Field theory based design and comparison of two kinds of quasi-planar bandpass filters

Field theory based design and comparison of two kinds of quasi-planar bandpass filters

Y.Tikhov J.H.Ko Y.K.Cho

Indexing umns: Quusi-planar bundpuss ,fillers, Electrodynuinirul unulysis, A 1goritlm.s

Abstract: The paper presents a rigorous algorithm for the electrodynamical analysis and design of two kinds of E-plane waveguide quasi-planar filters. The theoretical analysis is based on the generalised multi-mode scattering matrix techniques in conjunction with the spectral- domain approach and space-domain integral equation method. The principal feature of the method consists of utilisation of complete sets of adequate basis functions corresponding to the field behaviour near the metal and dielectric discontinuities of the filter structures. Numerical and experimental results are shown for K-band bandpass filters. The calculated results agree well with experimental data.

1 introduction

Waveguide networks for microwave and millimetre- wave applications require compact, high-electrical performance and low-cost bandpass filters. To meet these criteria, various quasi-planar filters [ 11 have been widely used. Such filters are composed of planar circuit elements sandwiched between the rectangular waveguide shells parallel to the E-plane. There are two principal classes of quasi-planar filters: one class (of conventional structure) containing several waveguide resonators separated by metal inductive strips, and the other class with non-touching capacitive diaphragms placed in the evanescent-mode waveguide. The conventional structures with inductive strips typically include dielectric substrate to support a thin metallisation (5- 35 pm thickness). Relatively thick inductive strips (50- 200pm thickness) do not require a supporting slab as long as the insert is clamped in both broad walls of the filter housing. On the other hand, quasi-planar structures of the evanescent-mode waveguide type always technologically include dielectric material to support the non-touching diaphragms.

Since both classes of filter are amiable to precision and low-cost photolithographic techniques, they are highly suitable for mass production, Accurate manufac- turing techniques require an adequate accurate filter 0 IEE, 1998 IEE Proceedings online no 19982256 Paper first received 6th Jdnuary 1998 and in revtsed form 26th June 1998 The authors are with Kyungpook National University, Department of Electronics, Taegu 702-701, Korea

design algorithm, which makes unnecessary the difficult task of physical fine-tuning. To allow accurate CAD, it is necessary to use rigorous full wave analysis of the filters. Moreover, the effectiveness of the rigorous analysis procedure should be maximised because of the iterative nature of any optimisation approach used for CAD of the filters.

Analysis algorithms have been reported based on electrodynamical theory [2-61. Among these methods, the mode-matching method has been applied only to metallic E-plane circuits without dielectric substrate [2] or to bilateral structures E3, 41. A residue-calculus technique and generalised scattering parameter method has been used [5] but for a case of infinite longitudinal size of dielectric slab only. In fact, there are always dielectric edges in real filter structures. A two-resonator evanescent-mode filter has been designed successfully by using the spectral-domain approach, mode-matching method and generalised scattering parameter technique [6]. However, the filter structure [6] contains capacitive slots located near the waveguide wall. In this case, it is technologically very difficult to obtain precise slot sizes. Evanescent-mode waveguide filter structures with centrally located slots are more advanced in the manufac- turing sense.

waveguide --<-fl

7- escent-mode

Fig, 1 c upacitive diaphragms

Structure uf evunescenf-mode bvaveguide $her with non-touthmg

This paper presents efficient CAD-oriented analysis algorithms developed for the two principal kinds of quasi-planar filter mentioned above. The model structure of the evanescent-mode filter containing diaphragms with the centrally located slot (Fig. l ) and the structure of the conventional filter (Fig. 2) include dielectric slab edges. The rigorous analysis for the two kinds of real filter structure is based on the generalised

44 1 IEE Proc -Microw Antennas Propug., Vol. 145, No. 6. December I998

multi-mode scattering matrix techniques, in conjuric- tion with the spectral-domain approach and the space- domain integral equation method. Since the basis furic- tions explicitly take into account the appropriate edge conditions near the discontinuities, such as conductor edge and dielectric edge, rapid convergence and high accuracy are guaranteed. The analysis and design can be applied for an arbitrary number of resonators and in a wide frequency range. Numerical and experimental results for the K-band bandpass filter of the two kinds of structure are presented.

Fig.2 Structure of conventional filter with inductive strips

2 Theory

Let us begin with a brief description of an application of the generalised multi-mode scattering matrix techniques for full wave analysis. The matrix descriptors for the structure's discontinuities chosen by decomposition scheme are then obtained. These discontinuities are the double-plane waveguide junction, the junction between the hollow and dielectrically loaded waveguides, the non-touching diaphragms placed on a dielectric substrate in the E-plane, and the strip connecting the top and bottom waveguide wall.

A key point in suggested theory is the choice of the weighted Gegenbauer polynomials as the basis fu nc- tions for the waveguide junction problems (space- domain integral equation method) and the basis Bessel functions for E-plane circuit element problems (spectral-domain approach). The principal feature of the chosen basis functions consist of utilisation of prior explicit information about field behaviour near the conducting and dielectric discontinuities of the filter structures.

2. I Application of generalised scattering matrices for full wave analysis: decomposition schemes The present electrodynamical analysis takes into account higher order mode interaction of the structure's discontinuities by use of generalised scattering matrix techniques, introduced by Pace and Mittra [7]. In fact, this technique is a mathematical description in matrix form of multiple-reflection phenomenon for all eigenmodes. The multi-mode generalised scattering matrix for each waveguide discontinuity relates all the excited modes to the incident modes. Theoretically, the generalised matrix is of infinite dimensions corresponding to the infinite number of eigenmodes. In practice, however, the matrix is truncated to finite size for numerical calculations.

A very important point in the application of this technique is the optimal choice of the decomposition scheme for the filter structure. Fig. 3 shows a longitudinal section and decomposition scheme of the conventional filter structure. As a general case, the structure consists of n - 1 resonators, whose physical lengths are

442

indicated by t2, t3, ..., in, These resonators are separated by printed inductive strips, whose widths are indicated by wl , w2, ..., w,. t l( tn+,) indicates the distance between the dielectric edge and first (nth) strip, respectively.

In the first step, referring to Fig. 3, we consider a substructure 1 which consists of strip wI , uniform waveguide segment t l and a junction between a semi- infinite hollow waveguide and a waveguide loaded by a dielectric slab. With the knowledge of the scattering parameters for single discontinuities and the uniform waveguide segment, the generalised scattering matrix techniques can be applied to obtain the matrix SI of the substructure 1. In the next step, we consider the substructure which consists of substructure 1 plus segment t2 with strip w2. The same procedure is repeated until the overall matrix S for filter structure is obtained.

waveguide broad walls

dielectric E-plane strip edge

Fig.3 Lunftudinal se!tionul view Cnd th? decomposition scheme of the conventional ilter with E plune inductive strips

double-plane junction non-touching diaphragms

1 ?

electric or magnetic wall Fig. 4 evunescent-mode waveguide filter

Longitudinul sectiorzul view and the decomposition scheme of the

A similar approach has been applied to the decomposition and the recomposition analysis of the evanescent- mode waveguide filter structure shown in Fig. 4. In this case, the double-plane waveguide junctions, hollow evanescent-mode waveguide portions, dielectric slab edges, E-plane non-touching diaphragms and dielectrically loaded waveguide uniform portions are included in the decomposition scheme. The analysis may be sim- plified considerably by the application of the symmetry principle. For the even TE( filter excitation case, desired multi-mode matrices can be obtained with respect to the new equivalent structure with the electric wall, as shown in Fig. 4. In the odd excitation case (where TE(TIW),~,+~ modes are incident), it is reasona- ble to consider the equivalent structure with the magnetic wall placed in the geometrical symmetry plane of the filter. Although the model with the electric wall symmetry for the standard TEl0 filter excitation has

IEE Proc -Micioic Antennas Propag , Vol 145, No 6 , Deceniber 1998

been used for practical calculations, the more general nonsymmetrical case is shown below for the arbitrary eigenmode excitation problem. Such an approach is of great universal value. For example, the structures with the nonsymmetrical input ports (double-plane waveguide nonsymmetrical junctions) can be used in the multiplexers as the channel filters.

It should be also noted that, unlike the general case, both the real filter structures are usually of symmetrical topology with respect to the input (output) waveguide ports. Thus, it is unnecessary to use the above procedure strictly step by step until the overall matrix to be desired is obtained. Independently of the odd or even number of the resonators, in the symmetrical case we can decompose the filter structure into three portions. Note that two portions at both ends are inverse. There- fore, the overall scattering matrix can be obtained by the recomposition procedure in considerably fewer operations than in the general case.

xT I I

y ? I I I

a b Fig. 5 , Double-plane wuveguide junction U Longitudinal xOz sectional view b Longitudinal yOz sectional view

2.2 plane waveguide junction The general case of nonsymmetrical double-plane waveguide junction is shown in Fig. 5a and 6, where the correlation between geometrical parameters of the structure is arbitrary and geometrical parameters c and d can be positive as well as negative. The generalised multi-mode scattering matrix of this junction can be obtained by solving the three-dimensional vector diffraction problems formulated using the following principle: ‘each of the eigenmodes excites the entire spectrum of eigenmodes’.

The electromagnetic field csn be derived frsm the x- components of the magnetic I Ih and electric ne Hertz- ian vector potential as follows:

E = -jwp,oV x eh + V x V x fie H =jweoV x fie + V x V x Gh

Scattering matrix descriptor of double-

f i h = ( r I h , O , O ) ; fie = (ne,o,o) (1)

where time dependence expoof) is omitted. The choice of the x-components of potentials (eqn. 1 ) means using the system of LSE and LSM eigenmodes. Note that the LSEmo (LSMon) modes are equal to TEmo (TE,,) modes. Hertzian potentials for each waveguide are given by linear sums of suitable eigenfunctions satisfying the Helmholtz equation and the appropriate boundary conditions. Neglecting losses for simplicity, the potentials in the waveguides are given by the

IEE Proc.-Microw. Antennas Propug.. Vol. 145, No. 6, December. 1998

following expressions: M N

rI: = [ A W ” X P ( - ~ Y v m n 4 m = l n=O

+ Bvrnn e X P ( j Y 7 J m n 4 ] c m n ( T Y) M N

n~ = [ C ~ ~ ~ ~ ~ P ( - - ~ Y , , ~ Z ) m=O n=l

+ Dwmn exp( jyWmnz)]Qvmn(z , Y) (2)

where subscript v = 1, 2 indicates the waveguide 1 or 2; A,,,, B,,l,l( Cum,, D,,,) are unknown LSE(LSM) eigenmode amplitudes of the forward and backward waves which are related to the scattering parameters of the generalised matrix; and P,,,( e,,) are the orthogonal eigenfunctions of the LSEvl,n(LSM,m,l) fields, respectively. The eigenfunctions are represented by

Pvmn(z, Y) = N,h , ,RJm(mJn(Y) Qumn(x, Y) = NUe,,Tvm(z)Fun(~) RI,(%) = sin(m7rx/al) R ~ ~ ( z ) = sin (m7r(x - d ) / a n ) Sl,(Y) = cos(n7ry/h) S2n(Y) = cos ( n 4 Y - c ) / b 2 )

T I m ( S ) = cos(m7rz/a1) T2,(z) = cos (m7r(2 - d ) / a 2 )

F l n ( Y ) = s i n ( n w / h ) F2%(y) = sin ( n d g - c ) / h )

( 3 ) The eigenfunctions (eqn. 3) are suitably normalised by coefficients N $g), so that the power carried by a given mode is 1 W for propagating waves or j W for evanescent modes. Wavenumbers for each eigenmode must satisfy the following dispersion equation:

k2 - r;,, = (m7r/atJ2 + (n7r/bw) (4) where k = 2nlA is the wavenumber in free space. The propagation constants are denoted in eqns. 2 and 4 as

The multi-mode scattering matrix for the junction can be obtained completely by solving the set of diffraction problems. The number of such problems can be considerably decreased by utilisation of the reciproc- ity property of the structure to be analysed. Each particular diffraction problem can be conventionally reduced to an integral equation whose unknown is the tangential electric field vector B(x, y ) upon the junction plane z = 0. The integral equations are obtained by imposing the continuity condition for the tangential magnetic field components upon the junction plane as follows:

Ywnn.

y-(-W%,(z’, Y’) m,n 7J>P

x J’ (% Y)E:Xn(z> 9 ) ) OtmndzdY G

- - (-1)w’-12Ht:&,n, (xi, y’)

(5) where index p = h(e) depicts the LSE(LSM) mode, v =

443

l(2) again indicates the waveguide 1(2), primed indices V I , p’ correspond to the incident fields, E;;,, and HKiln are the tangential electric and magnetic eigenfield vec- tors upon the z = 0 plane, Ormn are the normalised constants, and G is the junction aperture rectangular area.

Note that the choice of the LSE, LSM eigenmode system is analytically convenient because the integral equation method requires us to match transverse field components and LSE modes contain no E, field components and LSM modes contain no H , transverse components. The solution of eqn. 5 can be found by employing of Galerkin’s scheme with the choice of the

where C,“ (9) are Gegenbauer polynomials.

basis

(6) The

dependence on variables x,y in eqn. 6 is shown implic- itly by means of local co-ordinate system x*, y*, with the point x* = y* = 0 placed in the centre of the junction aperture area G = p x q (Fig. 6). Such choice of the basis functions in eqn. 6 explicitly takes into account the rectangular metal edge condition for the electric field:

E(z ,y ) - e , r i z r ; + e,rzr, 3

where T . ~ , rJ. are corresponding distances from the (lis- continuity edges.

2 1

( 7 ) - - _ 1 2

yT I ‘< a1 I

I >I

I I I - d

Fig. 6 Double-plane \cawguide junciion cross xOy sectional view

Now, the integral eqn. 5 can be reduced to the system of linear algebraic equations, from which the unknown expansion coefficients CY^, Pkjkl in eqn. 6 car1 be determined. This yields the tangential electric field k(x: y ) on the junction plane. Therefore, unknown eigenmode amplitudes and scattering matrix for the double- plane junction can be calculated.

444

2.3 Scattering matrix descriptor of hollow dielectrically loaded waveguide junction The scattering matrix for a junction between the hollow waveguide and a partially filled waveguide (Fig. 7) can be obtained by solving the set of diffraction problems. Each particular problem can be solved by the integral equation method again. The electromagnetic field can be derived from the x-components of the magnetic and electric Hertzian vector potentials in the form of eqn. 1. Thus, the system of LSE and LSM eigenmodes is used. The potentials in the waveguides are given by the expressions similar to eqn. 2 with the suitable choice of the orthogonal eigenfunction set. The integral equations are formulated relative to the unknown. dis- tribution functions for the tangential component H,(x, y ) of the magnetic field and the E r(x, y ) component of the electric field upon the junction plane z = 0. The continuity condition for the HJ. and E,, field components upon z = 0 plane is used to obtain the desired integral equations. The solution of these integral equations can be found by means of Galerkin’s method. The standard Galerkin’s procedure requires us to expand unknown field in terms of the set of basis functions:

a Y Fig. 7 Junction hetwren holloiv waveguide und pcrriially filled wuveguide

Since there is no y-dependent distortion in the junction structure (Fig. 7), the y-dependence in eqn. 8 is the same as in the incident LSEmtnf or LSM,78,1r mode. Thus, unlike the above problem, the problem here is two- dimensional. Moreover, in the case of the LSEnIro incident mode formulation, there is no x-component of the electric field and only LSEI,,,, modes are scattered. Note that the simplest such formulation is quite sufficient for the analysis of the conventional filters with inductive strips (Fig. 2). For the analysis of the evanescent-mode

IEC Pro< -Micron Antennu> Piupug , Vol 145, N o 6. December 1998

waveguide filters (Fig. I), it is necessary to consider arbitrary LSEmfnl or LSM,,,,,,, mode as incident.

The weighted Gegenbauer polynomials are chosen as the basis functions for Galerkin’s procedure as follows:

U&) = (1 - (./dl)2)v-+ c;(x/dl)

Vl,(.) = (1 - (z/d1)2)”-i CZ(./dl)

U22 (z) = v2, (.) = (1 - (2(. - dl - d 2 / 2 ) / d 2 ) 2 )

U&) = (1 - ((. - u) /d3)2)%; ((x - a ) / & )

Vh(2) = (1 - ((. - u ) / d 3 ) 2 ) ” - i c : ((x - U ) / & )

3 = 0,2 ,4 , . . .

z = 1 , 3 , 5 , . . ?- 1

x c3” (2(. - dl - d 2 / 2 ) / d 2 )

3 = 0 , 1 , 2 , . . .

3 = 0 , 2 , 4 , . . .

z = 1 , 3 , 5 , . . . (9)

where Cg (q) are Gegenbauer polynomials. The parameter q = 2/n arccos(& - 1)/(2(~ + 1)) - 1/2 in eqn. 9 is determined from the dielectric rectangular edge condition:

H z , E z - r 2 (10) “- 1

where r means a distance from the edge. Now, the integral equations can be reduced to the

systems of linear algebraic equations. The unknown expansion coefficients alj, aIj, a3j, Bli, &, (33i in eqn..8 can be determined from these systems. This yields H , and E , field components on the junction plane. There- fore, all unknown eigenmode amplitudes of the scattered field for the given arbitrary incident can be calculated. Thus, the multi-mode generalised scattering matrix for the hollow dielectrically loaded waveguide junction can be obtained completely.

I I I

d, fid2” dg ’ \ L \ L

Fig. 8 E-plane non-touching diaphragms placed on dielectric substrate

2.4 Scattering matrix descriptor of E-plane circuit elements supported by dielectric substrate The scattering matrix for capacitive diaphragms (Fig. 8) as well as for the inductive metal strip (Fig. 9) can be obtained by the spectral-domain method. Let us consider the non-touching diaphragms’ structure first. The diaphragms are assumed to be perfectly

fEE Pvor -Microw Antennu Ptopug Vol 145, No 6, December 199X

conducting and infinitesimally thin. Let the arbitrary LSE,,.,’ or LSMmrnr mode with unit amplitude be incident. For a given incident field, the field scattered by the diaphragms can be expressed by LSE, LSM modal expansion. Suitable orthogonal eigenfunctions are piecewise functions with respect to three-layer subregions of the dielectrically loaded rectangular waveguide. On the other hand, the scattered field can be expressed in terms of the induced current JY,= on the diaphragms:

b c o

q ( . , Y , 4 = J ’ / [G,,(.,Y-%/,z-2)Jyiy,2)

E,“(.> !I, .) = / / [G&, Y - Y ? - 4 J y ( Y , .)

0 -CO

+Gyz(z,y - y,z - z )J , (? I , z ) ]d2& b c c

0 -02

(11) where GYY, GYz, G,,, G,, are components of dyadic Green’s function.

I k * & & z

Fig. 9 E-plane strip connecting the top and bottom waveguide ~vaN

Once the current is known, the scattered field can also be obtained by substitution of Jy , , into eqn. 11. Each scattered mode amplitude may.be found from orthogonality of the eigenfunctions.

The tangential incident field E& and the scattered field E;,: are summed to give the total tangential electric field EY,: at x = dl + d2 (Fig. 8). In the space domain formulation, coupled integral equations for the unknown current distributions JY,= on the diaphragms can be derived from the condition that the total tangential electric field must be equal to zero on the perfectly conducting diaphragms:

Ey(., Y, 2 ) = E;(., y, .) + E,”(& Y, 2 ) = 0 E,(z,y,z) = Ej(z,y,z) +E,”(x,y,z) = 0

~ = d l + d 2 , I z ( < w / ~ , O < y < h , b - h < y < b

(12) According to the standard spectral-domain approach [SI, the corresponding formulation is carried out in the Fourier-transformed (spectral) domain. The only rea- son to use such an approach is that dealing with the Green’s function in the space domain is not so convenient. In fact, the Fourier-transformed Green’s function in the spectral domain is given in simple closed form

44s

[9]. In the Fourier transform domain where the Fourier transform is given by

b m

F ( p n , 7) = 1 / f(Y, 2 ) exp(jpnY) exp(j?z)dz4/

E,” = Eg + E; = Ggy Jy + Gyz jZ

-b -a

pn = n7r/b; n = 0, 1 , 2 , . . . (13) the above statements become

E,” = Ez + = Gz,jy + G z z j z (14) where _tilde jndicgtes Eourier transforms of functions and G,, G,, G,, GzS are components of dyadic Green’s function in spectral domain [9].

Eqn. 14 can be solved by applying Galetkin’s method. The unknown current densities, J and J ~, in the Fourier-transformed domain defined by eqn. 13 are expanded in terms of the following basis functions:

j y ( p n , Y)

where 6 Ik , 6,. are unknown coefficients. Most suitable basis functions for the Galerkin’s procedure in spec1 ral domain for the present problem are as follows:

.i,”l ( p n ) = J L ( p n h ) / ( p n h ) ; 1 = 1 , 3 , 5 , . . .

j;k (7) = Jk (rw/2); k = 0,1 ,2 , .

j z Y ( p n ) = J z ( p n h ) ; Z = l , 3 , 5 , . . .

’ t J 7 ) = J3(7w/2)/(7w/4; j = 1 , 2 , 3 , . . . (16)

Here &(q) are Bessel functions of the first kind. In the space domain formulation, the above expressions (ecps. 15 and 16) are equivalent to following expansion of unknown current density:

JY (Y , 2 ) = Qllc Jy”l (Y) J y k (2)

L,k

i , j

where a#,, Pii are unknown coefficients. The basis functions in space domain are given by

J,”,(z) = (1 - (2z/w)2)3 U j ( 2 4 W ) Iz( < w / 2 j =0,1,2, . . .

(18) where Tt(cp) and U d q ) are Chebyshev polynomials of the first and second kind, respectively. The expansion (eqns. 17 and 18, hence, eqns. 15 and 16) takes into account the edge condition near the sharp conducting edge for current densities:

1

J L N r t , Jll - r-2 (19) Here JI (Jll) denotes the current density normal (parallel) to the edge and Y is a distance from the edge.

The application of Galerkin’s method yields a system olf linear algebraic equations whose unknowns are 6 / k , Pji. In fact, this system consists of two independent subsystems corresponding to different conditions at the z = 0 plane (electric or magnetic wall). For a given incident field, we solve these systems and find expansion c_oefficient_s in eqn. 15, which give the current densities J , and J = in the spectral domain. Hence, the modal amplitudes which are related to the scattering parameters of the generalised matrix can be found by equating the residue of both sides of the key equations in eqn. 14. This step corresponds to the use of the orthogonality relationship to find the modal amplitudes in the space domain.

In the case of the E-plane strip connecting the top and bottom wall (Fig. 9), the above formulation is sim- plified. Unlike the three-dimensional vector problem about non-touching diaphragms, the diffraction problem for the inductive strip is two-dimensional. Since there is no field variation in the y-direction and only LSE,,o modes are scattered for a LSE,., excitation, instead of eqn. 14 we have only one equation:

E; (7) = EY (7) + E; (7) = G y y (7) j y (7) (20) where Fourier-transformed quantities are functions of y only.

3 Results and discussion

The solution convergence has been evaluated first. It is found that for practical calculations it is sufficient to take into account only three or four basis functions in eqns. 6 and 15 per co-ordinate for the double-plane junction and for the E-plane circuit element problems. The choice of about 20-40 for the parameters M and N in the modal expansion in eqn. 2 has been observed to give accurate results. For the dielectric edge problem, it is sufficient to take into account a maximum of seven basis functions in eqn. 8 for each subregion. In this case, the errors for scattering coefficients corresponding to the fundamental LSElo mode are less than O.S% The errors for elements of the generalised scattering matrix corresponding to the higher order mode excitation by the higher order incident modes (LSE,,,, LSM,, with the values of integers m and n of about 4) may be within a few percent.

Owing to the generalised matrix techniques (higher order mode interaction of the discontinuities), it is typically sufficient to take into account not more than 20 eigenmodes to yield a better than 1% accuracy (convergence value) of the filter response calculated for the fundamental mode in a wide frequency range. The rapid convergence shown above is achieved by a special choice of the basis functions for Galerkin’s method.

IEE Proc.-Microw Antennas Propug., Vol. 145, No. 6 , December 1998

This is because the weighted Gegenbauer polynomials in eqns. 6 and 9 (space domain formulation) as well as the Bessel functions in eqn. 16 (spectral domain) ade- quately describe the field behaviour not only very near the edges as expressed in eqns. 7, 10 and 19, but also in the region not so near the edges. In fact, the utilisation of prior explicit information about field behaviour near the conducting and dielectric discontinuities of the filter structures guarantees rapid convergence and high accuracy of numerical results.

To verify our analysis method, an evanescent-mode waveguide filter of four resonators for broad passband application has been designed and tested. This K-band filter has following dimensions (in mm): the size of input-output waveguide is 11 .O x 5.5; the size of smaller evanescent-mode waveguide is 6.2 x 4.0; substrate (RTI Duroid E = 2.2) thickness d2 = 0.254; t l = t7 = 0.1; t2 = t6 = 0.935; t 3 = t j = 5.45; t 4 = 6.1; ~1 = ~2 = ~3 = w 4 = 0.56; SI = ~4 0.7; ~2 ~3 = 0.6.

30.0 1 I \

18.0 20.0 22.0 24.0 frequency, GHz

Fig 10 moie waueguiLfi/tLr ~ theory ~ _ _ ~ measurement

Veri ication of the fheory with rneusurenzent for the evunescent-

Fig. 10 shows the comparison between the calculated and measured insertion loss for a designed filter. The calculated centre frequency is 20.8GHz, and the experimental value is 20.76GHz. The calculated 0.5dB band- width is 2.4GHz, and the experimental value is 2.34GHz. The calculated magnitude of the ripple in passband is 0.13dB, and the measured value is about 0.5 dB because of absorption losses. Therefore, good agreement between the measurement and the theory is observed.

The conventional filter with inductive strips for narrow passband application has also been designed and fabricated. This four-resonator K-band filter has the following dimensions (in mm): waveguide of cross section 10.67 x 4.32; resonator lengths t2 = t5 = 6.746; t3 = t4 = 6.811; distance between dielectric edge and strip t l = t6 = 1.800; strip widths wl = w j = 3.060; w2 = w4 = 7.788; w3 = 8.142; substrate (RT/Duroid E = 2.2) thickness d2 = 0.254 and dimension d, = 5.081. Fig. 11 shows the calculated and measured insertion loss for this filter. It is again found that the numerical results agree well with the experimental data. However, about 0.2% (48 MHz) frequency shift of measured passband downward is observed to take place. This shift is thought to be due to the neglect of topological grooves

IEE Proc.-Microw. Antennas Propug.. Vol. 145, No. 6, December 1998

for supporting the substrate in the broad walls of the waveguide and the finite thickness (1 7 pm) of metal strips. A relatively high level of measured insertion loss can again be explained in terms of neglect of the absorption losses.

a tZ ‘3 t4 t5 t6

160’0 i\- 8 120.0 4 Y w2 w3 w4 w5

- _ - _ _ _ . _ _ _ - _ _ _ _ . _ _ _ _ _ _ _ _ _

measuring limit

= 4 0 , 0 ~ 0 16.0 18.0 20.0 22.0 24.0

.-

frequency, GHz Ver&ition of the theory with measurmzent ,for the conven-

It is well known that conventional kinds of E-plane filter often have a too low and too narrow second stopband for many applications. This is because the frequency corresponding to the second stopband becomes higher than the cut-off frequency of the fundamental mode within the metal septum section, which is determined by the distance between the inductive metal strip and the waveguide sidewalls. Thus, the power is increasingly transported directly by propagating waves. In recent years, much effort has been devoted to improve stopband performance. A review of such solu- tions has been reported [lo]. However, usually improvements are not so satisfactory for some applications, such as for multiplexers.

Fig. 11 tional filter

n -I 120.0

r . 0 2 5 0 A 30.0 35.0 40.0

K-band Ka-band \ 1 frequency, GHz

Fig. 12 range between two kin& of E-plane quasi-planur filt ers __ evanescent-mode waveguide filter __.. conventional filter

C~~rnparison of calculated insertion losses for wide frequency

To compare the stopband performances of two kinds of E-plane quasi-planar filter, the evanescent-mode waveguide filter has been designed with the same passband width as for the conventional filter shown in

447

Fig. 11. Moreover, in order to compare the filter performances under the condition of same physical length, the evanescent-mode waveguide filter has been designed for the same given length as for the conventional filter. This designed evanescent-mode waveguide filter of four resonators has the following dimensions (in mm): the size of input-output waveguide is 10.67 x 4.32; the size of smaller waveguide is 6.2 x 4.0; substrate (RT/Duroid E = 2.2) thickness d2 = 0.254; t l = t7 = 0.2; t2 = t6 =

0.86; sl = s4 = 0.892; s2 = s3 = 0.900. The calculal.ed result of insertion loss as a function of frequency including Ka-band is shown in Fig. 12, as is the corresponding result for the conventional filter. It is found that the evanescent-mode waveguide filter has considerably improved the performance for the second stopband, which ranges from about 29GHz to 40GHz. It should be noted that any location of narrow undesired passbands can be controlled, because the desired passband characteristics of the same location can be achieved by the choice of different combination sets for values of slot width s, and lengths w, of the clia- phragms.

The calculated return losses in the working passband are shown in Fig. 13. Desired passband performance of the evanescent-mode waveguide filter is preferable here. Therefore, for the given equivalent passband performance, a more compact filter design can be achieved in comparison with the conventional E-plane filters.

5.40; t 3 = t 5 14.60; t4 = 15.80; W I = W* = ~3 = lli‘ 1

60’o 1

I 1

0 I I I I I 1 20.4 20.6 20.8 21 .o

frequency, GHz ,Coniparison of calculated return losses between two kinds oj E- Fig. 13

plune quusr-plunur filters

__.. conventional filter evanescent-mode waveguide filter

Both kinds of filter have good group-delay time clnar- acteristics (Fig. 14). The group delay has been calculated by determining the slope of the phase/frequcncy characteristic with frequency increments of 5 MHz.

4 Conclusions

An efficient CAD-oriented algorithm for the analysis of two kinds of quasi-planar E-plane waveguide filter

448

has been described. The rigorous electrodynamical method considered here guarantees rapid convergence and high accuracy of numerical results. The theoretical results agree well with the experimental data. The comparison of two kinds of filter shows that performance of the evanescent-mode waveguide filter can be preferable for various applications. Fabrication costs can be reduced to a minimum by the application of low-cost precision photolitographic techniques and the accurate CAD algorithm, which makes unnecessary the difficult task of physical fine-tuning.

10.0 - -

8.0 - -

E 6.0 - - 2 U a

- 9 0) 4.0 -

-

2.0 - -

20.4 20.6 20.8 21 .o frequency, GHz

Fig. 14 two kinds of E-plme quasi-planar ,filters

~~~ .. evanescent-mode waveguide filter . _ _ _ conventional filter

Coniparison of calculated group-delay rime behaviour between

5

1

2

3

4

5

6

7

8

9

10

References

VAHLDIECK, R.: ‘Quasi-planar filters for millimeter-wave applications’, IEEE Trans., 1989, MTT-37, (2), pp. 324-334 VAHLDIECK, R., BORNEMANN, J., ARNDT, F., and GRAUERHOLZ, D.: ‘Optimized waveguide E-plane metal insert filters for millimeterwave applications’, IEEE Trans., 1983, MTT- 31, (I), pp. 65-69 ARNDT, F., BORNEMANN, J . , GRAUERHOLZ, D,, and VAHLDIECK, R.: ‘Theory and design of low-insertion loss fin- line filters’, IEEE Trans., 1982, MTT-30, (2), pp. 155-163 ARNDT, F., BORNEMANN, J., VAHLDIECK, R., and GRAUERHOLZ, D.: ‘E-plane integrated circuit filters with improved stopband attenuation’, IEEE Trans., 1984, MTT-32,

SHIN, Y.C., ITOH, T., and BUI, L.Q.: ‘Computer-aided design of millimeter-wave E-plane filters’, IEEE Trans., 1983, MTT-31, (2), pp. 135-142 ITOH, T., and KONG, K.S.: ‘Computer-aided design of evanescent mode waveguide bandpass filter with non-touching E-plane fins’, IEEE Trans., 1989, MTT-37, (12), pp. 1998-2004 PACE, J., and MITTRA, R.: ‘Generalized scattering matrix analysis of waveguide discontinuity problem’ in ‘Quasi-Optics XIV’ (Brooklyn Press, New York, 1964), pp. 172-194 ZHANG, Q., and ITOH, T.: ‘Spectral-domain analysis of scattering from E-plane circuit elements’, IEEE Trans., 1987, MTT-35,

ITOH, T.: ‘Spectral-domain immittance approach for dispersion characteristics of generalized printed transmission lines’, IEEE Trans., 1980, MTT-28, (7), pp. 733-736 BUDIMIR, D.: ‘Optimized E-plane bandpass filters with improved stopband performance’, IEEE Trans.. 1997, MTT-45,

(lo), pp. 1391-1394

(2), pp. 138-150

(2), pp. 212-220

IEE Proc.-Microw. Anfennus P r o p q . Vol. 145, No. 6. Drcemher 1998

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Field theory based design and comparison of two kinds of quasi-planar bandpass filters