New Method for Optimum Design of Pyramidal Horn Antennas

This article introduces a procedure for determining the optimum design of pyramidal horn antennas. The efficiencies and phase errors in the optimum design are variables and depend on the design requirements. New equations are proposed for the optimum design, which can be solved numerically or analytically.


I. INTRODUCTION
Pyramidal horn antennas are widely used in various applications in the microwave range due to its high gain, moderate bandwidth and low voltage standing wave ratio VSWR. Its construction is relativity simple and they are often used as feeders of reflectors and currently in applications where wide bandwidth is required, such as the technology WiMAX.
The pyramidal horn antenna design consists of determining their dimensions where the gain, the operation frequency and waveguide to which the antenna will be attached are given. Fig. 1 shows the antenna and their dimensions. A horn antenna design seeks to determine the values of A, l H or R 1 , l E or R 2 and B, and these are related by [1] This work develops a new analytical design technique, using closed formulas based on equations that take into account the exact phase errors and optimize the gain relative to the E-and H-planes.
Thus, the procedure can be deemed as a consistent, closed form mathematical procedure, which can be solved numerically and/or analytically with high precision and accuracy.
New Method for Optimum Design of Pyramidal Horn Antennas antennas [1]. Thus, the designed antenna can yield a better stability in the gain for the frequency at which it was designed. Additionally, since the design is based on more accurate equations, the gain obtained from the design should be closer to reality.

II. ON-AXIS GAIN OF PYRAMIDAL HORN
There are several factors that influence the on-axis gain of pyramidal horns. When calculating the gain, it is usually assumed that only the dominant mode TE 10 is present and propagating within the antenna; however this does not occur, because when the dimensions of the antenna become larger, other higher-modes will appear in the antenna aperture, changing its gain [3].
Also, wave reflections are not included in the formulas as it propagates inside the antenna. With the increase of the aperture, the wave suffers reflections, increasing the value of VSWR. If the transition from waveguide to free space is made smoothly, there will be fewer reflections and impedance matching will perform better, resulting in higher operating bandwidth, for example. This occurs when the flare or aperture angle of the antenna is small and the value of the length of the antenna is large.
Another factor that is not normally taken into account is the oscillation of the gain of pyramidal with frequency due to multiple diffractions on the aperture edges of the E-plane with further reflections on the wall of the antenna [4]. These oscillations become increasingly smaller with increasing of the frequency.
Thus, it can be seen how complex may be the process of determining the gain of the antenna. The pyramidal antenna gain, ignoring the phenomena discussed above can be given by [ where where C( ) and S( ) functions are Fresnel integrals and The values of q, p 1 and p 2 are dependent on the values of phase errors t in the H-plane and s in the E-plane, given by Equations (3) to (12) were derived by Schelkunoff [5]. These expressions use the geometrical optics theory and single diffraction at the edges of the aperture antenna. They are used to calculate the onaxis gain of pyramidal horns and the H-plane sectoral horns [6]. For E-plane sectoral horns, Jull's formula for calculating the gain is more accurate [7]. In addition, the maximum possible error calculated for Schelkunoff's gain formula using (3) is about ± 0,3 dB [8], [9].
The values of exact phase errors [10] result in a gain close to the measured values, as shown by Fig. 2. They are given by Consequently, to obtain more accurate designs, it is necessary to use (13) and (14) instead of (11) and (12) in (8) to (10) to calculate the gain and analyzing the results, especially when the antenna length is small [10]. Jull's formula [7] gives more accurate result when is only used for E-plane sectoral antennas, because when it is used in the pyramidal antennas, the results are far from the measured, shown in Fig. 2. The effects of the fringe-currents at the edges on the axis-gain are different for pyramidal and E-plane sectoral horns, because the last have closely spaced edges whereas the former have widely spaced edges, which make these effects less significant on the gain of pyramidal antenna [7], [11].
Usually it is assumed that the optimum aperture efficiency ε ap in the design is 0.5 or 0.51 [1].
However, this value does not represent the optimum aperture efficiency, but is close to it. In this work, the optimal design is the result of the choice of the value of A that maximizes the gain in Hplane for a given axial length R 1 , and also by choosing the value of B that maximizes the gain in the E-plane for a given value of R 2 . It is important to notice that the design is optimum only for the desired frequency of operation, and does not necessarily also maximizes the aperture efficiency due to the fact that the gain curve is smooth around its peak.
Commonly the quadratic phase errors are used with t = 0.375 for the H-plane and s = 0.25 for the Eplane [1]. These values do not optimize the gain or directivity, since these are derived from approximations of the aperture dimensions that maximize the gain for a given axial length. This approach lead to It is also possible to define the optimum aperture efficiency as 0.49 and optimum phase errors t = 0.4 and s = 0.26 [12].

SOLUTIONS
There are many possibilities to design an antenna, which give several different results in the dimensions, but produce the same result in the gain. To find a single solution of the design, we have to obtain a minimum of equations relating the four dimensions that we want to achieve, given the value of the on-axis gain, the operating frequency and waveguide. The developed procedure uses (13) and (14) in (3) to (10), and for the antenna to be realizable, the following constrain has to be enforced: For the optimum design, it is chosen A for a fixed value of R 1 that maximizes the gain curve in the H-plane. This is done by deriving the expression of the gain or directivity (5) with respect to the where root ( ) is a numerical function that calculates the zeroes of its argument.
The same have to be done to the E-plane, where B is obtained according to the value of R 2 : However, these expressions do not have a single root, but many, as can be seen in Fig. 3. Its curve of directivity has several points where its derivative is zero. This also happens with the directivity of the H-plane, although the oscillations of the curve are less intense. To select the value of A and B that maximizes the directivity curve, the root can be chosen so that it is within the range [13]: where G is the pyramidal antenna gain. The range of B can be found by the limits of A when using We can estimate ε ap = 0.49 to find the range of root B, although it is possible to use other values close to this.
After that, equation (21) is used in order for the dimensions of the antenna to be realizable. As A is a function of R 1 and B is a function of R 2 , an equation is obtained with two independent variables, R 1 and R 2 . Thus, R 1 can be found as the root that satisfies (16) for a given value of R 2 : Using the previous result in (3), with G being the desired gain value, the numerical solution of the design is found through the R 2 value, since all other dimensions are dependent on R 2 : where G is given in absolute value. These equations can be used for standard antennas with gains between 10 and 25 dB. It can be seen that if the desired gain is high, t e and s e approach 0.3967 and 0.262, respectively. When using (11) and (13), one arrives at Similarly, with (12) and (14) The value of ε ap is exclusively determined by the use of (23) and (24). Equation (28) can be solved numerically or analytically. Hence, the approximated optimal design can be calculated using (28)  Normally, in traditional designs, the exact phase errors are not used in the equations of directivity for the H-and E-planes, but these were used in this work.
Cozzens's method is a design based on empirical measurements, which produces accurate results [2]. As the results are based on measurements, its design already includes all electromagnetic effects that can affect the gain. Tables I, II and III present the three designs respectively. In these three designs, it was sought to design short, long and intermediate antennas to examine different situations that may arise in practice.   In order to compare the new optimum design with a procedure more recently available in the literature, three news designs were performed and compared with a method using an improved set of formulas combined with Particle Swarm Optimization [14]. Tables IV, V e VI show the results. Accordingly to the results of Tables IV, V and VI, the A and B apertures for the proposed method is larger than the method in [14]. In all designs, the results are close. However, there are more differences in the values in the design of short antennas than with the design of long antennas, as seen by the Tables V and VI. This occurs because the method in [14] uses constant phase errors, unlike the proposed method, where these are variables to guarantee that the exact phase errors are accounted.
One of the advantages in using the optimum design is the possibility of the gain to be more stable with the aperture dimensions, if certain conditions on the directivity curve are satisfied, like the smoothness of the curve. For the values of R 1 and R 2 , if the values of A and B are increased or decreased, the gain is reduced, meaning that the pyramidal antenna gain is in its optimum point, where its derivative is zero.
In addition, analyses were done for all three techniques obtained for each design requirement.
Tables VII, VIII and IX present the results of the analyses, where we used (3) to (10) with (13) and  For Tables VII, VIII  In the optimum design, the quadratic phase errors are also variables. In Cozzens's method, the efficiencies and phase errors are also variables, determined by the procedure.
If (11) and (12)  antennas. In addition, as the gain calculated by (11) and (12) are smaller than when using exact phase errors, the designed antenna results in higher dimensions.
In For Table IX, it is seen that the exact and quadratic phase errors differ considerably. When the design requirements are for low gain antennas, the procedure results in short antennas, with flare angles that are large, and the exact and quadratic phase errors diverge with the aperture efficiency being dramatically reduced. Fig. 4 shows the variation of ε ap with the desired gain and Fig. 5 shows the variation of t e and t to the desired gain.  In the all designs, Cozzens's method resulted in negatives values of the derivatives of the gain in the H-plane. In Table IX, the derivative of the gain in the E-plane is also negative. This means that the designed apertures passed the optimum point of design in the curve of directivity.
Similarly, it can be achieved the optimum design using (23) to (28) instead of (17) to (22). With  Tables I, II and III were developed, whose results are shown in   Tables X, XI and XII. The analyses of these designs use (3) to (10) with (13) and (14). short antennas, the efficiency can reach values as low as 0.42. It is also shown that the optimum design proposed results in aperture dimensions larger than the method in [14], and in intermediate dimensions between Cozzens's method and the traditional design (t = 0.375, s = 0.25).