DESIGN RELATIONS FOR THE WIDE-BAND WAVEGUIDE FILTER

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!QS() PROCI:h.DI M.'S OF THE I.R.E.
709
Design Relations for the Wide-Band WaVeguide Filter*
SEYMOUR B. COHNt, MEMBER, IRE
Summary?Design formulas are derived and presented graphi-
cally for a wide-bemd waveguide filter structure analyzed in a previous tanh
paper. The design procedure is outlined and a brief example given.
Experiments indicate that the design cutoff frequencies and the in-
finite-attenuation frequency may be relied upon within one or two
per cent.
tin-1F
_
+ E [- -
2b IS0(6) Ii I
-1 jf, (5)
sin' irrib
X g 17-2 n>o -it(irii5)2 ?
HE WAVEGUIDE FILTER structure consid-
. ered in this paper is shown in Fig. 1. This struc-
ture is most suitable for pass-band widths of the
order of 1.5 to 1. The low-frequency cutoff./ is provided
by the natural cutoff of the waveguide, while the high-
frequency cutoff ft is provided by the constrictions and
cavities. The ? attenuation-versus-frequency response is
sketched in Fig. 2. An accurate solution for this filter is
given in a previous pnper.1 The formulas of that pub-
lication, however, are too unwieldly and contain too
many parameters to enable simple application to a filter
design problem. In this paper, it will be shown how the
original equations may be simplified and represented
graphically with very little loss of accuracy.
THE ORIGINAL FORMULAS
The following formulas for the image parameters are
given in the paper on the analysis of this filter.'
yr =
(1)
e .= a ji9 = 2 tanh-' (2)
where yl is the image admittance of the filter, ye,. and
yo, the short- and open-circuit admittances of a half
section, 0 the image transfer function of one section, a
the image attenuation function in nepers, and fl the
image phase constant in radians. All admittances are
normalized with respect to the characteristic admit-
tance of the rectangular-waveguide portions of the filter
whiCh are of height b and width a (Fig. 1). If a is held
constant, the characteristic admittance of the guide is
inversely proportional to b. Hence, the normalized char-
acteristic admittance of the terminating line is equal to
b/br. The half-section admittances are given by
yo. = tan [--wi' -1- tan-1
5 (?Ili
X, '
j by.,?)]
Yre = ? tan tan-1 ?_
6 LX0
where
(3)
(4)
tan?
*
A,
Decimal classification: R386.1 X R118. Original manuscript re-
ceived by the Institute, September 20, 1949.
t Sperry Gyros&pe Company, Great Neck, L. 1., N. Y.
S, B. Cohn, "Analysis of a wide-band waveguide filter," PROC.
IRE., vol. 37, P. 651; June, 1949.
FliTs ?s'
7r/
?j COL ?
X
2h
+j?.
X,
mill?
[coth -
b sin' irub
-----1 ? - - jf,, (6)
n [
F = 1
? 0.09b/X,
(7)
(8)
where b is the ratio b' /b, and h', and I are dimensions
shown in Fig. 1. X, is the guide wavelength of a uniform
rectangular guide of width a. .5 a.(6). is the Hahn function
of zero order, which is tabulated by Whinnery and
a
(a)
(b)
Fig. 1?The broad-band waveguide filter. The design information
applies to both (a) and (b).
Jamieson.' Their value Lot ti= 0.05 is in error, however,
and should be S.(0.05)=26.23. For h 0.15, the follow-
ing formula for US) is within 0.4 per cent.
1
ir" ( log, ? - 0.138). (9)
6
The correction term of (8) is sufficiently accurate if kis
less than about 0.15, which is the case in the design of
this type of filter. More accurate expressions for e are
given in (42) and (43) of footnote reference!.
FORMULA5, FtOlt:tHE FILTER PARAMETERS
In the design of .a filter, one must first decide where
to place the cutoff and infinite-attenuation frequencies,
J. R. Whinnery and Ft. W. Jamieson, "Equivalent circuits for
discontinuities in transmission lines," Paoc. IRE., vol. 32, pp. 98-
115; February, 1944.
- ? --
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80Q PkOCF.IWINGS THE IRE.
and w hat value of terminating resistance to use. For a
waveguide filter section, these parameters are more con-
VelliClIfly expressed by the wavelengths X, )10, and
and by the height br of the terminating guide. X,.( = 2a)
is the cu toll wavelength of a rectangular guide of width
a, X?1 the cutoff guide wavelength of the litter structure,
and X,? the intinite-rejection guide wavelength (Fig. 2).
When a filter is being designed, it is necessary to obtain
Fig. 2 -Attenuation-versus-frequency revorise.
the filter dimensions which correspond to these given
parameters. The difficulty of doing this by means of (1)
through (8) is obvious. Ilence it is necesgary to change
the form of these equations so that a straightforward de-
sign procedure will be possible. This is done below.
? Let b0 be the teiminating-guide height which would
match the filter at fe (i.e., for X--3.0). Then, by (1),
b
11111 (y1)' ?-? (Jim y?, lim
x?
When (3), (4), (5), and (6) are substitTited and the limit-
ing process arried out, one obtains'
bk) 2
I_ I \ i 1 + .I'__4 ____2_/) s?(6)
/ + be )1. 61 701 .
91) ,_., hit/i sin2 rItt5 NO
I i ? taut'- -. I- -- 1- - - ? . (10)
71-1 ,..-..0 b n(xitii)2 7r1
The actual height of the terminating guide need
not be equal to b.. It will generally be chosen Co give a
perfect match at some point. within the pass band, as
explained later in the Design Procedure section.
Next, an implicit relation for the cutoff wavelength
w ill be obtained. This cutoff occurs when By
means of (4) and (6), therefore, one obtains
b
cot it - - tan in ?
A,"
b A u1 b
I b I r' l b
it - - it' - ? -
b A?i b A111
tirl
coth \,(1 ? (b/ itA0)2-
)b b SI112 r116
i , E
rt ?_.0L V I - (6/ ttA ,,j)' J i1(irm5)2
3 Tht!Se ,acps are perfumed in detail iii "A Theoretical and Fx-
perimental Study of a Wayeerride Filter Structure," by S. It. Cohn,
Otlire of Naval lesearch, ('cult Laboratory, Harvard University,
Vt.vort. Nth 39, Apt it 25, 1948.
it
Agi
The largest value of Xo which satisfies this equation
yields the desired cutoff xya veleng
Lastly, an implicit relation for the infinite-attenua-
tion wavelength will be derived. By (2), the attenua-
tion a is infinite for y,,,=y?. When (3) and (4) are set
equal, and (5) and (6) substituted; One obtains'
csc (27r 1 ---1t)
b X?,G
Equations (10), (11), and (12) relate the critical wave-
lengths and the matching height to the filter dimensions,
but they are still too complex for convenient use. It will
now be shown how they may be put in graphical form
with very little loss of accuracy.
THE DESIGN GRAPHS
(12) contains three parameters, bA,?? l/b,
and b. It is plotted in Fig. 3, where it is seen to be inde-
pendent of b for 1/b>0.2. For //btc, and X0 of the above exam-
ple. The infinite-attenuation frequencies are, however,
from left to right 3,500, 4,500, and 5,500 Mc. By having
sections with different f?, values, the insertion loss may
'be kept high in the passband, and also the spurious
response of each section near b/Xo= 1 will fall in thc
stop band of another section. A further discussion on
the removal of spurious responses is given in the litera-
ture for a similar type of waveguide filter.'
6 See pp. 734-736 of footnote reference 4.
.
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Pk()('h:/.3)/.VGS 811i
ili'foro connccl in:, the end sections to I he iJli
gnitit', the discontimiiIN tap ta
at itic1' dill' 10 Ili(' jIII)( !in
n
of heights b and b' In compensated. This may be
done to a close approximation by shortening the length
of the ctmsta iction at each end pa the tiller front /'/2 to
l'/2 1\ here 61' is given by'
= (log, hr. ? 0 ? 38O) . (18)
6(b) shows the assembled filter, after correction of
?,?
the end constrictions.
t L
(a)
Vrtot,ICATIoN AccultAcv
Six individnal sections terminating in 2.75 X0.375-
inch guide were constructed wit'h a wide range of physi-
cal parameters. The J., cu oil' frequencies wei-e 'calculated
from (11), except for filter numb'er 5. and also from the
design graphs. They were measure(lt by two different
methods which hecked each other witi,tin 0.3 per cent.'
The various cutoff-frequency- values are listed in Table I.
The data in the "Tests" column are averages for the
two tests.
TABLE I
VALUES OF ft
Filter No. Equation (II) Design Curves
'rest
2
3
4
2,993 Mc
3,003
2,834
3,000
5
6 2,995
3,000 Mc
3,000
2,843
3,016
3,000
3,000
2,980 Mc
2,991
2,822
3,001
3,024
2,991
In each case, the spread of fi values is less than one
per cent. The f., values were calculated by (12) and also
measured, and these too agree within one per cent.
Since fa is the natural cutoff frequency of the wave-
guide, which is known exactly, it need not be checked.
The approximations used in obtaining (13) and Fig. 4
cause an error in the image admittance of up to 5 or JO
per
which is
too small to be of conse-
cent,
OM.
I
-4??
generally
quence. If the correction factor El -1-6/70/2 mentioned
above in the Design Graph section is used, this error can
be greatly reduced. ?
ACKNOWLEDGMENT
The writer wish-es to acknowledge the many helpful
suggestions of R. W. P. King and L. Brillouin toward
the preparation of this paper.
Fig. ??A three-section
(b) after assembly.
0340 ??4? 01611'
?las'
(b)
waveguide filter, (a) before assembly and
(ft =3,000 Mc, a=2.75 inches.) '
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