Progress In Electromagnetics Research M, Vol. 14, 71{83, 2010 ANALYSIS OF MICROWAVE CAVITY LOADED WITH LOSSY DIELECTRIC SLAB BY MEANS OF MODE MATCHING METHOD AND OPTIMIZATION OF LOAD LOCATION O. SÄule Department of Electronic Engineering Uludag University GÄorÄukle,Bursa 16059, Turkey S. Kent Department of Electronics & Communication Engineering Istanbul Technical University Maslak, Istanbul 34469, Turkey Abstract|An analysis is presented by means of a mode matching method for two microwave cavities of di®erent sizes which are fed by a TE 10 waveguide and loaded with lossy dielectric slab type material. The accuracy of the results obtained is presented together with a comparison of the results which are obtained by the HFSS numerical method. Optimization of the load location was performed in order to maximize the electrical ¯eld on the material. The principle of this optimization is based on ¯nding the existence of the positions in which the reflection coe±cient S11 is the lowest. When the feeding guide for the two di®erent microwave cavities was at the centre of the resonator, the change in the reflection coe±cient distribution was detected according to the di®erent positions of the material in the oven, and then the lowest positions were found. The changes in the electric ¯eld in the detected positions were recorded. Received 17 June 2010, Accepted 24 August 2010, Scheduled 31 August 2010 Corresponding author: S. Kent ([email protected]). 72 SÄuleand Kent 1. INTRODUCTION Microwave heating is widely used in industry due to its advantages over conventional heating methods [1]. However, it is rather di±cult to design an appropriate system according to the material which will be heated in spite of the present numerical methods, and very few studies have been carried out in this ¯eld [2{5]. The results found by means of the present numerical methods are related to checking the accuracy of the method rather than optimization. The mode matching method has been widely used in a large number of applications of electromagnetism [6]. A considerable number of studies have been presented on ¯nding the parameter S11 in an inhomogenously loaded microwave resonator. In the studies carried out, amplitudes of the reflection coe±cient related to the frequency have been obtained for the structures examined [7{11]. The parameter S11 changes depending on a considerable number of parameters such as frequency, the dimensions of the resonator, the position of the feeding guide on the resonator, the dielectric permittivity, the geometry of the load, and the position and size of the load in the microwave resonator. In the literature su±cient searches for the parameter S11 have not been carried out, especially from the point of view of microwave heating. In the studies of numerical and hybrid methods such as the ¯nite di®erence time domain, ¯nite element method, and mode matching, the changes in the parameter S11 in the structures examined have been presented in comparison with each other [12{16]. Di®erently in this work, the scattering parameter S11 has been obtained by changing the position of the material along the z axis while keeping stable the factors which a®ect the scattering parameter S11 in the resonator loaded with lossy slab type dielectric material, such as the frequency, the dimensions of the resonator, and the position of the feeding guide on the resonator. In this way, the change in the electric ¯eld has been found in the resonator at the lowest values of the reflection coe±cient. At these points the e±ciency of the microwave oven is at a maximum and loss power density on the material can be at a maximum too. 2. THEORETICAL BACKGROUND Basically, we can divide the resonator into four basic parts as shown in Figure 1. c1 is the starting point of the feeding guide's x coordinate and c2 Progress In Electromagnetics Research M, Vol. 14, 2010 73 B A waveguide 1 v v 2 1 c c 1 2 Multimode 2 cavity l Dielectric slab t 3 d 4 b a Figure 1. Three-dimensional rectangular cavity loaded with lossy slab. is the last point of the feeding guide's x coordinate; in the same way v1 is the starting point of the feeding guide's y coordinate and v2 is the last point of the feeding guide's y coordinate. t is the slab thickness, d and l are the empty spaces of the resonator, and a and b are the resonator's x and y dimensions respectively. When the basic equalities are taken into account and the equality of tangential electric ¯eld compounds (at the point of intersection of the ¯rst and the second area) on the aperture and mode matching equalities in the related boundaries (2) are used, the equation can be formulated as follows: ~ h ¡jkz;10z Es = ~et;10e (1) " ³ ´ # M N h +jkI z ¡jkI z h(I) X X A e z;mn + ±mn;10e z;mn ~e E~ I = mn t;mn (2) t e(I) I e +jkz;mnz m=0 n=0 +Amn~et;mne 2 ³ ´ 3 M N (I)I ¡jk(I)I z X X Ah e+jkz;mnz ¡ ± e z;10 ~hh H~ I = 4 mn mn;10 t;mn 5 (3) t e (I) e ~ +jkz;mnz m=0 n=0 +Amn ht;mne 20 13 h(+) ¡jk II z h(II) Bmn e mn ~emn XM XN 6B e(+) ¡jk II z e(II) C7 ~ II 6B ¡Bmn e mn ~emn C7 Et = 6B h(¡) II h(II) C7 (4) 4@ jkmn(z¡l) A5 m=0 n=0 +Bmn e ~emn e(¡) jk II (z¡l) e(II) +Bmn e mn ~emn 74 SÄuleand Kent 20 13 h(+) ¡jk II z h(II) ¡Bmn e mn ~hmn M N 6B e(+) II e(II) C7 X X ¡jkmnz~ ~ II 6B +Bmn e hmn C7 Ht = 6B h(¡) II h(II) C7 (5) 4@ jkmn(z¡l)~ A5 m=0 n=0 +Bmn e hmn e(¡) jk II (z¡l) e(II) +Bmn e mn ~hmn q I 2 2 2 kmn = ! ¹o"0 ¡ (m¼=A) ¡ (n¼=B) (6) q II 2 2 2 kmn = ! ¹o"0 ¡ (m¼=a) ¡ (n¼=b) (7) where k is the mode propagation constant, eh is the normal TE mode vector, hh is the normal TM mode vector; m and n are mode indices, Et is the tangential electric ¯eld, Ht is the tangential magnetic ¯eld. ~ II ~ I For z = 0Et = Et ³ ´ II h(I) h h(I) h(+) h(II) h(¡) ¡j¤kmnl h(II) (±mn;10~et;mn +Amn~et;mn)= Bmn ~emn +Bmn e ~emn (8) ³ ´ II e e(I) e(+) e(II) e(¡) ¡j¤kmnl e(II) (Amn~et;mn)= ¡Bmn ~emn +Bmn e ~emn (9) In ¯nding the reflection coe±cient in the second part which is indicated with a minus sign, the equation below which consists of the e®ects of the third and fourth parts can be used as follows: l3 ¡ Zer ¡23 = (10) l3 + Zer II II eikmnd ¡ e¡ikmnd l3 = Z0 II II (11) eikmnd + e¡ikmnd l2 ¡ Z0 ¡12 = (12) l2 + Z0 γt ¡γt e + ¡23e l2 = Zer γt ¡γt (13) e ¡ ¡23e !¹ kII Zh = ;Ze = mn (14) mn kII mn !" qmn 2 2 2 γ = (m ¢ ¼=a) + (n ¢ ¼=b) ¡ ! ¹o"0"r (15) II ¡ + ¡jkmnl + Bmn = Bmne ¡12 = Bmn ¢ ¡0 (16) ~ II ~ I To enable the condition z £ (Ht ¡ Ht ) = 0 on the aperture (z = 0 Progress In Electromagnetics Research M, Vol. 14, 2010 75 plane) 2³ ´³ ³ ´´ 3 h(+) h(II) II M N 1 1 ¡jkmnl 1 1 X X Bmn ~emn h(II) + h(I) ¡¡0 ¢ e h(II) ¡ h(I) ¡ 4³ ´³Zmn Zmn ³ Zmn Zmn ´´ 5 e(+) e(II) II 1 1 ¡jkmnl 1 1 m=0 n=0 Bmn ~emn e(II) + e(I) ¡¡0 ¢ e e(II) ¡ e(I) Zmn Zmn Zmn Zmn 1 = 2± e h(I) (17) 10 h(I) 10 Z10 We can write the approximate magnetic current as follows: ³ ´ Z ³ ´ h ~ h ±mn;10 + Amn = MS ¢ z^ £ ~et;mn ds (18) SZa ¡ ¢ e ~ e Amn = MS ¢ z^ £ ~et;mn ds (19) Sa Z 1 ³ ´ Bh(+) = M~ ¢ z^ £ ~e h ds (20) mn ¡jkII l S t;mn 1 + ¡0 ¢ e mn Sa Z 1 ¡ ¢ Be(+) = M~ ¢ z^ £ ~e e ds (21) mn ¡jkII l S t;mn ¡1 + ¡0 ¢ e mn Sa The magnetic current is approximated with a truncated set of independent basis functions: ³ ´ XP h ³ ´ ³ ´ i ¹ ~0 » ~0 ~0 MS ½ = apx upx ½ x^ + apy upy ½ y^ (22) p ³ ´ XM XN Z ³ ´ h h h ±mn;10 +Amn = ¡amnxumnxey;mn +amnyumnyex;mn ds(23) m=0 n=0 Sa XM XN Z e ¡ e e ¢ Amn = ¡amnxumnxey;mn +amnyumnyex;mn ds (24) m=0 n=0 Sa II h(+) h(¡) ¡jkmnl Bmn + Bmn ¢ e XM XN Z ³ ´ h h = ¡amnxumnxey;mn + amnyumnyex;mn ds m=0 n=0 Sa II e(+) e(¡) ¡jkmnl ¡Bmn + Bmn ¢ e XM XN Z ¡ e e ¢ = ¡amnxumnxey;mn + amnyumnyex;mn ds (25) m=0 n=0 Sa 76 SÄuleand Kent " # · ¸ h(+) ¡jkII l ¡1 Bmn 1 + ¡0 ¢ e mn 0 e(+) = ¡jkII l Bmn 0 ¡1 + ¡0 ¢ e mn 2 R R 3 h h ¡ umnxey;mnds umnyex;mnds · ¸ 6 SRa SRa 7 amnx 4 e e 5 (26) ¡ umnxey;mnds umnyex;mnds amny Sa Sa 20 1 0 1 3 1 1 1 1 h(II) + h(I) ¡ ¡0¢ R e(II) + e(I) ¡ ¡0¢ R Zmn ³Zmn ´ h(II) Zmn ³Zmn ´ e(II) 6@ II A wsx~emn ds ¡@ II A wsy~emn ds7 ¡jkmnl 1 1 ¡jkmnl 1 1 6 e h(II) ¡ h(I) Sa e e(II) ¡ e(I) Sa 7 6 Zmn Zmn Zmn Zmn 7 0 1 0 1 6 1 1 1 1 7 6 h(II) + h(I) ¡ ¡0¢ R e(II) + e(I) ¡ ¡0¢ R 7 4 Zmn ³Zmn ´ h(II) Zmn ³Zmn ´ e(II) 5 @ II A wsx~emn ds ¡@ II A wsy~emn ds ¡jkmnl 1 1 ¡jkmnd 1 1 e h(II) ¡ h(I) Sa e e(II) ¡ e(I) Sa Zmn Zmn Zmn Zmn · ¸ ¡jkII l ¡1 1+¡0 ¢ e mn 0 ¡jkII l 0 ¡1+¡0 ¢ e mn 2 R R 3 h h ¡ umnxey;mnds umnyex;mnds · ¸ 6 SRa SRa 7 amnx 4 e e 5 ¡ umnxey;mnds umnyex;mnds amny Sa Sa 2 3 2 R h(I) h(I) wsx~ex10 ds 6 Z10 7 = Sa (27) 4 2 R h(I) 5 wsy~e ds Z h(I) y10 10 Sa By solving the matrix equation above it is possible to ¯nd the coe±cients amnx and amny.