What Can We Expect From a 2-Element Beam?

Part 2
The Full-Size 2-Element Yagi

L. B. Cebik, W4RNL





The Full-Size 2-Element Yagi

Two-element Yagis have several variables which the designer can vary.

1. Spacing between elements;

2. Length of reflector; and

3. Length of driven element.

We can also handle these variables in a number of ways. for example, we can

1. Optimize gain at the design frequency;

2. Optimize front-to-back ratio at the design frequency;

3. Strive for resonance;

4. Strive for maximum operating bandwidth (perhaps as defined by a 2:1 SWR); and/or

5. Strive for a 50-ohm match.

The mix and match of design goals leads to an almost indefinitely large number of antenna designs, according to what compromises the designer reaches. A maximum gain design may yield a combination of elements leaving considerable reactance at the feedpoint. Altering the driven element toward resonance may yield an element combination, even when the reflector is remaximized for gain, that is slightly off peak. Similar compromises apply to any other combination of ingredients in the design goals.

We shall look at several models in free space using different spacings and optimized at each spacing for maximum front-to-back ratio and resonance. The degree of element lengthening needed for a gamma or Tee match or the degree of shortening needed for a beta match are too small to make a significant difference in performance. To see why designers lean toward the maximum front-to-back ratio frequency as the design center (or near- center), we shall examine some beams designed for maximum gain at the design center frequency. We shall also look at some models over real ground using one or two spacings and optimized for front-to-back ratio at antenna resonance to determine the operating bandwidth characteristics of the array.

In general, 2-element Yagis optimized for maximum front-to-back ratio have resonant feedpoint impedances in the mid-30-ohm range with spacings of about 1/8 wavelength and in the 50-ohm range with spacings in the vicinity of 0.16 wavelength. The represents a range of 4.1 to 5.4 feet at 29 MHz, which you can scale to any other frequency with an appropriate multiplier.

Let's take a more comprehensive look at this collection of antennas by specifying a sequence of spacing at 0.04 wl intervals from 0.08 wl through 0.24 wl (2.7' through 8.1'). The models will be in free space. The driven element will be resonated. Gain figures will be for 29 MHz, although that is the frequency of maximum front-to-back ratio. Maximum gain occurs somewhat lower in frequency.

NEC (2 or 4)  (Reference dipole gain in free space = 2.13 dBi)
Spacing   Gain (dBi)     Gain (dBdr)    F-B (dB)  Feed Z (R +/- jX)
0.08 wl   6.32           4.19           11.38          17.12 - j0.66
0.12      6.25           4.12           11.19          32.47 + j0.24
0.16      6.12           3.99           10.86          46.67 - j0.33
0.20      5.87           3.74           10.35          61.32 - j0.31
0.24      5.56           3.43            9.73          72.81 - j0.16

Let's make the same run with MININEC. Typically, MININEC yields dimensions that are about 0.04' (1/2") shorter than NEC at 29 MHz, although one version of MININEC (AO) has a correction factor to bring them into alignment as frequency increases.

MININEC  (Reference dipole gain in free space = 2.12 dBi)
Spacing   Gain (dBi)     Gain (dBdr)    F-B (dB)  Feed Z (R +/- jX)
0.08 wl   6.31           4.19           11.40          17.09 - j0.59
0.12      6.25           4.13           11.19          32.33 + j0.17
0.16      6.09           3.97           10.86          46.84 - j0.56
0.20      5.87           3.75           10.34          61.12 + j0.07
0.24      5.56           3.44            9.72          72.42 - j0.59

The differences between the two modeling systems are not great enough to make a difference under any practical circumstance.

To understand why designers tend to select spacings of 0.12 wl to 0.16 wl, we need one additional data table in hand: the SWR of the antennas across the band from 28 to 30 MHz (given here at 0.5 wl, using the SWR sweep facility of EZNEC). Each sweep is centered on the resistive component of the feedpoint impedance at the design center frequency. Values greater than 1.0 occur at that frequency because of the remnant reactance.

Spacing   SWR at    28        28.5      29        29.5      30   MHz
0.08 wl             31.2      5.88      1.04      2.91      5.52
0.12                7.12      2.33      1.01      1.81      2.76
0.16                3.56      1.71      1.01      1.47      1.99
0.20                2.30      1.42      1.01      1.31      1.65
0.24                1.82      1.29      1.00      1.23      1.49

Obviously, the widest spacing offers the greatest operating bandwidth, but at the cost of reduced gain and front-to-back ratio. Consequently, a design tends to compromise among the highest gain, the highest front-to- back ratio, adequate operating bandwidth, and feedpoint impedance. ).16 wavelength spacing offers the opportunity for a direct match to 50-ohm coax feedlines with a fairly useful bandwidth for most of the HF ham bands. (Remember to reduce the bandwidth by dividing the 10 meter figure by the ratio of 29 MHz to the frequency of interest for lower HF bands.) Spacings closer to 0.12 wl yield higher gains and front-to-back ratios, but over a narrower bandwidth.

There are two other design problems one must consider. First, the frequency of maximum gain is well below the frequency of maximum front-to- back ratio. The gain tapers gradually as the frequency increases within the operating bandwidth. Second, SWR increases rapidly below the design frequency and more slowly above it. When this factor is combined with the gain situation, one can design an illusion: an antenna with decent SWR but very little gain or front-to-back ratio in he upper half of its operating range.

To illustrate this situation, let's look at the models in more detail, examining their gain and front-to-back patterns across 10 meters.

Frequency      28        28.5      29        29.5      30  MHz
0.08 wl spacing
Gain (dBi)     6.37      6.92      6.32      5.77      5.37
Gain (dBdr)    4.24      4.79      4.19      3.64      3.24
F-B (dB)       1.82      8.65      11.38     10.12     8.57
0.12 wl spacing
Gain (dBi)     6.98      6.74      6.25      5.82      5.48
Gain (dBdr)    4.85      4.61      4.12      3.69      3.35
F-B (dB)       5.46      9.79      11.19     10.37     9.18
0.16 wl spacing
Gain (dBi)     6.88      6.55      6.12      5.74      5.43
Gain (dBdr)    4.75      4.42      3.99      3.61      3.30
F-B (dB)       6.66      9.84      10.86     10.29     9.32
0.20 wl spacing
Gain (dBi)     6.64      6.28      5.87      5.50      5.20
Gain (dBdr)    4.51      4.15      3.74      3.37      3.07
F-B (dB)       7.31      9.66      10.35     9.87      9.03
0.24 wl spacing
Gain (dBi)     6.37      5.98      5.56      5.18      4.86
Gain (dBdr)    4.24      3.85      3.43      3.05      2.73
F-B (dB)       7.36      9.22      9.73      9.29      8.51

From this and previous tables we can draw several conclusions applicable to 2-element Yagis on any band.

1. Reading across the tables, it is clear that the maximum gain frequency is within the sweep for the closest spaced beam, but at or beyond the lower frequency limit for the other models. The closer the spacing, the closer together are the frequencies of maximum gain and maximum front-to-back ratio.

2. The wider the spacing, the lower the overall values of gain for the entire sweep.

3. Gain falls off somewhat rapidly above the design center frequency. It rises even more rapidly below the design center frequency, although that curve is invisible in these tables.

4. Using the SWR table in conjunction with this table, it is clear that maximum gain occurs in a region of high SWR when the beam is designed for maximum front-to-back ratio.

5. Front-to-back ratio holds up best at spacings between 0.12 and 0.20 wavelengths, inclusive.

It is therefore possible to design a beam with a wide operating (2:1-SWR) bandwidth using spacings of 0.20 or 0.24 wl, but accrue little more than 3 dB gain over a dipole and a front-to-back ratio under 10 dB for most of that bandwidth. Equally, achieving more than 4 dB gain over a dipole and a front-to-back ratio greater than 10 dB for a large portion of the operating bandwidth is not feasible with a full size 2-element Yagi.

As a result of these limiting conditions, when a 2-element Yagi is designed for maximum front-to-back ratio, design compromises are necessary. When the bandwidth requirements are narrow, as on 17 and 12 meters, a spacing in the vicinity of 0.12 wl is often chosen for the best combination of gain and front-to-back ratio, along with a sufficiently high feedpoint impedance to assure efficiency. For wider bands, a spacing around 0.16 wl is favored, trading some gain and front-to-back ratio for operating bandwidth and an easy match to 50-ohm coax.

The alternative design strategy we might use is to design our beam so that the array resonates at or close to the frequency of maximum gain.

NEC (2 or 4)  (Reference dipole gain in free space = 2.13 dBi)
Spacing   Gain (dBi)     Gain (dBdr)    F-B (dB)  Feed Z (R +/- jX)
0.08 wl   7.02           4.89           6.29            9.54 - j0.06
0.12      6.99           4.86           6.12           19.50 - j0.14
0.16      6.88           4.75           5.97           30.67 + j0.70
0.20      6.68           4.55           5.77           43.45 + j0.16
0.24      6.43           4.30           5.75           55.84 - j0.43

The gain figure for the 0.08 wl spaced Yagi optimized for gain approaches the absolute maximum gain obtainable from a 2-element parasitical array. However, this gain is obtained at a cost: a severe reduction in the front- to-back ratio and a very low feedpoint impedance. As spacing is increased, the maximum obtainable gain also decreases, along with the front-to-back ratio at that gain figure.

The operating bandwidth picture is equally bleak:

Spacing   SWR at    28        28.5      29        29.5      30   MHz
0.08 wl             40.6      16.4      1.01      6.87      1452
0.12                14.7      5.20      1.01      3.03      5.43
0.16                7.12      2.83      1.02      2.11      3.27
0.20                4.22      2.03      1.00      1.65      2.29
0.24                2.99      1.68      1.01      1.42      1.83

Only at a spacing of 0.24 wl do we obtain any significant operating bandwidth, and by that spacing, gain and front-to-back ratio have fallen severely. In fact, gain has decreased to the levels of more closely spaced maximum front-to-back designs.

As if these factors were insufficient reasons for designers to move the operating point of the array toward the maximum front-to-back region, an additional problem emerges if one examines the beam's properties across a frequency span.

Frequency      28        28.5      29        29.5      30  MHz
0.08 wl spacing
Gain (dBi)     7.15 R    6.01 R    7.02      6.59      6.02
Gain (dBdr)    5.02 R    3.88 R    4.89      4.46      3.89
F-B (dB)       10.29     0.72      6.29      10.61     10.54
0.12 wl spacing
Gain (dBi)     6.94 R    6.13      6.99      6.70      6.24
Gain (dBdr)    4.81 R    4.00      4.86      4.57      4.11
F-B (dB)       4.74      1.05      6.12      9.80      10.87
0.16 wl spacing
Gain (dBi)     6.31 R    6.35      6.88      6.63      6.23
Gain (dBdr)    4.18 R    4.22      4.75      4.50      4.10
F-B (dB)       2.23      1.97      5.97      9.12      10.52
0.20 wl spacing
Gain (dBi)     5.53 R    6.36      6.68      6.45      6.07
Gain (dBdr)    3.40 R    4.23      4.55      4.32      3.94
F-B (dB)       0.55      2.59      5.77      8.44      9.89
0.24 wl spacing
Gain (dBi)     5.45      6.29      6.43      6.17      5.77
Gain (dBdr)    3.32      4.16      4.30      4.04      3.64
F-B (dB)       0.69      3.16      5.75      7.96      9.22

The entries labeled "R" indicate gain in the reverse direction from that of the remainder of the entries. The maximum gain point in the geometry of a 2-element Yagi occurs just above the frequency at which the parasitical element functions as a reflector. Below a certain critical frequency that varies with spacing, the parasitical element becomes a director, even though it is physically longer than the driven element. (It would be shorter if the driven element were lengthened to resonance.) We may note in passing, with an eye on the 0.08 wl spaced beam, that the driven- element-director configuration is capable of slightly higher gain than the driven-element-reflector arrangement.

As spacing is increased, the frequency at which the beam flips directions grows more distant from the frequency of maximum gain. However, performance of the beam in the range between reversal and maximum gain is marginal at best.

The purpose of these latter tables is twofold. First, they demonstrate the maximum gain of which a full size 2-element driven-element-reflector Yagi is capable, and the conditions surrounding that achievement. Second, they also illustrate why designers tend to give up maximum gain in favor of maximum front-to-back ratio: adequate gain, wider operating band width, higher feedpoint impedances, and higher front-to-back ratios. We may reiterate that above the frequency of maximum front-to-back ratio, the feedpoint SWR (referenced to the impedance at the design center frequency) decreases more slowly than below the maximum F-B frequency, but both gain and F-B ratio decrease together. Hence, specifying the peak values of gain, front-to-back ratio, and operating bandwidth does not always give a fair indication of beam performance. We may also note that in no case of normal directional operation does the driven-element-reflector reach 5 dBdr.

The characteristics of a given 2-element Yagi design are not constant with height above ground until the beam is well above 1 wavelength high. Note the changing gain, front-to-back ratio, and feedpoint impedance of the 0.12 and 0.16 wl spaced beams at heights between 1/8 and 1 wavelength. The gain in dBdr is calculated using the TO angle (or elevation maximum radiation) of the beam, not the dipole, as reported in an earlier table.


0.12 wl spaced Yagi
Height    TO Angle  Gain dBi  Gain dBdr F-B dB    Feed Z Ohms
FS        --         6.25      4.12     11.19     32.47 + j0.24
1/8 wl    55         5.10      1.44      6.08     25.67 - j5.28
1/4       43         8.58      3.34      9.79     27.79 + j4.15
3/8       33         9.84      3.89     13.34     34.40 + j4.35
1/2       27        10.80      3.58     13.37     36.14 - j1.18
5/8       21        11.23      3.50     10.22     21.21 - j2.56
3/4       18        11.27      3.98     10.08     30.29 + j1.04
7/8       16        11.37      4.22     11.81     32.96 - j0.20
1         14        11.61      3.98     12.51     34.30 + j0.24

0.16 wl spaced Yagi
Height    TO Angle  Gain dBi  Gain dBdr F-B dB    Feed Z Ohms
FS        --         6.12      3.99     10.86     46.67 - j0.33
1/8 wl    56         5.19      1.46      5.84     34.94 - j6.52
1/4       44         8.56      3.26      9.66     40.69 + j6.57
3/8       33         9.79      3.84     13.05     50.55 + j5.25
1/2       27        10.74      3.52     13.49     51.69 - j3.31
5/8       22        11.14      3.38     10.10     44.17 - j4.15
3/4       18        11.16      3.87      9.75     43.66 + j1.35
7/8       16        11.27      4.12     11.39     47.84 + j2.39
1         14        11.50      3.87     12.30     49.25 - j1.40

Overall, there is not much to choose between the two models. As noted, the wider spaced beam has a somewhat greater operating bandwidth, but slightly lower gain and front-to-back figures. The seemingly anomalous front-to- back ratio at a height of 1/2 wl is real and indicates that not all properties of antennas can be predicted by simple curves. The higher gain of both beams over a dipole at 7/8 wavelengths height is also accurate, since at that height a dipole hits a gain minimum which the multi-element design largely overcomes.

Below a height of 1/2 wavelength, feedpoint impedance and other values wander fairly far from free space values. Indeed, for many purposes, a height of 1/2 wavelength (or at least 3/8 wavelength) represents a practical minimum recommended height for the 2-element Yagi. The variability of values also extends upward beyond the 1 wl cut-off for this study. Gain tends to reach a maximum for many antennas in the 1 1/8 to 1 1/4 wavelength height region, and this region has become popular among some manufacturers for specifying gain. This and related height phenomena were reported upon extensively in "The Effects of Antenna Height on Other Antenna Properties: A Computer Study," Communications Quarterly, 2 (Fall, 1992), 57-79.

Needless to say, when fine shades of performance comparison are at stake, mere numbers for gain, front-to-back ratio, and operating bandwidth are normally meaningless without a complete specification of their derivation. Even summaries of typical cases of derivation can make comparison elusive, since they often leave it ambiguous as to which derivation was used for a particular antenna. Until buyers of amateur radio antennas are provided with the same detailed information that can be demanded by military, government, and private corporations for contract fulfillment, caveat emptor must still rule the marketplace.



Updated 5-5-97. L. B. Cebik, W4RNL. Data may be used for personal purposes, but may not be reproduced for publication in print or any other medium without permission of the author.



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