Propagation in Lossy Rectangular Waveguides

Transcription

1 10 Propagation in Lossy Retangular Waveguides Kim Ho Yeap 1, Choy Yoong Tham, Ghassan Yassin 3 and Kee Choon Yeong 1 1 Tunku Abdul Rahman University Wawasan Open University 3 University of Oxford 1, Malaysia 3 United Kingdom 1. Introdution )In millimeter and submillimeter radio astronomy, waveguide heterodyne reeivers are often used in signal mixing. Wave guiding strutures suh as irular and retangular waveguides are widely used in suh reeiver systems to diret and ouple extraterrestrial signals at millimeter and submillimeter wavelengths to a mixer iruit (Carter et al., 004; Boifot et al., 1990; Withington et al., 003). To illustrate in detail the appliations of waveguides in reeiver systems, a funtional blok diagram of a typial heterodyne reeiver in radio telesopes is shown in Fig. 1 (Chattopadhyay et al., 00). The eletromagneti signal (RF signal) from the antenna is direted down to the front end of the reeiver system via mirrors and beam waveguides (Paine et al., 1994). At the front end of the reeiver system, suh as the sideband separating reeiver designed for the ALMA band 7 artridge (Vassilev and Belitsky, 001a; Vassilev and Belitsky, 001b; Vassilev et al., 004), the RF signal is hannelled from the aperture of the horn through a irular and subsequently a retangular waveguide, before being oupled to the mixer. In the mixer iruit, a loal osillator (LO) signal whih is generally of lower frequeny is then mixed with the RF signal, to down onvert the RF signal to a lower intermediate frequeny (IF) signal. Here, a superondutor-insulator-superondutor (SIS) heterodyne mixer is ommonly implemented for the proess of down onversion. At the bak end of the system, the IF signal goes through multiple stages of amplifiation and is, eventually, fed to a data analysis system suh as an aousto-opti spetrometer. The data analysis system will then be able to perform Fourier transformation and reord spetral information about the input signal. The front-end reeiver noise temperature T R is determined by a number of fators. These inlude the mixer noise temperature T M, the onversion loss C Loss, the noise temperature of the first IF amplifier T IF, and the oupling effiieny between the IF port of the juntion and the input port of the first IF amplifier IF. A omparison of the performane of different SIS waveguide reeivers is listed in Table 1 (Walker et al., 199). It an be seen that the value of T R for the 30 GHz system is a fator of 3 to 4 less than that ahieved with the 49 GHz system. The derease in system performane at 49 GHz is due to the inrease of C Loss and T M by a fator of approximately 3.

2 56 Eletromagneti Waves Propagation in Complex Matter Fig. 1. Blok diagram of a heterodyne reeiver. SIS Juntion Nb Pb Nb Center Frequeny (GHz) T R (K) T M (K) C Loss (db) T IF (K) Table 1. Comparison of SIS reeiver performane. Sine the input power level of the weak millimeter and submillimeter signals is quite small i.e. of the order of to 10 0 W (Shankar, 1986), it is therefore of primary importane to minimize the onversion loss C Loss of the mixer iruit. One way of doing so, is to ensure that the energy of the LO and, in partiular, the RF signals is hannelled and oupled from the waveguides to the mixer iruit in a highly effiient manner. It is simply too time onsuming and too expensive to develop wave guiding strutures in a reeiver system on a trial-anderror basis. To minimize the loss of the propagating signals, the availability of an aurate and easy-to-use mathematial model to ompute the loss of suh signals in wave guiding strutures is, of ourse, entral to the development of reeiver iruits.. Related work Analysis of the propagation of wave in irular ylindrial waveguides has already been widely performed (Glaser, 1969; Yassin et al., 003; Clarioats, 1960a; Clarioats, 1960b; Chou and Lee, 1988). The analyses by these authors are all based on the rigorous method formulated by Stratton (1941). In Stratton s formulation, the fields at the wall surfae are made ontinuous into the wall material. Assumption made on the field deaying inside the wall material yields relations whih allow the propagation onstant to be determined. Due to the diffiulty in mathing the boundary onditions in Cartesian oordinates, this approah, however fails to be implemented in the ase of retangular waveguides. A similar rigorous tehnique to study the attenuation of retangular waveguides is not available hitherto. The perturbation power-loss method has been ommonly used in analyzing wave attenuation in lossy (Stratton, 1941; Seida, 003; Collin 1991; Cheng, 1989) and superonduting (Winters and Rose, 1991; Ma, 1998; Wang et al., 1994; Yalamanhili et al., 1995) retangular waveguides; respetively. This is partly due to its ability to produe simple analytial solution, and also partly beause it gives reasonably aurate result at frequenies f well above its utoff frequeny f. In this method, the field expressions are derived by assuming that the walls to be of infinite ondutivity. This allows the solution to

3 Propagation in Lossy Retangular Waveguides 57 be separated into pure Transverse Eletri (TE) and Transverse Magneti (TM) modes. For a waveguide with finite ondutivity, however, a superposition of both TE and TM modes is neessary to satisfy the boundary onditions (Stratton, 1941; Yassin et al., 003). This is beause, unlike those of the lossless ase, the modes in a lossy waveguide are no longer mutually orthogonal to eah other (Collin, 1991). To alulate the attenuation, ohmi losses are assumed due to small field penetration into the ondutor surfae. Results however show that this method fails near utoff, as the attenuation obtained diverges to infinity when the signal frequeny f approahes the utoff f. Clearly, it is more realisti to expet losses to be high but finite rather than diverging to infinity. The inauray in the powerloss method at utoff is due to the fat that the field equations are assumed to be idential to those of a lossless waveguide. Sine a lossless waveguide behaves exatly like an ideal high pass filter, signals ease to propagate at f below f. It an be seen that the assumption of lossless fields fail to give an insight or deeper understanding on the mehanism of the propagation of wave in pratial lossy waveguides. Moreover, at very high frequeny espeially that approahes the millimeter and submillimeter wavelengths the loss tangent of the onduting wall dereases. Therefore, suh assumption turns out to be inaurate at very high frequeny. Although Stratton (1941) has developed a truly fundamental approah to analyze waveguides, his approah is only restrited to the ase of irular waveguides and ould not be applied to retangular waveguides. The workhorse of this hapter is, therefore, to develop a novel and aurate formulation i.e. one that does not assume lossless boundary onditions to investigate the loss of waves in retangular waveguides. In partiular, the new method shall be found more aurate and useful for waveguides operating at very high frequenies, suh as those in the millimeter and submillimeter wavelengths. In Yeap et al. (009), a simple method to ompute the loss of waves in retangular waveguides has been developed. However, the drawbak of the method is that different sets of harateristi equation are required to solve for the propagation onstants of different modes. Here, the method proposed in Yeap et al. (009) shall be developed further so that the loss of different modes an be onveniently omputed using only a single set of equation. For onveniene purpose, the new method shall be referred to as the boundary-mathing method in the subsequent setions. Fig.. A waveguide with arbitrary geometry.

4 58 Eletromagneti Waves Propagation in Complex Matter 3. General wave behaviours along uniform guiding strutures As depited in Fig., a time harmoni field propagating in the z diretion of a uniform guiding struture with arbitrary geometry an be expressed as a ombination of elementary waves having a general funtional form (Cheng, 1989) 0 xy, exp[j( t kz)] (1) where ψ 0 (x, y) is a two dimensional vetor phasor that depends only on the ross-setional oordinates, = πf the angular frequeny, and k z is the propagation onstant. Hene, in using phasor representation in equations relating field quantities, the partial derivatives with respet to t and z may be replaed by produts with j and jk z, respetively; the ommon fator exp[j(t + k z z)] an be dropped. Here, the propagation onstant k z is a omplex variable, whih onsists of a phase onstant z and an attenuation onstant z z k j () z z z The field intensities in a harge-free dieletri region (suh as free-spae), satisfy the following homogeneous vetor Helmholtz s equation z k kz z ( ) 0) (3) where z is the longitudinal omponent of ψ, is the Laplaian operator for the transverse oordinates, and k is the wavenumber in the material. For waves propagating in a hollow waveguide, k = k 0, the wavenumber in free-spae. It is onvenient to lassify propagating waves into three types, in orrespond to the existene of the longitudinal eletri field E z or longitudinal magneti H z field: 1. Transverse eletromagneti (TEM) waves. A TEM wave onsists of neither eletri fields nor magneti fields in the longitudinal diretion.. Transverse magneti (TM) waves. A TM wave onsists of a nonzero eletri field but zero magneti field in the longitudinal diretion. 3. Transverse eletri (TE) waves. A TE wave onsists of a zero eletri field but nonzero magneti field in the longitudinal diretion. Single-ondutor waveguides, suh as a hollow (or dieletri-filled) irular and retangular waveguide, annot support TEM waves. This is beause aording to Ampere s iruital law, the line integral of a magneti field around any losed loop in a transverse plane must equal the sum of the longitudinal ondution and displaement urrents through the loop. However, sine a single-ondutor waveguide does not have an inner ondutor and that the longitudinal eletri field is zero, there are no longitudinal ondution and displaement urrent. Hene, transverse magneti field of a TEM mode annot propagate in the waveguide (Cheng, 1989). 4. Fields in artesian oordinates For waves propagating in a retangular waveguide, suh as that shown in Fig. 3, Helmholtz s equation in (3) an be expanded in Cartesian oordinates to give

5 Propagation in Lossy Retangular Waveguides 59 z z x k k z z 0 ) (4) y By applying the method of separation of variables, z an be expressed as XxYy ( ) ( ) (5) z Equation (4) an thus be separated into two sets of seond order differential equations, as shown below (Cheng, 1989) dx(x) k 0 x X(x) (6) dx dy(y) k 0 y Y(y) (7) dy where k x and k y are the transverse wavenumbers in the x and y diretions, respetively. The longitudinal fields an be obtained by solving (6) and (7) based on a set of boundary onditions and substituting the solutions into (5). Fig. 3. The ross setion of a retangular waveguide. The transverse field omponents an be derived by substituting the longitudinal field omponents into Maxwell s soure free url equations E -jh (8) H je (9) where ε and μ are the permittivity and permeability of the material, respetively. Expressing the transverse field omponents in term of the longitudinal field omponents E z and H z, the following equations an be obtained (Cheng, 1989)

6 60 Eletromagneti Waves Propagation in Complex Matter j dhz dez H x k z kx ky dx dy j dhz dez H y k z kx ky dy dx j dez dhz E x k z kx ky dx dy (10) (11) (1) j de dh E y (13) k z z k z x ky dy dx 5. Review of the power-loss method In the subsequent setions, analysis and omparison between the perturbation power-loss method and the new boundary mathing method shall be performed. Hene, in order to present a omplete sheme, the derivation of the onventional power-loss method is briefly outlined in this setion. The attenuation of eletromagneti waves in waveguides an be aused by two fators i.e. the attenuation due to the lossy dieletri material z(d), and that due to the ohmi losses in imperfetly onduting walls z() (Cheng, 1989) z = z(d) + z(), (14) For a onduting waveguide, the inner ore is usually filled with low-loss dieletri material, suh as air. Hene, z(d) in (14) shall be assumed zero in the power-loss method and the loss in a waveguide is assumed to be aused solely by the ondution loss. It ould be seen later that suh assumption is not neessary in the new boundary-mathing method. Indeed, the new method inherently aounts for both kinds of losses in its formulation. The approximate power-loss method assumes that the fields expression in a highly but imperfetly onduting waveguide, to be the same as those of a lossless waveguide. Hene, k x, k y, and k z are given as (Cheng, 1989) m kx (15) a k y k z n (16) b (17) where a and b are the width and height, respetively, of the retangular waveguide; whereas m and n denote the number of half yle variations in the x and y diretions, respetively. Every ombination of m and n defines a possible mode for TE mn and TM mn waves. Condution loss is assumed to our due to small fields penetration into the ondutor surfaes. Aording to the law of onservation of energy, the attenuation onstant due to ondution loss an be derived as (Cheng, 1989) z

7 Propagation in Lossy Retangular Waveguides 61 PL z ) (18) P where P z is the time-average power flowing through the ross-setion and P L the timeaverage power lost per unit length of the waveguide. Solving for P L and P z based on Poynting s theorem, the attenuation onstant z for TM and TE modes i.e. z(tm) and z(te), respetively, an thus be expressed as (Collin, 1991) z ztm ( ) 3 3 Rs( m b n a ) f ab 1 m b n a f (19) b 1 f Rs b f b f m ab n a zte ( ) 1 1 a f a f f ( mb) ( na) where R s is the surfae resistane, f the utoff frequeny, and the intrinsi impedane of free spae. (0) 6. The new boundary-mathing method It is apparent that, in order to derive the approximate power-loss equations illustrated in setion 5, the field equations must be assumed to be lossless. In a lossless waveguide, the boundary ondition requires that the resultant tangential eletri field E t and the normal derivative of the tangential magneti field Ht an to vanish at the waveguide wall, where a n is the normal diretion to the waveguide wall. In reality, however, this is not exatly the ase. The ondutivity of a pratial waveguide is finite. Hene, both E t and Ht an are not exatly zero at the boundary of the waveguide. Besides, the loss tangent of a material dereases in diret proportion with the inrease of frequeny. Hene, a highly onduting wall at low frequeny may exhibit the properties of a lossy dieletri at high frequeny, resulting in inauray using the assumption at millimeter and submillimeter wavelengths. In order to model the field expressions loser to those in a lossy waveguide and to aount for the presene of fields inside the walls, two phase parameters have been introdued in the new method. The phase parameters i.e x and y, are referred to as the field s penetration fators in the x and y diretions, respetively. It is worthwhile noting that, with the introdution of the penetration fators, E t and Ht an do not neessarily deay to zero at the boundary, therefore allowing the effet of not being a perfet ondutor at the waveguide wall. 6.1 Fields in a lossy retangular waveguide For waves propagating in a lossy hollow retangular waveguide, as shown in Figure 3, a superposition of TM and TE waves is neessary to satisfy the boundary ondition at the wall (Stratton, 1941; Yassin et al., 003). The longitudinal eletri and magneti field omponents E z and H z, respetively, an be derived by solving Helmholtz s homogeneous equation in

8 6 Eletromagneti Waves Propagation in Complex Matter Cartesian oordinate. Using the method of separation of variables (Cheng, 1989), the following set of field equations is obtained E E sin k x sin k y (1) z 0 x x y y H H os k x os k y () z 0 x x y y where E 0 and H 0 are onstant amplitudes of the fields. The propagation onstant k z for eah mode will be found by solving for k x and k y and substituting the results into the dispersion relation z 0 x y k k k k (3) Equations (1) and () must also apply to a perfetly onduting waveguide. In that ase E z and Hz an are either at their maximum magnitude or zero at both x = a/ and y = b/ i.e. the entre of the waveguide, therefore ka x kb y sin xsin y 1 or 0 Solving (4), the penetration fators are obtained as, (4) m kxa x (5) n kyb y (6) For waveguides with perfetly onduting wall, k x = mπ/a and k y = nπ/b, (5) and (6) result in zero penetration and E z and H z in (1) and () are redued to the fields of a lossless waveguide. To take the finite ondutivity into aount, k x and k y are allowed to take omplex values yielding non-zero penetration of the fields into the waveguide material k k j (7) x x x j (8) y y y where x and y are the phase onstants and x and y are the attenuation onstants in the x and y diretions, respetively. Substituting the transverse wavenumbers in (7) and (8) into (3), the propagation onstant of the waveguide k z results in a omplex value, therefore, yielding loss in wave propagation. Substituting (1) and () into (10) to (13), the fields are obtained as jkzkxh 0 0kE y 0 sin( kx x x)os( ky y y) H x k x ky (9)

9 Propagation in Lossy Retangular Waveguides 63 H y jkzkyh0 0kE x 0 os( kx x x)sin( ky y y) k x ky (30) jkzkxe0 0kH y 0 os( kx x x)sin( ky y y) E x k x ky (31) jkzkye0 0kH x 0 sin( kx x x)os( ky y y) E y k x ky (3) where μ 0 and ε 0 are the permeability and permittivity of free spae, respetively. 6. Formulation At the wall, the tangential fields must satisfy the relationship defined by the onstitutive properties μ and ε of the material. The ratio of the tangential omponent of the eletri field to the surfae urrent density at the ondutor surfae is represented by (Yeap et al., 009b; Yeap et al., 010) Et a H n t (33) where μ and ε are the permeability and permittivity of the wall material, respetively, and is the intrinsi impedane of the wall material. The dieletri onstant is omplex and ε may be written as (34) 0 j where is the ondutivity of the wall. In order to estimate the loss of waves in millimeter and submillimeter wavelengths more aurately, a more evolved model than the onventional onstant ondutivity model used at mirowave frequenies is neessary. Here, Drude s model is applied for the frequeny dependent ondutivity (Booker, 198) (1 j ) (35) where is the onventional onstant ondutivity of the wall material and the mean free time. For most ondutors, suh as Copper, the mean free time is in the range of to s (Kittel, 1986). At the width surfae of the waveguide, y = b, E z /H x = E x /H z = (9), and (31) into (33), the following relationships are obtained. Substituting (1), (),

10 64 Eletromagneti Waves Propagation in Complex Matter j E x 0 E kk z x k 0 ytan kb y y z kx ky H0 H j H x 0 H kk z x k 0 yot kb y y Ez kx ky E0 (36a) (36b) Similarly, at the height surfae where x = a, we obtain E y /H z = E z /H y = (1), (), (30), and (3) into (33), the following relationships are obtained. Substituting Ey j E0 kk z y 0 k xtan ka x x Hz kx ky H0 Hy j H0 kk z y 0 k xot ka x x Ez kx ky E0 (37a) (37b) In order to obtain nontrivial solutions for (36) and (37), the determinant of the equations must be zero (Yeap et al., 009a). By letting the determinant of the oeffiients of E 0 and H 0 in (36) and (37) vanish the following transendental equations are obtained ot j0kytan kb y y j 0ky kb y y kk z x kx ky kx ky kx ky (38a) j0kx tan ka x x j0kx ot ka x x kk z y kx ky kx ky kx ky (38b) Sine the dominant TE 10 mode has the lowest utoff frequeny among all modes and is the only possible mode propagating alone, it is of engineering importane. In the subsequent setions, omparison and detail analysis of the TE 10 mode shall be performed. For the TE 10 mode, m and n are set to 1 and 0, respetively. Substituting m = 1 and n = 0 into the penetration fators in (5) and (6), the transendental equations in (38) for TE 10 mode an be simplified to b b j0kytan ky j0kyot ky jkzk x kx ky kx ky kx ky a a j0kxot kx j0kxtan kx jkzk y kx ky kx ky kx ky (39a) (39b)

11 Propagation in Lossy Retangular Waveguides 65 In the above equations, k x and k y are the unknowns and k z an then be obtained from the dispersion relation in (3). A multi root searhing algorithm, suh as the Powell Hybrid root searhing algorithm in a NAG routine, an be used to find the roots of k x and k y. The routine requires initial guesses of k x and k y for the searh. For good ondutors, suitable guess values are learly those lose to the perfet ondutor values i.e. x = mπ/a, y = nπ/b, x = y = 0. Hene, for the TE 10 mode, the initial guesses for k x and k y are π/a and 0 respetively.) 6.3 Experimental setup To validate the results, experimental measurements had been arried out. The loss as a funtion of frequeny for a retangular waveguide was measured using a Vetor Network Analyzer (VNA). A 0 m opper retangular waveguide with dimensions of a = 1.30 m and b = 0.64 m suh as that shown in Fig. 4 were used in the measurement. To minimize noise in the waveguide, a pair of hokes had also been designed and fabriated as shown in Fig. 5. A detail design of the hoke drawn using AutoCAD are shown in Fig. 6(a) and Fig. 6(b). In order to allow the waveguide to be onneted to the adapters whih are of different sizes, a pair of taper transitions had also been used as shown in Fig. 7. Fig. 8 depits the omplete setup of the experiment where the retangular waveguide was onneted to the VNA via tapers, hokes, oaxial ables, and adapters. Before measurement was arried out, the oaxial ables and waveguide adapters were alibrated to eliminate noise from the two devies. The loss in the waveguide was then observed from the S 1 or S 1 parameter of the sattering matrix. The measurement was performed in the frequenies at the viinity of utoff. Fig. 4. Retangular waveguides with width a = 1.30 m and height b = 0.64 m. Fig. 5. A pair of hokes made of aluminum.

12 66 Eletromagneti Waves Propagation in Complex Matter ) Fig. 6. Parameters of the (a) ross setion and (b) side view of the hoke. (a) (b) Fig. 7. Taper transitions. Fig. 8. A 0 m retangular waveguide onneted to the VNA, via tapers, hokes, adapters, and oaxial ables.

13 Propagation in Lossy Retangular Waveguides Results and disussion As shown in Fig. 9, a omparison among the attenuation of the TE 10 mode near utoff as omputed by the new method, the onventional power-loss method, and the measured S 1 result was performed. Clearly, the attenuation onstant z omputed from the power-loss method diverges sharply to infinity, as the frequeny approahes f, and is very different to the simulated results, whih show learly that the loss at frequenies below f is high but finite. The attenuation omputed using the new boundary-mathing method, on the other hand, mathes very losely with the S 1 urve, measured using from the VNA. As shown in Table, the loss between GHz and GHz omputed by the boundarymathing method agrees with measurement to within 5% whih is omparable to the error in the measurement. The inauray in the power-loss method is due to the fat that the fields expressions are assumed to be lossless i.e. k x and k y are taken as real variables. Analyzing the dispersion relation in (3), it ould be seen that, in order to obtain z, k x and/or k y must be omplex, given that the wavenumber in free spae is purely real. Although the initial guesses for k x and k y applied in the new boundary-mathing method are assumed to be idential with the lossless ase, the final results atually onverge to omplex values when the harateristi equations are solved numerially. Fig. 9. Attenuation of TE 10 mode at the viinity of utoff. the new boundary mathing method. power loss method. S1 measurement. Fig. 10 shows the attenuation urve when the frequeny is extended to higher values. Here, the loss due to TE 10 alone ould no longer be measured alone, sine higher-order modes, suh as TE 11 and TM 11, et., start to propagate. Close inspetion shows that the loss

14 68 Eletromagneti Waves Propagation in Complex Matter predited by the two methods at higher frequenies is in very lose agreement. It is, therefore, suffied to say that, although the power-loss method fails to predit the attenuation near f aurately, it is still onsidered adequate in omputing the attenuation of TE 10 in lossy waveguides, provided that the frequeny f is reasonably above the utoff f. As depited in Fig. 11, at frequenies beyond millimeter wavelengths, however, the loss omputed by the boundary-mathing method appears to be muh higher than those by the power-loss method. The differenes an be attributed to the fat that at extremely high frequenies, the loss tangent of the wall material dereases and the field in a lossy waveguide an no longer be approximated to those derived from a perfetly onduting waveguide. At suh high frequenies, the wave propagating in the waveguide is a hybrid mode and the presene of the longitudinal eletri field E z an no longer be negleted. % Frequeny GHz Experiment Boundary-mathing method Table 1. Attenuation of TE 10 at the viinity of the utoff frequeny. Unlike the power-loss method whih only gives the value of the attenuation onstant, one other advantage of the boundary-mathing method is that it is able to aount for the phase

15 Propagation in Lossy Retangular Waveguides 69 Attenuation Np/m Frequeny GHz Fig. 10. Attenuation of TE 10 mode from 0 to 100 GHz. mathing method. power loss method. the new boundary 0.13 Attenuation Np/m Frequeny GHz Fig. 11. Attenuation of TE 10 mode from 100 GHz to 1 THz. boundary mathing method. power loss method. the new

16 70 Eletromagneti Waves Propagation in Complex Matter onstant of the wave as well. A omparison between the attenuation onstant and phase onstant of a TE 10 mode is shown in Fig. 1. As an be observed, as the attenuation in the waveguide gradually dereases, the phase onstant inreases. Fig. 1 illustrates the hange in the mode i.e. from evanesent below utoff to propagating mode above utoff. Attenuation Np/m Frequeny GHz Fig. 1. Propagation onstant (phase onstant and attenuation onstant) of TE10 mode in a lossy retangular waveguide. phase onstant. attenuation onstant. 7. Summary A fundamental and aurate tehnique to ompute the propagation onstant of waves in a lossy retangular waveguide is proposed. The formulation is based on mathing the fields to the onstitutive properties of the material at the boundary. The eletromagneti fields are used in onjuntion of the onept of surfae impedane to derive transendental equations, whose roots give values for the wavenumbers in the x and y diretions for different TE or TM modes. The wave propagation onstant k z ould then be obtained from k x, k y, and k 0 using the dispersion relation. The new boundary-mathing method has been validated by omparing the attenuation of the dominant mode with the S1 measurement, as well as, that obtained from the powerloss method. The attenuation urve plotted using the new method mathes with the powerloss method at a reasonable range of frequenies above the utoff. There are however two regions where both urves are found to differ signifiantly. At frequenies below the utoff f, the power-loss method diverges to infinity with a singularity at frequeny f = f. The new method, however, shows that the signal inreases to a highly attenuating mode as the frequenies drop below f. Indeed, suh result agrees very losely with the measurement result, therefore, verifying the validity of the new method. At frequenies above 100 GHz, the attenuation obtained using the new method inreases beyond that predited by the power-loss method. At f above the millimeter wavelengths, the field in a lossy waveguide Phase Constant rad/m

17 Propagation in Lossy Retangular Waveguides 71 an no longer be approximated to those of the lossless ase. The additional loss predited by the new boundary-mathing method is attributed to the presene of the longitudinal E z omponent in hybrid modes. 8. Aknowledgment K. H. Yeap aknowledges Boon Kok, Paul Grimes, and Jamie Leeh for their advise and disussion. 9. Referenes Boifot, A. M.; Lier, E. & Shaug-Petersen, T. (1990). Simple and broadband orthomode transduer, Proeedings of IEE, 137, pp Booker, H. (198). Energy in Eletromagnetism. 1st Edition. Peter Peregrinus. Carter, M. C.; Baryshev, A.; Harman, M.; Lazareff, B.; Lamb, J.; Navarro, S.; John, D.; Fontana, A. -L.; Ediss, G.; Tham, C. Y.; Withington, S.; Terero, F.; Nesti, R.; Tan, G. -H.; Sekimoto, Y.; Matsunaga, M.; Ogawa, H. & Claude, S. (004). ALMA front-end optis. Proeedings of the Soiety of Photo Optial Instrumentation Engineers, 5489, pp Chattopadhyay, G.; Shleht, E.; Maiwald, F.; Dengler, R. J.; Pearson, J. C. & Mehdi, I. (00). Frequeny multiplier response to spurious signals and its effets on loal osillator systems in millimeter and submillimeter wavelengths. Proeedings of the Soiety of Photo-Optial Instrumentation Engineers, 4855, pp Cheng, D. K. (1989). Field and Wave Eletromagnetis, Addison Wesley, ISBN , US. Chou, R. C. & Lee, S. W. (1988). Modal attenuation in multilayered oated waveguides. IEEE Transations on Mirowave Theory and Tehniques, 36, pp Clarioats, P. J. B. (1960a). Propagation along unbounded and bounded dieletri rods: Part 1. Propagation along an unbounded dieletri rod. IEE Monograph, 409E, pp Clarioats, P. J. B. (1960b). Propagation along unbounded and bounded dieletri rods: Part. Propagation along a dieletri rod ontained in a irular waveguide. IEE Monograph, 409E, pp Collin, R. E. (1991). Field Theory of Guided Waves, John Wiley & Sons, ISBN , New York. Glaser, J. I. (1969). Attenuation and guidane of modes on hollow dieletri waveguides. IEEE Transations on Mirowave Theory and Tehniques (Correspondene), 17, pp Kittel, C. (1986). Introdution to Solid State Physis, John Wiley & Sons, New York. Ma, J. (1998). TM-properties of HTS s retangular waveguides with Meissner boundary ondition. International Journal of Infrared and Millimeter Waves, 19, pp Paine, S.; Papa, D. C.; Leombruno, R. L.; Zhang, X. & Blundell, R. (1994). Beam waveguide and reeiver optis for the SMA. Proeedings of the 5th International Symposium on Spae Terahertz Tehnology, University of Mihigan, Ann Arbor, Mihigan. Seida, O. M. A. (003). Propagation of eletromagneti waves in a retangular tunnel. Applied Mathematis and Computation, 136, pp Shankar, N. U. (1986). Appliation of digital tehniques to radio astronomy measurements, Ph.D. Thesis. Raman Researh Institute. Bangalore University.

18 7 Eletromagneti Waves Propagation in Complex Matter Stratton, J. A. (1941). Eletromagneti Theory, MGraw-Hill, ISBN , New York. Vassilev, V. & Belitsky, V. (001a). A new 3-dB power divider for millimetre-wavelengths. IEEE Mirowave and Wireless Components Letters, 11, pp Vassilev, V. & Belitsky, V. (001b). Design of sideband separation SIS mixer for 3 mm band. Proeedings of the 1th International Symposium on Spae Terahertz Tehnology, Shelter Island, San Diego, California. Vassilev, V.; Belitsky, V.; Risaher, C.; Lapkin, I.; Pavolotsky, A. & Sundin, E. (004). Design and haraterization of a sideband separating SIS mixer for GHz. Proeedings of the 15th International Symposium on Spae Terahertz Tehnology, Hotel Northampton, Northampton, Masahusetts. Walker, C. K.; Kooi, J. W.; Chan, M.; Ledu, H. G.; Shaffer, P. L.; Carlstrom, J. E. & Phillips, T. G. (199). A low-noise 49 GHz SIS waveguide reeiver. International Journal of Infrared and Millimeter Waves, 13, pp Wang, Y.; Qiu, Z. A. & Yalamanhili, R. (1994). Meissner model of superonduting retangular waveguides. International Journal of Eletronis, 76, pp Winters, J. H. & Rose, C. (1991). High-T superondutors waveguides: Theory and appliations. IEEE Transations on Mirowave Theory and Tehniques, 39, pp Withington, S. (003). Terahertz astronomial telesopes and instrumentation. Philosophial Transations of the Royal Soiety of London, 36, pp Yalamanhili, R., Qiu, Z. A., Wang, Y. (1995). Retangular waveguides with two onventional and two superonduting walls. International Journal of Eletronis, 78, pp Yassin, G., Tham, C. Y. & Withington, S. (003). Propagation in lossy and superonduting ylindrial waveguides. Proeedings of the 14th International Symposium on Spae Terahertz Tehnology, Tuson, Az. Yeap, K. H., Tham, C. Y., and Yeong, K. C. (009a). Attenuation of the dominant mode in a lossy retangular waveguide. Proeedings of the IEEE 9th Malaysia International Conferene on Communiations, KL., Malaysia. Yeap, K. H., Tham, C. Y., Yeong, K. C. & Lim, E. H. (010). Full wave analysis of normal and superonduting mirostrip transmission lines. Frequenz Journal of RF-Engineering and Teleommuniations, 64, pp Yeap, K. H., Tham, C. Y., Yeong, K. C. & Yeap, K. H. (009b). A simple method for alulating attenuation in waveguides. Frequenz Journal of RF-Engineering and Teleommuniations, 63, pp

19 Eletromagneti Waves Propagation in Complex Matter Edited by Prof. Ahmed Kishk ISBN Hard over, 9 pages Publisher InTeh Published online 4, June, 011 Published in print edition June, 011 This volume is based on the ontributions of several authors in eletromagneti waves propagations. Several issues are onsidered. The ontents of most of the hapters are highlighting non lassi presentation of wave propagation and interation with matters. This volume bridges the gap between physis and engineering in these issues. Eah hapter keeps the author notation that the reader should be aware of as he reads from hapter to the other. How to referene In order to orretly referene this sholarly work, feel free to opy and paste the following: Kim Ho Yeap, Choy Yoong Tham, Ghassan Yassin and Kee Choon Yeong (011). Propagation in Lossy Retangular Waveguides, Eletromagneti Waves Propagation in Complex Matter, Prof. Ahmed Kishk (Ed.), ISBN: , InTeh, Available from: InTeh Europe University Campus STeP Ri Slavka Krautzeka 83/A Rijeka, Croatia Phone: +385 (51) Fax: +385 (51) InTeh China Unit 405, Offie Blok, Hotel Equatorial Shanghai No.65, Yan An Road (West), Shanghai, 00040, China Phone: Fax: