The difficulty of the task is due to the complex non-linear relationship of mutual coupling between close-spaced elements, defying efforts to finding analytical solutions. Numerical solutions only became effective in the 1980s with the evolution of computing technology and the concurrent evolution of algorithms and their software implementations.

My first exposure to the problem of yagi design and optimization was in the late 1980s. I and several other hams interested in building more competitive contest stations were unhappy with many of the commercial antennas we were using. Some with downright dreadful, relying more on myth and reputation than on measured performance. Yet the tools available to design something better were lacking. Then along came John Lawson W2PV and his excellent book

*Yagi Antenna Design*(out of print), formalizing his analytical and experimental work on yagi design over many years.

With that book in hand I implemented several of his algorithms and designed realistic yagis for several of my friends. Aside from performance was the difficult problem of scaling the design to the mechanical specification of telescoped tube elements and the effects of clamps and boom. W2PV's algorithms provided an excellent analytical approach to solve those difficulties.

**Stepped Diameter Correction (SDC)**

There has been progress in the past 30 years. MiniNEC and then NEC2 engines arrived and were incorporated into many commercial and non-commercial antenna modelling applications. However neither engine correctly models tapered elements using telescoping tubes. NEC4 does handle it but it is an expensive solution for most hams. I have not seen fit to pay up for it.

It was long ago determined that NEC2 can be manipulated to give correct results for tapered elements. W2PV did it long ago (I don't know if he was the first) and then W6NL codified it for NEC2. EZNEC and others incorporate W6NL's SDC algorithms. They have been amply verified in the field so that we can confidently employ those algorithms and know that the results will closely match NEC4 and real antennas. The algorithms do have limitations on their application (e.g. loading coils). EZNEC 6, for one, has improvements in this area but I do not know how reliable those are.

**Problem statement**

Someone approached me with a request to model an optimized dual-band yagi for their custom tubing schedule and element-to-boom clamps. The antenna is a long boom, wide band interlaced yagi for 15 and 20 meters with 6 active elements on each band designed by NW3Z. The antenna is an intriguing one. It is in the class of OWA yagis that achieves excellent performance along with exceptional low SWR across both bands.

There were ample design challenges which you can read about in that document. In fact you

*must read*the details of that antenna there since I will not repeat any of it. Otherwise the discussion that follows could be confusing in several important aspects. You can read some commentary about this type of antenna by Cebik. There is another discussion of the design principles starting on page 18-22 of this book extract.

The antenna has separate feeds for each band. Simultaneous operation on both bands absolutely requires very good high power filters. Otherwise you can use a remote switch to use one transmission line for both bands. No matching is needed for 50 Ω coax.

Certainly an antenna of this type would be best modelled with NEC4, and that is what NW3Z did. NEC2 with SDC can give accurate results although not without additional effort and some residual uncertainty. I use EZNEC+ version 5 (NEC2 engine) with the standard W6NL SDC algorithm. There were many challenges to overcome yet in the end the result matches the performance quoted by NW3Z.

This article is about how I did it. There is nothing here that is novel -- it's all been done before. The point is to guide readers along should they wish to do something similar. Being hams we are often likely to want to build yagis based on a proven design while using hardware that is locally available or cheaper than that specified in the design. Once you know how to scale the design to the chosen hardware you can proceed with construction confident that your actual performance will be a close match.

For me the interest was the interlacing of yagis. You cannot do this with tools such as YW, YO and some others, and is still quite challenging with a comprehensive modelling tool such as EZNEC. There was more to my initiative than doing someone a favour: I wanted to learn something about a subject I care about.

**The antenna is already optimized**

One important thing to state is that the NW3Z design is already thoroughly optimized. That is, it is optimized to its performance objectives. The antenna is not a maximum gain design, giving up around 0.5 db on 20 meters and perhaps 1.0 db on 15 meters. Its F/B is similar or better than mono-band designs of similar size. The SWR is exceptionally low by design, and is in part responsible for the lower gain. Despite giving up some gain it is a great antenna and will fare better than a multi-band antenna using traps or other element loading techniques.

Model view and currents when excited at 14.100 MHz |

Other operators, even competitive DXers, can get by without this feature. For them a slightly higher SWR at the band edge (say, 2 to 2.5, or even 3) should be an acceptable trade off for an additional decibel of gain. If that's you there are other designs from which you can choose. Do not try to "optimize" this antenna since you are almost certain to make it worse. Small departures from the published dimensions will do just that. I experienced this when the ham asking me to do the model made a small calculation error on one of those 12 elements.

In accord with these points I am accepting the NW3Z antenna as is; I am scaling the antenna, not optimizing it or changing it into an antenna with different performance metrics.

**What needs to be scaled**

In this article I will not publish dimensions of the scaled antenna. That would be pointless since every ham is likely to use their own set of materials, in each case producing unique scaling results. It's the scaling procedure that is at issue here. If you want this antenna and do not want to bother with scaling you are best advised to adhere to the exact dimensions provided by NW3Z.

- The tubing sections must decrease in diameter toward the element ends, the element halves must be identical and no loading elements. This is easy to achieve in a mono-band yagi, and EZNEC will warn you when you make a mistake. Clamps at the tube boundaries can be ignored at HF.
- Element-to-boom clamps must be converted to an equivalent diameter which is then specified in the model.
- Depending on the clamp style the effect of the boom may need to be included. For example, in Hy-gain yagis where the element effectively pierces the boom. In homemade yagis with the more typical rectangular plates and u-bolts (with or without a saddle) the boom effect can be ignored.

**Effective diameter of plate style element-to-boom clamps**

The yagi I modelled uses plate style element-to-boom clamps. W2PV in his book presents equations for this style of clamp and for those where the element pierces the boom. I will only cover the first. However the boom effect for the latter style is small, being the equivalent of electrically shortening the element by about 10% the boom diameter.

I am using W2PV's equations despite more accurate models that are more recent. For example, there is the improved model promoted by W6NL. Unfortunately I don't his book

*Physical Design of Yagi Antennas*(out of print) where this is discussed. The difference from what I can tell from my limited ability to compare results is within ±2% for HF size material, which is not significant. Note that the error is in the

*effective diameter of the clamp*, not the element length.

There is lots of software around that will calculate the effective diameter of several common element-to-boom clamps if you insist on that degree of accuracy. One I feel confident recommending even though I don't own a copy (or at least not yet) is AutoEZ by AC6LA, which does this for you and much more when used in combination with EZNEC.

Example clamp here |

However the W6NL and related models don't appear to model clamps where a u-bolt saddle is placed between the tube and plate, or at least not that I know of. W2PV's equation does, and that factor is not insignificant. So I put his model into a spreadsheet and used it in this antenna design, with the resulting effective diameter placed into the taper schedule in EZNEC.

The screen capture of the spreadsheet with my implementation of the W2PV equation for a tube over plate style clamp includes the specs of the antenna I am scaling. Although in this case the units are centimeters any units can be used provided it is used consistently throughout. The cell with the W2PV equation is shown so that you can replicate it.

The saddle height is equal to the distance between the tube and plate. The height is zero when there is no saddle. The calculated values are: a1, radius of the tube; S1, circumference of the tube; a2, effective radius of the plate (width / 4); S2, perimeter of the plate cross-section; d, centre-to-centre distance between tube and plate. The effective diameter is twice the calculated effective radius. This is the number to use in the EZNEC wires table. The wire length is simply the length of the plate.

In all case the effective diameter should be intermediate between the tube diameter and plate width. If it isn't you've made a mistake.

The effective diameter calculation does not work for elements clamps that electrically isolate the element. This is common in driven elements in many antenna, including the NW3Z design. Although the calculated effective diameter, or no correction at all, will be in error it is not of serious consequence. The reason is that tuning of the driven element(s) in a yagi does not affect gain and F/B performance. Once constructed the driven elements can be adjusted, if needed, to get the desired match. By using the actual tube diameter rather than the effective diameter the required adjustment should be less.

**Segmentation and tubing schedule**

Nearly end-on view showing the segment and tube alignment |

I used a segment length of 10 cm (4"). This adds up to over 1,000 segments in the model and so can be quite slow to calculate on older computers. The length works well for 20 and 15 meters and being a round number it is relatively easy to align tube junctions, which is also desirable for model accuracy. Luckily the builder provided a detailed tube schedule with all this taken care of. I believe his intent was cost and convenience, yet it also helped make for a good model.

The element-to-boom clamps must also follow the plan since it is in effect the centre tube. Happily the clamps in this instance are 20 cm long -- two segments. However I used one 20 cm segment for the driven elements in order to avoid using a split source, which in my experience can introduce errors.

To build the model I worked up the wires for one element on each band and copied it until I had a full complement of elements, moving each into position. Then it is a matter of adjusting the tip lengths on the elements to match the spec. It is necessary to be inspect the segment length of the tip so that each is as close as possible to, in my model, 10 cm. The work is bothersome yet necessary, and must be repeated several times during the scaling procedure.

Even with the extensive segmentation work there was still a residual error in the model. This shows up using the average gain test that W7EL describes in the EZNEC manual. Since all other potential error sources were covered to the best of my knowledge the solution is to adjust the gain figures by the average gain. That is, if the average gain is -0.32 db you subtract this value from the calculated gain. For example if the calculated gain in a particular direction is 3.79 dbi the true gain should be 4.11 dbi.

Making this adjustment brought my model's gain almost precisely equal to what NW3Z got with NEC4. That's a good indication that my model is correct. Unfortunately the average gain adjustment is a function of frequency so the true gain must be uniquely adjusted at several points on each band. The F/B does

*not*require this adjustment, so you can read F/B directly from the pattern plot. The reason should be clear when you realize the average gain error affects every point on the far field plot.

**Scaling the element lengths**

In the discussion of segmentation I said that it is only the element tip lengths that are adjusted during the scaling procedure. I took the element length spec from the ham I did this for and simply ran the model once I had everything else taken care of. I compared the SWR, gain and F/B curves with those published by NW3Z to see by how much the antenna's frequency range.

However it wasn't quite that easy. I eventually discovered why on one band the performance was unexpectedly poor: the length of one element was miscalculated. I adjust this to conform with NW3Z's spec and the expected performance immediately emerged. Just at the wrong frequency range.

When scaling a yagi all lengths are geometrically adjusted, not arithmetically. This means all elements for one band are multiplied by a constant. The constant is determined by the ratio of the calculated frequency to the desired frequency. Never adjust elements by adding or subtracting. The geometrical adjustment ensures that the resonant frequency ratios of any two elements is unaffected by scaling. That relationship must be preserved to maintain the performance metrics.

It can be argued that even this scaling factor includes an inaccuracy since we are only scaling the tips of the elements and not each tube in the schedule. This is a quibble since the introduced error is very small for the degree of scaling we are doing, which is only 1% to 2%. The error can be very significant should you attempt to scale the antenna to a different HF band.

Now we can proceed. Assume, for example, the calculated frequency of maximum F/B is 14.250 MHz. Yet it ought to be 14.100 MHz. To pull the yagi's frequency range down by 150 kHz all six 20 meter elements must be lengthened. The scaling constant is 14250 / 14100 = 1.01064. You can round this off to 1.01. After every scaling operation remember to adjust the segment count of the tip sections to keep it close to the selected value, and then confirm that the gain, F/B and SWR are where they should be. If not, repeat.

It is important that in a multi-band antenna like this that you first scale the elements for the lowest frequency band (the longest elements) and then do each next higher band until you're done. Do it the other way and the higher band will be incorrect after scaling the lower one. In this antenna that means you first scale the 20 meter elements. Even so, check 20 meters again after scaling the 15 meter elements since there is a possibility that another adjustment is necessary. If it is you will of course also have to redo the 15 meter elements.

**SDC works on only one band at a time**

The W6NL SDC algorithm only works within 15% of the resonant frequency. EZNEC will perform SDC on the 20 meter elements or the 15 meters elements, but not both at the same time. Resonant frequency is the frequency you select in the main EZNEC window. Check in the wires window that SDC is being applied as it should.

To measure the antenna you must set that frequency for the band you are calculating. This is in addition to moving the source to the corresponding driven element. It's a bother but you must set the frequency for the correct band when you do an SWR plot since SDC is not selected per the frequency range of the plot. Otherwise some error will be introduced into the impedance calculations.

The failure to perform SDC on both bands simultaneously affects the higher band of a two band antenna more than the lower band. Therefore the 20 meter results are very reliable while there may be some error on 15 meters. As far as I can tell for an antenna of this type the error ought to be very small, and so I ignored this limitation of the SDC algorithm. That isn't always advisable since every antenna is a unique case.

**If you have NEC4...**

SDC corrections are not needed in NEC4. That eliminates many of the modelling precautions I've described above. But not all. You must still calculate the effective diameter of the mast clamp and, if a concern, adjust for the boom.

NEC4 is not perfect, nothing is. Its usable domain is greater than NEC2, which is very helpful provided you keep in mind its limitations and constraints. For straightforward mono-band yagis it is certainly easier than NEC2 with the supplementary extensions in EZNEC and some other software tools.

Should you happen to have a friend with a NEC4 license by all means ask them run the model for you. It is a good way to see how well you've scaled the antenna and the accuracy of NEC2 plus SDC and other modelling precautions.

**Comparing results**

NW3Z modelled his antennas using NEC4. How close can we come to his results using EZNEC? Very close indeed as it turns out. It is so close that I was hard pressed to find any differences after completing the scaled model and tuning it so that the frequency ranges matched the curves in his document.

Here are a couple of examples. The first is an azimuth plot at the 20 meter frequency where F/B is greatest. Due to the average gain issue described earlier it is necessary to subtract -0.31 db from the gain in this particular plot. This brings the actual gain to 10.3 dbi. The F/B is correct as is.

The second example is the SWR curve across 15 meters. The impedances are a good match. This is telling since even small deviations from equal segmentation and total element length and position can cause significant miscalculation of impedance by NEC2.

For an antenna of this type I would aim to have the measurement correct to less than 1 cm (½"), which works out to 0.1% on 20 meters and 0.13% on 15 meters. In many cases that may be more accuracy than needed since the presence of cables, guy wires and other antennas even some distance away will introduce errors of at least this amount.

Beyond this modelling you can only build the antenna and do a field strength test to measure the performance. Since few hams will undertake that amount of work it is important that the scaling be done correctly and that the antenna is built as exactly as possible. However there is the alternative of finding the frequency of maximum F/B with the assistance of a friend within ground range. Don't try this with stations via ionospheric propagation since signal strengths change faster than you can rotate the antenna.

**Conclusion**

For my primary interest of HF contesting this type of antenna is a poor choice. Using one effectively would require a tri-plexer, just as one would use with a tri-bander shared among two or more operating positions. I prefer to aim for more separation and directional diversity which requires independent mono-banders.

Where it does enter my plans is for the WARC bands: 12, 17 and 30 meters. These bands are not used for contests but fit well into my DXing activity, either for country chasing or casual operation at any time. I will use my experience with scaling the NW3Z to play with some configurations that provide up to 3 elements on 30 and 17 meters on a single boom.

The importance of this is that I can get good performance on these bands while occupying the minimum amount of tower space, space that is a priority for the HF bands used for contests. Maybe not this year, yet I will have to do something eventually.