This is the last lesson in Module 2, and it closes the loop the module opened with. You can now predict an antenna’s pattern from its geometry, simulate it, and measure it with a quantified statement of how much of the measurement to believe. Everything today is a gain claim we already know how to check.

Every high-gain antenna ever built is a different scheme for assembling a big, coherent aperture: a reflector collects area with a mirror, a Yagi collects it from currents induced on its parasitic elements, and an array adds the area one element at a time. Read the physics before the algebra — \(A/\lambda^2\) counts how many square wavelengths the antenna spans, \(\eta_{\text{ap}}\) is the fraction of that area the antenna uses, and \(A_e = \eta_{\text{ap}}A = G\lambda^2/4\pi\) is the number Friis cares about.

The formula is the one Lesson 13 derived for the horn, and nothing about it is specific to horns. Straight ahead of a flat aperture every phase factor in the radiation integral is 1, so the radiation vector is the plain sum of the aperture field \(E_a\), and Lesson 2’s directivity is

with

A uniform, in-phase field makes the fraction exactly 1. Every loss term in this lesson is a way of making the numerator smaller than the denominator: a darker rim, a phase error from a bumpy surface, a patch of the aperture blocked by the feed. Hold on to this ratio; the next five frames each evaluate a piece of it.

For a circular dish, \(A = \pi D^2/4\) and the aperture formula collapses to the one to memorize. The second consequence is why the satellite industry keeps moving up in frequency: the same dish on the roof gains 6 dB every time the band doubles, until the surface tolerance limits it.

Lesson 6 established the Fourier logic: a wider aperture is a narrower beam, and the beamwidth of an aperture of size \(D\) scales as \(\lambda/D\). Power the antenna no longer sends sideways goes to boresight instead.

The parabola is the surface that does this exactly, and Lesson 9’s compact range is the same geometry used indoors — there the collimated wave is aimed across a chamber instead of at a satellite.

Put the vertex at the origin with the axis along \(z\), so the surface is \(z = \rho^{2}/4f\) and the focus sits at \(z = f\). The distance from the focus to a point \(P\) on the surface, at radius \(\rho_P\) and height \(z_P\), follows from Pythagoras and the surface equation \(\rho_P^2 = 4fz_P\):

so \(\overline{FP} = f + z_P\). That is the parabola’s defining property: every point on it is as far from the focus as from the line \(z = -f\) behind the vertex. A ray from the focus to \(P\) then travels parallel to the axis to an aperture plane at \(z = z_a\), covering a further \(z_a - z_P\). The figure says the same thing without algebra: replace the leg \(\overline{FP}\) with its equal, the dashed run from \(P\) back to the line, and every ray’s path becomes the straight distance from that line to the aperture plane. That distance does not depend on which ray you follow, so every point of the aperture plane is in phase.

The half-angle to the rim follows from the equal-path property. Measure a ray from the focus by its length \(r\) and its angle \(\theta'\) from the axis, pointing back toward the vertex. The point it reaches sits at height \(z_P = f - r\cos\theta'\), and \(\overline{FP} = f + z_P\) becomes

The point’s distance from the axis is then

and the rim is where \(\rho = D/2\):

The same \(r(\theta')\) says the rim is farther from the feed than the vertex is, by the factor \(2/(1 + \cos\theta_0)\), so the rim is dimmer than the center even before the feed’s own pattern rolls off. For \(f/D = 0.5\) that spreading alone puts the rim 1.9 dB below the center.

The first trade is the feed’s own pattern. Aim a narrow feed at the dish and the rim sits 20 dB down: the outer part of the dish contributes little, and the effective area shrinks. Widen the feed and the rim brightens — but now power passes the rim entirely and is lost. On receive, spillover also adds noise. Lesson 12’s antenna temperature \(T_A\) is the temperature of whatever the pattern sees, weighted by the pattern. The main beam of an earth-station dish sees sky at a few kelvin to a few tens of kelvin, but the spilled part of the feed pattern looks past the rim at ground near 290 K, and it raises \(T_A\) and with it the noise \(kT_AB\) the antenna delivers.

Spillover and taper are two of the factors inside \(\eta_{\text{ap}}\), and nothing else in the gain formula depends on the feed, so their product is the gain, up to a constant: the edge taper that maximizes \(\eta_\text{s}\eta_\text{t}\) is the edge taper that maximizes the dish’s gain.

It is the house rule of thumb for reflector feeds, and it is why a dish’s aperture distribution always looks like one of the tapers from Lesson 6, with lower sidelobes as a side effect. A uniform circular aperture has its first sidelobe at −17.6 dB; a feed that follows the rule pushes it to about −25 dB.

We can compute the optimum. Model the feed’s power pattern as \(\cos^n\theta'\) and sweep \(n\). Two efficiencies pull against each other. The spillover efficiency is the fraction of the feed’s power that lands on the dish at all,

and the taper efficiency \(\eta_\text{t}\) is the ratio from the frame on the one idea, \(\left\vert\int E_a\ dS'\right\vert^2/\left(A\int\vert E_a\vert^2\ dS'\right)\), with the aperture field set by the feed pattern and the \(r(\theta')\) spreading from the frame on \(f/D\):

A broad feed (small \(n\)) gives a nearly uniform aperture and \(\eta_\text{t}\) near 1, but spills. A narrow feed (large \(n\)) puts everything on the dish but leaves the rim dark. For \(f/D = 0.5\) the product peaks at an edge taper of 10.7 dB, where \(\eta_\text{s} = 0.92\), \(\eta_\text{t} = 0.89\), and \(\eta_\text{s}\eta_\text{t} = 0.82\).

The peak is broad: anything from about 8 to 14 dB is within 0.03 of the best efficiency, which is why a rule of thumb is good enough.

On a 3 m dish a 15 cm feed is negligible. On a 45 cm consumer dish it is not, which is why the dish on the roof looks oval and has its arm hanging off the bottom: the reflector is a slice cut off-axis from a much larger imaginary paraboloid, giving zero blockage and cleaner sidelobes, and the tilted slice is what makes the panel look taller than it is wide.

The blockage factor is the aperture-efficiency ratio again. The feed still radiates the power aimed at the blocked disk of area \(A_b\), so the denominator keeps all of \(A\), but the blocked disk scatters that power instead of adding it in phase on boresight, so the numerator loses the blocked area. For a uniform field,

The square is the whole story: blocking a fraction \((d/D)^2\) of the area costs that fraction of the boresight field, and gain goes as field squared, while the power the feed pays for does not shrink at all. A 15 cm feed loses 0.02 dB on a 3 m dish and 1.0 dB on a 45 cm one.

Phase errors from a bumpy surface reduce the gain exponentially. A bump of height \(\epsilon\) on a shallow dish lengthens the path in and back out by \(2\epsilon\), a phase error \(\delta = 2k\epsilon\), so an RMS surface error \(\sigma\) is an RMS phase error \(\delta_\text{rms} = 4\pi\sigma/\lambda\). In the aperture-efficiency ratio the magnitude of \(E_a\) is unchanged, so the denominator is too, but the numerator now sums \(e^{j\delta}\) over the aperture. For random, Gaussian-distributed errors that sum averages to \(e^{-\delta_\text{rms}^2/2}\), and squaring it gives

In decibels the constant is \(10\log_{10}(e)\ (4\pi)^2 = 685.8\). Note what it means for a fixed piece of hardware: a dish held to 0.5 mm RMS is essentially perfect at 6 GHz and has thrown away 1.7 dB by 30 GHz. Big dishes at short wavelengths are a machining problem, not an electromagnetics problem.

In the figure, each blue bar is the efficiency left after that term and the red cap on it is what the term took away; each bar starts where the one before it ended, because the terms multiply. That product is where the \(\eta_{\text{ap}} \approx 0.65\) we have been assuming comes from. It is an accounting result, not a constant of nature, and every row of it is a design decision someone made. The spillover and taper rows sit a little below the 0.92 and 0.89 of the 10 dB rule’s optimum, because a real feed’s pattern is not exactly \(\cos^n\theta'\). The cross-polarization in “everything else” is power the dish radiates in the polarization orthogonal to the intended one, which Lesson 3 showed a matched receiver cannot collect.

The driven element is a dipole near \(0.47\lambda\). Every other element has no feed line and no source — just a rod that the driven element’s near field induces a current on. That induced current re-radiates, and the total pattern is the superposition of all of them. There is nothing new physically: this is Lesson 6’s radiation integral with several current filaments instead of one.

Why it matters is the cable. An array, at the end of this lesson, gets its gain by feeding every element and setting each one’s phase with electronics. A Yagi gets most of that gain with one feed line and a row of rods, which is why it is on every rooftop. The price is that nothing sets a parasite’s phase directly. The next three frames are about that: watching the parasites switch on, what phase they need, and how a rod with no feed line gets it.

The lag and lead are Lesson 7’s input impedance at work. A parasite is a dipole with no feed, driven by the voltage \(V\) the driven element’s field induces on it, so its current is \(I = V/Z\) with \(Z = R + jX\). Lesson 7 found \(X = +42.5\ \Omega\) at exactly \(\lambda/2\), with resonance a little shorter. Longer than resonance, \(X > 0\) and the current lags \(V\) by \(\arctan(X/R)\); shorter, \(X < 0\) and it leads.

Why that matters is the previous frame: the reflector has to carry its current 108° from the driven element’s, and a rod with no feed line has only one way to be told. Most of that phase comes from the coupling, since a parasite’s induced current re-radiates roughly in opposition to the field that drove it, delayed by the trip across the gap. The detuning trims the rest. The reflector runs about \(0.5\lambda\), and its added lag is what tips the balance toward canceling behind. The directors run around 0.40 to \(0.45\lambda\), and their detuning sets the phase lag from one director to the next at slightly more than the free-space \(kd\). That is a slow traveling wave: a wave along the boom whose phase velocity is slightly less than the speed of light, which is what narrows an endfire beam below what elements phased at exactly light speed would give.

The plotted points are typical published designs:

The result is an endfire beam, main lobe along the boom (the line of the elements) and away from the reflector, rather than broadside to it. Its front-to-back ratio, Lesson 2’s main-lobe peak over the lobe directly behind, is 15 to 25 dB.

Why the boom sets the gain is Lesson 6’s line source turned on its end. Put a current \(I(z') = I_0e^{-jkz'}\) along a boom of length \(L\), phased to travel forward at the speed of light. The radiation vector is the sinc of Lesson 6 with its argument shifted,

so the beam peaks at \(\theta = 0\), along the boom. The half-power point \(\vert u\vert = 1.39\) now falls at \(\theta^2 \propto \lambda/L\), not \(\theta \propto \lambda/L\) as it did broadside, and the beam is a cone, that narrow in every plane. Its solid angle goes as \(\theta^2 \propto \lambda/L\), so the directivity is proportional to \(L\): the exact result is \(D \approx 4L/\lambda\), and a slow wave pushes it toward \(7L/\lambda\). That is +3 dB per doubling of the boom, whatever the number of elements on it. Those are the figure’s two ideal lines: the lower one is a current phased to travel at exactly light speed (\(4L/\lambda\)), the upper one the slightly slow wave of the previous frames (\(7L/\lambda\)). The horizontal axis doubles at every step, so a straight line on it means a fixed number of dB per doubling.

Real Yagis fall short of that slope. The table climbs about 1.8 dB per doubling: a short Yagi beats the line-source estimate (\(7L/\lambda\) gives 8.5 dBi at \(1\lambda\) against the table’s 10) because its elements are dipoles with gain of their own, and a long one falls behind it (15.0 dBi at \(4.5\lambda\) against 14.5) because the current dies away on the far directors. A Yagi needs only one reflector, because the first already sees very little field behind it. Adding more directors to the same boom adds almost nothing. Practical single Yagis range from 8 to 15 dBi; beyond that, designers stack several and let the stack act as an array.

That is fine for a fixed-channel TV, amateur, or point-to-point link, which is where Yagis appear, and a poor fit for anything wideband. It is also the easiest prediction in this lesson to check on the analyzer from Lesson 10.

The Yagi’s narrow band comes from having one set of lengths: every element is detuned by a fixed amount at one frequency only. The log-periodic dipole array keeps the Yagi’s physics and gives up the single set of lengths. Each dipole is a fixed fraction \(\tau\) (typically 0.8 to 0.95) of the one behind it, and so is its spacing. At any frequency, the two or three elements within roughly 15% of \(\lambda/2\) carry almost all the current; that is the active region. The longer element just behind it is inductive and acts as a reflector, the shorter ones just ahead are capacitive and act as directors, exactly the rule from the phasing frame. Raise the frequency and a shorter group becomes the resonant one, so the active region slides toward the short end and the antenna looks electrically the same. The usable band runs roughly from where the longest element is about \(\lambda/2\) to where the shortest is.

That capability is worth a module of its own, and it gets one. Module 3 develops the array factor, pattern multiplication, beam steering, grating lobes, and tapering, and you will steer a real beam on the ADALM-PHASER hardware. For today, file the array alongside the reflector and the Yagi as the third approach to the same goal: coherent area.

The \(10\log_{10}N\) is the radiation vector once more. Fed in phase, \(N\) identical elements sum to \(N\) times one element’s radiation vector on boresight, as Lesson 12’s element-and-image pair summed two:

The third line holds only under a condition. Total radiated power adds element by element only when the elements are far enough apart, about \(\lambda/2\), that they do not share aperture; crowd them closer and their mutual coupling changes each one’s radiated power. So the gain grows with \(N\) only if the aperture grows with it, which is the one idea again: a \(\lambda/2\) grid of \(N\) elements has area \(N\lambda^2/4\), and \(4\pi(N\lambda^2/4)/\lambda^2 = N\pi\).

In that order, those five settle almost every real selection. Excess gain adds cost and tightens the pointing requirement. Aperture antennas get small and cheap as \(\lambda\) shrinks while wire antennas get fragile. Reflectors and horns are broadband; Yagis and patch arrays are not. A dish steers mechanically and slowly, an array electronically and instantly, a Yagi mostly not at all. And a 20 dBi antenna that cannot survive local wind and ice loading is not a usable answer.

Cubesat at 1000 km, 2 W transmitter (33.0 dBm) into a 0 dBi antenna, and our 20 dBi dish on the ground, at \(\lambda = 0.125\ \text{m}\). The path loss is Lesson 2’s Friis factor \((\lambda/4\pi R)^2\) in decibels, with \(4\pi R/\lambda = 4\pi(10^6)/0.125 = 1.005\times10^{8}\).

The noise floor is Lesson 12’s \(kT_AB\) with the reference temperature \(T_0 = 290\ \text{K}\), which gives \(kT_0 = -174\ \text{dBm/Hz}\), plus the receiver’s noise figure:

in a 100 kHz channel with a 3 dB noise figure. A dish pointed at cold sky sees a \(T_A\) well below 290 K, so this floor is conservative. The link closes, and it closes because of the 20 dB the dish contributed. Swap the dish for a 0 dBi antenna and \(P_r = -127.0\ \text{dBm}\), 6 dB under the noise.

Today the illumination was a side effect: the feed’s pattern set it, and the 10 dB rule picked it for efficiency. The figure is what it did to the beam on the way, a first sidelobe pushed from −17.6 to −25 dB at the price of widening the beam from 59° to \(67^\circ\lambda/D\). Lesson 15 turns that around and asks which illumination to choose when the sidelobes are the requirement, and what each choice costs in beamwidth and efficiency.

Lesson 16 builds the third approach from today. Sample that same aperture at discrete points, put an element at each, and the pattern becomes the array factor; the illumination becomes a set of element weights you can set yourself. By Lesson 18 those weights carry phase as well as amplitude, and the beam moves with nothing mechanical moving: you will steer the eight-element ADALM-PHASER across the bench and measure what it costs at each angle.