Tuesday, June 30, 2015

Where Gain Comes From

In pursuit of antenna gain it's physical basis is sometimes overlooked. Of course most hams already know that element coupling and phase relationships are among the critical elements. However the details hold some insights. As I write this, it is raining, band conditions are poor and ARRL Field Day is occupying the bands (and in which I chose not to participate). This seems a good time to reflect on fundamentals.

This article is not a rigorous or mathematical look at gain. There are ample texts available for that, both in the amateur radio and professional literature. Instead I will give my approach to understanding gain, with the hope that a few readers will learn something new. You can always delve deeper into the technical literature if you are sufficiently intrigued to want to learn more. While I did double-check the mathematics I present in this article I'll ask you to forgive any minor errors. Major errors are the ones I want to know about.

Update July 1: Errors were found and corrections made. Nothing major, although one equation was bungled. Edits are not marked since they are not earth shaking.

Update July 9: The opening paragraph of the Radiation Resistance section said that phase and current equalization alone limits the gain in an ideal 2-element yagi to 3 db. This is wrong. The error has been corrected, as has the rest of that section. Even though the ultimate conclusions are correct the path there should not mislead. Therefore the correction is warranted.

Conservation of energy

Gain is inseparable from directivity. The mythical radio antenna with no gain is the isotropic radiator which, by definition, has no directivity. Real antennas have directivity, whether accidental or designed. Since it is a fundamental law of physics that energy be conserved, gain in one direction must come at the expense of gain in another. Gain is commonly expressed in dbi, relative to an isotropic radiator.

High gain is therefore strongly associated with high directivity. Typically one strong main lobe with minor side and back lobes.

Radiation is proportional to antenna current

Antenna operating parameters at the granular level include voltage, phase, resistance and current. But in the end it is antenna current that determines the radiation field. The other items are our tools to get there, in that they determine the current magnitude and where it flows. Therefore antenna designers must pay attention to all those parameters.

It is not only current magnitude we care about. The distance over which the current flows matters. For antennas not too long (relative to wavelength) a similar current profile over a longer antenna results in more radiation. Short antennas require a higher current (and lower loss) to match the gain of a longer antenna. In mathematical terms, it is the integral of the current over the element that determines the radiation.

When an antenna has multiple elements the fields from the elements interfere in the near field (mutual impedance) and in the far field (superposition), which determines the pattern. Interference does not destroy energy. There is a difference in how this manifests for near-field coupled elements (e.g. yagis) and far-field coupled elements (e.g. wide-spaced vertical arrays and stacked yagis).

Objectives for gain

With gain as our objective we want to:
  • Maximize current magnitude
  • Maximize the element length over which high current is maintained
  • Phase currents from multiple elements so they reinforce in desired directions and cancel in others
Unfortunately we can't have it all. Compromises are necessary. Let's delve into a few details.

Perfect Front-to-Back (F/B)

The criteria for maximizing F/B are stringent, far more than for gain. This can be illustrated with a hypothetical example.

Let's assume a simple 2-element yagi, with the parasite configured as a reflector, designed for an infinite F/B in the backward direction -- 0° degrees elevation and 180° azimuth, in free space. This is in fact not possible to achieve in a real 2-element yagi, but let's set that aside for the moment for this thought experiment.

We'll pick a typical boom length of 60° (0.167λ). Let's imagine that the current phase on the reflector is 120° ahead (or 240° behind) the driven element (source). When the rearward radiation from the driven element reaches the reflector the reflector current has advanced to 180° (120° + 60°) so the radiation cancels. This is an infinite F/B.

Well, not quite. Complete cancellation also requires the fields to have equal magnitude. This requires equal current in both elements. It is akin to balancing a pencil on its point since the slightest variation in magnitude or phase results in a finite F/B, or the null will skew to a different direction. That's in addition to element diameter, wind, etc. You also can't QSY since these conditions can only be achieved at one frequency; an infinite F/B on CW would not be infinite on LSB or USB! High, yes. Infinite, no.

As we can see, even in the ideal case a perfect F/B is elusive, and one cannot come close is a real 2-element yagi with conventional dipole elements. It is possible to do better in yagis with 3 or more elements, though with some confounding design criteria.

The gain associated with this idealized 2-element yagi with infinite F/B is not perfect. In the forward direction the phase of the field from the reflector does not equal that of the driven element, which is necessary for maximum reinforcement. The phase difference is 60°. This yagi would have less than the ideal 3 db gain for equal and in-phase element currents. One saving grace is that there still is gain when the magnitudes are unequal and the phase relationship is not ideal.

Yet this is only for two narrow directions: exactly rearward and forward. To calculate the complete 3-dimensional pattern of F/B and gain it is necessary to do vector addition of the fields over the full sphere since the far-field phase difference between elements is direction-dependent.

Radiation resistance

If current equalization and phasing are all there is to yagis we could not get more than 6 dbd gain (8.1 dbi) from a 2-element yagi. Yet a real gain-optimized 2-element yagi can only achieve a peak gain of about 5 dbd (7.1 dbi) since element currents cannot be equal and ideal element phase is elusive. Even in a Moxon, where currents are nearly equal, phase inevitably favours F/B at the expense of gain.

Indeed, currents and phasing alone can do no better than ~5.1 dbi forward gain, about 3 db less than the ideal maximum gain. Clearly there must be more to the story. The answer lies in the radiation resistance and its impact on antenna currents.

To get an idea of what is happening it is helpful to modify the source behaviour in our software modelling systems. EZNEC, by default, normalizes the source current to 1 A. It does this by adjusting the source power after calculating the antenna feed point (source) impedance. This is usually the preferred approach since it is helpful to work with a constant reference current when analyzing models. That is, wire and segment currents relative to a constant source current. EZNEC allows us to set the power to a constant and let the source current float. This is how we normally operate our transmitters, provided the SWR does not cause output power to be rolled back. It is also what we need to do to better understand yagis and gain.

When we do this and feed 100 watts to a dipole, a 2-element yagi (60° boom) and a Moxon rectangle we are presented with a set of element currents and impedances per the table at right. The yagi is the 2-element 6 meter yagi I designed and installed a few weeks ago, and the dipole is the driven element of the same yagi. The Moxon rectangle is an optimized design for 40 meters. All are modelled in free space.

There is some variation in feed impedances due to the different wire length to diameter ratio for these 6 and 40 meter antennas, but otherwise scale appropriately for our purposes. The table is a hybrid of EZNEC source data and segment currents.

The dipole impedance is less than 73 Ω due to being made from tubing (low length to diameter ratio). The slight shortness of the dipole (to accommodate a gamma match of the yagi it comes from) does not appreciably affect the current distribution versus a true λ/2 length. Ignore the reactance and you'll see that the current follows from the usual relationship: I = (P / R).

Adding back the reflector we have a 2-element yagi for 6 meters, gain optimized for 50.1 MHz. Notice what has changed:
  • Driven element current is 39.5% higher than the dipole.
  • There is a reflector current that is 76.9% of the driven element. It is 153.9° ahead of the driven element.
  • Feed point impedance has dropped by about half. R and X are suitable for the designed gamma match.
First, let's consider the driven element. The current increase represents a field increase of 1.95 (1.395²), which is close to 3 db. The impedance reduction is due to the mutual impedance with the reflector element.

This arrangement of parallel identical dipoles is a standard study for mutual impedance since it is a simple case. The mathematics involve solving a matrix (2 by 2 for two elements, and scales with additional elements) of complex impedances. One complication is that the induced current on the parasite induces a current back onto the driven element. A good mathematical derivation targetting the ham audience is by the late Les Moxon in his book HF Antennas For All Locations (Chapter 5). If you don't have his book (why not?) there is also a good description on Wikipedia. I'll simply note that for this 2-element yagi the mutual impedance (Zm) is about 60 - j10 Ω, for both elements. Zm is responsible for both decreased radiation resistance and increased Q.

Real yagis

The currents and phases in the yagi elements maximize neither gain nor F/B. There is in fact no 2-element yagi design that can accomplish both, or either one for that matter. The 3 key performance parameters of yagi design -- gain, F/B and impedance -- cannot all be optimized in the same antenna.

In yagis with 3 or more elements the adjustable parameters rapidly increase yet the same constraint applies. For example, if you focus on gain you will find that the impedance drops sharply, resulting in increased I²R conductor loss and require a matching network that will introduce additional loss.

For our 2-element yagi the reflector current relative phase (154°+60°=214°) and relative magnitude (0.77) differ from the ideal to achieve maximum F/B. This can be seen in the elevation plot below. The relative phase (154°-60°=94°) is also not ideal for maximum gain. In fact most of the gain is from the increased current (~2.9 db, due to the lower radiation resistance) than the element phasing (~1.8 db)!

The Moxon rectangle does better at optimizing all performance parameters. While the gain is 1 db worse than the yagi it does better on F/B and impedance. The free space elevation plots of both antennas are overlaid in the adjacent elevation plot.

The high F/B is due to the near equal currents (0.95 relative current in the reflector) and near-ideal rearward relative phase: -4.6°-(48°+124.9°) = 182.5°.

Current gain in the Moxon rectangle delivers only ~1 db, because the radiation resistance is higher than the yagi. The rest (~2.7 db) comes from reflector current and phase: 124.9°-(-4.6+48°) = 81.5° in the forward direction.

Note: For simplicity I am ignoring the forward gain and overall pattern impact of the inward-turned ends of the Moxon rectangle elements. The effective element lengths are shorter than in a conventional yagi, which reduces the broadside field strength. The effect is small since there is little current at the element ends.

It is not surprising that many hams prefer this compromise for small antennas. Larger antennas that incorporate high element coupling for optimum F/B and match include the Spiderbeam style of yagi and the OWA yagis used in some contest super-stations.

Stacking gain

Although I've written about stacking gain in an earlier article, I want to revisit it in the context of the current discussion. Let's place two identical yagis in free space, stacked far enough apart that their mutual impedance is 0 Ω. In the forward direction (centre of the main lobe) the far-field gain will be 3 db. I will break this down, step by step, to highlight interesting relationships between antenna current, radiation resistance and field strength.

Start with feeding just one of the yagis with 1 A source current. We then imagine a distant centre-fed dipole that has induced on it a current of 1 μA. For a receive antenna radiation resistance, transmission line and load (receiver) all 50 Ω, the voltage at the antenna terminals is 50 μV (S9). This follows from the Ohm's Law relationship: E = IZ.

Now we hook up the second yagi and split the transmitter power between them. The source current in both yagis will now be 0.707 A since we've halved the power and I, E ∝ P. That is, I' = I / 2.

In the far field, directly in the centre of the forward lobe, the phases and magnitudes of the fields from both yagis are identical and can be summed (superposition) as scalars. Field strength will be proportional to 2 x 0.707 A = 1.414 A, or 2 A. At the receiving antenna the induced current will be 1.414 μA and the terminal voltage 70.7 μV. The equivalent power of the induced power is doubled, or S9 + 3 db, since for a constant load impedance P' = (E'2)(I'(2).

As always, conservation of energy applies, so that power must come from elsewhere in the antenna pattern. For vertically stacked yagis the main lobe is narrower in the elevation plane than for each individual yagi.

Ground reflections

It is widely known that over real ground a horizontally-polarized antenna exhibits gain over free space due to the sum of the space wave and ground reflection of equal angles. We are achieving gain by, in effect, constraining the free space pattern to one hemisphere. However, the doubling is only seen when integrating over the full hemisphere, and if the ground is perfect (zero loss). The gain and pattern depend on the antenna's free space pattern, height above ground, ground characteristics and local terrain.

One interesting difference from the stacking case is that in the case of perfect ground reflection the nominal gain is 6 db, not 3 db. This has two causes which we will look into. The first is height above ground.

For an antenna like a dipole or yagi the free space elevation pattern is symmetrical (mirror image) above and below the horizontal plane containing the antenna. For each ray of the upward space wave (positive elevation angle) the gain of the antenna is the same as the ray headed ground-ward at the same, but negative, elevation angle. Since the incident and reflection angles are equal these two waves will interfere in the far field.

For a phase difference of 0° the two rays deliver maximum gain (lobe centre). For a phase difference of 180° they will completely cancel (null). At all other phase angles the field strength will be intermediate between those values. Thus are lobes and nulls formed. Were it possible for the reflection to combine in phase throughout the hemisphere the broadside gain of the dipole would be 2.15 + 3 = 5.15 db at all elevation angles. Since the actual pattern has lobes and nulls and we are assuming perfect ground at least one lobe must have a gain in excess of 5.15 db.
Primary is a dipole 1.25λ above perfect ground; blue is a height of 1λ

A comparison of a dipole at different heights above perfect ground illustrates what occurs.

Since perfect ground inverts the phase of the reflected ray by 180°, in-phase superposition requires the path lengths of the space and reflected waves to differ by ½λ or every 1λ increment beyond that. Similarly, the first null has a path length difference of 0λ and there is a null at every 1λ increment. We'll place the dipole at heights of 1λ and 1.25λ, then compare their elevation patterns.

The dipole has a null at an angle of 0°, no matter its height, since the paths lengths are equal and the reflected wave is phase shifted 180°. All lobes have a peak gain of 8.15 dbi, which is the expected 6 db above that of the dipole in free space. The displayed 0.1 db gain difference of the two patterns is due to a plotting anomaly.

For the dipole up 1λ there are 2 lobes, for path length differences of 0.5λ and 1.5λ. More lobes require more height since the maximum possible path difference is 2λ, towards the zenith. For the dipole up 1.25λ there are 3 lobes, for path length differences of 0.5λ, 1.5λ and 2.5λ.

Clearly the 6 db gain in the lobes exceeds the simple reflection average of 3 db, and must be due to the pattern gaps caused by those deep nulls. It is the precise mechanism I want to focus on with respect to in-phase superposition of the space and reflected waves in the centre of those lobes.

Unlike stacking there is no power splitting. If the antenna current is 1 A the current in the dipole's image (ground reflection) is also 1 A. When the space and reflection rays are in phase the field strength due to superposition is now proportional to their scalar sum: 2 A. Thus the gain is 2² = 4, or 6 db. This is why the lobe peaks of the dipole are 8.15 dbi.

Over real ground there is some absorption of downward directed radiation. Ground absorption increases at low angles and poor ground.Also, phase inversion is no longer 180°. Phase increasingly shifts from that ideal as ground quality worsens.

Additional factors are that ground is non-homogeneous and terrain can be far from flat. Tools like HFTA are employed by many hams to get a true picture of pattern formation and ground reflection gain using detailed topographic data for their QTH. Fresnel zones for low (DX friendly) elevation angles come into play. At low incidence angles, reflection can be from a large area, not at a single point on the ground.

Yagi ground reflection gain can be very sensitive to ground quality and terrain since so much of the total radiation is concentrated in a narrow beam width. Since the (height dependent) lowest angle at which the space and reflected waves are in phase is not at the centre of the free space main lobe (the main lobe for a yagi in free space peaks at 0° elevation) the gain due to ground reflection in the first lobe will be less than 6 db. The higher the yagi, the closer one comes to the ideal. However, higher angle lobes are small since the yagi's main lobe has a narrower elevation beam width. Therefore picking an ideal height to reduce high angle radiation is less critical for yagis than for dipoles or other simple antennas.


The physical mechanism by which yagis develop gain and F/B are easily understood with the help of some mathematics and geometry. I believe this is useful knowledge to keep in mind when we design antennas or plan our antenna farms, whether large or small. Software modelling tools and the ready availability of the optimized designs can blind us from an understanding of important details of how real antennas behave, and how to best optimize gain.

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