This paper presents a method to simplify the analysis of a square patch microwave sensor built on a multi-layer substrate. The main feature of this method is the simplification of a multi-layer sensor structure into a sensor structure with a single homogeneous layer. The equivalent dieletric constant for the new single homogenous layer was discriminated using the method proposed in this paper. Effectively it eliminates the problem of mode matching between layers and thus reduces the amount of calculation and computing time. The method was compared with the experimental results and was found to give an error less than 4% in the worse case

Wan Ali, Wan Khairuddin

English

Universiti Teknologi Malaysia Institutional Repository

AsMENSE 2003 SENSOR. pg. 257 - 262 A Theoretical Development to Simplify the Analysis of Rectangular Patch Microwave Sensor Built on Multi-layer Substrate Wan Khauuddm Wan Ah For@ OfMcrbonrrOl Eng"g Unmrnh Tehohg~ M&m 81310 U l M S k d a  ]ohor, M a 5 n e  Absvscr -This papet presents a method to simplify the analysis of a square patch microwave scnsoc built on a multi-layer substrate. T h e  main feature o f  this method is the simplification of  a multi-layer sensor smcture into a sensor s " c t u e  with a single homopous layer. The equivalent dielecmc conrmt for the new single homogenous layer was dcrcrmincd using the method proposed in this paper. Effeecdvcly it eliminates the problem o f  mode matching between layers and thus reduces the amount of calculation and computing time. T h e  method was compared with the cxpdmental results and was found to pivc an cnor less than 4 % in the worse cme. 1. Introduction Microwave sensors can be designed and according to the given specifications can take any form and shapes. In many applications the microwave sensor is in the form of a patch resonator. Some of these resonators were fabricated onto a substrate or onto multi-layer substrates. In order to accurately predict the characteristic of these resonator several method of analyses were employed. In most applications, the analyses require to solve the electromagnetic field in each layer. A considerable number of tmhniques to analyse the fields have been repotted in the published literature [I -61 and to date there are at least five techniques available with varying degrees of accuracy [G] .  In the patch resonator applications, some of the frequently used techniques of analysis indude the transmission line theory, modal-expansion cavity model, spectral domain integral equation and finite difference h e  domain, While rigornus methods are more complete in that they offer accurate radiation models, surface wave effect and the field internal to the patch, they are quite complicated and cannot be readily implemented especially for preliminary design work. It was reported that the accuracy of the rigorous approach 0-7803-8102-5/03/$17.00 8 2003 IEEE may not be much better than that of the simplc transmission line or cavity models provided the latter are used within their range of v&dity [6]. One draw back of these simple models i.e. transmission line and cavity model, is that they are not efficient when applied to a patch resonator built on a multi-layer substrates as shown in figure 1. Each mode in any particular layer has to be solved and matched with an adjacent layer [7 - 81. This paper presents a theoretical development backed with experimental work in an attempt to simplify the cavity model analysis for microwave sensot built on multi- layer substrate. Ground plane Figure 1 A single square patch microsmp resonator AsiaSENSE 2. Equivqlent permittivity analysis For microwave sensor built as a single patch resonator on a multi-kyer structure as shown in figure 1, the cavity under consideration is the space between the patch resonator or the conducting patch and the ground plane. As a result, the usual assumptions of the cavity model, which are applicable for the patch resonator on a single dielectric layer, have to be modified. The modified assumptions are as follows; 1. Due to the closc. proximity between the conducting patch and the ground plane, only transverse magnetic pl$ modes are assumed to exist. T h e  z component of the v x  v x  &, = V(V .Ed)-V'B, = - j o p , V x R ,  =,(V'+k:)E, =jop,J,(5) Where k, = the wave number in the dth dielectric layer. Solving the inhomogeneous wave equation (5) is done by firstly determining the eigenfuction (p, , which satisfy the following 'homogeneous wave equation field is a since the cavity is a multi-layered. ne cavity is assumed bounded by the bottom and by a perfect magnetic wall along the edge. 3. Across the dielectric-dielectric Where the second letter in the subsuipt, I, is the mode number. Without lost of generality and to simplify the analysis, let consider a shape consrmcted on two layers substrate as shown in figure 2. 2. perfect electtic walls on the top and microstrip patch ,&, cl.,.& interfaces, the tangential electric field and the normal electric flux density are continuous. Bared on these assumptions and the resonator as shown in figure 1, the Maxwell's equation in the cavity region take the form v X E ,  =-jopodffd (1) v x f f ,  =j+jos,,&, (2) V.E,  - 0  (3) v.ff ,=o (4) Where subscript, d, indtcates the corrcs onding 'd'th dielectric layer and fi respectively. The current source, J, in equation (2) is assumed to be z independent because of the thinness of the dielecuic layers. As a result, the continuity equation gives V . J = - j o p = O .  As a consequence of this, the volume charge density distribution is identically zero in the rcgion, which is the reason for writing V.E,  - 0  in equation (3). Taking the curl of equation (1) pives, and w are the Electric and Magnetic fields Figure 2. A microstrip patch resonator with circular shape constructed on twolayers substrate. The field solution to equation (6) at a point (p, 'p, z) would be as follows, In dielectric I; E,, =C,J,(k,p)cosh pcos@,(h-z)) (7 ED, = C$,k;'J;(ktp)cosh'p sin(P,(h-z)) (8) E# =~s,p-'k:J.(k,p)sinlrpsil@,(h-z)) (9) 4 = ~ j . ~ k ; ' j ~ ( k , p ) c o s l c o ~ ( h - z ) )  (11)% = ~ 9 l p ~ ~ ~ J . ( k , d s i ~ o ~ ( h - z ) )  (10) 258 where k: =o~B,E, -p: (12) In dielectric 2 E,, =C,J,(k,p)coshcpcosP,z (13) Epl =-C,P,k,'J,(k,p)coshlpsinP,z 14) E, =C,np,p-'k:J,(k,p)sinhcpsinp,z (1 5) Ho> = C j o ~ , d ~ J , ( k , p ) s i n h c o ~ z  (16)H, = -C2jOE2k;'l"(klp)cosh~cos~,z (I?) where k: $ 0 3  p,~; -p, . 2 2 (18) On applying the magnetic wall condition at p = a to equations (8) and (14) gives Where xnm is the mth root of Jb which is the dctivative of the Bessel function of the fust kind of order n. Applying the condition that the tangential E and normal D must be conMuous across z = tz, gives Using equations (12), (IS), (19), (22) and (23) the expression far resonant frequency is given by, Compadng equation (25) with the resonant frequency formula for a single layer substrate, it can be said that the equivalent permittivity of the two-layers substrate is given by For multi-layer substrates, the same technique as above can be applied by considering two layers at each time. In this case equation (27) becomes: where tn = height of the nth substrate E" = permittivity of the nth layer substrate layer. 3. The proposed simplified multi- layer model Extending the idea of equivalent permitdvity to the multi layer substrate and calculating it using equation (ZS), the author proposed to simplify the resonator S ~ C N I C  as shown in figure 1 into a resonator constructed on a singlc equivalent homogenous substrate as shown in figore 3 where the cavity model is best applied. 259 Figure 3. A single patch microstrip resonator constructed on multi-layer substrate is simplified into a resonator on single layer homogenous substrate. The equivalent substrate is considered lossless and has a permittivity given (28). from the same equation (28) the equivalent dielectric constant is given by The height of the equivalent substrate is taken to equal to the total height of the substrates and is given by N t, =ct. (30). "21 The rest of the dimensions such as the width and the length of the patch were unchanged. 4. Experimental work Rectangular patch resonators were constructed on several multi-layer substrates to test the proposed method. In all resonesor structurcs, the total substrate heights were several time smaller than the operadng wavelength in order to satisfy the assumptions of the proposed method. For t, much less than the operating wavelength, the cavity model solution for the z- component elccmc field underneath the square patch is given by [9] For simplicity in fabricating the coupling structure of the resonators, direct conductor connections were chosen in this experimental work, This simply means that either direct microstrip line .or direct prohe connection to the patch. From equation (31) the input impedance is given by Several resonators were fabricated to test the simplified model and figire 4(a) and 4(b) shows two of them. 1 70" 4, - 2.2.1, 4co) Figure 4 Miwosmp patch resonators used in this investigation 260 5. Experimental and Calculated Results . .  A network analyser was employed to measure the remm loss versus frequcnq characteristic of' these resonators. Table 1 and table 2 summarised the mult  obtained either from calculation or experiment. These compadson resulu show a dose agreement between calculated and measured data. The wont case shows a discrepancy of less than 4% for mode resonant kequencies TMIo, TMoi,Tt& T M z o  andTMii. 6. Conclusion It CM be concluded that using the simplified multi-layer model the calculation of equivalent permittiYity is easily done and this value can he used to analyse a simple resonator s m c t u e  on multi-layer substrate. The problem of solving the modes in each Table 2 Summalv of the results o b k c d  for resonator 4@) 7Resonator 4@) t 2  = 0.02 mm Resonator 4@) t2 = 0.04 mm dielectri: having height calculated using equation (29). Table 1 A Summary of the results obtained for resonator 4(a) For resonator 4(a) t, is equal to 1.59 mm For resonator 4(b) t, and 4 are equal to 1.59 mm and 0.25 mm repectively Mode frequencies (GHz) Calculated v layer of substrate is simplihed to solving just the modes in a single layer. The problem with mode matchine is eliminated 1. 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(Editor), ‘Numerical Techniques for Microwave and Millimetre-wave Passive Smccures’, John Wiley & Sons Inc., New York (1989). m Lee, K.F., Ho, K.Y. and Dahele, J.S., ‘Circular-Disk Microsmp Antenna With an Airgap’, IEEE Trans. Anrenoas and Propagation, Vol. AP-32, No. 8, pp 880-884, (1984). [8] Shen, Z. and MacPhie, R.H., Waveguide Modal Analysis of Single and Stacked Probe-fed microstrip Antennas with Circular Geomeuies’, IEEE Ant&“ and Prooaearion Sodetv International . -  Symposium Digest, Vol. 2, pp 598-601, 1199~1 \ -  I [9] Harisawa, K. and Henashi, M. (Editors), ‘Analysis, Design and Measurement of Small And Low-Pro6le Antennas, Ancch House Inc., Norwood, USA, (1992). 0 262 

A theoretical development to simplify the analysis of rectangular patch microwave sensor built on multi-layer substrate

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A theoretical development to simplify the analysis of rectangular patch microwave sensor built on multi-layer substrate

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