Figure 10. Analyzed frequency response of a constant diameter single rod filter analysis by MICROSTRIPE w4x., a s 22.86 mm, b s 10.16 mm; rod radii: r s 3.21

mm; h1 s h7 s 5.384 mm, h2 s h6 s 8.355 mm, h3 s h5 s 8.963 mm, h4 s 9.053 mm; distances between

rods: l1 s 22.170 mm, l2 s 24.609 mm, l3 s 24.893 mm.

impossible, because the structure does not permit any kind of pole extraction. We subsequently analyzed the filter using the mode matching technique w12x. Figure 11 shows the results of mode matching analysis.

In the preceding examples the K inverters were modeled based on FDTD analysis MICROSTRIPE w4x., but the designed filters were analyzed using the finite element method and the mode matching method. This establishes our con-

Figure 11. Analyzed frequency response of a constant diameter single rod filter analysis by the mode matching method w11x., a s 22.86 mm, b s 10.16 mm; rod radii: r s 3.21 mm; h1 s h7 s 5.215 mm, h2 s h6 s 8.172 mm, h3 s h5 s 8.863 mm, h4 s 8.971 mm; distances between rods: l1 s 21.895 mm, l2 s 24.461 mm, l3 s 24.797 mm.

ACKNOWLEDGMENTS

The authors are grateful to the Natural Sciences and Engineering Research Council of Canada and Motorola Ceramic Products, USA for their support in this work.

APPENDIX

One rod filter with constant diameter,

a q c ln z q e ln z .2

y s 1 q b ln z q d ln z .2 q f ln z .3 ,

where y is hra and z is K. Thus,

as a1 q b1 ln x q c1 ln x .2 q d1 ln x .3 q e1 ln x .4 q f1 ln x .5 q g1 ln x .6 q h1 ln x .7 q i1 ln x .8 q j1 ln x .9

q k1 ln x .10 ,

bs a2 q b2 ln x q c2 ln x .2 q d2 ln x .3 q e2 ln x .4 q f2 ln x .5 q g2 ln x .6 q h2 ln x .7 q i2 ln x .8 q j2 ln x .9

q k2 ln x .10 ,

cs a3 q b3 ln x q c3 ln x .2 q d3 ln x .3 q e3 ln x .4 q f3 ln x .5 q g3 ln x .6 q h3 ln x .7 q i3 ln x .8 q j3 ln x .9

q k3 ln x .10 ,

d s a4 q b4 ln x q c4 ln x .2 q d4 ln x .

qe4 ln x .4 q f4 ln x .5 q g4 ln x .6

qh4 ln x .7 q i4 ln x .8 q j4 ln x .9

qk4 ln x .10 ,

e s a5 q b5 ln x q c5 ln x .2 q d5 ln x .

qe5 ln x .4 q f5 ln x .5 q g5 ln x .6

qh5 ln x .7 q i5 ln x .8 q j5 ln x .9

qk5 ln x .10 ,

f s a6 q b6 ln x q c6 ln x .2 q d6 ln x .

qe6 ln x .4 q f6 ln x .5 q g6 ln x .6

qh6 ln x .7 q i6 ln x .8 q j6 ln x .9

qk6 ln x .10 ,

where x is frfc. See Table A.I. Phase w one rod filter.,

3

3

3

y s a q bz q cz 2 q dz3 q ez 4 q fz 5 q gz6 ,

where y is f and z is hra. Thus,

where x is frfc. See Table A.II.

Double rod filter with constant diameter,

a q c ln z q e ln z .2

y s 1 q b ln z q d ln z .2 q f ln z .3 ,

where y is dra and z is K. Thus,

a1 q c1 x q e1 x 2 q g1 x 3

a s 1 q b1 x q d1 x 2 q f1 x 3 q h1 x 4 ,

a2 q c2 x q e2 x 2 q g2 x 3

b s 1 q b2 x q d2 x 2 q f2 x 3 q h2 x 4 ,

a3 q c3 x q e3 x 2 q g3 x 3

c s 1 q b3 x q d3 x 2 q f3 x 3 q h3 x 4 ,

a4 q c4 x q e4 x 2 q g4 x 3

d s 1 q b4 x q d4 x 2 q f4 x 3 q h4 x 4 ,

a5 q c5 x q e5 x 2 q g5 x 3

e s 1 q b5 x q d5 x 2 q f5 x 3 q h5 x 4 ,

a6 q c6 x q e6 x 2 q g6 x 3

f s 1 q b6 x q d6 x 2 q f6 x 3 q h6 x 4 ,

where x is frfc. See Table A.III. Phase w double rod filter.,

y s a q bz q cz 2 q dz3 q ez 4 q fz 5 q gz6 ,

q h1 x 4 q i1 q j1 x 5 , x 4

where x is frfc. See Table A.IV.

One rod filter with variable diameter,

y s a q b ln z q c

2

d1

ln x .3

d2

ln x .3

where x is frfc. See Table A.V.