Title:

United States Patent 3577104

Abstract:

A compact microwave band-pass filter utilizes a sequence of thick capacitive irises stationed along the interior of a waveguide to form a series of directly coupled resonant chambers. The spacing between consecutive irises is λ_{g} /4 or less, where λ_{g} is the upper cutoff frequency of the filter. The close iris spacing causes the second harmonic pass band for the fundamental mode to be far above the main pass band.

Application Number:

04/786967

Publication Date:

05/04/1971

Filing Date:

12/26/1968

Export Citation:

Assignee:

Microwave Development Laboratories, Inc. (Needham Heights, MA)

Primary Class:

Other Classes:

333/175, 333/202, 333/251, 333/252

International Classes:

Field of Search:

333/76,73,73 (W)

View Patent Images:

US Patent References:

3196339 | Microwave harmonic generator and filter element therefor | July 1965 | Walker et al. | |

3027525 | Microwave frequency selective apparatus | March 1962 | Salzberg | |

2787766 | Filter for electric line system | April 1957 | Scheftelowitz | |

2740094 | Wave-guide impedance elements | March 1956 | Fox |

Primary Examiner:

Lieberman, Eli

Claims:

I claim

1. In a microwave band-pass filter having a sequence of irises positioned in a waveguide to form a series of directly coupled resonant cavities, the improvement wherein

1. In a microwave band-pass filter having a sequence of irises positioned in a waveguide to form a series of directly coupled resonant cavities, the improvement wherein

Description:

This invention relates to frequency selective passive apparatus for filtering signals which are in the microwave portion of the electromagnetic frequency spectrum. More particularly, the invention pertains to a structure utilizing thick, capacitive irises which are closely spaced along the interior of a waveguide to form a compact band-pass filter that exhibits good harmonic rejection.

DISCUSSION OF THE PRIOR ART

Frequency filters formed by capacitive irises spaced along the interior of a waveguide to form a sequence of resonant cavities are described in "Maximally Flat Filters in Waveguide," by W. W. Mumford, Bell System Technical Journal, Vol. 27, Oct. 1948 and in "Design of Tunable Resonant Cavities With Constant Bandwidth" by L. D. Smullen, Proc. IRE, Vol. 37, Dec. 1949. The capacitive iris waveguide filters are principally considered to be variants of the more common filters having inductive irises in the waveguide which form resonant cavities that are each approximately λ_{g} /2 in length, where λ_{g} is the wavelength in the guide. The resonant cavities in inductive iris filters are somewhat shorter than 180° in electrical length (that is, somewhat less than λ_{g} /2 ) whereas the resonant cavities in a capacitive iris filter are somewhat longer than 180°. In the capacitive iris filter, the bandwidth remains fairly constant when the filter is tuned over a wide frequency range, and the rate of cutoff is more rapid at the high frequency edge of the band compared to the low frequency band edge cutoff rate. For inductive iris filters, the reverse situation is encountered both for the cutoff rate at the band edges and for tuning over a wide frequency range. The disadvantages of a half-wavelength capacitive iris filter arise from the difficulty of producing large capacitive susceptances in waveguide. Because of that difficulty, a structure employing capacitive irises to form direct-coupled resonant cavities is not generally deemed to be a feasible type of narrow band filter and in a capacitive iris, direct-coupled, resonant cavity, waveguide filter of moderate bandwidth the harmonic rejection capability is little better than in the inductive iris filter because the resonant cavities are long and consequently possess harmonic "windows" (viz, harmonic pass bands) at comparatively low frequencies above the main pass band.

Where two irises form a resonant cavity in a conventional waveguide filter, the shortest length of the cavity is between one-half to three-quarters of the guide wavelength λ_{g}. It is known from theoretical considerations that the spacing between the capacitive irises may be between zero and one-fourth λ_{g} but the interacting fringing effects associated with the close spacing of the two irises is deemed to make such close spacing impracticable. See, for example, the discussion in Principles and Applications of Waveguide Transmission, by George C. Southworth, pp. 286 to 287, published by Van Nostrand.

In U.S. Pat. No. 3,027,527, there is disclosed a waveguide band-pass filter, invented by Edward Salzberg, which has capacitive irises spaced along the waveguide at one-quarter wavelength (λ14) intervals. In the Salzberg filter, a resonant iris is situated midway between adjacent capacitive irises. The reduction thus achieved in the length of the cavity has resulted in an improvement in harmonic rejection in addition to providing a more compact structure.

THE INVENTION

The invention here disclosed provides better performance than can be obtained with the Salzberg arrangement while retaining the compactness of the filter. In the invention, thick capacitive irises are disposed along the interior of a waveguide to form a series of directly coupled resonant chambers. The spacing between consecutive irises is approximately between one-quarter (1/4 ) to one-eighth (1/8 ) of λ_{g} , the wavelength in the guide at the upper cutoff frequency of the filter. Because of the small spacing between adjacent irises, the second harmonic pass band for the fundamental mode is in a frequency region that is far above the main pass band. Despite the close spacing of the irises, mutual interacting fringing effects have, in general, been found to be quite small and such effects are taken into account by means of a proximity factor. The performance of the filter can be realistically predicted by simple cascade analysis.

THE DRAWINGS

The invention, both as to its construction and mode of use, can be better understood from the following exposition when considered in conjunction with the accompanying drawings in which:

FIG. 1 symbolizes a cavity formed in a waveguide by capacitive irises spaced apart by an electrical length Θ;

FIG. 2A is a diagram of a filter having n directly coupled cavities formed by a sequence of capacitive irises spaced along a waveguide;

FIG. 2B symbolizes the distributed low-pass prototype of FIG. 2A in impedance inverter form;

FIG. 3 represents an n -cavity filter with generalized symmetrical lossless networks;

FIGS. 4A and 4B depict a thick capacitive iris in a rectangular waveguide;

FIG. 4C symbolizes the equivalent electrical circuit of the thick capacitive iris;

FIG. 5 depicts various capacitive iris configurations;

FIG. 6 is a view of an embodiment of the invention with part of the waveguide broken away to expose capacitive irises;

FIG. 7 is a sectional view taken along the plane 7-7 in FIG. 6;

FIG. 8 is a top view of the FIG. 6 embodiment with the upper broad wall of the waveguide removed to expose the interior of the filter; and

FIG. 9 is a graph comparing actual filter performance with theoretically predicted performance.

THE EXPOSITION

FIG. 1 symbolically depicts a single cavity formed by a pair of capacitive irises 1 and 2 spaced in a waveguide 3. Each capacitive iris is characterized by a normalized susceptance jb and the irises are spaced by an electrical length Θ. The insertion loss of the cavity is given by

where P_{i} is the power available from the power source

P_{o} is the power transmitted through the cavity to a load. Resonant frequencies of the cavity occur when

Because b is positive for a capacitive iris, the first resonance occurs where

For expository purposes, the filters considered herein have been given susceptance values in the range 0.5 to 4 and the corresponding resonant lengths are therefore in the range approximately 25° to 75°. The second harmonic resonances consequently are in the 205° to 255° range which is far removed from the fundamental range.

Consider now the waveguide filter in FIG. 2A having n cavities formed by capacitive irises spaced along the interior of the waveguide to form cavities that are π/2 or less in electrical length. The theory presented in my monograph "Theory of Direct-Coupled Cavity Filters," IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-15, No. 6 June 1967, is not completely applicable here because the cavities in the FIG. 2A filter are appreciably less than a half-wavelength and that filter is low-pass rather than band-pass. The susceptances of the irises in the FIG. 2A filter are, however, determined in a manner similar to that set forth in the aforesaid monograph. The susceptances are determined by equating the VSWR (the voltage standing wave) ratio of each reactive coupling discontinuity of the filter (i.e., each iris) to the corresponding junction VSWR V_{} i of a distributed low-pass prototype filter. See my monograph "Table of Element Values For the Distributed Low-Pass Prototype Filter," IEEE Transactions on Microwave Theory and Technique, Vol. MTT-13, No. 5, Sept. 1965. FIG. 2B shows the distributed low-pass prototype filter of FIG. 2A in impedance inverter form. For a discussion of the concept of the impedance inverter see "Direct-Coupled Resonator Filters" by S. B. Cohn, Proc. IRE, Vol. 45, Feb. 1957. The FIG. 2B prototype leads to the equation

This gives an exact realization of the upper cutoff frequency of the filter, if the susceptance values are chosen to satisfy equation (3) at that particular frequency and if the spacings between irises are properly chosen. The spacings between irises should all be equivalent to an effective electrical length of Θ_{} o at the upper cutoff frequency. The actual electrical length between any pair of adjacent irises is greater than Θ_{} o because it necessary to measure the effective spacings from reference planes on either side of the irises as indicated in FIG. 2A to account for the phase shift introduced by the irises.

The transfer matrix of any iris of normalized susceptance jb shunting a line of electrical length φ at its midpoint is given by ##SPC1## Equation (9) can be extended to cover the case of capacitive irises whose thickness is such that the irises cannot be represented by simple shunt susceptances. The extension covers the general case represented by the filter shown in FIG. 3 which has n cavities and in which each iris has been replaced by a network T_{} i which may be any symmetrical reciprocal lossless two-port network. The transfer matrix of any such network T, including adjacent lengths of line φ/2 on either side is given by ##SPC2##

The condition that this shall be equivalent to an impedance inverter, as in equation (6), is that

giving an inverter impedance

K= l- l-1 (12)

where

is the insertion loss of the symmetrical two-part network when terminated in a normalized characteristic impedance of unity. In obtaining this result, the reciprocity condition

A^{} 2 +bc= 1 (14) has been used and it should be noted that equation (12) may be reduced to the form given by equation (8). When T is a simple shunt susceptance jb, as in FIG. 2A, then

A=1; B=0; C=b (15)

and equation (11) reduces to equation (5), as required. At this point, it is convenient to introduce the concept of an equivalent shunt susceptance b, defined as having a value giving the same insertion loss as that given by equation (13) for the general network. We have, therefore,

where B and C are normalized to the terminating immitance.

FIG. 4A depicts a thick capacitive iris in a rectangular waveguide and FIG. 4B is a sectional view taken along the parting plane 4B-4B in FIG. 4A. In FIG. 4A, the narrow internal dimension of the waveguide is denoted by b in accordance with conventional notation and should not be confused with a normalized susceptance. The equivalent circuit of the thick capacitive iris is symbolized in FIG. 4C and the parameters of that circuit are given in pages 250 to 255 of the Waveguide Handbook, MIT Radiation Laboratory Series, Vol. 10, published by McGraw Hill. With slight changes in nomenclature, the parameters are as follows ##SPC3##

and g is a parameter lying in the range 1<g <1.67 for

and is plotted on page 253 of the Waveguide Handbook with l= t. A sufficiently accurate value of g for a given t/d may be obtained from linear interpolation between the specific points in the following table.

The design of a filter employing thick capacitive irises proceeds by applying the generalized impedance inverter theory. The transfer matrix of the thick iris equivalent circuit (FIG. 4C) is

so that the equivalent normalized susceptance as defined by equation (16) is

where b is related to the corresponding junction VSWR of the distributed low-pass filter by the equation

These equations give a unique determination of the dimensions of the thick capacitive iris for a given thickness t. Simple iterative techniques may be used to solve equations (17), (18), and (19). The distance between facing surfaces of adjacent thick irises is given by the generalized form of equation (9) via equation (11); that is, by replacing b_{i} in equation (9) as follows

FIG. 4A is merely representative of one iris configuration that is capacitive, i.e., which is characterized by positive susceptance. Other capacitive iris configurations commonly used in rectangular waveguide are shown in FIG. 5. The invention is applicable to capacitive irises in general and is not restricted, to any particular iris configuration.

FIG. 6 depicts an embodiment of the invention with parts of the waveguide broken away to show the capacitive irises.

FIG. 7 is a sectional view taken along the parting plane 7-7 in FIG. 6 and FIg. 8 is a top view of the invention with the upper broad wall of the waveguide removed to expose the interior of the filter. In the illustrated embodiment, the waveguide 10 is partitioned by the capacitive irises 1, 2,...7 to provide six cavities. In an actual embodiment of the invention that was constructed, the six-cavity filter was designed for an upper cutoff frequency of 7650 MHz. and was based on a distributed low-pass prototype having N=6, VSWR=1.05 and BW 0.50 (see my monograph Table of Element Values For The Distributed Low-Pass Prototype Filter, cited supra). In that actual embodiment WR 137 waveguide was employed, the inside dimensions of the waveguide being a=1.372 and b=0.622. The spacings measured in inches between facing surfaces of adjacent irises are shown in FIg 8 and are tabulated below as cavity lengths

As the design is physically symmetrical the cavity at either end can be considered as the first cavity. In the embodiment, two types of capacitive irises are employed alternately. The first iris type is a conductive plate which extends across the guide to form an aperture with the top waveguide broad wall and another aperture with the bottom waveguide broad wall. The second type of capacitive iris employs a pair of aligned conductive plates which extend from the top and bottom broad walls of the waveguide and have a gap between them. The first type is represented in FIG. 5 by the plate 51 while the second type is represented in that FIG. by plates 52 and 53.

From FIG. 7 it can be seen that irises 1, 3, 5, and 7 are of the first type and irises 2, 4, and 6 are of the second type. All the irises are symmetrically disposed in the interior of the waveguide, as that disposition is a manufacturing convenience. All the iris plates are preferably rectangular in cross section and in the example here described are 0.062 inch thick. For irises 2 and 6, the gap between the 0.062 inch thick plates is 0.192 inch and for iris 4 the gap between the 0.062 inch thick plates is 0.118 inch. The height of the plate forming iris 1 or 7 is 0.255 inch and the height of the plate forming iris 3 or 5 is 0.493 inch. The filter has no tuning screws although such screws can be provided if desired.

FIG. 9 is a graphical comparison between the theoretically predicted results, computed by multiplication of transfer matrices, and the results obtained from actual embodiments of the six-cavity filter. The theoretical VSWR is reproduced quite well at frequencies below 7.3 GHz., but the insertion loss curve shows that the upper cutoff frequency F_{} c is 600 MHz. higher than predicted. The cutoff rate is in good agreement with theory, and the attenuation in the stop band above the upper cutoff was ascertained to be greater than 40 db. to 16 GHz. and greater than 25 db. to 18 GHz. Closer agreement between theoretical predicted and actual results has been obtained for several later filter designs and it has been found that in the later designs the theoretical upper cutoff frequency is usually more closely approached.

The order of magnitude of the interacting effects arising from the close spacing of the irises can be estimated by calculating the proximity factor P by which the adjacent capacitances must be multiplied to obtain a more correct result, as described, for example, by H. E. Green in "The Numerical Solution of Some Important Transmission Line Problems," IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-13, Sept. 1965. It has been found that even for the shortest cavity in the described six-cavity filter, the proximity factor is greater than 0.9 and hence the proximity effect is small. Much of the capacitance of a thick iris is due to the thickness itself, that is, is due to the parallel plate capacitance of the facing surfaces, and as the parallel plate capacitance is not affected by the proximity factor, the mutual interacting effects of the closely spaced thick irises need not, in many instances be considered.

A thick capacitive iris, within the context of this exposition, means an iris whose thickness is such that the series inductance of the iris is not negligible compared with its shunt capacitance. In terms of the lumped equivalent circuit of a thick iris, the series inductive component is appreciable compared with the shunt capacitive component.

The lower cutoff frequency of the filter is determined by the normal cutoff frequency of the waveguide. The filter may, therefore, be considered a low-pass filter in the sense that of all the frequencies that can propagate in the waveguide, those frequencies below the upper cutoff of the filter are all passed.

DISCUSSION OF THE PRIOR ART

Frequency filters formed by capacitive irises spaced along the interior of a waveguide to form a sequence of resonant cavities are described in "Maximally Flat Filters in Waveguide," by W. W. Mumford, Bell System Technical Journal, Vol. 27, Oct. 1948 and in "Design of Tunable Resonant Cavities With Constant Bandwidth" by L. D. Smullen, Proc. IRE, Vol. 37, Dec. 1949. The capacitive iris waveguide filters are principally considered to be variants of the more common filters having inductive irises in the waveguide which form resonant cavities that are each approximately λ

Where two irises form a resonant cavity in a conventional waveguide filter, the shortest length of the cavity is between one-half to three-quarters of the guide wavelength λ

In U.S. Pat. No. 3,027,527, there is disclosed a waveguide band-pass filter, invented by Edward Salzberg, which has capacitive irises spaced along the waveguide at one-quarter wavelength (λ14) intervals. In the Salzberg filter, a resonant iris is situated midway between adjacent capacitive irises. The reduction thus achieved in the length of the cavity has resulted in an improvement in harmonic rejection in addition to providing a more compact structure.

THE INVENTION

The invention here disclosed provides better performance than can be obtained with the Salzberg arrangement while retaining the compactness of the filter. In the invention, thick capacitive irises are disposed along the interior of a waveguide to form a series of directly coupled resonant chambers. The spacing between consecutive irises is approximately between one-quarter (1/4 ) to one-eighth (1/8 ) of λ

THE DRAWINGS

The invention, both as to its construction and mode of use, can be better understood from the following exposition when considered in conjunction with the accompanying drawings in which:

FIG. 1 symbolizes a cavity formed in a waveguide by capacitive irises spaced apart by an electrical length Θ;

FIG. 2A is a diagram of a filter having n directly coupled cavities formed by a sequence of capacitive irises spaced along a waveguide;

FIG. 2B symbolizes the distributed low-pass prototype of FIG. 2A in impedance inverter form;

FIG. 3 represents an n -cavity filter with generalized symmetrical lossless networks;

FIGS. 4A and 4B depict a thick capacitive iris in a rectangular waveguide;

FIG. 4C symbolizes the equivalent electrical circuit of the thick capacitive iris;

FIG. 5 depicts various capacitive iris configurations;

FIG. 6 is a view of an embodiment of the invention with part of the waveguide broken away to expose capacitive irises;

FIG. 7 is a sectional view taken along the plane 7-7 in FIG. 6;

FIG. 8 is a top view of the FIG. 6 embodiment with the upper broad wall of the waveguide removed to expose the interior of the filter; and

FIG. 9 is a graph comparing actual filter performance with theoretically predicted performance.

THE EXPOSITION

FIG. 1 symbolically depicts a single cavity formed by a pair of capacitive irises 1 and 2 spaced in a waveguide 3. Each capacitive iris is characterized by a normalized susceptance jb and the irises are spaced by an electrical length Θ. The insertion loss of the cavity is given by

where P

P

Because b is positive for a capacitive iris, the first resonance occurs where

For expository purposes, the filters considered herein have been given susceptance values in the range 0.5 to 4 and the corresponding resonant lengths are therefore in the range approximately 25° to 75°. The second harmonic resonances consequently are in the 205° to 255° range which is far removed from the fundamental range.

Consider now the waveguide filter in FIG. 2A having n cavities formed by capacitive irises spaced along the interior of the waveguide to form cavities that are π/2 or less in electrical length. The theory presented in my monograph "Theory of Direct-Coupled Cavity Filters," IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-15, No. 6 June 1967, is not completely applicable here because the cavities in the FIG. 2A filter are appreciably less than a half-wavelength and that filter is low-pass rather than band-pass. The susceptances of the irises in the FIG. 2A filter are, however, determined in a manner similar to that set forth in the aforesaid monograph. The susceptances are determined by equating the VSWR (the voltage standing wave) ratio of each reactive coupling discontinuity of the filter (i.e., each iris) to the corresponding junction VSWR V

This gives an exact realization of the upper cutoff frequency of the filter, if the susceptance values are chosen to satisfy equation (3) at that particular frequency and if the spacings between irises are properly chosen. The spacings between irises should all be equivalent to an effective electrical length of Θ

The transfer matrix of any iris of normalized susceptance jb shunting a line of electrical length φ at its midpoint is given by ##SPC1## Equation (9) can be extended to cover the case of capacitive irises whose thickness is such that the irises cannot be represented by simple shunt susceptances. The extension covers the general case represented by the filter shown in FIG. 3 which has n cavities and in which each iris has been replaced by a network T

The condition that this shall be equivalent to an impedance inverter, as in equation (6), is that

giving an inverter impedance

K= l- l-1 (12)

where

is the insertion loss of the symmetrical two-part network when terminated in a normalized characteristic impedance of unity. In obtaining this result, the reciprocity condition

A

A=1; B=0; C=b (15)

and equation (11) reduces to equation (5), as required. At this point, it is convenient to introduce the concept of an equivalent shunt susceptance b, defined as having a value giving the same insertion loss as that given by equation (13) for the general network. We have, therefore,

where B and C are normalized to the terminating immitance.

FIG. 4A depicts a thick capacitive iris in a rectangular waveguide and FIG. 4B is a sectional view taken along the parting plane 4B-4B in FIG. 4A. In FIG. 4A, the narrow internal dimension of the waveguide is denoted by b in accordance with conventional notation and should not be confused with a normalized susceptance. The equivalent circuit of the thick capacitive iris is symbolized in FIG. 4C and the parameters of that circuit are given in pages 250 to 255 of the Waveguide Handbook, MIT Radiation Laboratory Series, Vol. 10, published by McGraw Hill. With slight changes in nomenclature, the parameters are as follows ##SPC3##

and g is a parameter lying in the range 1<g <1.67 for

and is plotted on page 253 of the Waveguide Handbook with l= t. A sufficiently accurate value of g for a given t/d may be obtained from linear interpolation between the specific points in the following table.

The design of a filter employing thick capacitive irises proceeds by applying the generalized impedance inverter theory. The transfer matrix of the thick iris equivalent circuit (FIG. 4C) is

so that the equivalent normalized susceptance as defined by equation (16) is

where b is related to the corresponding junction VSWR of the distributed low-pass filter by the equation

These equations give a unique determination of the dimensions of the thick capacitive iris for a given thickness t. Simple iterative techniques may be used to solve equations (17), (18), and (19). The distance between facing surfaces of adjacent thick irises is given by the generalized form of equation (9) via equation (11); that is, by replacing b

FIG. 4A is merely representative of one iris configuration that is capacitive, i.e., which is characterized by positive susceptance. Other capacitive iris configurations commonly used in rectangular waveguide are shown in FIG. 5. The invention is applicable to capacitive irises in general and is not restricted, to any particular iris configuration.

FIG. 6 depicts an embodiment of the invention with parts of the waveguide broken away to show the capacitive irises.

FIG. 7 is a sectional view taken along the parting plane 7-7 in FIG. 6 and FIg. 8 is a top view of the invention with the upper broad wall of the waveguide removed to expose the interior of the filter. In the illustrated embodiment, the waveguide 10 is partitioned by the capacitive irises 1, 2,...7 to provide six cavities. In an actual embodiment of the invention that was constructed, the six-cavity filter was designed for an upper cutoff frequency of 7650 MHz. and was based on a distributed low-pass prototype having N=6, VSWR=1.05 and BW 0.50 (see my monograph Table of Element Values For The Distributed Low-Pass Prototype Filter, cited supra). In that actual embodiment WR 137 waveguide was employed, the inside dimensions of the waveguide being a=1.372 and b=0.622. The spacings measured in inches between facing surfaces of adjacent irises are shown in FIg 8 and are tabulated below as cavity lengths

As the design is physically symmetrical the cavity at either end can be considered as the first cavity. In the embodiment, two types of capacitive irises are employed alternately. The first iris type is a conductive plate which extends across the guide to form an aperture with the top waveguide broad wall and another aperture with the bottom waveguide broad wall. The second type of capacitive iris employs a pair of aligned conductive plates which extend from the top and bottom broad walls of the waveguide and have a gap between them. The first type is represented in FIG. 5 by the plate 51 while the second type is represented in that FIG. by plates 52 and 53.

From FIG. 7 it can be seen that irises 1, 3, 5, and 7 are of the first type and irises 2, 4, and 6 are of the second type. All the irises are symmetrically disposed in the interior of the waveguide, as that disposition is a manufacturing convenience. All the iris plates are preferably rectangular in cross section and in the example here described are 0.062 inch thick. For irises 2 and 6, the gap between the 0.062 inch thick plates is 0.192 inch and for iris 4 the gap between the 0.062 inch thick plates is 0.118 inch. The height of the plate forming iris 1 or 7 is 0.255 inch and the height of the plate forming iris 3 or 5 is 0.493 inch. The filter has no tuning screws although such screws can be provided if desired.

FIG. 9 is a graphical comparison between the theoretically predicted results, computed by multiplication of transfer matrices, and the results obtained from actual embodiments of the six-cavity filter. The theoretical VSWR is reproduced quite well at frequencies below 7.3 GHz., but the insertion loss curve shows that the upper cutoff frequency F

The order of magnitude of the interacting effects arising from the close spacing of the irises can be estimated by calculating the proximity factor P by which the adjacent capacitances must be multiplied to obtain a more correct result, as described, for example, by H. E. Green in "The Numerical Solution of Some Important Transmission Line Problems," IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-13, Sept. 1965. It has been found that even for the shortest cavity in the described six-cavity filter, the proximity factor is greater than 0.9 and hence the proximity effect is small. Much of the capacitance of a thick iris is due to the thickness itself, that is, is due to the parallel plate capacitance of the facing surfaces, and as the parallel plate capacitance is not affected by the proximity factor, the mutual interacting effects of the closely spaced thick irises need not, in many instances be considered.

A thick capacitive iris, within the context of this exposition, means an iris whose thickness is such that the series inductance of the iris is not negligible compared with its shunt capacitance. In terms of the lumped equivalent circuit of a thick iris, the series inductive component is appreciable compared with the shunt capacitive component.

The lower cutoff frequency of the filter is determined by the normal cutoff frequency of the waveguide. The filter may, therefore, be considered a low-pass filter in the sense that of all the frequencies that can propagate in the waveguide, those frequencies below the upper cutoff of the filter are all passed.