Microwave Ring Circuits and Related Structures phần 9 pdf

CHAPTER ELEVEN Ring Antennas and Frequency- Selective Surfaces 297 Microwave Ring Circuits and Related Structures, Second Edition, by Kai Chang and Lung-Hwa Hsieh ISBN 0-471-44474-X Copyright © 2004 John Wiley & Sons, Inc. 11.1 INTRODUCTION The ring antenna has been used in many wireless systems. The ring resonator is constructed as a resonant antenna by increasing the width of the microstrip [1–4]. As shown in Figure 11.1, a coaxial feed with the center conductor extended to the ring can be used to feed the antenna. The ring antenna has been rigorously analyzed using Galerkin’s method [5, 6]. It was concluded that the TM 12 mode is the best mode for antenna applications, whereas TM 11 mode is best for resonator applications.Another rigorous analysis of probe-feed ring antenna was introduced in [7]. In [7], a numerical model based on a full-wave spectral-domain method of moment is used to model the connection between the probe feed and ring antenna. The slot ring antenna is a dual microstrip ring antenna. It has a wider imped- ance bandwidth than the microstrip antenna. Therefore, the bandwidth of the slot antenna is greater than that of the microstrip antenna [8–10]. By intro- ducing some asymmetry to the slot antenna, a circular polarization (CP) radi- ation can be obtained.The slot ring antenna in the ground plane of a microstrip transmission line can be readily made into a corporate-fed array by imple- menting microstrip dividers. Active antennas have received great attention because they offer savings in size, weight, and cost over conventional designs. These advantages make them desirable for possible application in microwave systems such as wireless communications, collision warning radars, vehicle identification transceiver, self-mixing Doppler radar for speed measurement, and microwave identifica- tion systems [11, 12]. Frequency-selective surfaces (FSSs) using circular or rectangular rings have been used as the spatial bandpass or bandstop filters. This chapter will briefly discuss these applications. Also, a reflectarray using ring resonators will be described in this chapter. 11.2 RING ANTENNA CIRCUIT MODEL The annular ring antenna shown in Figure 11.1 can be modeled by radial trans- mission lines terminated by radiating apertures [13, 14]. The antenna is con- structed on a substrate of thickness h and relative dielectric constant e r .The inside radius is a, the outside radius is b, and the feed point radius is c. This model will allow the calculation of the impedance seen from an input at point c. The first step in obtaining the model is to find the E and H fields supported by the annular ring. 11.2.1 Approximations and Fields The antenna is constructed on a substrate of thickness h, which is very small compared to the wavelength (l). The feed is assumed to support only a z- 298 RING ANTENNAS AND FREQUENCY-SELECTIVE SURFACES FIGURE 11.1 The annular ring antenna configuration. directed current with no variation in the z direction (d/dz = 0). This current excitation will produce transverse magnetic (TM) to z-fields that satisfy the following equations in the (r, f, z) coordinate system [15]: (11.1) (11.2) (11.3) where (11.4) (11.4) f n (f) is a linear combination of cos(nf) and sin(nf), A n and B n are arbitrary constants, J n is the nth-order Bessel function, and Y n is the nth-order Neumann function. The equations for E z (r) and H f (r), without the f dependence, are (11.5) (11.6) where J n ¢ (kr) is the derivative of the nth-order Bessel function and Y n ¢ (kr) is the derivative of the nth-order Neumann function with respect to the entire argument kr. These fields are used to define modal voltages and currents. The modal voltage is simply defined as E z (r). The modal current is -rH f (r) or rH f (r) for power propagating in the r or -r direction, respectively.This results in the fol- lowing expressions for the admittance at any point r: (11.7) (11.8) Y H E c z r rr r r f () = - () () >, Y H E c z r rr r r f () = () () <, H jk AJ k BY k nn nnf r wm rr () =- ¢ () +¢ () [] 0 EAJkBYk znn nn rrr () = () + () y we rrf wmee w m e = () + ()()() = = = = j k AJ k BY k f k j nn nn n r 2 00 frequency in radians per second permeability of free space permittivity of free space = – 1 0 0 H f d dr =- Y H r r d df = 1 Y E k j z = 2 we Y RING ANTENNA CIRCUIT MODEL 299 11.2.2 Wall Admittance Calculation As shown in Figure 11.2 the annular ring antenna is modeled by radial trans- mission lines loaded with admittances at the edges. The s subscript is used to denote self-admittance while the m subscript is used to denote mutual admit- tance. The admittances at the walls (Y m (a, b), Y s (a), Y s (b)) are found using two approaches. The reactive part of the self-admittances (Y s (a), Y s (b)) is the wall susceptance.The wall susceptances b s (a) and b s (b) come from Equations (11.7) and (11.8), respectively. The magnetic-wall assumption is used to find the con- stants A n and B n in Equation (11.6). The H f (r) field is assumed to go to zero at the effective radius b e and a e . The effective radius is used to account for the fringing of the fields. ¢= Ê Ë ˆ ¯ +++ + () x a h h a rr ln . . . . 2 1 41 1 77 0 268 1 65ee x b h h b rr = Ê Ë ˆ ¯ +++ + () ln . . . . 2 1 41 1 77 0 268 1 65ee aa hx a e r =- ¢ 1 2 pe bb hx b e r =+1 2 pe 300 RING ANTENNAS AND FREQUENCY-SELECTIVE SURFACES FIGURE 11.2 The annular ring antenna modeled as radial transmission lines and load admittances [13]. (Permission from IEEE.) It is easily seen that Equations (11.7) and (11.8) will be purely reactive when the magnetic-wall assumption is used to calculate A n and B n . This results in the expressions (11.9) (11.10) The mutual admittance Y m (a, b) and wall conductances g s (a) and g s (b) are found by reducing the annular ring structure to two concentric, circular, copla- nar magnetic line sources. The variational technique is then used to determine the equations [15]. The magnetic line current at r = a was divided into differential segments and then used to generate the differential electric vector potential dF.The electric field at an observation point is found from (11.11) Then the magnetic field at r = b and z = 0 can be found from Maxwell’s equation: (11.12) The total H f component of the magnetic field at r = b and z = 0 due to the current at r = a is (11.13) where a is the angle representation for the differential segments. The mutual admittance will obey the reciprocity theorem, that is, the effect of a current at a on b will be the same as a current at b on a. The reaction concept is used to obtain where E a = the radial electric fringing aperture field at a E b = the radial electric fringing aperture field at b The mutual admittance is then found to be Yab H hbE d hE E m b ab , cos () = Ú f p ff p 0 2 HdH a ff p = = Ú 0 2 dH dE=- —¥ 1 0 jwm dE dF=-—¥ ba ka J ka Y ka Y ka J ka J ka Y ka Y ka J ka s nnenne nnenne () =- ¢ () ¢ () -¢ () ¢ () () ¢ () - () ¢ () wm 0 bb kb J kb Y kb Y kb J kb J kb Y kb Y kb J kb s nn nne nnenne () = ¢ () ¢ () -¢ () ¢ () () ¢ () - () ¢ () wm 0 3 RING ANTENNA CIRCUIT MODEL 301 302 RING ANTENNAS AND FREQUENCY-SELECTIVE SURFACES where This equation can be reduced to a single integral equation by replacing the coefficient of the cos f term in the Fourier expansion of H f with the sum of all the coefficients and evaluating at f = 0: (11.15) and The self-conductance at a or b can be found by substituting a = b in Equa- tion (11.15) and extracting only the real part: (11.16) (11.17) where rb b = 2 2 sin a ra a = 2 2 sin a gb bh r kr kr kr kr kr d s b bb b b b () = ¥+ Ê Ë ˆ ¯ - () - È Î Í ˘ ˚ ˙ Ú 2 0 3 0 2 2 00 0 0 22 2 0 2 1 22 pwm a aa a p cos cos sin cos sin sin ga ah r kr kr kr kr kr d s a aa a a a () = ¥+ Ê Ë ˆ ¯ - () - È Î Í ˘ ˚ ˙ Ú 2 0 3 0 2 2 00 0 0 22 2 0 2 1 22 pwm a aa a p cos cos sin cos sin sin rab ab=+- 22 2 cosa Yab jabh e r jk r baba r kr jkr d m jk r , cos cos cos cos () = ¥+ () + - () - () () È Î Í ˘ ˚ ˙ - Ú pwm a a aa a p 0 3 0 2 0 2 0 22 0 0 21 3 3 rab ab k =+- - = 22 000 2 cosfa wem Yab jabh e r jk r baba r kr jkr dad m jk r , cos cos cos cos cos () = Ê Ë ˆ ¯ - () + (){ È Î Í + - () - () ()() ¥- - () ¸ ˝ ˛ ˘ ˚ ˙ - ÚÚ 2 21 33 2 0 3 0 2 0 0 2 2 0 22 0 0 pwm fa fa fa fa f pp (11.14) This completes the solutions for the admittances at the edges of the ring: 11.2.3 Input Impedance Formulation for the Dominant Mode The next step is to transform the transmission lines to the equivalent p- network. This is accomplished by finding the admittance matrix of the two- port transmission line. The g-parameters of a p-network can then easily be found: where For r = a, r 1 is replaced by c and r 2 is replaced with a.When r = b, r 1 is replaced with b and r 2 by c. Figure 11.3 shows the equivalent circuit and the simplified circuit. From simple circuit theory, the input impedance is seen to be: (11.18) where s n == >2010for n for n; h = thickness of the substrate Z hZRZRZRZZRZR ZRZR ZRZRZR n AABBCCCAABB nAABB CCAABB in = + () + () ++++ () + () + () ++ () +++ () ps ps D 112 1 2 1 2 rr r r r r, () =¢ ()() -¢ ()() Jk Yk Yk Jk nn nn D rr r r r r 12 1 2 1 2 , () = ()() - ()() Jk Yk Yk Jk nn nn g j k 3 021 21 2 1 2 r wm r r rrr p () = () () + È Î Í ˘ ˚ ˙ D D , , g j 2 021 2 r pwm r r () = - () D , g j k 1 012 11 1 2 2 r wm r r rrr p () = - () () + È Î Í ˘ ˚ ˙ D D , , Y b g b jb b ss s () = () + () Y a g a jb a ss s () = () + () RING ANTENNA CIRCUIT MODEL 303 Z gb ga C s = () + () 1 1 Z Ya Y ab ga B sm = () - () + () 1 1 , Z Ya Y ab ga A sm = () - () + () 1 3 , 304 RING ANTENNAS AND FREQUENCY-SELECTIVE SURFACES FIGURE 11.3 The complete circuit model of the annular ring antenna: (a) circuit model with g-parameters; (b) simplified circuit model [13]. (Permission from IEEE.) RING ANTENNA CIRCUIT MODEL 305 The h/(ps n ) term arises from the discontinuity of the H f field at c. 11.2.4 Other Reactive Terms The equation for Z in , Equation (11.18), given earlier assumes that the domi- nant mode is the only source of input impedance. The width of the feed probe and nonresonant modes contribute primarily to a reactive term. The wave equation is solved using the magnetic walls, as stated earlier, to find the non- resonant mode reactance: (11.19) X h J kc Y ka Y kc J ka J kb Y ka Y kb J ka J kc Y kb Y kc J kb md c md c M m m mn a mmemme meme meme mmemme = () ¢ () - () ¢ () ¢ () ¢ () -¢ () ¢ () È Î Í ˘ ˚ ˙ ¥ () ¢ () - () ¢ () [] () = π Â wm s 0 0 2 2 2 sin / / (() È Î Í ˘ ˚ ˙ 2 R g Yabga gb Yab ga Yabgb gb Yab gbga gb ga Yab c b m m m m m = + ()() () + () + () + ()() () + () Ê Ë Á Á Á - () () () + () + () ˆ ¯ ˜ ˜ ˜ () 1 2 11 1 2 2 2 2 2 2 22 22 , , , , , R gbga gb ga Yab gb Yabga ga Yab ga Yabgb gb Yab B m m m m m = () () () + () + () + () + ()() () + () Ê Ë Á Á Á - () + ()() () + () ˆ ¯ ˜ ˜ ˜ 1 2 11 1 22 22 2 2 2 2 2 2 , , , , , R gbga gb ga Yab gb Yabga ga Yab ga Yabgb gb Yab A m m m m m = () () () + () + () - () + ()() () + () Ê Ë Á Á Á + () + ()() () + () ˆ ¯ ˜ ˜ ˜ 1 2 11 1 22 22 2 2 2 2 2 2 , , , , , s m = 2 for m = 0; 1 for m > 0 d = the feed width n = the resonant mode number The reactance due to the probe is approximated from the dominate term of the reactance of a probe in a homogeneous parallel-plate waveguide [16]: (11.20) where u c is the speed of light. 11.2.5 Overall Input Impedance The complete input impedance is found by summing the reactive elements given earlier. The final form of Z input is (11.21) where Re and Im represent the real and imaginary parts of Z in , respectively. The reactive terms are summed because X M and X p contribute very little to the radiated fields. 11.2.6 Computer Simulation A computer program was written in Fortran to find the input impedance. The program followed the steps shown in Figure 11.4. The results shown in Figure 11.5 were checked well with the published results of Bhattacharyya and Garg [13]. ZZjZXX MPinput in in Re= {} + {} ++ [] Im X hv d p c r = wm p we 0 2 4 1 781 ln . 306 RING ANTENNAS AND FREQUENCY-SELECTIVE SURFACES FIGURE 11.4 Flow chart of the input impedance calculation. [...]... 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CHAPTER ELEVEN Ring Antennas and Frequency- Selective Surfaces 297 Microwave Ring Circuits and Related Structures, Second Edition, by Kai Chang and Lung-Hwa Hsieh ISBN 0-471-44474-X. stacked annular ring microstrip antenna [18]. (Permis- sion from IEEE.) 310 RING ANTENNAS AND FREQUENCY-SELECTIVE SURFACES FIGURE 11 .9 Comparison of (a) microstrip ring and (b) slot ring structures. . of 2.45 and height of 0.762 mm. The widths of the microstrip (w m ) and slot ring (w s ) were 2.16 mm and 2 .9 mm, respectively. The mean circumference of the slot ring is 93 .3 mm. Ignoring the