TRAN SACTI 0 NS ON
`
`NTENNASAND
`BOPAGATION
`
`A PUBLICATION OF THE IEEE ANnNNAS AND PROPAGATION SOCIETY
`
`APRIL 1998
`
`VOLUME 46
`
`NUMBER 4
`
`IETPAK
`
`(ISSN 0018-926X)
`
`Pi\PI RS
`
`Phy..,ical Interpretation of the Phase Asymmetry of a Slant-Path Transmission Matrix ................................... .
`. . ..... ...................... ............. ....... ......... N. J. McEwan, Z. A. A. Rashid, and S. M. R. Jones
`A Wide-Band Single-Layer Patch Antenna ................................................................... N. Herscovici
`Multiport etwork Model for CAD of Electromagnetically Coupled Microstrip Patch Antennas ........................ .
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. P. P(lrrikar and K. C. Gupta
`Simultaneous Time- and Frequency-Domain Extrapolation ............... . ......... . . ........ R. S. Adve and T. K. Sarkar
`Scattering from Structures Formed by Resonant Elements ............... ...... ... .............. V. Veremey and R. Mittra
`Normalization and Interpretation of Radar Images ......................................................................... .
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. P. Skinner. B. M. Ke111. R. C. Wittmann, D. L. M ensa, and D. J. Andersh
`Perfectly Matched Layer Mesh Terminations for Nodal-Based Finite-Element Methods in Electromagnetic Scattering ...
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Tang. K. D. P(lulsen, and S. A. Haider
`On the Behavior of the Sierpinski Multiband Fractal Antenna .............. .......... ................... ................. .
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Puente-8(1/iarda. J. Romeu. R. Pow;. and A. Cardama
`On \todeling and Personal Dosimetry of Cellular Telephone Helical Antennas with the FDTD Code .................. .
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. La-::,:,i and 0. P. Gandhi
`Study of Impedance and Radiation Properties of a Concentric Microstrip Triangular-Ring Antenna and Its Modeling
`Techniques Using FDTD Method ........ . ................................ ...... .. . .... /. S. Mism and S. K. Chowdhury
`Analysis of Stripline-Fed S lot-Coupled Patch Antennas with Vias for Parallel-Plate Mode Suppression ..... . .......... .
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Bhattacharyya, 0. Fordham, and Y. Liu
`Blindness Removal in A1Tays of Rectangular Waveguides Using Diclcctrically Loaded Hard Walls ....... ............ . .
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. P. Skobelel' and P.-S. Kildal
`Near-Field Scanering by Physical Theory of Diffraction and Shooting and Bouncing Rays ................... S.-K. Jeng
`~ cthod of Moments Solution for a Wire Attached to an Arbitrary Faceted Surface ......... I. Tekin and E. H. Newman
`Fon,ard-Scaneriog Analysis in a Focused-Beam System ....... . ....................... R. Slravit, T Wells, and A. Collen
`High-Frequency Analysis of an Array of Line Sources on a Truncated Ground Plane ........................ .. ......... .
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Capolino, M. Albani. S. Maci. and R. Tiberio
`A l f D Solution for the Scarte ring by a Wedge with Anisotropic Impedance Faces: Skew Incidence Case .... . .... . ... .
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Pelosi. G. Manara, and P. Nepa
`On lhe Use of Cavity Modes as Basis Functions in the Full-Wave Analysis of Printed Antennas ....................... .
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Vecchi, P. Pirino!i, and M. Orefice
`
`1.t:.1 fERS
`
`ear-Field Formulas in the Case Where the Radial Components of the Electromagnetic Field
`Time-Domain Spherical
`Are Measured ............................................................................................. I. Christiansen
`A Compact PIFA Suitable for Dual-Frequency 90011800-MHz Operation ........ . ....... C. R. Rowell and R. D. Murch
`Scauering by a Dielectric-Loaded Nonplanar Slit- TM Case .............. .. ..... . ........... J. -W. Yu and N.-H. Myung
`Pan rnl Control for Wide-Band Interference Suppression Using Eigen Approach ......... M. ./. Mismar and T. H. Ismail
`A Boundary Element Approach to the Scattering from Inhomoge neous Dielectric Bodies ........ G. Pelosi and G. Toso
`
`465
`471
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`475
`484
`494
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`502
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`507
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`517
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`525
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`53 1
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`538
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`546
`551
`559
`563
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`570
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`579
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`589
`
`595
`596
`598
`600
`602
`
`ZTE v. Fractus
`IPR2018-01461
`
`ZTE
`Exhibit 1003.0001
`
`

`

`IEEE ANTENNAS AND PROPAGATION SOCIETY
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`IPR2018-01461
`
`ZTE
`Exhibit 1003.0002
`
`

`

`IEEE TRANSAC110NS ON ANTENNAS AND PROPAGATION. VOL. 46. NO. 4, APRIL 1998
`
`531
`
`f
`
`Study of Impedance and Radiation Properties of a
`Concentric Microstrip Triangular-Ring Antenna and
`Its Modeling Techniques Using FDTD Method
`
`Iti Saha Misra and S. K. Chowdhury, Senior Member, IEEE
`
`Abstract- A concentric microstrip triangular-ring antenna
`structure using the log-periodic principle for increasing the
`impedance bandwidth of the microstrip patch antenna is
`described. The finite-difference time-domain (FDTD) method is
`applied to analyze the proposed structure. A special technique
`to model the slanted metallic boundaries of the triangular ring
`has been used in the general FDTD algorithm to avoid the
`staircase approximation. The method improves the accuracy of
`the original FDTD algorithm without increasing the complexity.
`The radiation patterns at different frequencies over wide-band
`width are obtained experimentally.
`
`flldex Terms-FDTD methods, microstrip antennas.
`
`I. INTRODUCTION
`
`A three-element concentric microstrip triangular-ring an(cid:173)
`
`tenna (CMTRA) has been designed using a log-periodic
`principle and fabricated on a polytetra fturoethylene (PTFE)
`substrate. The clements o f the CMTRA are fed electromagnet(cid:173)
`ically by a 50-H microstrip line. The impedance and radiation
`characteristics have been measured and compared with those
`of two single triangular-ring antennas (STRA) having the
`largest and the smallest clement dimensions of the CMTRA.
`The impedance variation for the CMTRA has been measured
`at different feed locations. Results show that the bandwidth of
`the CMTRA is increased with respect to the STRA and the
`maximum bandwidth obtained for a particular feed location
`away from the center. The measured radiation characteristics at
`different feed locatio n show its invariant nature. The measured
`impedance pattern for two feed locations are verified by FDTD
`method. This method is similar to the method described for
`concentric microstrip circular-ring antenna (CMCRA) in [I].
`A special technique to model the slanted metallic boundaries
`of the triangular ring has been used in the general FDTD
`algorithm to avoid the staircase approximation (2), f3]. The
`method improves the accuracy of the original FDTD algorithm
`without increasing the complexity.
`
`(a)
`
`(b)
`
`(c)
`
`Fig. 1. Geometry of (a) 1hree-elemeat CMTRA, (b) STRA investigation of
`the smallcs1 clement di mensions, and (c) STRA investigation of the largest
`elcmcn1 dimensions.
`
`this figure. First. the innermost element was chosen with side
`ct = 1.4 cm and width w = 0.2 cm. The spacing between the
`adjacent elements and their widths are then chosen maintaining
`the following relation:
`
`( I )
`
`II. DESIGN OF THREE-ELEMENT CMTRA
`The three-clement CMTRA is shown in Fig. 1 (a). The width
`"w" and spacing ··<1·· between the elements are specified in
`
`Mnnuscripl received Scph:mber 6. 1996; revised May 30. 1997.
`The au1 hon-: arc with the Department of Electro11ics and Telecommunication
`Engineering. fadavpur Univcn,ity, Calcutta. 700 032 India.
`Publisher llcm lden1 ifi er S 00 I 8-lJ'.?6XC98)02680-5.
`
`where n is the suffix of the n th number of patch as indicated
`in Fig. I (a). The ring width and spacing decrease from the
`innermost c lement to the outermost element. Maintaining this
`re lation the outermost ring has the sides a = 3.0 cm and width
`w = 0.128 cm. The STRA's investigated have the smallest
`00 I 8-926X/98$ I 0.00 © 1998 IEEE
`
`ZTE v. Fractus
`IPR2018-01461
`
`ZTE
`Exhibit 1003.0003
`
`

`

`532
`
`IEEE TRANSAcnONS ON ANTENNAS AND PROPAGATION, VOL. 46, NO. 4. APRU. 19'18
`
`\ll'l
`
`Re f ere nce plane
`
`y
`
`1J•x
`
`--~
`
`c
`
`Fig. 2. Modeling of u·iangular ring in FDTD domain.
`
`and £he largest element dimensions of the CMTRA as given
`in Fig. l(b) and (c), respectively.
`
`lll. ANALYSIS OF CMTRA AND SlNGLE
`TRIANGULAR RJNG BY FDTD METHOD
`In this investigation one CMTRA and two triangular rings
`having the largest and smallest element dimensions of the
`CMTRA are analyzed by FDTD method. These antennas are
`fed electromagnetically by a 50-12 microstrip line which was
`fabricated on PTFE substrate with dielectric constant 2.55
`and thickness 0.159 cm (Fig. I). The application of FDTD
`method to these antennas are similar to those of CMCRA
`[I]. However, the modeling of the triangular-ring element
`is different. ln the following sections, we will describe the
`modeling techniques of triangular ring in the FDTD domain.
`
`A. Modeling of Lar1:er Triangular Ring
`
`Fig. 2 shows the actual dimension of the triangular ring
`antenna in the X-Y plane of the FDTD domain. Since the
`ting is an equilateral triangle the angle between the X axis
`and slanting plane is 30°. All the slanting sides of this ring
`can be represented by the equation of a line as
`
`y = m.r+C · ··
`
`(2)
`
`where rn = tan 8, 8 = angle between the X axis and slanting
`plane.
`Putting the value of 8 and any coordinate (x, y) passing
`through the line, the value of C can be found out.
`Now, if one can select the proper aspect ratio of each cell in
`the FDTD domain, then the slanted plane wall can be modeled
`exactly, i.e., it is possible to locate the boundary nodes of the
`mesh exactly on the slanted planes [3). In this particular case if
`we select 6.x = 0.()4 mm and, hence, y = 6.x tan 8 then for
`every space increment along X and Y direction the equation
`of t11e line will be well maintained.
`
`Impedance plot of largest lriangular ring of Fig. J(c) at different feed
`Fig. 3.
`loca1ion-sc1 I for center feed, set 2 for 0.35 cm off-center feed.
`
`Fig. 4.
`
`Impedance plot of smallest triangular ring of Fig. l(b) for center feed.
`
`The tangential electric field components Er and Ey and
`normal component of magnetic field Hz were made equal to
`zero on the triangular-ring patch bounded by the six lines.
`The FDTD parameters for this ring were: !lx = 0.G4 mm,
`A1J = 0.577356.x = 0.3694 mm, and flz = 0.5926 mm.
`flt = D.11/2Co (Co = free-space velocity of light), and
`Guassian half-width r = 18 ps. Tbe distance between the
`source plane to microstrip antenna was 39/lx and the reference
`plane was set at a distance 19/l:c from the source plane. The
`
`lit?.
`a1 ~
`
`50-
`
`B.
`1
`in F
`O.!i
`an
`mo
`l'.U
`
`ZTE v. Fractus
`IPR2018-01461
`
`ZTE
`Exhibit 1003.0004
`
`

`

`MISRA AND CllOWDllURY: IMPt;;DANC~ /\ND RADIATION PROPERTIES OF TRIANGULAR RIN(i /\NI hNNA
`
`(a)
`
`(b)
`
`Fig. 5.
`(a) and (bl Impedance plol of three-clement CMTRA al different frequency band for center feed. (c) Impedance plot of lhrcc-clement CMTRA
`at diffcrem frcqucnC} b:tnd for center feed.
`
`(c)
`
`50 n micro-;trip line width and the ring width were modeled
`as 12J.y and 2~.r. respectively.
`
`8. Modeling of Small Triangular Ring
`The actual dimcn<;ions of the smaJ I triangular ring is shown
`in Fig. I (b). The FDTD parameters were D.:r = 1.0 mm, 6.:i1 =
`0.577 :~52\.r = 0.577 :3.:;, ~z = 0.5296 mm. D.f. = D.z/2Go.
`and Gaussian half-width = 16 ps. Microstrip line width wa<;
`modeled as 7 6.y, the reference plane was set at a distance
`126.:r from the source plane.
`
`C. Modeling of 171ree-Element CMTRA
`
`The application of FDTD method is similar to the applica(cid:173)
`tion or single rings. The mesh size was considered such that it
`can model the actual dimensions or the CMTRA. The FDTD
`parameters were i\.r = 0.4 mm. i\y = 0.230 94ilz = 0.5WG
`mm, uf = uy/2C'u. and Gaussian half-width = 20 ps.
`In all three cases Mur's first-order absorbing boundary
`condition had been applied at the end. side, and top walls
`141. The total and incident E field were stored at the ref(cid:173)
`erence plane. Frequency-dependent scattering parameter was
`
`ZTE v. Fractus
`IPR2018-01461
`
`ZTE
`Exhibit 1003.0005
`
`

`

`534
`
`IEEE TRANSACTIONS ON ANTENNAS Ai'il) PROPAGATION. VOL 46, NO . .t, APR IL 1998
`
`MIS~
`
`TABLE 1
`Cm1PARISO:-.I OP 2: I VSWR ,.,cBW OP STRA AXD CMTRA
`
`Antenna
`Smallest
`STRA
`Largest
`STRA
`
`Feed location
`CF
`
`Freq. Range in GHz %BW
`5.950. 6.180 = 0.230
`3.790
`
`2.730- 2.770 = 0.040
`CF
`0.35 cmOCF 2.730 • 2.755 E 0-025
`0.60cm0CF 2.725 - 2.735 = 0.010
`
`•J.4540
`0.9tl0
`0.3660
`
`CMTRA
`
`CF
`
`0.35 cm OCF
`
`2.730 - 2.790 = 0.060
`4.600- 4.710 = 0.110
`6.650 - 6.920 = 0.270
`
`2.1700
`2.3630
`3.9700
`
`2.660 - 2.730 = 0.o70 .. 2.5900
`3.700 - 3.830 = 0.130 *3.4520
`6.340 - 6.620 = 0.280 *4.3200
`
`0.60cm0CF 2.660. 2.690 = 0.030
`2.720. 2.745 = 0.025
`3.690 - 3.830 = 0.140
`6.400 - 6.600 = 0.200
`
`1.1200
`0.9149
`3.7200
`3.0760
`
`CF • Center Feed, OCF = Off Center Feed,
`• a 2.2: t VSWR, •• = 2.4: I VSWR
`
`Fig. 7. Measured radiation patterns for largest triangular-ri ng anten na at
`different feed locations.
`
`locations to observe the effect or feed location. The impedance
`patterns for the largest and smallest the single ring antennas
`are given in Figs. 3 and 4 and for the CMTRA are given
`in Figs. 5 and 6. The 2: I YSWR circles are shown in the
`corresponding figures.
`
`(a)
`
`(b)
`
`(a) and (b) Impedance plot of three-element CMTRA at different
`Fig. 6.
`frequency band for 0.35-cm off-center feed.
`
`B. Bandwidth
`
`computed following the procedure as described in [1 ] and [5]
`and from this input impedance of the antennas were calculated.
`
`IV. MEASUREMENTS AND REsULTS
`
`The computed 2: I YSWR bandwidth for single ring and
`CMTRA at different band and feed locations are given in
`Table I. From this table it is seen that increase in bandwidth
`is obtained for CMTRA as compared to a single ring and
`maximum total bandwidth is obtained at a feed location 0.35
`cm away from center for this particular CMTRA.
`
`A Impedance Pattern
`
`The variation of input impedance for the two STRA and
`CMTRA were measured by HP 84JOB network analyzer at
`different band of frequenc ies. For the large single ring and
`CMTRA impedance patterns are measured at different feed
`
`C. Radiation Pat/ems
`The radiation patterns for STRA and three-element CMTRA
`have been measured for both <p = 90° and cp = 0° planes
`over the entire bandwidth of the antenna and are plotted in
`Figs. 7- 10. Comparison of the radiation pattern of a sing le
`
`. ~
`
`I
`
`rin1
`nat
`of
`froi
`rent
`
`c~
`off·
`pat
`pat
`one
`ex1
`FD
`cor
`(
`tha
`Thi
`eel
`
`ZTE v. Fractus
`IPR2018-01461
`
`ZTE
`Exhibit 1003.0006
`
`

`

`MISRA At\D CHOWOHl'RY: ll\IPEDANCI' I\ '10 Rl\OIATIO:\ PROPF.RTll·S 01- TRIMGL'LAR Rl'\Ci l\'11 E:..;;o.; .\
`
`535
`
`e
`
`( a )
`
`e
`o•
`
`~ ,o·
`
`-20
`
`- 25
`
`(b )
`
`'8.
`
`6 -
`
`(ti)
`
`e
`
`( h )
`
`,// /
`.
`
`/
`
`/
`
`I
`
`I /
`I
`I
`• !/
`!
`'I , .
`I
`
`I
`
`I
`
`I
`I
`
`0
`
`... .
`
`Fig. 8. Mea~ured radiation patterns for three clement CMTRA for center
`feed (a) at .,: = !)0° plane and (b) at -r' = o0 plane.
`
`Fig. 9. Mea~urcd radiation pattem~ for three-clement CMTRA for 0.35-c:ii
`off-center feed (a) at .; = !)()0 ph111r and !h) at .; = o0 plane.
`
`ring and CMTRA shows that over the entire bandwidth the
`nature of the radiation pattern is qualitatively similar to that
`of the single ring operating at the fundamental mode. Again
`from Figs. 8- 10 it is seen that radiation patterns for CMTRA
`remain unchanged with the change of feed location.
`
`V. NUMERICAL RESULTS
`The FDTD impedance pattern<, for the single rings and
`CMTRA are computed for the center feed and for 0.35-cm
`off-center feed and are plotted in Figs. 3- 6 with the measured
`patterns. From these figures it is seen that the computed
`patterns arc in reasonable agreement with the experimental
`one. The difference between the computed data and the
`experimental one may be due to the consideration of original
`FDTD algorithm for the slanted planes. A simple method of
`correcting this algorithm is described below [3).
`Consider a rectangular cell, located at a metallic boundary
`that crosses the cell along its diagonal as shown in Fig. I l .
`The proper FDTD formulation for the fields relevant to the
`cell is obtained from the integral form of Maxwell's equations
`
`J H.dl = J ls ( E r/E/ dl + " E ).d.<1
`J E.dl = j l - (1/(lH/ dt).ds .
`
`(3a)
`
`(3b)
`
`Applying (3b) to the cell of Fig. 11. we obtain
`
`E;(i I 1/2. j. k).::u + E~(i + l. j + 1/ 2. kp.y
`h,'~inut (i. + 1 / 2, j + 1 /2, k) J p .. r 2 ; ~.1/2)
`= (11D.:r6.:1!/2~t)[H ~1 +- 1 1 2 (i +1/2. j + 1/ 2. k)
`f{~ - 11 2 0 + 1/2 . j ~ 1/ 2. k)]
`-
`
`(4)
`
`where ~.r, ~!J. ~ z are the <.pace steps. j,,/ is the time steps
`according to the Yee· s [ 6 J formulation. E , 1mit is the E field
`along the diagonal of the cell and is zero on the metallic wall.
`From (4) J/ . can be easily cak:ulated as follows:
`
`u;•+112u + 1/ 2. J + t/2. k )
`= H~·- 1 !2 (i + 1/2. j + 1/ 2. k )
`(2~t/11~.r~y ) .I W~ U + 1/ 2. j. k)~.r
`-
`+ R;(i -t l. j I 1/ 2. k)uy].
`
`(5 )
`
`The modified (5) for the triangular cell differs from the
`conventional one onl) by a factor of two in the right-hand side.
`This simply corresponds to the cell having half the surface.
`This technique can be used lo conform the mesh to any slanted
`plane wall by choosing the proper aspect ratio of the cell as
`described in this paper for the triangular-ring antenna.
`
`ZTE v. Fractus
`IPR2018-01461
`
`ZTE
`Exhibit 1003.0007
`
`

`

`536
`
`IEEE TRANSACTIONS ON ANTHNNAS AND PROPAGATION. VOL. 46, NO. 4. APRIL 1998
`
`MISR
`
`121
`
`)3)
`
`161
`
`the a
`
`Case 2:
`
`w;-112(i + 1/2, j + L/ 2. k )
`= H~1- 1 12(i + 1/ 2. j + 1/ 2, k)
`- (2D.l/ 1iD..r:J.y) * [E~(i + 1/ 2. j + l. k)D..t
`+ t:;'(i. j + 1/ 2. k)~JJ]
`
`(6b)
`
`Case 3:
`
`H~·+ 112 (i + 1/2. j + 1/2, k)
`= H~- 11 2(i + 1/2. j + l/'2. k)
`- (2L:lt./1iD..rD.y) * [E~ (i+ 1/2, j . k)ilx
`- E~'(i, :i + 1/2, k)D.y]
`
`(6c)
`
`Case 4:
`
`H;•+ 112(i + L/2, .i + 1/2. k)
`= n~·- 112 c; + i;2. J + 1/2. k)
`- (26.f /1iD.:r6.y) * [E;(i + 1/2, .i + 1. k)D.:c
`- t:;(i + 1. j + 1/2. k):J.y].
`
`(6d)
`
`These modified equations for the H:: field have been imple(cid:173)
`mented on the smaller triangular-ring antenna for impedance
`calculation. The calculated impedance panem for this antenna
`is given in Fig. 4 with the measured and general FDTD
`algorithm patterns and Fig. 4 reveals the accuracy of the new
`system of equations. This method was applied Lo CMTRA
`considered earlier for center feed and the result shows less
`deviation from the experimental value.
`
`VI. CONCLUSION
`
`From the above investigations it may be said that the
`concentric micorslrip triangular ring antenna bas a multiple
`band effect with increase in total % bandwidth with respect
`to the single ring having the largest and the smallest physical
`dimension of the CMTRA. Changing the location of the feed
`the total % bandwidth is increased and with the changed feed
`location radiation patterns of the CMTRA remain unaltered.
`Comparison of the experimental impedance panems with the
`FDTD patterns shows the validity of the modeling of the
`triangular-ring structures as described in this paper. The most
`attractive feature of this structure is the increase of impedance
`and radiation band width without losing the advantage of small
`size of microstrip antenna.
`
`ACKNOWLEDGMENT
`The authors would like to thank Council of Scientific and In(cid:173)
`dustrial Research (CSIR). India. and Microwave Laboratory of
`Electronics and Telecom Engineering Department of Jadavpur
`University, Calcutta. India. for providing resources required
`for the research.
`
`(a)
`
`e -
`
`(b)
`
`Fig. 10. Measured radiation J'anems for lhree-elemenl CMTRA for 0.6-cm
`off-center feed (a) :u ..; = 90 plane and (b) at -r = o0 plane.
`
`Ey(i+1 1 i+-'12 ,k)
`
`E,,, (i+•tz,i,k)
`
`E11.
`
`, ..
`Ey~Ey fy~fy EymEy e,~er
`
`E"
`
`E.,
`
`Ex
`
`E,.
`
`2
`
`E,.
`3
`
`Ex
`4
`
`Fig. 11. Enlargement of a cell close 10 the slaated metallic surface.
`
`Consider the four cases (Fig. I I), which occur in the metal(cid:173)
`lic boundary of the triangular ring as indicated by the line
`number I, 2, 3, and 4 in Fig. 2. Applying (5) in these four cases
`the modified equations for H:: are given by the following:
`
`Case I:
`
`H~+lf 2('i + 1/2, j + 1/2, k)
`= H~1- 11 2 ( i + 1/2, .i + 1/2, k)
`- (2D.l/JtD.:i:D.y) * [E~(i + 1/2, j, k)D..x
`+ E;(i + 1, .i + 1/2, k)A:IJ]
`
`REFERENCES
`
`[ 1] T. S. Misra and S. K. Chowdhury, "Concentric mkrostrip ring antenna:
`Theory and cxperimen1." J. Elec1romag11. Wt1Ves Applical., vol. 10, pp.
`439-450, 1996.
`
`(6a)
`
`ZTE v. Fractus
`IPR2018-01461
`
`ZTE
`Exhibit 1003.0008
`
`

`

`MISRA AND CHOWl)JiURY: IMPEDANCE AND RADIATION PROPERTIRS OP TRIANGULAR RING AN-n!NNA
`
`5.17
`
`[2] J . Fang and J. Ren. "A locally conformed finite-difference time-domain
`algori1hm of modeling arbitrary shape planar me1al •trips;· IEEE Trw1s.
`Micrmn11·e Theory Tech.. vol. 41. pp. 830-838. May 1993.
`(3] P. Meuancne, L. Ro-.clli, and R. Sorrentino, "A simple way to model
`curved me1al boundaries in FDTD algorithm avoiding staircase approxi(cid:173)
`mation:· IF.££ Micn111·m·e G11i1led \\~ll'e LR11 .• vol. 5. pp. 267- 269. Aug.
`1995.
`(4) G. Mur, ""Absorbing boundary condition~ for the finite difference ap(cid:173)
`proximation of1hc time domain electromagnetic equations;· IEEE Trans.
`£/ec1romagn. C11111plll .. \Ol. EMC-2. pp. J77-382. Apr. 1981.
`(51 D. M. Sheen. S. M. Ali. M. D. Abournhr.i. and J. A. Kong. ··Application
`of the thrce-dimcn\ional finite difference lime-domain method to the
`analysis of planar micro,trip circuit<" IE€£ Trans. Microll'm·e Theory
`Tech.. vol. 38. pp. 849-857. July 1990.
`[61 K. S. Yee. "Numerical M>lution of initial boundary value problems in(cid:173)
`volving Maxwell's cqua1io11~ in isotropic media." IEEE Tra11s. A111e1111M
`Propagm .. vol. AP- 14. pp. 302· 307. Apr. 1966.
`
`S. K. C howdhury (M'79-SM'83) received Lhe
`B.Tcl.E.. M.E.Tel.E.. and Ph.D.
`(engineering)
`degrees from Jadavpur University. India. in 1964,
`1968. and 1972, rcspecti,·cty.
`He joined the Departme01 of Electronics and
`Jadavpur
`Tclccommunicatjon Engineering of
`University in 1969 as a Lecturer. He is now
`a Profcs~or in the same departmcm. He has
`written approximately 50 papers
`for national
`and inlemalional journals. His current interest;;
`include engineering education. electromagne1ics.
`and micro\\:l\C<;. in general, and microstrip antennas and component<;, in
`panicular.
`Dr. Chowdhury i' a Felio\\ IE (India).
`
`lti Saba M isra was born in Lndi:1 on Sep1cmbcr
`25, 1964. S he received the B.Sc. (Hons.) degree
`in phy,.ics, in 1986. the B.Tech degree in rad io
`phy,.ics and electronics, in 1989. from the Calcutta
`Univer~ity. India, and the M.E. and Ph.D. degrees
`in electronic~ and telecommunications engineering.
`in 199 1 and 1997, respec1ively, from the Jadavpur
`Univer~ity. India.
`She is now a Lecturer in the Electronics and
`Telecommunications Engineering Department m the
`Jada\pur Uni,·e~ity. Her research interests arc in
`the area~ of clec1romagnc1ics in gene