US2669658A - Phase correction of asymmetric dual feed horns - Google Patents

Phase correction of asymmetric dual feed horns Download PDF

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US2669658A
US2669658A US234701A US23470151A US2669658A US 2669658 A US2669658 A US 2669658A US 234701 A US234701 A US 234701A US 23470151 A US23470151 A US 23470151A US 2669658 A US2669658 A US 2669658A
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horn
oscillations
waveguide
section
conductor
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US234701A
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Hagan L Jackson
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CBS Corp
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Westinghouse Electric Corp
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    • HELECTRICITY
    • H01ELECTRIC ELEMENTS
    • H01QANTENNAS, i.e. RADIO AERIALS
    • H01Q19/00Combinations of primary active antenna elements and units with secondary devices, e.g. with quasi-optical devices, for giving the antenna a desired directional characteristic
    • H01Q19/06Combinations of primary active antenna elements and units with secondary devices, e.g. with quasi-optical devices, for giving the antenna a desired directional characteristic using refracting or diffracting devices, e.g. lens
    • H01Q19/08Combinations of primary active antenna elements and units with secondary devices, e.g. with quasi-optical devices, for giving the antenna a desired directional characteristic using refracting or diffracting devices, e.g. lens for modifying the radiation pattern of a radiating horn in which it is located
    • HELECTRICITY
    • H01ELECTRIC ELEMENTS
    • H01PWAVEGUIDES; RESONATORS, LINES, OR OTHER DEVICES OF THE WAVEGUIDE TYPE
    • H01P1/00Auxiliary devices
    • H01P1/16Auxiliary devices for mode selection, e.g. mode suppression or mode promotion; for mode conversion
    • H01P1/161Auxiliary devices for mode selection, e.g. mode suppression or mode promotion; for mode conversion sustaining two independent orthogonal modes, e.g. orthomode transducer
    • HELECTRICITY
    • H01ELECTRIC ELEMENTS
    • H01QANTENNAS, i.e. RADIO AERIALS
    • H01Q13/00Waveguide horns or mouths; Slot antennas; Leaky-waveguide antennas; Equivalent structures causing radiation along the transmission path of a guided wave
    • H01Q13/02Waveguide horns
    • H01Q13/025Multimode horn antennas; Horns using higher mode of propagation
    • H01Q13/0258Orthomode horns
    • HELECTRICITY
    • H01ELECTRIC ELEMENTS
    • H01QANTENNAS, i.e. RADIO AERIALS
    • H01Q19/00Combinations of primary active antenna elements and units with secondary devices, e.g. with quasi-optical devices, for giving the antenna a desired directional characteristic
    • H01Q19/10Combinations of primary active antenna elements and units with secondary devices, e.g. with quasi-optical devices, for giving the antenna a desired directional characteristic using reflecting surfaces
    • H01Q19/12Combinations of primary active antenna elements and units with secondary devices, e.g. with quasi-optical devices, for giving the antenna a desired directional characteristic using reflecting surfaces wherein the surfaces are concave
    • H01Q19/13Combinations of primary active antenna elements and units with secondary devices, e.g. with quasi-optical devices, for giving the antenna a desired directional characteristic using reflecting surfaces wherein the surfaces are concave the primary radiating source being a single radiating element, e.g. a dipole, a slot, a waveguide termination
    • H01Q19/132Horn reflector antennas; Off-set feeding

Definitions

  • WITNESSES adv/2kg
  • My invention relates to the propagation of electromagnetic oscillations, and more particularly to phase correction in a conductor for electromagnetic oscillations.
  • waveguides have been designed for use with two sets of oscillations polarized at right angles to each other.
  • a waveguide of this type is to be used as a feed horn for a radar transmitting apparatus
  • this phase front relationship is critical because it is desirable that the major lobes of the two sets of oscillations emitted from the feed horn substantially coincide.
  • an object of my invention to provide a waveguide wherein the phase front relationship between two sets of oscillations polarized at right angles to each other is controlled.
  • Another object of my invention is to provide a radar feed horn for use with two sets or" oscillations polarized at right angles to each other wherein the phase front of one of the sets of oscillations is changed without substantially changing the phase front of the other set of oscillations, so that ridge lines of the two sets of oscillations coincide.
  • Still another object of my invention is to provide a. radar antenna system for use with two sets of oscillations polarized at right angles to each other.
  • Still another object of my invention is to provide a waveguide having therein means .for changing the phase front of'oscillations therein having a predetermined polarization.
  • Fig. 1 is a plan view of apparatus embodying my invention.
  • Fig. 2 is an elevational view of the apparatus shown -in Fig. 1.
  • Fig. 3 is a sectional view of the apparatus shown in Fig. 1 taken along the line IIL-III of Fig. 1.
  • Fig. 4 is a view of the apparatus shown in Fig. 1 taken along the line IV--IV of Fig. 1.
  • Fig. 5 is a bottom plan view of the embodiment of my invention shown in Fig. 1.
  • Fig. 6 is an elevational view of an apparatus embodying my invention in which the feed horn is used to feed a parabolic reflector, and
  • Fig. 7 is a front view of the reflector employed in the embodiment shown in Fig. 6.
  • I provide a feed horn comprising a waveguide, a first end 6 of which is designed for connection to a standard waveguide l and the opposite end 8 is open for omitting electromagnetic oscillations such as for projecting electromagnetic energy against a parabolic antenna I6.
  • the standard waveguide has connections for feeding thereto oscillations polarized in one direction at one place 9 and for feeding oscillations polarized in another direction at another place l3.
  • the open end of the feed horn is cut at an angle as desired so as to direct the electromagnetic oscillations emitted therefrom toward a parabolic reflector in such a manner that a maximum length of the line maximum radiation intensity lies in the reflector.
  • the elevation flared out portion of the feed horn extends across the horn horizontally at an acute angle with a cross-section of the horn.
  • the elevation flared out section does not extend across the waveguide at right angles to the axis of the horn but rather the elevation flared out portion extends across the wave uide at an angle as to a line which is. perpendicular to the axis of the horn.
  • the flaring out occurs in two directions at the same place along the length of the horn, however, in accordance with another embodiment of my invention the horn might have only a flare upward and not have a flare downward. Also, in accordance with still another embodiment of my invention, the two horizontal flares or the two elevation flares need not occur at the same place along the length of the feed horn.
  • the horizontal and vertical flaring sections need not be in the order shown. Instead, it may be desirable in some instances that the elevation flaring section be further from the antenna than is the horizontal flaring section.
  • the cross-sectional area is determined by the size and shape of the reflector to be fed by the horn. Ihe open end 8 of the feed horn is cut at an angle 41 to a cross-section of the feed horn.
  • the angle the height F of the feed horn on the transmitter side of the flared out section, and the height E on the antenna side of the flared out portion are so selected that:
  • 9A, 03, 0c and on are measured in electrical degrees, and represent distances in electrical degrees for the oscillations to be employed.
  • 0A represents the electrical degrees corresponding to the distance W tan a as measured in air.
  • 03 represents the electrical degrees corresponding to the distance W tan c5 as measured in the feed horn.
  • 00 represents the electrical degrees corresponding to the distance along the short side of the horn between an arbitrary line [4, extending across the horn and located between the elevation flare and the horizontal flare, and the mouth of the horn, plus the electrical degrees corresponding to the distance W tan as measured in air.
  • 01) represents the electrical degrees corresponding to the distance along the long side of the horn between the arbitrary line H, extending across the horn and located between the elevation flare 1and the horizontal flare, and the mouth of the Since the distance in electrical degrees between two points inside the horn is a function of the cross-sectional dimensions of the horn the distance in electrical degrees can be varied by varying the location of the flaring out portions along the length of the horn. By placing the flaring section at an angle the electrical degrees along one side of the horn can be changed. with respect to the electrical degrees along the other side of the horn.
  • the dimension E does not affect the cutoff of F1.
  • the dimension W does not affect the cutoff of the oscillations with a frequency F2. I am thus able to adjust the. main lobe of oscillations of one frequency without substantially affecting the main lobe of oscillations of another frequency which are polarized at right angles to the oscillations of the first frequency.
  • the major lobe of the first set of oscillations may be caused to coincide with the major lobe of the second set of oscillations.
  • a reflector i6 is employed which is a section of a parabola, a section of a parabola is employed where it is desired that, the beam have a longer cross-sectional dimension in the vertical direction than in the horizontal direction or vice versa.
  • a reflector i6 is employed which is a section of a parabola, a section of a parabola is employed where it is desired that, the beam have a longer cross-sectional dimension in the vertical direction than in the horizontal direction or vice versa.
  • the feed horn is then so oriented with respect to the reflector I6 that as much as possible of the line of maximum radiation intensity It lies in the reflector I6.
  • a hollow conductor for electromagnetic oscillations having a substantially rectangular cross-section, said conductor having an end thereof which is cut at an acute angle to the axis of said conductor, said conductor flaring outward in the vertical direction at a sharp angle over a part of its length, the flaring portion where said conductor flares out sharply ending at a different place along said conductor at one side of said conductor than at the other side of said conductor.
  • a hollow conductor for electromagnetic oscillations having a rectangular cross-section, said conductor having a flared out portion in one of the sides thereof, said flared out portion extending across said one side at an acute angle to the axis of said conductor.
  • a hollow conductor for electromagnetic oscillations having a rectangular cross-section, said conductor having a-flared out portion in oneof the sides thereof, said flared out portion-extending cross said one side at an acute angle to the axis of said conductor, said conductor having an end thereof which is at an acute angle with the axis of said conductor.
  • a hollow conductor for electromagnetic oscillations having two sides, a top and a bottom and being of substantially rectangular cross section, said top having a portion thereof where said top rises sharply, said portion extending across said top at an acute angle to a cross section of said conductor.
  • a radar feed horn having two sides, a top and a bottom, said top having a portion thereof where the distance between the top and bottom of said feed horn increases sharply, said portion extending across said top at an angle so that said portion lies at a different place along the length of said horn at one side of said horn from the place where said portion lies on the other side of said horn.
  • a radar antenna apparatus comprising a waveguide, connections for applying oscillations of one polarization to said waveguide at one point along the length of said waveguide and connections for applying oscillations of another polarization to said waveguide at another place along the length of said waveguide, a feed horn, said waveguide being connected so as to supply oscillations to said feed horn, said feed horn having a fiaring out portion where one dimension of said horn increases sharply, said flaring out portion having one end thereof located at a difierent point along the length of said horn from the other end thereof.
  • a radar antenna apparatus comprising a waveguide, connections for applying oscillations of one polarization to said waveguide at one point along the length of said waveguide and connections for applying oscillations of another polarization to said waveguide at another place along the length of said waveguide, a feedhorn, said waveguide being connected so as to supply oscillations to said feed horn, said feed horn having a flaring out portion where one dimension of said horn increases sharply. said flaring out portion having one end located at a different point along the length of said horn from said other end, and a parabolic reflector of electrically conducting material, said feed horn being so oriented that oscillations emitted therefrom are directed against said reflector.
  • Apparatus as described in claim 7 characterized in that the mouth of said feed horn is cut at such an angle that oscillations emitted therefrom produce a line of maximum radiation intensity across said reflector, said line of maximum intensity being so located that substantially a maximum length of said line lies in said reflector.

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  • Aerials With Secondary Devices (AREA)
  • Variable-Direction Aerials And Aerial Arrays (AREA)

Description

Feb. 16, 1954 H. 1.. JACKSON PHASE CORRECTION OF ASYMMETRIC DUAL FEED HORNS 3 Sheets-Sheet 1 Filed July 2, 1951 Fig.4.
Fig.5
WITNESSES: adv/2kg,
j X f ATTORN EY Feb. 16, 1954 ,H. L. JACKSON 2,569,658
PHASE CORRECTION OF ASYMMETRIC DUAL FEED HORNS Filed July 2, 1951 3 Sheets-Sheet 2 WITNESSES: INVENTOR 147%? HogonL.Jockson.
' ATTORNEY Feb. 16, 1954 JACKSON 2,669,658
PHASE CORRECTION OF ASYMMETRIC DUAL FEED HORNS Filed July 2, 1951 5 Sheets-Sheet 3 Fig.7.
/ INVENTOR Hogan L.Jockson.
Patented F eb. 16, 1954 PHASE CORRECTION OF ASYMMETRIC DUAL FEED HORNS Hagan L. Jackson, Baltimore, Md., assignor to Westinghouse Electric Corporation, East Pittsburgh, Pa., a corporation of Pennsylvania Application July 2, 1951, Serial No. 234,701
9 Claims. (01. 250-3353) My invention relates to the propagation of electromagnetic oscillations, and more particularly to phase correction in a conductor for electromagnetic oscillations.
In accordance with the prior art of which I am aware, waveguides have been designed for use with two sets of oscillations polarized at right angles to each other. For some purposes, as for example, where a waveguide of this type is to be used as a feed horn for a radar transmitting apparatus, it is desirable that the two waves which are polarized at right angles to each other have a certain predetermined phase front relationship. Where the waveguide is being used as a feed horn, this phase front relationship is critical because it is desirable that the major lobes of the two sets of oscillations emitted from the feed horn substantially coincide.
It is, accordingly, an object of my invention to provide a waveguide wherein the phase front relationship between two sets of oscillations polarized at right angles to each other is controlled.
It is another object of my invention to provide a radar feed horn for use with two sets of oscillations polarized at right angles to each other whereby the phase front relationship of the two sets of oscillations is controlled.
till another object of my invention is to provide a radar feed horn for use with two sets or" oscillations polarized at right angles to each other wherein the phase front of one of the sets of oscillations is changed without substantially changing the phase front of the other set of oscillations, so that ridge lines of the two sets of oscillations coincide.
Still another object of my invention is to provide a. radar antenna system for use with two sets of oscillations polarized at right angles to each other.
Still another object of my invention is to provide a waveguide having therein means .for changing the phase front of'oscillations therein having a predetermined polarization.
The novel features which I consider characteristic of my invention are set forth with more particularity in the appended claims. The invention, however, with respect to both the organization and the operation thereof, together with other objects and advantages may be best understood from the following description of specific embodiments when read in connection with the accompanying drawings, in which:
Fig. 1 is a plan view of apparatus embodying my invention.
Fig. 2 is an elevational view of the apparatus shown -in Fig. 1.
Fig. 3 is a sectional view of the apparatus shown in Fig. 1 taken along the line IIL-III of Fig. 1.
Fig. 4 is a view of the apparatus shown in Fig. 1 taken along the line IV--IV of Fig. 1.
Fig. 5 is a bottom plan view of the embodiment of my invention shown in Fig. 1.
Fig. 6 is an elevational view of an apparatus embodying my invention in which the feed horn is used to feed a parabolic reflector, and
Fig. 7 is a front view of the reflector employed in the embodiment shown in Fig. 6.
- In accordance with my invention, I provide a feed horn comprising a waveguide, a first end 6 of which is designed for connection to a standard waveguide l and the opposite end 8 is open for omitting electromagnetic oscillations such as for projecting electromagnetic energy against a parabolic antenna I6. The standard waveguide has connections for feeding thereto oscillations polarized in one direction at one place 9 and for feeding oscillations polarized in another direction at another place l3. Preferably, the open end of the feed horn is cut at an angle as desired so as to direct the electromagnetic oscillations emitted therefrom toward a parabolic reflector in such a manner that a maximum length of the line maximum radiation intensity lies in the reflector. Between the point 6 of connection with a standard waveguide and the open end 8 of the horn, there is a horizontal flaring out section [0 wherein the feed horn flares out in width gradually for a short distance while the thickness or elevation remains constant. This flaring out in width should be sufiiciently gradual as not to cause substantial reflections of the oscillations being propagated through the horn. Adjacent the horizontal flaring section in which the feed horn flares out in width, there is a short section of rectangular waveguide ll having a constant cross-section. An elevation flared out section i2 is provided between the section of constant-section rectangular waveguide II and the open end 8 of the horn wherein the width of the horn remains constant while the elevation or thickness increases rapidly as one proceeds along the horn toward the open end 8.
The elevation flared out portion of the feed horn extends across the horn horizontally at an acute angle with a cross-section of the horn. In other words, the elevation flared out section does not extend across the waveguide at right angles to the axis of the horn but rather the elevation flared out portion extends across the wave uide at an angle as to a line which is. perpendicular to the axis of the horn.
In the preferred embodiments of my invention, the flaring out occurs in two directions at the same place along the length of the horn, however, in accordance with another embodiment of my invention the horn might have only a flare upward and not have a flare downward. Also, in accordance with still another embodiment of my invention, the two horizontal flares or the two elevation flares need not occur at the same place along the length of the feed horn.
It is also understood that in accordance with another embodiment of my invention the horizontal and vertical flaring sections need not be in the order shown. Instead, it may be desirable in some instances that the elevation flaring section be further from the antenna than is the horizontal flaring section.
At the open end 8 of the horn, which is the end opposite that to which the standard waveguide 1 is attached, the cross-sectional area is determined by the size and shape of the reflector to be fed by the horn. Ihe open end 8 of the feed horn is cut at an angle 41 to a cross-section of the feed horn.
The following is a mathematical explanation of the manner in which it is believed that an apparatus built in accordance with my invention operates.
When a feed horn is to be used with a flrst set of oscillations having a frequency F1, which are polarized vertically, i. e., polarized so that the electric vector extends in a vertical direction, and a second set of oscillations having a frequency F2 polarized horizontally, the angle the height F of the feed horn on the transmitter side of the flared out section, and the height E on the antenna side of the flared out portion are so selected that:
(0A for F1--0B for F1) (0D for F20c for F2) :0
where 9A, 03, 0c and on are measured in electrical degrees, and represent distances in electrical degrees for the oscillations to be employed.
0A represents the electrical degrees corresponding to the distance W tan a as measured in air.
03 represents the electrical degrees corresponding to the distance W tan c5 as measured in the feed horn.
00 represents the electrical degrees corresponding to the distance along the short side of the horn between an arbitrary line [4, extending across the horn and located between the elevation flare and the horizontal flare, and the mouth of the horn, plus the electrical degrees corresponding to the distance W tan as measured in air.
01) represents the electrical degrees corresponding to the distance along the long side of the horn between the arbitrary line H, extending across the horn and located between the elevation flare 1and the horizontal flare, and the mouth of the Since the distance in electrical degrees between two points inside the horn is a function of the cross-sectional dimensions of the horn the distance in electrical degrees can be varied by varying the location of the flaring out portions along the length of the horn. By placing the flaring section at an angle the electrical degrees along one side of the horn can be changed. with respect to the electrical degrees along the other side of the horn.
When the conditions of the above equation are satisfied, the main lobe of the first set of oscillations and the main lobe of the second set Q1 os- '4 cillations will coincide. This may be better understood by noting the effect of the following equations:
( hair where the wavelength of oscillations in the guide is Ag, the cutoff wavelength in the guide for F1 is A and the cutoff wavelength in the guide for F2 is M. From these equations it is seen that the dimension E does not affect the cutoff of F1. and the dimension W does not affect the cutoff of the oscillations with a frequency F2. I am thus able to adjust the. main lobe of oscillations of one frequency without substantially affecting the main lobe of oscillations of another frequency which are polarized at right angles to the oscillations of the first frequency. Thus, by adjustment of the respective dimensions the major lobe of the first set of oscillations may be caused to coincide with the major lobe of the second set of oscillations.
In accordance with one embodiment of my in vention as shown in Figs. 6 and '7, a reflector i6 is employed which is a section of a parabola, a section of a parabola is employed where it is desired that, the beam have a longer cross-sectional dimension in the vertical direction than in the horizontal direction or vice versa. In accordance with a preferred embodiment of my invention,
' the feed horn is then so oriented with respect to the reflector I6 that as much as possible of the line of maximum radiation intensity It lies in the reflector I6.
I have described primarily an embodiment of my invention which is in the form of a feed horn for feeding a parabolic reflector of a radar ap paratus. However, it is understood that the invention may be employed in other situations where it is desirable to change the relative phase front of two sets of oscillations passing through a waveguide, which oscillations are polarized at substantial angles to each other.
Although I have shown and described a specific embodiment of my invention, I am aware that other modifications thereof are possible. My invention, therefore, is not to be restricted except insofar as is necessitated by the prior art and the spirit of the invention.
I claim as my invention:
1. A hollow conductor for electromagnetic oscillations having a substantially rectangular cross-section, said conductor having an end thereof which is cut at an acute angle to the axis of said conductor, said conductor flaring outward in the vertical direction at a sharp angle over a part of its length, the flaring portion where said conductor flares out sharply ending at a different place along said conductor at one side of said conductor than at the other side of said conductor.
2. A hollow conductor for electromagnetic oscillations having a rectangular cross-section, said conductor having a flared out portion in one of the sides thereof, said flared out portion extending across said one side at an acute angle to the axis of said conductor.
.3. A hollow conductor for electromagnetic oscillations having a rectangular cross-section, said conductor having a-flared out portion in oneof the sides thereof, said flared out portion-extending cross said one side at an acute angle to the axis of said conductor, said conductor having an end thereof which is at an acute angle with the axis of said conductor.
4. A hollow conductor for electromagnetic oscillations having two sides, a top and a bottom and being of substantially rectangular cross section, said top having a portion thereof where said top rises sharply, said portion extending across said top at an acute angle to a cross section of said conductor.
5. A radar feed horn having two sides, a top and a bottom, said top having a portion thereof where the distance between the top and bottom of said feed horn increases sharply, said portion extending across said top at an angle so that said portion lies at a different place along the length of said horn at one side of said horn from the place where said portion lies on the other side of said horn.
6. A radar antenna apparatus comprising a waveguide, connections for applying oscillations of one polarization to said waveguide at one point along the length of said waveguide and connections for applying oscillations of another polarization to said waveguide at another place along the length of said waveguide, a feed horn, said waveguide being connected so as to supply oscillations to said feed horn, said feed horn having a fiaring out portion where one dimension of said horn increases sharply, said flaring out portion having one end thereof located at a difierent point along the length of said horn from the other end thereof.
'7. A radar antenna apparatus comprising a waveguide, connections for applying oscillations of one polarization to said waveguide at one point along the length of said waveguide and connections for applying oscillations of another polarization to said waveguide at another place along the length of said waveguide, a feedhorn, said waveguide being connected so as to supply oscillations to said feed horn, said feed horn having a flaring out portion where one dimension of said horn increases sharply. said flaring out portion having one end located at a different point along the length of said horn from said other end, and a parabolic reflector of electrically conducting material, said feed horn being so oriented that oscillations emitted therefrom are directed against said reflector.
8. Apparatus substantially as claimed in claim 7 characterized in that said parabolic reflector comprises a section of a parabola.
9. Apparatus as described in claim 7 characterized in that the mouth of said feed horn is cut at such an angle that oscillations emitted therefrom produce a line of maximum radiation intensity across said reflector, said line of maximum intensity being so located that substantially a maximum length of said line lies in said reflector.
HAGAN L. JACKSON.
References Cited in the file of this patent UNITED STATES PATENTS Number Name Date 2,398,096 Katzin Apr. 9, 1946 2,423,073 Willoughby June 24, 1947
US234701A 1951-07-02 1951-07-02 Phase correction of asymmetric dual feed horns Expired - Lifetime US2669658A (en)

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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US2982961A (en) * 1957-03-20 1961-05-02 Calvin C Jones Dual feed antenna
US3034118A (en) * 1957-05-28 1962-05-08 Gen Electronic Lab Inc Omnipolarized antenna horn
WO1987007440A1 (en) * 1986-05-19 1987-12-03 Hughes Aircraft Company Horn antenna array phase matched over large bandwidths

Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US2398096A (en) * 1940-12-12 1946-04-09 Rca Corp Two frequency electromagnetic horn radiator
US2423073A (en) * 1941-06-13 1947-06-24 Standard Telephones Cables Ltd Electromagnetic wave radiator

Patent Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US2398096A (en) * 1940-12-12 1946-04-09 Rca Corp Two frequency electromagnetic horn radiator
US2423073A (en) * 1941-06-13 1947-06-24 Standard Telephones Cables Ltd Electromagnetic wave radiator

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US2982961A (en) * 1957-03-20 1961-05-02 Calvin C Jones Dual feed antenna
US3034118A (en) * 1957-05-28 1962-05-08 Gen Electronic Lab Inc Omnipolarized antenna horn
WO1987007440A1 (en) * 1986-05-19 1987-12-03 Hughes Aircraft Company Horn antenna array phase matched over large bandwidths
US4758842A (en) * 1986-05-19 1988-07-19 Hughes Aircraft Company Horn antenna array phase matched over large bandwidths

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