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This article was originally written to canvass issues related to the calculation of recommended stacking distances for DL6Wu long boom yagis by the design tool DL6WUGG.EXE. A revised program dl6wu2006.exe which uses the methods proposed in this article has been published and should be used in preference to DL6WUGG.EXE. 
Of particular interest is prediction of the beamwidths (E plane and H plane) of a single Yagi, for the purpose of calculation of optimal stacking distances.
This article explores NEC models of a DL6WU Yagi designed with DL6WUGG.EXE, and compares the predictions of DL6WUGG.EXE and those of the NEC models. (91 DL6WU antennas with 3mm aluminium elements and a nonmetallic boom were modelled in NEC2 at 432MHz.
Comparison of results of DL6WUGG.EXE reveals that while the methods in DL6WUGG.EXE are fairly accurate in the mid range of boom lengths, the differences are larger for the smaller and larger Yagis.
This article further explores how well the algorithms in DL6WUGG.EXE predict NEC2 model predictions, and whether there is a better, but simple way to make such predictions.
DL6WUGG.EXE is a program to design DL6WU Yagi antennas and to predict their performance. It produces designs for required forward gain from 11.8dBd to 21.6dBd.
Table 1 sets out summary statistics of the differences between the DL6WUGG.EXE method (lets call it DL6WU) and the NEC models as a baseline.
Gain  BWe  BWh  
Avg  0.2dB  18.3%  19.1% 
SD  0.1dB  16.0%  15.6% 
Min  0.3dB  48.1%  48.3% 
Max  0.1dB  1.8%  0.2% 
The differences of up to 48% in the estimates of beamwidths may frustrate efforts to optimally stack Yagis designed with this tool. To be fair, differences of that magnitude only occur in the larger Yagis.
In examining the code in DL6WUGG.BAS, the following snippet reveals the algorithms for calculating beamwidth and stacking distance from gain (dBd).
2630 BH = 30  3.14 *
(G1  14) 'Correlation from published patterns of DL6WU
yagis 2640 PRINT #1, USING " Horizontal beamwidth = ##.# deg"; BH 2650 BV = BH / COS(BH / (2 * 57)) 'Overestimates BV for shorter yagis 2660 PRINT #1, USING " Vertical beamwidth = ##.# deg"; BV 2670 SH = 51 / BH 2680 PRINT #1, " Suggested stacking distances for 2 yagis:" 2690 PRINT #1, USING "Horizontal = #### mm = ###.# inches = #.## wavelengths"; SH * MM; SH * INCH; SH 2700 SV = 51 / BV 2710 PRINT #1, USING "Vertical = #### mm = ###.# inches = #.## wavelengths"; SV * MM; SV * INCH; SV 
The expression at line 2630 can be rearranged to BH=73.963.14*gain, which suggests that eplane beamwidth is linearly related to gain in dBd which is unlikely to be true. Note that with this algorithm, beamwidth is negative for gain greater than 23.55dBd. This algorithm is mainly responsible for the gross errors shown in Table 1.
Line 2650 calculates hplane beamwidth based on the flawed eplane beamwidth, and using an algorithm that does not compare well with NEC model predictions.
Line 2670 and 2700 calculate suggested stacking distances using an algorithm that seems traceable to W1JR.
The notion that gain is related to capture area suggests that the product BWe*BWh*Gain should have similar values for all of the Yagi sizes. BWe*BWh*Gain was calculated (gain as a dimensionless ratio, and beamwidths in degrees) from the 91 NEC model predictions. The average value is 37632 and the SD of 10*log(BWe*BWh*Gain) is 0.05dB. Note that the value 37632 incorporates the loss in the modelled antennas.
The main lobe of a Yagi is not circular in cross section, the Hplane beamwidth is greater than the Eplane beamwidth, and the ratio of these decreases as the gain increases. Exploring the ratio BWh/BWe against Gain from the 91 NEC model predictions suggests a solution to prediction of BWh/BWe. Fitting a curve to the data points gives an expression of BWh/BWe=1+4.03306*EXP(0.264152*GaindBi) with a correlation coefficient of 0.99948, which is is a pretty good fit considering the model predictions are not stated to great precision, so there is noise introduced by the rounding process in the model output. This curve should be well behaved (in fact the regression is constrained to be asymptotic to BWh/BWe=1).
So, using the constant BWe*BWh*Gain and the expression BWh/BWe=1+4.03306*EXP(0.264152*GaindBi) , it should be possible to predict BWe and BWh from Gain. Table 2 sets out summary statistics of the differences between this method (lets call it VK1OD) and the NEC models.
BWe  BWh  
Avg  0.00%  0.01% 
SD  0.54%  0.58% 
Min  0.94%  1.03% 
Max  1.07%  1.15% 
Figure 1 shows the predicted E plane bandwidth from the DL6WU method (DL6WUGG.EXE) and the VK1OD estimator described above, against the NEC model predictions.
Figure 2 shows the predicted H plane bandwidth from the DL6WU method (DL6WUGG.EXE) and the VK1OD estimator described above, against the NEC model predictions.
The VK1OD method proposed should be a sufficiently good estimator of the model results for the purpose of calculating stacking distances.
Fig 3 shows the suggested stacking distances for optimal gain, against gain using the two methods for estimation of beamwidth. Both methods use DL6WU's recommendation of SDx=1/(2*sin(BWx/2))λ. (Note that DL6WUGG.EXE does not use SDx=1/(2*sin(BWx/2))λ, but it uses SDx=51/BWX which yields about 89% of the stacking distance given by the first equation.)
(Optimal gain is not necessarily maximum gain, it is possible to increase stacking distance and obtain slightly higher gain (tenths of a dB) but at the expense of artificially splitting the main lobe and creating a very narrow main lobe with relatively high amplitude second lobes. Interested readers can explore this using NEC.)
The remaining questions are "are the NEC models a good estimator of real antennas, and if so, is the analytical method described above for predicting the beamwidths in the models a better way than the current algorithms to predict the beamwidths in DL6WUGG.EXE and similar tools?"
My attention has been drawn to the fact that the stacking distance for maximum gain can be quite a deal greater than DL6WU's recommendation (Dopt). On the NEC models that I tried, it is possible to achieve gains of up to nearly 0.5dB better than with DL6WU Dopt, but maximum gains is not necessarily optimum, it is a single parameter quest for optimisation, without regard for instance to the physical size of the array and the level of side lobes, especially those nearest to the main lobe which may be only 4dB below the main lobe in some maximum gain cases. In the small number of test cases that I tried, stacking in the hplane at the Dopt distance gave near the 3dB increase in gain over the single array.
In summary:
The differences in beamwidth calculations are relatively small for Yagis of size less than about 19dBi (~30 elements), so most implementations based on DL6WUGG.EXE will be in the reasonably accurate range. The consequence of the W1JR distance algorithm is a slight reduction in gain (~0.3dB) for the benefit of a smaller physical array and reduced first side lobe intensity.
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