome effects on an electromagneticallt coupled dipole, in: IEEE Antennas and Propagation Symposium Digest, 1989, pp. 612—615. [24] N.K. Das and D.M. Pozar, Analysis and design of series-fed arrays of printed-dipoles proximity-coupled to a perpendicular microstripline, IEEE Trans. Antennas Propag. 37 (1989) 435. [25] J. Herd, Scanning impedance of electromagnetically coupled rectangular microstrip patch arrays, in: IEEE Antennas and Propagation Symposium Digest, 1989, p. 1150. [26] D.M. Pozar, A reciprocity method of analysis for printed slot and slot-coupled microstrip antennas, IEEE Trans. Antennas Propag. 34 (1986) 1439. [27] D.M. Pozar, Analysis of an infinite array of aperture coupled microstrip patches, IEEE Trans. Antennas Propag. 37 (1989) 418. [28] R.Q. Lee and K.F. Lee, Effects of parasitic patch sizes on multi-layer electromagnetically coupled patch antenna, in: IEEE Antennas and Propagation Symposium Digest, 1989, p. 624. [29] C.H. Tsao, Y.M. Hwang, F. Kilburg and F. Dietrich, Aperture coupled patch antennas with wide-bandwidth and dual polarization capabilities, in: IEEE Antennas and Propagation Symposium Digest, 1988, p. 936. [30] J.F. Zurcher, The SSFIP: a global concept for high-performance broadband planar antennas, Electron. Lett. 24 (1988) 1433. [31] N.K. Das and D.M. Pozar, A generalized spectral-domain Green’s function for multilayer dielectric substrates with applications to multilayer transmissionlines, IEEE Trans. Microwave Theory Tech. 35 (1987) 326. [32] M.C. Bailey and M.D. Deshpande, Integral equation formulation of microstrip antennas, IEEE Trans. Antennas Propag. 30 (1982) 651. [33] N.K. Das and D.M. Pozar, Full-wave spectral-domain computation of material, radiation and guided wave losses in infinite multilayered printed transmission lines, IEEE Trans. Microwave Theory Tech. 39 (1991) 54. [34] N.K. Das and D.M. Pozar, UNIfied Computer Aided Analysis of Multilayer Transmissionline, UNICAAMT, a computer code for generalized planar transmissionline analysis, in: Proc. Int. Symp. Electronic Circuits, Devices and Systems, ISELDECS ‘87, vol. I, Kharagpur, India, December 1987, p. 85. [35] N.K. Das and D.M. Pozar, PCAAMT, Personal Computer Aided Analysis of Multilayer Transmission Lines, Version 1.0, User’s Manual (June 1990). [36] T. Itoh, Spectral domain immittance approach for dispersion characteristics of generalized printed transmission lines, IEEE Trans. Microwave Theory Tech. 28 (1980) 733. [37] A. Farrar and A.T. Adams, Multilayer microstrip transmission lines, IEEE Trans. Microwave Theory Tech. 22 (1974) 889. [38] E.J. Denlinger, A frequency dependent solution for microstrip transmission lines, IEEE Trans. Microwave Theory Tech. 19 (1971) 30. [391J.B. Davies and D. Mirshekar-Syahkal, Spectral domain solution of arbitrary coplanar transmission line with multilayer substrates, IEEE Trans. Microwave Theory Tech. 25 (1977) 143. [40] H.G. OIlman and D.A. Huebner, Electromagnetically coupled microstrip dipoles, IEEE Trans. Antenas Propag. 29(1981)151. [411 N.K. Das and D.M. Pozar, Infinite array analysis of an aperture coupled stacked microstrip antenna: considerations for bandwidth enhancement, in: Proc. Symp. Progress in Electromagnetic Research, MIT, Cambridge, July 1989, p. 510. [42] N.K. Das, J.T. Aberle and D.M. Pozar, Scan and frequency performance of circularly polarized microstrip antenna arrays, Proc. Symp. Progress in Electromagnetic Research MIT, Cambridge, July 1989, p. 512. [43] N.K. Das and D.M. Pozar, Printed antennas in multiple layers: general considerations and infinite array analysis using a unified method, in: Proc. lEE Int. Conf. Antennas and Propagation, ICAP, Univ. Warwick, UK, Part 1, April 1989, p. 364. [44] N.K. Das, A study of multilayered printed antenna structures, Ph.D. Dissertation, Dept. Electrical and Computer Engineering, Univ. Massachusetts, Amherst, September 1989 (Microfilm International, Michigan, 1989). [45] D.M. Pozar, Radiation and scattering from a microstrip patch on a uniaxial substrate, IEEE Trans. Antennas Propag. 35 (1987) 613. [46] P.L. Sullivan and D.H. Schaubert, Analysis of an aperture coupled microstrip antenna, IEEE Trans. Antennas Propag. 34 (1986) 977.