along the vertical dimension of the aperture. The antenna meets all of the specifications aimed at.

ACKNOWLEDGMENT

The authors wish to express their thanks to Mr. N. Divakar, Director, DLRL, for constant encouragement and guidance during the work. The authors thankfully acknowledge the valuable assistance given by Mrs. M. Bharath Jyothi in the measurement of the antenna characteristics. Thanks are due to the QRMF staff for mechanical assembly of the antenna and fabrication of the test jigs. Neat typing of the paper by Mrs. N. Rama Devi and Mrs. P. R. Subba Lakshmi is also acknowledged.

REFERENCES

1.D.G. Kiely, Parabolic cylinder aerials design for very small side lobes, Wireless Eng ŽMar. 1951., 73 78.

2.A.W. Rudge and N. Williams, Design of offset reflector antennas at 30 GHz, MSN ŽMay 1978., 56 64.

3.A.W. Rudge and N.A. Adatia, Off-set parabolic reflector antennas: A review, Proc IEEE 66 Ž1978., 1592 1618.

4.M.I. Skolnik, Radar handbook, McGraw-Hill, New York, 1970.

5.L.V. Blake, Antennas, Artech House, Norwood, MA, 1984.

2001 John Wiley & Sons, Inc.

SYNTHESIS AND DESIGN OF N-ORDER FILTERS WITH N-TRANSMISSION ZEROS BY MEANS OF SOURCE – LOAD DIRECT COUPLING

Jose´ R. Montejo-Garai1 and Jesus´ M. Rebollar1

1 Departamento de Electromagnetismo y Teorıa´ de Circuitos Universidad Politecnica´ de Madrid

Ciudad Universitaria S / N Madrid 28040, Spain

Recei ed 28 No ember 2000

ABSTRACT: An extension is made to the synthesis of microwa e filters with cross couplings between nonadjacent ca ities to the case where there is a direct coupling between the input and output ports, i.e., the source and the load of the de ice. This coupling extends to N-order filters the possibility to accomplish responses with N finite transmission zeros. In addition, symmetric and asymmetric rejection slopes out of band and equalized group delay can be obtained simultaneously. Se eral examples of filter synthesis are presented to alidate the proposed synthesis proce- dure. Finally, to show the applicability of the load source direct cou- pling, a second-order Ku-band filter with two transmission zeros has been designed using three physical structures: two different rectangular wa e- guide E-plane configurations and one H-plane. 2001 John Wiley & Sons, Inc. Microwave Opt Technol Lett 29: 248 252, 2001.

Key words: filter synthesis; source load direct coupling; elliptic response

I. INTRODUCTION

The specifications for microwave filters employed for mobile communications or satellite systems are more and more demanding. These filters are characterized by transfer functions with equiripple response in the bandpass and transmission zeros at finite frequencies. If these frequencies are real Žlocated in the imaginary axis., the out-of-band rejection is controlled. On the contrary, if they are complex Žlocated in the real axis or in a complex quad., the group delay is equalized. There are different kinds of networks that imple-

ment these transmission zeros, such as cross coupling between nonadjacent resonators, extracted poles, or trisections. Nevertheless, if the synthesis of nonminimum transfer functions Žtransmission zeros in the right-half complex plane. is desired, the first kind, i.e., the cross-coupling networks, must be employed because they provide more than one path between the input and output ports 1 .

The general folded cross-coupled network, that makes it possible to implement symmetrical and both minimum and nonminimum phase responses, was first introduced by Rhodes2 . To consider asymmetrical responses, Cameron 3 added to the above network diagonal cross couplings and frequency-invariant shunt susceptances to deal with an asynchronously tuned network. It was proved by Gross in 4 that the introduction of a cross coupling produces the same number of transmission zeros as resonance circuits are bypassed. Consequently, the N-order cross-coupled network can realize a maximum number of transmission zeros Nz equal to N 2.

Several contributions that include source Žor load. to resonator coupling can be found in the technical literature: 1. a rectangular waveguide third-order filter 5 , 2. a cylindrical cavity triple-mode filter 6 , and 3. a dielectric resonator planar filter 7 . All of the above designs provide the possibility to implement N 1 transmission zeros. Furthermore, the parasitic source resonator coupling was analyzed in 8 to improve the out-of-band response.

Canonical symmetric and asymmetric circuits including the source-to-load direct coupling were suggested in 9, 10 , but no examples were shown incorporating it. Examples of N-order filters with N transmission zeros by means of load source direct coupling also have been published. A secondand a fourth-order dual-mode dielectric filter with two and four transmission zeros, respectively, were presented in 11 . Likewise, a second-order dual-mode ŽTE311 TE113. cylindrical cavity filter was realized in 12 . In the first case, empirical techniques were employed to obtain the components of circuital models. In the second one, an optimization process was followed to calculate the response. Nevertheless, no synthesis procedure was described in any of the above references to accomplish the response.

Following the statement in 4 , it is apparent that the introduction of a direct coupling between the source and the load yields the possibility of two additional transmission zeros. A synthesis procedure for N even-order symmetric filters with N transmission zeros that included the source load direct coupling was proposed in 13 .

The present work further extends the above synthesis procedure to the case of asymmetric filters by means of the generalized cross-coupled network ŽFig. 1.. The paper also shows the applicability of the source-to-load direct coupling, showing different physical structures to accomplish it, in a rectangular waveguide.

In Section II, the theoretical procedure for the filter synthesis will be described. Two filters Ževen and odd orders. have been synthesized, and their circuital values and coupling matrices calculated in Section III. Finally, in Section IV, three different structures for a second-order filter centered at 12 GHz with two transmission zeros at real frequencies have been carried out in rectangular waveguide cavities, showing the applicability of the proposed source load direct coupling.

II. THEORETICAL SYNTHESIS

The first task in the filter’s synthesis is to place in the complex plane the N transmission zeros at finite frequencies

Figure 1 Generalized cross-coupled network with source-to-load direct coupling included

to satisfy its rejection and group delay requirements. This may be done through an optimization routine, the only restriction being that they may be located symmetrically about the imaginary axis Ž j . 14 .

Once the transmission zero values are known, they are introduced in the generalized Chebyshev function 15 , and rational expressions for S21Ž s. and S11Ž s. are accomplished. The next step is to calculate the polynomials AŽ s., BŽ s., CŽ s., DŽ s., FŽ s. of the ABCD matrix from them 3 .

Figure 1 represents the generalized cross-coupled network that includes the source and the load Ža. for even-order filters and Žb. for odd-order filters . The black points represent shunt capacitors Ci and invariant susceptances Bi, the circles the input output ports, i.e., the source and the load, the continuous lines connecting the nodes represent unity admittance inverters, and the dashed lines represent admittance inverters. The operations to extract the components of this network are the following:

1.extract parallel inverter KSL

2.extract a unit element

3.extract shunt capacitor C1 and susceptance B1

4.extract a parallel inverter K1L

5.turn network

6.extract a unit element

7.extract shunt capacitor CN and susceptance BN

8.extract parallel inverter K1N

9.turn network, etc.

As can be seen, in this process, the extraction must begin with a parallel inverter. To generate the elements of the symmetric coupling matrix from the elements of the network, with all of the capacitor values normalized to unity, the following expressions have been employed:

The second one is a five-order filter with three transmission zeros located in the imaginary axis for obtaining high rejection: s1 5.1 j, s2 3.2 j, s3 2.6 j, and the other two are located symmetrically about the imaginary axis: s4, 52.2 0.15j. This combination has been chosen for ob-

taining high rejection and equalized group delay simultaneously. The required return loss level is again 25 dB. Figure 3 shows the response of the filter with a theoretical attenuation at infinity of 45.57 dB. The coupling matrix of the low-pass prototype is Ž9.:

It is interesting to note the great difference between the values of the different couplings in the matrix. This is a remarkable difference in comparison with the values synthesized when the load-to-source direct coupling is not included. For this reason, since the ratio Mi j max Mi j min increases with the order of the filter, the applicability of high-order filters may be smaller.

IV. DESIGN OF A Ku-BAND FILTER

In order to show the applicability of the proposed synthesis procedure, a second-order Ku-band filter with two transmission zeros at real frequencies has been designed. Their positions have been chosen to increase the rejection level symmetrically out of the passband. This filter can be employed as a beacon in multiplexer satellite applications. The electrical requirements are

central frequency fo: 12 GHz bandwidth: 100 MHz

return loss: 23 dB

transmission zeros: fo 300 MHz.

In accordance with the above data, the normalized transmission zeros must be located at s1, 2 j6. Figure 4 repre-

Figure 2 Coupling and routing schematic, and low-pass prototype response for the fourth-order filter

sents the low-pass prototype network for the case N 2 and a symmetrical response, where there are no invariant shunt susceptances. Figure 5 shows its theoretical response.

After applying Eqs. Ž1. Ž6., the coupling matrix is calculated, where all of the values in the antidiagonal are negatives. However, they all may be changed to positive values without modifying the electrical response. Therefore, the final coupling matrix of the low-pass prototype is Eq. Ž10.

As can be observed from this matrix Eq. Ž10. and Figure 5, an elliptical response is obtained without any change in the sign of the couplings. This is a very curious and interesting result, with practical consequences from the point of view of the physical realization of the filter because only one type of electromagnetic coupling, i.e., inductive or capacitive, must be implemented to achieve the transmission zeros.

Using the mode-matching technique and an optimization process that takes into account the partial circuital responses, three different structures have been designed using only one type of electromagnetic coupling. The first two are H-plane and E-plane folded structures, with the input output ports placed on the same side. Figure 6 shows the dimensions for both structures, where the coupling between the source and the load is easily observed Žaperture C3.. The third one is an E-plane structure, with the input and output ports placed on the opposite sides of the filter ŽFig. 7..

Figure 8 shows the theoretical response, the full-wave responses for the three implementations calculated by mode matching, and the response for the H-plane folded structure calculated by the commercial software Agilent-HFSS. The agreement is excellent.

V. CONCLUSION

The introduction of direct coupling between the source and the load of the generalized cross-coupling network makes it possible to synthesize N-order microwave filters with N finite transmission zeros. The only restriction on the location of these transmission zeros in the complex plane is that they must be symmetrically placed about the imaginary axis.

Figure 3 Coupling and routing schematic, and low-pass prototype response for the fifth-order filter

Figure 4 Low-pass prototype for the second-order filter

Figure 5 Coupling and routing schematic, and low-pass prototype response for the second-order filter

Figure 6 Longitudinal section of the folded H-plane and E-plane filters. Folded H-plane filter dimensions Žin millimeters.: A 19.05, B 9.525, L1 4.54, L2 2.29, L3 0.27, t1 2.99, t2 3.8, S 2.57, C1 9.74, C2 11.5, C3 20.17. Folded E-plane filter dimensions Žin millimeters.: A 19.05, B 6.50, L1 3.84, L2 15.61, L3 1.76, t1 1.07, t2 1.34, S 5.57, C1 0.73, C2 0.38, C3 9.53

Figure 7 Longitudinal section of the in-line E-plane filter Ždimensions in millimeters.: A 19.00 Žwaveguide width., B 13.00, B1 14.75, B2 10.07, t1 3.80, t2 1.60, C1 1.91, S1 0.93, C2 8.96, S2 1.10, C3 3.74, L1 7.72, L2 15.22

Figure 8 Comparison between the full-wave responses of the three designed filters and the theoretical response: theoretical Ž ., folded H-plane filter Ž ., folded E-plane filter Ž ., in-line E-plane filter Ž ., Agilent-HFSS folded H-plane filter Ž .