Patent Publication Number: US-2023138687-A1

Title: Wideband monopole antenna

Description:
FIELD OF INVENTION 
     This application is generally related to wideband monopole antennas, and, more particularly, to wideband monopole antennas having curved outer surfaces. 
     BACKGROUND 
     Monopole antennas are radio antennas widely used in both transmit and receive contexts. Typically, monopole antennas are fabricated as straight rods positioned perpendicular to, and electrically referenced to, a flat, often circular, ground plane. This geometry—a straight rod arranged perpendicular to a flat ground plane—offers a consistent and symmetric pattern across azimuth but has a bandwidth that is limited to approximately 10%. At higher frequencies, the pattern can start to mode, resulting in radiation in undesired directions and nulls in desired directions. There is, then, a need in the art for very wide bandwidth monopole antennas that also maintain well behaved, rotationally symmetric patterns throughout the band of interest. 
     SUMMARY 
     All examples and features mentioned below can be combined in any technically possible way. 
     According to an aspect, a monopole antenna includes: a radiating element, wherein the radiating element has a first curved outer surface that is rotationally symmetric about a longitudinal axis extending through the monopole antenna, wherein a first diameter at a first end of the radiating element is less than a second diameter at a second end of the radiating element; and a ground plane disposed opposite the radiating element such that an electric field is generated by the radiating element using the ground plane as a counterpoise, wherein the ground plane has a second curved outer surface that is rotationally symmetric about the longitudinal axis, wherein a first diameter at a first end of the ground plane is less than a second at a second end of the ground plane, wherein the first end of the radiating element is disposed adjacent the first end of the ground plane. 
     In an example, at least a part of the first curved outer surface is substantially defined by a truncated circular paraboloid. 
     In an example, at least a part of the first curved outer surface is substantially defined by a truncated hyperboloid. 
     In an example, at least a part of the first curved outer surface is substantially defined by a truncated circular paraboloid joined to the truncated hyperboloid. 
     In an example, at least a part of the second curved outer surface is substantially defined by a truncated circular paraboloid. 
     In an example, the monopole antenna, during operation, has a VSWR of at least 3:1 over a predetermined bandwidth. 
     In an example, the first curved outer surface is defined, across height, by a first polynomial existing between an inner boundary and an outer boundary, wherein the inner boundary is defined by a second polynomial having the coefficients: 
       [−3.3697 e+ 05,1.2311 e+ 05,−1.6052 e+ 04,846.7565,−17.8756,1.1249,−0.0091],
 
     wherein the outer boundary is defined a third polynomial having the coefficients: 
       [−3.3697 e+ 05,1.2311 e+ 05,−1.6052 e+ 04,846.7565,−17.8756,1.1249,0.0062]
 
     wherein the predetermined bandwidth is at least 350-6,000 MHz. 
     In an example, the first polynomial has the coefficients: 
       [−3.3697 e+ 05,1.2311 e+ 05,−1.6052 e+ 04,846.7565,−17.8756,1.1249,0.0024].
 
     In an example, the second curved outer surface is defined, across height, by a fourth polynomial existing between a second inner boundary and a second outer boundary, wherein the second inner boundary is defined by a fifth polynomial having the coefficients: 
       [−1.6988 e+ 05,6.8727 e+ 04,−1.0044 e+ 04,626.0154,−16.5014,−0.1925,0.0096],
 
     wherein the second outer boundary is defined a sixth polynomial having the coefficients: 
       [5.6431 e+ 04,−6.3725 e+ 04,1.9482 e+ 04,−2.6004 e+ 03,166.6723,−5.4307,0.0656].
 
     In an example, the fourth polynomial has the coefficients: 
       [−1.6988 e+ 05,7.2610 e+ 04,−1.1390 e+ 04,789.2496,−24.5701,−0.0372,0.0100].
 
     In an example, the first curved outer surface is defined, across height, by a first polynomial existing between an inner boundary and an outer boundary, wherein the inner boundary is defined by a second polynomial having the coefficients: 
       [−1.8260 e+ 07,3.8143 e+ 06,−2.9036 e+ 05,9.2911 e+ 03,−99.4808,0.3379,0.0052]
 
     wherein the outer boundary is defined a third polynomial having the coefficients: 
       [−1.7632 e+ 07,3.7703 e+ 06,−2.9922 e+ 05,1.0539 e+ 04,−158.8708,1.5534,0.0075]
 
     wherein the predetermined bandwidth is at least 750-6,000 MHz. 
     In an example, the first polynomial has the coefficients: 
       [−1.7632 e+ 07,3.7703 e+ 06,−2.9922 e+ 05,1.0539 e+ 04,−158.8708,1.5534,0.0011].
 
     In an example, the second curved outer surface is defined, across height, by a fourth polynomial existing between a second inner boundary and a second outer boundary, wherein the second inner boundary is defined by a fifth polynomial having the coefficients: 
       [5.3043 e+ 07,−1.6244 e+ 07,2.0181 e+ 06,−1.2998 e+ 05,4.5683 e+ 03,−83.7362,0.6242]
 
     wherein the second outer boundary is defined a sixth polynomial having the coefficients: 
       [−6.0187 e+ 05,−2.0001 e+ 05,8.0273 e+ 04,−9.2436 e+ 03,482.7150,−12.5322,0.1321].
 
     The monopole antenna of claim  13 , wherein the fourth polynomial has the coefficients: 
       [−6.0187 e+ 05,−2.2295 e+ 05,7.3559 e+ 04,−7.2884 e+ 03,325.5185,−7.4394,0.0697].
 
     In an example, the first curved outer surface is defined, across height, by a first polynomial existing between an inner boundary and an outer boundary, wherein the inner boundary is defined by a second polynomial having the coefficients: 
       [−4.93500 e+ 35,6.42410 e+ 34,−3.84685 e+ 33,1.40487 e+ 32,−3.49753 e+ 30,6.28527 e+ 28,−8.42139 e+ 26,8.56859 e+ 24,−6.68061 e+ 22,3.99837 e+ 20,−1.82823 e+ 18,6.31088 e+ 15,−1.61039 e+ 13,2.93397 e+ 10,−3.59697 e+ 07,2.65028 e+ 04,−8.83888 e+ 00]
 
     wherein the outer boundary is defined a third polynomial having the coefficients: 
       [−4.55838 e+ 35,5.95257 e+ 34,−3.57632 e+ 33,1.31063 e+ 32,−3.27491 e+ 30,5.90804 e+ 28,−7.94839 e+ 26,8.12231 e+ 24,−6.36156 e+ 22,3.82570 e+ 20,−1.75807 e+ 18,6.10037 e+ 15,−1.56504 e+ 13,2.8669 e+ 10,−3.534904 e+ 07,2.61810 e+ 04,−8.7756 e +00]
 
     wherein the predetermined bandwidth is at least 9,000-10,000 MHz. 
     In an example, the first polynomial has the coefficients: 
       [−4.55837 e+ 35,5.95256 e+ 34,−3.57631 e+ 33,1.31063 e+ 32,−3.27491 e+ 30,5.90803 e+ 28,−7.94838 e+ 26,8.12229 e+ 24,−6.36155 e+ 22,3.82569 e+ 20,−1.75806 e+ 18,6.10036 e+ 15,−1.56504 e+ 13,2.86693 e+ 10,−3.53408 e+ 07,2.61809 e+ 04,−8.77787 e+ 00].
 
     In an example, the second curved outer surface is defined, across height, by a fourth polynomial existing between a second inner boundary and a second outer boundary, wherein the second inner boundary is defined by a fifth polynomial having the coefficients: 
       [−8.4064 e+ 10,−1.7844 e+ 09,−8.1603 e+ 06,3.3145 e+ 04,247.4060,−0.1378,0.0094]
 
     wherein the second outer boundary is defined a sixth polynomial having the coefficients: 
       [3.1457 e+ 11,7.6263 e+ 09,6.7617 e+ 07,2.6504 e+ 05,364.4879,−0.6138,0.0115].
 
     In an example, the fourth polynomial has the coefficients: 
       [−5.1635 e+ 10,−1.5511 e+ 09,−1.5377 e+ 07,−6.8894 e+ 04,−223.2867,−0.9833,0.0094].
 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In the drawings, like reference characters generally refer to the same parts throughout the different views. Also, the drawings are not necessarily to scale, emphasis instead generally being placed upon illustrating the principles of the various aspects. 
         FIG.  1 A  depicts a cross-section view of a monopole antenna having a radiating element and a ground plane that have a rotationally symmetric curved surface, according to an example. 
         FIG.  1 B  depicts a three-dimensional model of the monopole antenna of  FIG.  1 A , having a radiating element and a ground plane that each have a rotationally symmetric curved surface, according to an example. 
         FIG.  2 A  depicts a plot of a range of potential curves of the radiating element, according to an example. 
         FIG.  2 B  depicts a plot of a range of potential curves of the ground plane, according to an example. 
         FIG.  3    depicts a three-dimensional model of a monopole antenna having a radiating element and a ground plane that each have a rotationally symmetric curved surface, according to an example. 
         FIG.  4 A  depicts a plot of a range of potential curves of the radiating element, according to an example. 
         FIG.  4 B  depicts a plot of a range of potential curves of the ground plane, according to an example. 
         FIG.  5    depicts a three-dimensional model of a monopole antenna having a radiating element and a ground plane that each have a rotationally symmetric curved surface, according to an example. 
         FIG.  6 A  depicts a plot of a range of potential curves of the radiating element, according to an example. 
         FIG.  6 B  depicts a plot of a range of potential curves of the ground plane, according to an example. 
         FIG.  7    depicts a plot of the power of a continuous-wave signal handled by different connectors over frequency. 
     
    
    
     DETAILED DESCRIPTION 
     Applicant has recognized that a monopole antenna with a radiating element and ground plane that have curved surfaces rotationally symmetric about a longitudinal axis, can result in improved performance across azimuth for very wide bandwidths. In various examples described below, the curved surface of the monopole antenna includes a radiating element that widens along the longitudinal axis such that a diameter of the radiating element at the top of the monopole antenna is greater than the diameter at its bottom. In various examples, the radiating element substantially follows a truncated circular paraboloid. In other examples, the radiating element substantially follows a truncated hyperboloid combined with a truncated circular paraboloid. Similarly, the ground plane features a curved surface that widens along the longitudinal axis such that the diameter of the at the top of the ground plane is less than the diameter at its bottom. In various examples, the ground plane substantially follows a truncated circular hyperboloid. (Throughout this disclosure, the term “ground plane” is used to conform with the conventional name of the conductive surface positioned below the radiating element to reflect the radio waves emitted by it; as shown in the various examples, “ground plane” need not be planar.) 
       FIG.  1 A- 1 B  depicts a first examples of a monopole antenna featuring rotationally symmetric curved outer surfaces. More particularly, a cross-section view of a monopole antenna  100  disposed beneath a radome  102  is shown. Monopole antenna  100 , as shown in the view of  FIGS.  1 A and  1 B , comprises a radiating element  104  and a ground plane  106 , each of which includes a curved surface that is rotated about a longitudinal axis A-A extending through the center (when viewed from above) of monopole antenna  100 . 
     As described above, the curved outer surface of monopole antenna  100  is rotationally symmetric about the longitudinal axis, enabling the monopole antenna to maintain consistent patterns in azimuth, that is, to be omni-directional. More particularly, the curved outer surface of radiating element  104  widens along to the longitudinal axis to substantially follow a truncated (as the bottom, relative to the orientation of the drawing, is “cut off”) circular paraboloid. (For the purposes of this disclosure, “substantially follow” or “substantially defined by” means that the outer surface deviates from a best-fit shape—e.g., truncated circular paraboloid, truncated hyperboloid—by 20% or less.) Thus, the top of the radiating element  104  has a diameter greater than its bottom. Similarly, ground plane  106  widens along the longitudinal axis to substantially follow a truncated (as the top is “cut off”) circular paraboloid, such that the bottom of the ground plane  106  has a diameter greater than its top. 
     Again, as a result of the shape of the outer surfaces of radiating element  104  and ground plane  106 , monopole antenna  100  is omni-directional (i.e., has 360° azimuth coverage), and features elevation coverage of −10 to 50°. Further, monopole antenna  100  features a VSWR of 2:1 from 350-6,000 MHz and can transmit up to a 500 W continuous wave drive signal, dependent on the power handling of the N-type connector. In various alternative examples, as described below, other types of connectors, besides an N-type connector, can be used. Various types of connectors will handle different amounts of power over frequency.  FIG.  7    depicts a plot of the power of a continuous-wave signal handled by different connectors over frequency. 
       FIG.  2 A  depicts the curve  200  of the outer surface of radiating element  104 . Curve  200  is rotated about the Y-axis (i.e., longitudinal axis A-A) to form the outer surface of radiating element  104  and is defined by a polynomial having the following coefficients: 
       [−3.3697 e+ 05,0.2311 e+ 05,−1.6052 e+ 04,846.7565,−17.8756,1.12490.0024]
 
     The above polynomial, and polynomials provided below, define the X value (width) with the Y value (height) as the input (i.e., the polynomial defines the curve across height). Thus, in pseudocode, each polynomial can be written as follows: 
       element_ X _value=polyval(polynomial_coefficients,element_ Y _value) 
     While curve  200  results in optimized performance in the frequency range of interest—a VSWR of 2:1 from 350-6,000 MHz—variations of curve  200  result in acceptable performance, i.e., a VSWR of at least 3:1, within the same frequency band. Generally speaking, multiple variations of curve  200  are shown in  FIG.  2 A , defined between inner boundary  202  and outer boundary  204 . These examples, when paired with a suitably shaped ground plane (e.g., a suitable ground plane from the set of ground planes defined in connection with  FIG.  2 B ), result in a VSWR of at least 3:1 over at least the 350-6,000 MHz bandwidth. 
     As shown, inner boundary can be defined by the polynomial coefficients: 
       [−3.3697 e+ 05,1.2311 e+ 05,−1.6052 e+ 04,846.7565,−17.8756,1.1249−0.0091]
 
     And the outer boundary coefficients can be defined by the following coefficients: 
       [−3.3697 e+ 05,1.2311 e+ 05,−1.6052 e+ 04,846.7565,−17.8756,1.12490.0062]
 
     It should be understood that not all alternative examples of curves of monopole antenna  100  depicted in  FIG.  2 A  will substantially follow a truncated circular paraboloid. Indeed, the curve of inner boundary  202  substantially follows a truncated hyperboloid (at the bottom of the curve) and a truncated circular paraboloid above the truncated hyperboloid. In other words, the curve of inner boundary  202  substantially follows a truncated hyperboloid adjoined with a truncated circular paraboloid, akin to a wineglass in shape. For the purposes of this disclosure, a truncated hyperboloid is truncated to remove at least a half of the hyperboloid such that only a trunk remains that widens outward, along the longitudinal axis, at one end; rather than a hyperboloid that is not truncated, which widens at both ends. (Another example of such curved outer surface will be discussed in conjunction with  FIG.  36   .) 
       FIG.  2 B , similar to  FIG.  2 A , depicts the curve  206  of the outer surface of ground plane  106 . Curve  206  is rotated about the Y-axis (i.e., longitudinal axis A-A) to form the outer surface of ground plane  106  and is defined by a polynomial having the following coefficients: 
       [−1.6988 e+ 05,6.8727 e+ 04,−1.0044 e+ 04,626.0154,−16.5014,−0.1925,0.0096]
 
     As shown, curve  206 , in various alternative examples, can be defined between inner boundary  208  and outer boundary  210 . Inner boundary can be defined by the polynomial coefficients: 
       [5.6431 e+ 04,−6.3725 e+ 04,1.9482 e+ 04,−2.6004 e+ 03,166.6723,−5.4307,0.0656]
 
     And the outer boundary coefficients can be defined by the following coefficients: 
       [−1.6988 e+ 05,7.2610 e+ 04,−1.1390 e+ 04,789.2496,−24.5701,−0.0372,0.0100]
 
     Before moving to  FIG.  3   , which shows an alternative example of a monopole antenna with rotationally symmetric curved outer surfaces, the various structural features of monopole antenna  100  will be briefly discussed in connection with  FIG.  1 A . As shown, radiating element  104  receives a drive signal via a threaded pin extending from a flange mount N-type connector  110 , disposed within ground plane  106 . (In alternative examples, a press fit pin can be used—press fit pins can potentially avoid the situation where the radiating element pulls the pin out of the connector.) Flange mount N-type connector  110  is connected to an N to SMA adapter  112 , which receives the drive signal from an SMA connector  114 , which, in turn, is connected to a flange-mount N-type connector  116 . This construction, with these connector types, is provided only as an example, and any suitable RF type connectors and/or adapters can be employed to deliver the drive signal to the radiating element. 
     Further, as shown in  FIG.  1 A , radiating element is electrically insulated from the ground plane  106  via, in this example, a Polytetrafluoroethylene (PTFE) spacer  118 . Ground plane  106  is further seated on baseplate  120 . In addition, radome  102  fits into a cup-shaped indentation  122  on the top of radiating element  104  and creates a seal with radiating element  104  using O-ring  124 , with ground plane  160  using O-ring  126 , and using baseplate  120  with O-ring  128 . An alignment mechanism in the radome fits into a hole in the radiating element and compresses a spring or a compression gasket to stabilize the antenna without additional screws. 
     The remaining figures depict three-dimensional models of the monopole antennas, or the range potential curves for such antennas, to focus on the curves of the outer surfaces of the radiating element and the ground plane, rather than structural features such as those highlighted in  FIG.  1 A . It should, however, be understood that the additional examples of monopole antennas having curved outer surfaces described in this disclosure can likewise include structural elements, such as RF connectors, adapters, spacers, O-rings, etc., necessary to receive a drive signal and to operate. Further, it should be understood that, in alternative examples, any type of suitable RF connector, adapter, spacer, O-ring, etc., can be used to receive a drive signal, to insulate the radiating element from the ground plane, to seal the radome, etc. 
     As mentioned above,  FIG.  3    shows an alternative example of a monopole antenna featuring rotationally symmetric curved outer surfaces. More particularly, a perspective view of a monopole antenna  300  disposed beneath a radome  302  is shown. Similar to monopole antenna  300 , monopole antenna  300  comprises a radiating element  304  and a ground plane  306 , each of which includes a curved surface that is rotated about a longitudinal axis B-B extending through the center (when viewed from above) of monopole antenna  300 . 
     Like monopole antenna  100 , the curved outer surface of monopole antenna  300  that is rotationally symmetric about the longitudinal axis enables the monopole antenna to maintain consistent patterns in azimuth, that is, to be omni-directional. In this example, however, the curved outer surface of radiating element  304  widens along to the longitudinal axis to substantially follow a truncated hyperboloid (at the bottom) joined to a truncated circular paraboloid (at the top). The top of the radiating element  304 , thus, has a diameter greater than its bottom. Similarly, ground plane  306  widens along the longitudinal axis to substantially follow a truncated circular paraboloid, such that bottom of the ground plane  306  has a diameter greater than its top. 
     Again, as a result of the shape of the outer surfaces of radiating element  304  and ground plane  306 , monopole antenna  300  is omni-directional (i.e., has 360° azimuth coverage), and features elevation coverage of −10 to 50°. Further, monopole antenna  100  features a VSWR of 2:1 from 750-6,000 MHz and can transmit up to a 500 W continuous wave drive signal over the operating band (depending on the type of connector used, as described above in connection with  FIG.  7   ). 
       FIG.  4 A  depicts the curve  400  of the outer surface of radiating element  304 . Curve  400  is rotated about the Y-axis (i.e., longitudinal axis B-B) to form the outer surface of radiating element  304  and is defined by a polynomial having the following coefficients: 
       [−1.7632 e+ 07,3.7703 e+ 06,−2.9922 e+ 05,1.0539 e+ 04,−158.8708,1.5534,0.0011]
 
     Similar to the example described in connection with  FIG.  2 A , curve  400  results in optimized performance in the frequency range of interest—a VSWR of 2:1 from 750-6,000 MHz—but variations of curve  400  result in acceptable performance, i.e., a VSWR of at least 3:1, within the same frequency band. Multiple variations of curve  400  are shown in  FIG.  4 A , defined between inner boundary  402  and outer boundary  404 . These examples, when paired with a suitably shaped ground plane (e.g., a suitable ground plane from the set of ground planes defined in connection with  FIG.  4 B ), result in a VSWR of at least 3:1 over at least the bandwidth of 750-6,000 MHz. 
     Inner boundary can be defined by the polynomial coefficients: 
       [−1.8260 e+ 07,3.8143 e+ 06,−2.9036 e+ 05,9.2911 e+ 03,−99.4808,0.3379,0.0052]
 
     And the outer boundary coefficients can be defined by the following coefficients: 
       [−1.7632 e+ 07,3.7703 e+ 06,−2.9922 e+ 05,1.0539 e+ 04,−158.8708,1.5534,0.0075]
 
       FIG.  4 B  depicts the curve  406  of the outer surface of ground plane  306 . Curve  406  is rotated about the Y-axis (i.e., longitudinal axis B-B) to form the outer surface of ground plane  306  and is defined by a polynomial having the following coefficients: 
       [−6.0187 e+ 05,−2.2295 e+ 05,7.3559 e+ 04,−7.2884 e+ 03,325.5185,−7.4394,0.0697]
 
     As shown, curve  406 , in various alternative examples, can be defined between inner boundary  408  and outer boundary  410 . Inner boundary can be defined by the polynomial coefficients: 
       [5.3043 e+ 07,−1.6244 e+ 07,2.0181 e+ 06,−1.2998 e+ 05,4.5683 e+ 03,−83.7362,0.6242]
 
     And the outer boundary coefficients can be defined by the following coefficients: 
       [−6.0187 e+ 05,−2.0001 e+ 05,8.0273 e+ 04,−9.2436 e+ 03,482.7150,−12.5322,0.1321]
 
       FIG.  5    shows yet another example of a monopole antenna  500  featuring rotationally symmetric curved outer surfaces. More particularly, a perspective view of a monopole antenna  500  disposed beneath a radome  502  is shown. Similar to monopole antennas  100  and  300 , monopole antenna  500  comprises a radiating element  504  and a ground plane  506 , each of which includes a curved surface that is rotated about a longitudinal axis C-C extending through the center (when viewed from above) of monopole antenna  500 . 
     Similar to the above examples, the curved outer surface of monopole antenna  500 , rotationally symmetric about the longitudinal axis, enables the monopole antenna to maintain consistent patterns in azimuth, that is, to be omni-directional. In this example, the curved outer surface of radiating element  504  widens along to the longitudinal axis to substantially follow a truncated hyperboloid (at the bottom) joined to a truncated circular paraboloid (at the top). (Similar to the example of  FIG.  3   ) The top of the radiating element  504 , thus, has a diameter greater than its bottom. Similarly, ground plane  506  widens along the longitudinal axis to substantially follow a truncated circular paraboloid, such that bottom of the ground plane  506  has a diameter greater than its top. 
     Again, as a result of the shape of the outer surfaces of radiating element  504  and ground plane  506 , monopole antenna  500  is omni-directional (i.e., has 360° azimuth coverage), and features elevation coverage of −10 to 50°. Further, monopole antenna  100  features a VSWR of 2:1 from 9,000-10,000 MHz and can transmit up to a 100 W continuous wave drive signal at X-band (depending on the type of connector used, as described above in connection with  FIG.  7   ). 
       FIG.  6 A  depicts the curve  600  of the outer surface of radiating element  504 . Curve  600  is rotated about the Y-axis (i.e., longitudinal axis C-C) to form the outer surface of radiating element  504  and is defined by a polynomial having the following coefficients: 
       [−4.55837 e+ 35,5.95256 e+ 34,−3.57631 e+ 33,1.31063 e+ 32,−3.27491 e+ 30,5.90803 e+ 28,−7.94838 e+ 26,8.12229 e+ 24,−6.36155 e+ 22,3.82569 e+ 20,−1.75806 e+ 18,6.10036 e+ 15,−1.56504 e+ 13,2.86693 e+ 10,−3.53408 e+ 07,2.61809 e+ 04,−8.77787 e+ 00]
 
     Curve  600  results in optimized performance in the frequency range of interest—a VSWR of 2:1 from 9,000-10,000 MHz—but variations of curve  600  result in acceptable performance, i.e., a VSWR of at least 2:1, within the same frequency band. Multiple variations of curve  600  are shown in  FIG.  6 A , defined between inner boundary  602  and outer boundary  604 . These examples, when paired with a suitably shaped ground plane (e.g., a suitable ground plane from the set of ground planes defined in connection with  FIG.  6 B ), result in a VSWR of at least 3:1 over at least the bandwidth of 9,000-10,000 MHz. 
     Inner boundary can be defined by the polynomial coefficients: 
       [−4.93500 e+ 35,6.42410 e+ 34,−3.84685 e+ 33,1.40487 e+ 32,−3.49753 e+ 30,6.28527 e+ 28,−8.42139 e+ 26,8.56859 e+ 24,−6.68061 e+ 22,3.99837 e+ 20,−1.82823 e+ 18,6.31088 e+ 15,−1.61039 e+ 13,2.93397 e+ 10,−3.59697 e+ 07,2.65028 e+ 04,−8.83888 e+ 00]
 
     And the outer boundary coefficients can be defined by the following coefficients: 
       [−4.55838 e+ 35,5.95257 e+ 34,−3.57632 e+ 33,1.31063 e+ 32,−3.27491 e+ 30,5.90804 e+ 28,−7.94839 e+ 26,8.12231 e+ 24−6.36156 e+ 22,3.82570 e+ 20,−1.75807 e+ 18,6.10037 e+ 15,−1.56504 e+ 13,2.86694 e+ 10,−3.53409 e+ 07,2.61810 e+ 04,−8.77586 e+ 00]
 
       FIG.  6 B  depicts the curve  606  of the outer surface of ground plane  506 . Curve  606  is rotated about the Y-axis (i.e., longitudinal axis C-C) to form the outer surface of ground plane  506  and is defined by a polynomial having the following coefficients: 
       [−5.1635 e+ 10,−1.5511 e+ 09,−1.5377 e+ 07,−6.8894 e+ 04,−223.2867,−0.9833,0.0094]
 
     As shown, curve  206 , in various alternative examples, can be defined between inner boundary  208  and outer boundary  210 . Inner boundary can be defined by the polynomial coefficients: 
       [−8.4064 e+ 10,−1.7844 e+ 09,−8.1603 e+ 06,3.3145 e+ 04,247.4060,−0.1378,0.0094]
 
     And the outer boundary coefficients can be defined by the following coefficients: 
       [3.1457 e+ 11,7.6263 e+ 09,6.7617 e+ 07,2.6504 e+ 05,364.4879,−0.6138,0.0115]
 
     For the purposes of this disclosure, with respect to the above examples, it should be understood that not all curves between an inner boundary and an outer boundary will result in good performance across the frequency range of interest. Indeed, not all potential curves between the inner boundary and outer boundary will result in good performance. Thus, only those curves existing between inner boundary and outer boundary that result in a monopole antenna exhibiting a VSWR of at least 3:1 are considered acceptable. It would be understood that these curves are typically a smoothly varying curve, defined by a polynomial having an order less than or equal to nine and that is a monotonically increasing function in the vertical dimension (Z dimension in the figures). 
     In alternative examples, each of the above-described antennas can be only partially rotationally symmetric (i.e., the curved surface is not rotated fully about the longitudinal axis). These examples will not be as omni-directional as the above examples, however. Further, in alternative examples, the monopole antennas can be scaled down or up in size. 
     While several inventive embodiments have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and/or structures for performing the function and/or obtaining the results and/or one or more of the advantages described herein, and each of such variations and/or modifications is deemed to be within the scope of the inventive embodiments described herein. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and/or configurations will depend upon the specific application or applications for which the inventive teachings is/are used. Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, many equivalents to the specific inventive embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described and claimed. Inventive embodiments of the present disclosure are directed to each individual feature, system, article, material, and/or method described herein. In addition, any combination of two or more such features, systems, articles, materials, and/or methods, if such features, systems, articles, materials, and/or methods are not mutually inconsistent, is included within the inventive scope of the present disclosure.