Leif Asbrink, SM5BSZ

Computer-Assisted Design of High-Gain Yagi Aerials

VHF Communications 3/1998

For some time now many radio amateurs have had the option available of designing antennas with the aid of computers as a standard application.

Computer programs with automatic optimisation can also be obtained, and are frequently used by radio amateurs. However, designing the optimum antennas for a special application is still anything but simple

1. INTRODUCTION

Unfortunately, the optimisation programs available at present have a problem - they are not convergent. The optimum antenna which the computer finally discovers depends on which antenna has been pre-set as the initial antenna.

A more convergent method is set out in this article. This method is even fully convergent if used on long Yagis or groups of long Yagis.

Naturally, it depends on the intended application which antenna is optimum. Thus, normally, the best transmission antenna is the one which supplies maximum radiation in a specific direction. It is thus the antenna with maximum gain.

However, in many cases the optimum transmission antenna is the one which permits the greatest possible output in various directions, without generating BCI or TVI.

A very good signal / noise ratio (S/N) is desirable at the receiver. The signal, S, is proportional to the gain, but it often makes more sense to reduce the noise, N, instead of increasing S. The noise can be thermal, due to ohmic losses, or can stem from side lobes which indicate noise sources. The method which is set out here can be used to find the optimum antenna for many different situations.

Since the early sixties, I have had a preference for the 2m band. In the country, where there is practically no interference fog generated by people, this band is something special: the ground, the sky and the antenna have the same temperature, while the noise temperature of the receiver is considerably lower. The optimum antenna here for both reception and transmission operation is the one with maximum gain.

The antenna which provides maximum gain for the given number of elements is certainly of more than theoretical interest, it is the practical solution to my problem: how can we assemble a competitive EME station, in spite of restrictions on antenna size?

2. THE YAGI MODEL

In 1967, Roger F.Harrington published an article with the title: "Matrix Methods for Field Problems" [1]. A few years later, computer programs based on Harrington's methods came on the market. I use such a program, compiled by D.C.Kuo and B.J.Strait [2].

This program is very widely used and can operate with arbitrarily bent wires. The speed suffers, of course, and the inputs required are rather complicated. However, the program already constitutes a more efficient version of earlier programs [3, 4].

To simplify matters and save time on calculation, I have modified the program to the extent that all it will accept are geometric configurations which stem from groups of Yagi antennas.

I know of no other geometrical configuration which can hold its own with the excellent performance data from this class of antennas.

Although the article which follows deals exclusively with Yagi antennas, the principles can naturally be transferred to other types of antenna. But no further reference will be made to this. The model functions in the following way. All those elements which make up the complete antenna system are conceptually broken down into a series of short sub-sections or segments which are connected to one another. It is assumed that the current in each element runs linearly from one end to the other. This leads to delta functions for the current, which means a common impedance matrix can be calculated. With this matrix it is possible to calculate the current in each segment if a voltage is applied in the middle of the radiator. The currents can then be used to calculate the impedance of the feeding point, the radiated field patterns and the ohmic losses.

James L.Lawson, W2PV [5], has published a simplified method. Since only one element per segment is specified, this method is very fast. Although it is not explicitly stated, it is assumed that the current distribution is the same on all elements. So the impedances can be taken from tables. If the computer-assisted development of antennas is still not clear, it would make sense to read the article by W2PV first.

3. CALIBRATION OF MODEL

If the program from Kuo and Strait is used to calculate the model, the result obviously depends on the number of functions which are specified per element. This can be illustrated using a very critical antenna configuration, the optimised 6-element Yagi from Chen and Cheng [6] (Table 1). On the basis of Table 1, it appears that about 21 functions would be needed for each element to obtain an acceptable level of accuracy. Even with a Pentium processor, it would take far too long to do the calculations for an EME aerial of medium size in this way. The error which arises because not enough functions are included can be compensated for by corrections to the element lengths. After prolonged experiments, the following correction factor was established:

(Correction factor)

M is the number of functions for the current on each element, and R/l the element radius in wavelengths.

(Table omitted)

If all element lengths are multiplied by the correction factor before any further calculations using the Kuo and Strait program, we obtain the results from Table 2.

Table 2 shows that the error which arises because too few functions are specified can be eliminated by means of a simple correction factor. Since all elements are of approximately the same length, a corrector summand would give similar results. But the use of a correction factor seems more appropriate, since the effect of there being too few segments per element for shorter elements is less significant when the segments are smaller as well.

A further correction is required for Yagi antennas. This necessary correction has been empirically determined [7]. It states that all theoretically determined element lengths are too long by one and the same amount of the element thickness. My explanation for this "final effect correction" is that the theory of it is based on the fact that the current at the tips of the elements is precisely zero - although this is not quite true.

A small current flows from the cylindrical area of the element onto the flat end surface of the element, and then into the air, through a very small capacity at the tip of the element against infinity. The elements of the model must thus be slightly shortened to compensate. As an initial estimate, it was assumed that the section of the element to be removed must have approximately the same area - and thus the same capacity against infinity - as the flat end of the element.

There are certainly other bases for an empirical correction of the element length. Thus approximations are used in the calculation method which could cause errors which are eliminated again using these corrections. The decisive point is the high correlation between the model and the experimentally determined radiated field patterns and impedance values, even for long Yagi antennas with a very high Q, if the corresponding corrections are carried out.

4. OPTIMISING GAIN

Normally, VHF amateurs would compare an optimised-gain Yagi with a Yagi with a specific boom length which has more gain than all other Yagis with the same boom, or with a shorter boom. This task can be replaced if we pose a significantly simpler problem - for a given element diameter, we have to find a Yagi with N elements which gives more gain than all other designs. The model is used to calculate the gain, to do which the element lengths and the positions of the initial antenna are altered until no further improvement can be made. Various calculation methods are used here which are available for defining this problem, and it emerges from this in the end that the antenna finally obtained depends on the antenna originally pre-selected.

Thus, for example, it is assumed in [8] that the reason for this lies in the fact that the gain "hypersurface" has many maxima, and that it is thus impossible to determine whether a specific maximum is the best possible.

I have another opinion on this - the reason for the difficulties is simply that the gain "hypersurface" is too flat for the extremely small rise in the gradient to the global maximum to be determined using the methods previously used. It is true that the antenna with maximum gain can also be determined using traditional methods, with sufficient computational accuracy and adequate computing time; but to be satisfied with a gain increase of 0.1dB per iteration step [8] would surely not be enough. In computer programs, the gain is normally calculated by means of numerical integration of the radiated fields pattern. The integral supplies the mean value for the power density radiated in all directions. The directive efficiency is then the power density in the forward direction divided by the mean power density. The gain is finally obtained as the efficiency multiplied by the directive efficiency. The efficiency, h, is the power radiated, divided by the sum of the power radiated and the thermal output caused by ohmic losses. The key to a convergent optimisation method is thus to work on the radiated field pattern, instead of considering only the gain. The optimisation program uses the antenna calculation package as a sub-program, CALC. The inputs required for CALC are as follows:

N = Number of elements
D = Element diameter (the same for all)

Pi for all i's between 1 and 2 N-1
P1 = Length of first element
P2 = Length of second element
PN + 1 = Co-ordinates of second element
PN + 2 = Co-ordinates of third element
P2N-1 = Co-ordinates of Nth element
K = Number of identical Yagis which are stocked

Xj, Yj is the stocking configuration of Yagis for all j's between 1 and K.

The result obtained using CALC is a radiated field pattern in steps of, for example, 2 degrees. The radiated field pattern can be a two-dimensional configuration of 90 x 180 elements, which consists of 16,200 complex numbers. Each number stands for an electrical field strength in a specific direction. Now, to obtain the gain all these numbers must be squared, weighted and added up, which gives us the power density. The power density in the forward direction is then divided by this value, and the result is multiplied by the efficiency.

These calculations naturally take time, but a lot of time can be saved here by another sequence of operations.

The radiated field pattern for a Yagi group can be very closely approximated as the radiated field pattern of a half-wave dipole, multiplied by the H-diagram of the Yagi, and further multiplied by the radiated field pattern which would be obtained for the corresponding stocking of isotropic radiators. For the optimisation procedure, this means that the CALC sub-routine very often has to be run with other parameters for Pi in each case. Alterations to P relate to the H-diagram of the Yagi alone. Everything else essentially need be calculated only once, and stored as a weighting factor, Wk. In the H-diagram, HK is a configuration of, for example, 90 complex values for 0 to 180 in 2-degree steps. The formula for the gain then gives us:

(Equations omitted)

The problem is now different, in that we are no longer looking for the maximum gain but for the least square sum for the squares of the numbers in array B. The CALC sub-routine has been suitably re-written to give B as the result, with P as the initial parameter. All other parameters are kept constant. This now gives us a well-known problem - the non-linear "least squares method", according to which B has to be approximated to 0.

If you read about this in mathematics textbooks, you will soon discover that this method has a bad reputation. You probably can find a minimum, but as a rule you can not be sure that no better minimum exists. But this is precisely the problem for which the procedures for Yagi optimisation have become well-known! Of course, the least squares method is a special case for Yagi antennas. In my experience, it will provide only a few minima, irrespective of the original design from which we begin. You simply seek out these minima and then select the best.

HE NON-LINEAR LEAST SQUARES PROBLEM

If you are not interested in mathematical relationships, you can simply skip this section. A comparison of various methods was published by the US Department of Industry National Physical Laboratory [9]. Corresponding Fortran programs are available in the NPL Algorithm Library. The method with which I work is very linear. I do not know how it comes off in comparison with other methods in relation to the speed of processing and its ability to find a minimum for a poorly defined problem, but I start from the assumption that my method is not especially good. The least squares method looks like this: We are looking for an x+, which minimises F(x):

(Equation omitted)

Fi(x) is B as in formula (2), the standardised electrical Frauenhofer region multiplied by a weighting function. The real sections and imaginary sections of B correspond to different values of i. If we calculate B in 2-degree steps from 1 to 179 degrees, we obtain 89 complex values corresponding to i = 1 to 178. If x is altered at any time by only one element value, xk, calculating F(x) anew gives us the corresponding change in the m point of the Frauenhofer region.

This route is inefficient but simple, and leads to J(x), the (m x n) Jacobi matrix of f(x), the ith row of which is:

(Equation omitted)

For the case where all fi's are linear functions of x, we have the least squares method, and x+ can be found in one step using standard procedures. If the linear least squares computing routine is applied to the solution of the non-linear Yagi calculation, the result is a long way from the original antenna. Moreover non-linear effects are at work which cause the gain to decrease instead of increasing.

If a further n rows are added to J(x): Dfi + 1(x) = (a,0,0,,,0), Dfi + 2(x) = (0,a,0,,,0), D fi + n(x) = (0,0,0,,,a) , a new x with the steepest descent for F(x) ("steepest descent solution") comes out of the linear least squares solution for this new non-linear problem if a is large enough.

If we repeat the procedure and reduce a step by step, the non-linear problem can certainly be solved. To prevent false minima from being obtained due to numerical difficulties, you should alternate between different ways of calculating J(x). I use four different computing routes: In the first computing path, J1(x) is calculated with every xj which goes in a positive direction. For the second option, J2(x) uses new X variables, which are linear combinations of the original X variables. The linear combinations are the eigenvectors of the J1(x)TJ1(x) square matrix. In the third method, J3(x) is calculated for every xj which goes in a negative direction. In the fourth method, J4(x) is calculated correspondingly to J2(x), using J3 (x)TJ3(x). It is not necessary to calculate J(x) anew for each iteration. If the non-linearity is monitored, it is a simple matter to establish when J(x) must be calculated anew. An appropriate name for this Yagi optimisation method is the "brute force method". The option of obtaining J(x) from the differences between complete calculations is very inefficient, but nonetheless good enough when the efficiency of modern computers is taken into account. The way in which the least squares problem is solved is probably another reason for this name.

6. ELEMENT DIAMETER AND OHMIC LOSSES

It is very important for Yagi models to take the ohmic losses into account, especially if very thin elements are used, because a Yagi with optimised gain takes the form of a superdirective antenna. "Superdirectivity" means that the currents within the antenna itself are very high, and that the radiation levels in the different parts of the antenna more or less cancel each other out in all directions - naturally considerably less in the forward direction, and in contrast more to the sides. In this way, we obtain the desired radiated field pattern. The near field is considerably stronger, since nothing is deleted in the immediate vicinity of the antenna.

A superdirective antenna is very sensitive with regard to metal components in the near field, and so you should be very careful about selecting the locations for the installation and erection of highly-optimised antennas.

"Superdirectivity" also means that the antennas store energy, which causes the antenna to have a high Q and a low band width.

The half-wave element can store more energy with the same currents if the element is thinner. The magnetic energy is LI2 and the inductance per unit of length is higher for a thinner conductor. A higher value is obtained for Q with thin elements, and at the same time the band width is reduced. Table 3 lists a series of 6-element antennas which have been optimised without ohmic losses. Antennas are listed with ideal "superconductors" and with ohmic losses.

Antennas which have been optimised without the ohmic losses being taken into account have low impedances and high levels of current. The convergence of the optimisation procedure is very slow and the optimum is very flat. If ohmic losses are disregarded, the gain is thus independent of the element diameter. These antennas do not have reasonable characteristics except for diameters exceeding 10 mm. if the losses are taken into account. Table 3 shows that it is necessary to bring ohmic losses into the optimisation procedure. I would even make the assumption that such disregard is the main reason for convergence problems in the optimisation of Yagi antennas. To make it possible to calculate the ohmic losses approximately, a resistor was positioned in the middle of each element. The resistance value in Ohms was assumed to be 0.44/d, where d is the element diameter in mm.. The 0.44 value is an experimentally determined value, which was obtained from trials using coax resonators, the Q values of which were measured. The formula appears variously as k/d or k/d2 in the literature. The formula k/d is correct, and gives reasonable values for the losses with element diameters between 5 and 10 mm.. Table 4 gives the optimised data for 6-element antennas with aluminium elements with varying diameters where f = 144.1 MHz. It can be recognised from Table 4 that the rule printed in the ARRL manual: "Avoid element diameters of less than 4 mm.!" is thoroughly well founded. The specifications in the VHF Handbook (Orr, 1956), in contrast, are of doubtful validity. It has proved to be impossible in all cases, with both thick and thin elements, to obtain a good gain value in the forward direction. Thin elements have a high Q value. But it is a deceptive conclusion to believe that a high Q value automatically leads to high gain. On the other hand, it is impossible to obtain high gain from a small antenna without obtaining a high Q level [10]. If the elements are somewhat thicker than usual, a slightly higher gain value is obtained on a somewhat shorter boom. This is especially important for those designing Yagis for relatively high bands, at which low atmospheric temperatures make low losses especially important.

7. COMPARISON WITH RESULTS OF PREVIOUS PUBLICATIONS

The simple and straightforward brute force method, as described above, has been used to improve the design of Chen and Cheng [6]. This design was optimised for loss-free elements with a diameter of 10 mm. [6]. The dimensions of these antennas are listed in Table 5 for f = 144.1 MHz. The optimisation was carried out with and without ohmic losses, and the results produced those antennas with 10-mm. elements which can be found in Tables 3 and 4. The theoretical gain is shown in each case as a function of the frequency for the original and final antennas in Fig.1. The gain value from Chen and Cheng is 13.356 or 11.26dB, and is thus lower than mine, which is 11.56dB. The YO program, version 1.00, from K6STI, gives 11.50dB for the 10-mm. Chen and Cheng design, so the value from Chen and Cheng is thus too low.

It is very probable that the difference between the individual antennas was calculated very precisely, even if there is a certain uncertainty concerning the absolute gain value. For loss-free antennas - shown as a continuous line in Fig.1 - the increase in gain is 0.22dB for a boom length of 3.22 m., as against 3.51 m. in the Chen and Cheng design. Expressed in terms of gain per boom length, the improvement is 15%, or 0.6dB.

If ohmic losses are also taken into account - shown as a dotted lines in Fig.1 - the increase in gain is 0.16dB, and the gain per boom length is improved by 7.5% or 0.3dB. The reason for the significant improvement, as against the design from Chen and Cheng, can be found in the fact that at that time the element intervals were optimised first for fixed element lengths, and only then were the element lengths optimised for fixed intervals. Obviously, simultaneous matching of element lengths and intervals is more flexible and thus better suited to finding the real optimum.

Another optimised antenna is presented by Longsomboom, Green and Cashman [8]. In this antenna, which uses elements only 5.2 mm. thick, I have included the ohmic losses in all calculations. In Fig.2, the gain is plotted against the frequency, and the dimensions are given in Table 6. If we start with this antenna as the original antenna, the brute force method gives a false optimum, which comes out with an element smaller than the original antenna of Fig.2 and Table 6. The iterations are continued until no further improvement is possible, which is the case if the unused element is short enough to have no further influence. If we remove the very short element, the 8-element antenna with a false optimum becomes an optimised 7-element Yagi. If we now add a director, in a normal position in front of this 7-element Yagi, we obtain a new "initial antenna".

If we use the brute force method to optimise this antenna again, we obtain a "long-optimised antenna" (Fig.2, Table 2). The increase in gain, compared to Longsomboom's design, is 0.8dB. This comparison is naturally inappropriate, since the antenna has become considerably longer. For long Yagis, the optimum antenna which can be realised in practice is the best antenna for a specific boom length, and not the antenna with the smallest number of elements. However, the antenna referred to above is the optimum 8-element Yagi for an element diameter of only 5.2 mm.. Slightly more gain can be obtained if the element diameter is increased and / or the aluminium elements are replaced by copper elements.

Another "initial antenna" is obtained if a normal director is positioned 0.1 m. in front of the antenna radiator, with a false optimum, and the unused element is removed. The brute force method then gives a Yagi described as a "short-optimised antenna" in Table 2 and Fig.2.

A comparison with Longsomboon's design shows an increase in gain of 0.19dB - i.e. about 4% - an increase in band width of 25%, and a simultaneous reduction in the boom length of 4%. Attempts to insert additional elements between the existing ones did not succeed. Thus, for the initial antenna with an inserted ninth element, we kept obtaining the "short-optimised antenna" with only eight elements.

8. CHECKING IMPEDANCE AND OTHER CHARACTERISTICS

The optimisation procedure encompasses the simultaneous minimising of various functions of antenna geometry. Naturally, we are free to introduce still more terms into the sum of the squares. If we provide these terms with suitable weighting factors, it becomes possible to influence the contribution made by these terms to the sum of the equations. Two obvious factors which can be added to F(x) are cz(Re(Z)-50)2 and cz.Im(Z) 2. These terms are zero if the 50-Ohm base point is resistive. The optimisation will thus always go in the direction of 50-Ohm antennas.

For the case in which we wish to have only a small reduction in gain for an impedance of 50 Ohms, only a very low value is required for the coefficient Cz. Different values for Cz lead to various compromises between gain and base point resistance. The 50-Ohm antenna in Table 2 was developed in this way. The gain reduction amounts to only 0.01dB, and the frequency range is approximately 7% narrower. A further term which I would like to add to Fu(x) is c.(ohmic losses)2. The reason for this is that I would like to provide a certain safety margin for the deterioration of the characteristics due to ohmic losses, small errors in the theory (model errors) and changes in the element surfaces due to ageing / corrosion.

With the help of the weighting factors, Cz = 0.005 and Cl = 10, with which I prefer to work, I can obtain an approximate optimum value using the brute force method. For this "pretty" antenna, the gain is only 0.04dB below the maximum with a boom which is just 4% longer.

These alterations make it possible to reduce the losses by 35% and to increase the band width by 39%! Should antennas be designed for bands other than 144 MHz, the equations for maximum G/T or for any other value combination can be appropriately altered. It is thus possible to calculate any antenna geometry you wish with this model.

9. GAIN TO BOOM LENGTH RATIO IN YAGI ANTENNAS

A large number of calculations were carried out using the "pretty" parameters referred to above, for antennas with varying numbers of elements and with varying stocking configurations. A uniform diameter of 10 mm. was selected for all calculations.

Thus, for example, the 8-element design gave a gain of 12.47dBd with a 4.387 m. long boom. Fig.3 shows the gain for this antenna, and for a pair of others [11] with different boom lengths. The "pretty" antennas referred to above are on average approximately 0.5dB better than typical EME antennas with boom lengths of 3 to 5 lambda. For extremely long or short boom tubes, the gain is approximately 1dB above the normal line for good antennas [11].

For over two years, I have been operating a quadruplet of 14-element cross-Yagi antennas which were calculated in accordance with the above procedures. I am very satisfied with the results. Likewise, good positions in EME contests demonstrate that these antennas, perfected to a high degree, do also actually function in practice. The assembly naturally requires great care and precision.