Lesson 15 treated an aperture as a continuous sheet of current and read its beamwidth from the aperture length and its sidelobes from the taper. Now we replace that continuous aperture with \(N\) discrete elements on a line, each fed its own copy of the signal. The far fields still add as phasors, so the same reasoning applies, and the sum reduces to one compact expression, the array factor, which predicts every feature you will measure on the PHASER for the rest of this module.

We put \(N\) identical elements on a straight line, spaced a distance \(d\) apart, and feed element \(n\) a copy of the same signal with complex weight \(a_n\). Every element radiates the same pattern; only the weights and the positions differ.

A far-field observer sees the array from a single direction, so the rays leaving the elements are parallel. Element \(n\) sits \(nd\) farther along the array axis than element 0, and along a ray heading off at angle \(\theta\) that position shortens the path to the observer by \(\Delta r_n = n\ d\sin\theta\). The wavenumber \(k = 2\pi/\lambda\) converts distance to phase, so a path shorter by \(\Delta r_n\) is a phase lead of \(k\ \Delta r_n\), and element \(n\) arrives at the observer carrying

relative to element 0.

Every term carries the same step per element, \(kd\sin\theta\), so we give it a name: \(\psi = kd\sin\theta\) is the inter-element phase, the phase by which each element leads its neighbor as seen from direction \(\theta\). It is not the progressive phase \(\beta\) that the phase shifters apply, and not the ADAR1000’s 2.8125° phase step; it is what the observer sees, the geometry less any applied \(\beta\). It is the only way the angle enters the sum. Each direction gives one value of \(\psi\): broadside, \(\theta = 0\), gives \(\psi = 0\), where every element arrives in phase, and endfire, \(\theta = \pm 90^\circ\), gives the largest step, \(\psi = \pm kd\). For the PHASER, \(kd = 173^\circ\). The rest of the lesson works in \(\psi\) and converts back to \(\theta\) through this one relation.

The same path differences change the amplitude \(1/r\) by a negligible amount. The PHASER is 112 mm long, so its far field begins at \(2D^2/\lambda = 0.86\ \text{m}\) (Lesson 5). At a 30 m range, 112 mm of path changes \(1/r\) by less than 0.4%, but it spans 3.85 wavelengths, or nearly 1400° of phase. This is the parallel-ray approximation of Lesson 6: we keep the path difference in the phase and drop it from the amplitude.

The total far field is the sum of the element fields, and because every element has the same pattern, the common factor \(E_0(\theta)\ e^{-jkr}/r\) comes out in front of the sum. The array factor \(AF(\theta)\) is the sum that remains. It depends only on how many elements there are, where they sit, and what weights we feed them. It does not depend on the type of element.

The array factor is the discrete form of the space factor \(S(\theta)\) of Lessons 6 and 15. There the current, or the aperture field, was continuous, and \(S(\theta)\) was its integral weighted by \(e^{\ jkx\sin\theta}\). Here the current exists only at \(N\) points, so the integral becomes a sum over the elements. The normalized pattern \(\vert F(\theta)\vert\) is \(\vert E\vert\) divided by its peak, and the \(1/r\) and \(e^{-jkr}\) drop out of it, so \(\vert F\vert\) is the element’s own pattern times \(\vert AF\vert\). That is Lesson 6’s element factor times space factor, and Part 3 names the element’s part \(EF\).

Setting every weight to 1 turns the sum into a finite geometric series in \(e^{\ j\psi}\). To close it, we factor half of each exponent out of the numerator and the denominator, which leaves a difference of conjugate exponentials in each bracket. Euler’s formula turns each bracket into \(-2j\sin(\cdot)\), and the \(-2j\) factors cancel between top and bottom.

The leading factor \(e^{\ j(N-1)\psi/2}\) has magnitude 1, so it changes the phase of \(AF\) and never its size. It comes from where we put the phase reference. We measured every element’s phase from element 0, at one end of the array. The center of the array sits \((N-1)d/2\) from element 0, and \((N-1)\psi/2\) is exactly the phase that point carries. Pulling that phase out of every term counts each element from the center instead:

The centered sum pairs each element with its mirror image across the center, and each pair adds to \(e^{\ jm\psi} + e^{-jm\psi} = 2\cos(m\psi)\), so the sum is real: it is the ratio of sines. Measured from the center, the leading factor is simply absent. Measured from element 0, it is there, but the pattern is \(\vert AF\vert\) and \(\vert e^{\ j(N-1)\psi/2}\vert = 1\), so it drops out either way. Dividing by \(N\) then normalizes the peak to 1. The main lobe, the nulls, and the sidelobes all follow from this expression.

Main lobe. At \(\psi = 0\) both numerator and denominator vanish, and the limit is 1. The peak therefore sits where \(\sin\theta = \sin\theta_0\), which at broadside is \(\theta = 0\). The function is periodic in \(\psi\) with period \(2\pi\), so the peak repeats every \(2\pi\); Part 4 finds the spacing at which a repeat reaches a real angle.

Nulls. The numerator vanishes when \(N\psi/2 = m\pi\), and the denominator is nonzero as long as \(m\) is not a multiple of \(N\). The nulls depend on \(N\) and \(d\) only through the total length \(Nd\).

Between consecutive nulls the numerator swings back to ±1, so a sidelobe sits in each gap: \(N - 2\) of them across one period. Their heights fall off as \(1/\sin(\psi/2)\), so the tallest is the one nearest the main lobe. The numerator reaches its peak magnitude halfway between its first two zeros, at \(N\psi/2 = 3\pi/2\), and evaluating \(AF_N\) there gives \(2/(3\pi) = 0.212\) for large \(N\), or −13.5 dB.

The denominator is still growing across that gap, so the true peak sits slightly closer to the main lobe, where the denominator is smaller. Locating it exactly gives −13.3 dB, which is the uniform line-source number from Lesson 6. At \(N = 8\) the value is −12.8 dB. The figure plots the exact level against \(N\): a short array has a higher first sidelobe, and by \(N = 16\) the level is within 0.2 dB of the line source. The estimate runs a fraction of a decibel low at every \(N\), for the reason above. Uniform excitation leaves the first sidelobe about 13 dB below the peak in a discrete array, as it did in the continuous aperture of Lesson 15, and Lesson 22 lowers it with a taper.

The figure marks the three features on the PHASER’s own array, \(N = 8\) and \(d/\lambda = 0.481\): a half-power width of 13.2°, a first null at 15.1°, and a first sidelobe at −12.8 dB, with six sidelobes, \(N - 2\), between the nulls.

The half-power row uses the line-source constant of Lessons 6 and 15 with \(L = Nd\), and Lesson 20 derives it for the array, along with the \(1/\cos\theta_0\) broadening under steering. The constant is close but not exact for a short array: the PHASER’s exact half-power width is 13.3°, against 13.2° from \(0.886\ \lambda/Nd\).

Nothing in Part 1 required the elements to be isotropic. It required them to be identical and identically oriented, which let \(E_0(\theta)\) come out of the sum. That factoring is pattern multiplication.

The element factor \(EF\) is the pattern one element would produce by itself. The array factor \(AF\) is what the arrangement adds. Their product is the array’s pattern. In decibels the two curves add, which is why a log plot is the natural place to check the result.

This is the Lesson 6 split, pattern equals element factor times space factor, with the continuous space factor replaced by a discrete sum. The element factor comes from the physics of the radiator; the array factor accounts for where the elements sit and how we feed them.

The figure zooms in on that sidelobe. Adding the two levels at the array factor’s peak gives the amber point on the product curve; the product’s own peak, in red, sits 3° closer to broadside and 0.2 dB higher, because the array factor is flat at its own peak while the element factor keeps rising toward broadside.

The three-panel figure shows the general rule. A directive element reshapes the skirts of the pattern and suppresses whatever the array factor puts at wide angles, but the array factor sets the main lobe. It sets the beamwidth, and the element and the array share the wide-angle behavior.

\(AF_N(\psi)\) repeats every \(2\pi\) in \(\psi\), but \(\psi\) is not free: \(\psi = kd(\sin\theta - \sin\theta_0)\) ties it to a real angle. As \(\theta\) sweeps the visible half-space from −90° to +90°, \(\sin\theta\) covers −1 to +1, and \(\psi\) covers a window of total width \(2kd = 4\pi d/\lambda\) that slides as we steer. That window is the visible region, Lesson 6’s window \(-k \le k_z \le +k\) written in \(\psi\). Values of \(\psi\) outside it correspond to no real angle, so the array never radiates them.

The width of the window alone does not produce a second beam. At \(d = \lambda/2\) the window is exactly one period wide, and at broadside it runs from \(-\pi\) to \(+\pi\), between the repeats. A second copy of the main lobe appears only when an edge of the window reaches a repeat at \(\psi = \pm 2\pi\). At broadside the edges sit at \(\pm kd\), which reach \(\pm 2\pi\) when \(d = \lambda\). A copy that enters is a grating lobe, and because \(AF_N\) is exactly periodic it is as tall as the main lobe and radiates as much power as the intended beam.

Grating lobes appear where \(\psi\) is a nonzero multiple of \(2\pi\). At \(\psi = 2\pi\) the path difference between neighbors, less the applied progressive phase, is exactly one wavelength, so every term \(e^{\ jn\psi} = e^{\ j2\pi n} = 1\) and the elements add in phase exactly as they do at the main lobe. That is why a grating lobe is full height.

The grating lobe nearest real space is \(m = 1\) on the side away from the steer. For \(\theta_0 \ge 0\) it sits at \(\sin\theta_g = \sin\theta_0 - \lambda/d\), and it stays out of real space as long as \(\sin\theta_g < -1\):

A beam steered to negative \(\theta_0\) gives the mirror image, so the criterion carries \(\vert\sin\theta_0\vert\). At broadside it is \(d < \lambda\). Scanning to ±90° tightens it to \(d < \lambda/2\), which is where the familiar half-wavelength spacing comes from. The figure plots the limit against the largest scan angle a design needs: a ±45° requirement allows \(0.586\lambda\), and the PHASER’s \(0.481\lambda\) sits below the curve at every angle, so it can scan anywhere without a grating lobe. Lesson 23 treats grating lobes in full, alongside beam squint and phase quantization; we use the criterion as stated until then.

Near the main lobe and the first few sidelobes, \(\psi\) is a small fraction of a period when \(N\) is large, so the denominator \(N\sin(\psi/2)\) is close to \(N\psi/2\). The array factor then becomes \(\sin u/u\), with

for \(L = Nd\). That is the \(u\) of Lesson 6’s uniform line source, written in scan angle. A uniform line source of length \(L\) has a beamwidth \(\approx 0.886\ \lambda/L\) and a first sidelobe of −13.3 dB, and a uniform array of \(N\) elements spaced \(d\) has a beamwidth \(\approx 0.886\ \lambda/(Nd)\) and a first sidelobe near −13 dB. The array is that line source sampled every \(d\).

The figure shows where the approximation holds. Plotted against \(u\), in units of the line source’s null spacing, the 4-, 8-, and 32-element patterns all match the sinc near the main lobe. They part from it where \(\psi\) is no longer small: an \(N\)-element array repeats its main lobe at full height \(N\) nulls out, so the 4-element pattern returns at \(u/\pi = 4\) and the 8-element one at 8, while the 32-element pattern follows the sinc across the whole plot.

The sampled view assigns each parameter its job. Two arrays with the same \(Nd\) have nearly the same main lobe whether 8 elements or 32 fill that length. Uniform excitation gives about −13 dB in the array exactly as in the aperture, and Lesson 22 tapers it.

The spacing decides what repeats. The array samples the current only at multiples of \(d\), so its phases \(e^{\ jn\psi}\) cannot distinguish \(\psi\) from \(\psi + 2\pi\), and the pattern repeats with that period. Lesson 6 showed that the pattern is the Fourier transform of the current (“The Radiation Integral as a Fourier Transform”), and sampling a function makes its transform periodic. A continuous aperture has a contribution from every position, so no shift of \(\psi\) leaves all of them unchanged, and it has no grating lobes.

Today we assumed the feed: element \(n\) simply received weight \(a_n\). Lesson 17 opens the hardware that produces those weights, the ADAR1000 beamformer chips, the per-element gain and phase controls, and the receive chain that turns eight elements into two digitized channels. Lesson 18 then sets the weights to a progressive phase ramp and steers the beam, which supplies the \(\theta_0\) that has gone unused in \(\psi\) all lesson.

Every lab in this module measures some feature of today’s array factor. The L20 Lab measures the beamwidth, the nulls, and the sidelobe level of the uniform 8-element pattern. The L22–L23 Lab changes the weights and watches the sidelobes drop. The L24 Lab places a null where you want one. Before the next lesson, be able to write \(AF_N\) from memory and locate its nulls, because every one of those labs starts by predicting from it.