The Antenna Pattern Measurement midterm project, assigned in Lesson 9, measured all of this from the other direction. The quantities you extracted from a measured cut — the half-power width, the null positions, the sidelobe level referenced to the peak — are exactly the quantities this lesson computes from theory, so keep your project data at hand.

In the L18 Lab you steered the array across the room and watched the beam widen as it went. The main lobe was about 13° across at boresight and noticeably wider at 45°, and the pattern sat on a floor of sidelobes that moved with the beam. Lesson 16 gave us the closed form that produces all of that. Today we extract numbers from it: the half-power beamwidth of a steered array, the angle of every null, the directivity, and the design step in reverse, which takes a beamwidth specification and finds the array size.

Lesson 16 summed \(N\) equal-amplitude elements with a progressive phase and collapsed the geometric series into the normalized uniform array factor

Everything about the shape of the pattern is in that one function of \(\psi\). The array’s physical parameters enter only through the map from angle to \(\psi\), and the beam peak is wherever \(\psi\) = 0, which is \(\theta = \theta_0\).

Near the peak, \(\psi\) is small, so \(\sin(\psi/2) \approx \psi/2\) and

That is the same \(\sin u / u\) we found in Lesson 6 for a uniform line source, and the reason is physical: an \(N\)-element uniform array with spacing \(d\) is a sampled aperture of length \(L = Nd\). Near the peak, where the sampling is too fine to matter, the array’s pattern matches the continuous aperture’s. The length is \(Nd\) rather than the \((N-1)d\) between the end elements because each element accounts for one spacing \(d\) of aperture, the sampled line source of Lesson 16.

Power is half the peak where \(\lvert \sin u/u\rvert = 1/\sqrt{2}\), which happens at \(u\) = 1.392. Solving outward from there:

The last step fixes the half-power width in \(\sin\theta\), and that width is the same at every scan angle, because the array factor depends only on \(\sin\theta - \sin\theta_0\). The scan angle enters when we convert it back to an angle. The derivative of \(\sin\theta\) is \(\cos\theta\), so near the beam peak the first term of its Taylor series gives

At broadside \(\sin\theta\) is steepest, with slope 1, and a step in \(\sin\theta\) is the same step in angle. Toward the horizon \(\sin\theta\) flattens, and the same step in \(\sin\theta\) covers more angle: at 60° the slope is 0.5, so the beam is twice as wide. The figure shows the course array’s half-power band in \(\sin\theta\) mapped back to angle at both scan angles. For a beam that is narrow compared with its distance from broadside, the half-width is

and the full width is twice that:

The same factor has a physical reading, shown in Lesson 18’s figure: seen from the beam direction, the array’s projected length is \(Nd\cos\theta_0\), and a shorter aperture gives a wider beam.

A null is a zero of the numerator that is not also a zero of the denominator. The numerator vanishes when

except when \(m\) is a multiple of \(N\), because then \(\sin(\psi/2)\) vanishes as well and the ratio returns to unity. Those are the main lobe and its repeats, the grating lobes of Lesson 23, not nulls. Converting through \(\psi = kd(\sin\theta - \sin\theta_0)\):

The nulls are equally spaced in \(\sin\theta\), not in \(\theta\). The figure plots the same broadside pattern twice. The top panel uses \(\sin\theta\) as its axis, and there the nulls are a comb with teeth every \(\lambda/Nd\) = 0.260. The bottom panel uses angle, and the dashed lines carry each null down to it: the gaps grow from 15.1° to 16.3° to 19.9°, because \(\sin\theta\) flattens toward the horizon, the same slope argument that widened the steered beam. A null exists only if the right-hand side lands inside the visible region \(\lvert \sin\theta\rvert\) ≤ 1; values outside it correspond to no real direction, and there is no null.

Taking \(m\) = 1 on both sides of a broadside beam gives the first-null beamwidth

The null-to-null width runs about 2.3 times the half-power width for a uniform array. That ratio is worth remembering when you read a measured sweep, because the nulls are sharp features that a coarse angle grid can miss entirely, while the half-power points are easy to interpolate.

Lesson 2 defined directivity as the peak radiation intensity over its average over the sphere, \(D = 4\pi U_\text{max}/P_\text{rad}\). For an array of isotropic elements the intensity follows \(\lvert AF\rvert^2\), so we need its peak and its average. Write the unnormalized array factor as a sum over elements, with \(\gamma\) the angle from the array axis (so \(\cos\gamma = \sin\theta\) in the plane of the array), and multiply it by its conjugate:

At the peak every term is 1, so the peak of \(\lvert AF\rvert^2\) is \(N^2\): the \(N\) fields add in phase. The average over the sphere goes term by term. A term with \(m = n\) is 1 everywhere. A cross term averages to a sinc of the separation \(x = d(m-n)\):

The ratio is the directivity:

At \(d = \lambda/2\) every cross term contains \(\sin(\pi(m-n))\), which is zero, and the double sum collapses to the \(N\) diagonal terms:

Written in that last form, the result carries over to other spacings: for any uniform broadside array with \(d\) below a wavelength,

which is the long uniform line-source directivity from Lesson 6 with \(L = Nd\) — the same sampled-aperture argument that produced the 0.886. Directivity grows linearly with aperture in wavelengths. Doubling the element count at fixed spacing doubles \(D\) and adds 3 dB, and it does so by halving the beamwidth, not by making any element radiate harder.

Directivity under steering. The array-factor directivity of a linear array barely changes as you steer, even though the beam visibly widens. The main lobe of a linear array is a cone about the array axis; steering off broadside thickens the cone wall by \(1/\cos\theta_0\) and shrinks its circumference by \(\cos\theta_0\), leaving the solid angle nearly unchanged. For the course array the array-factor directivity is 8.86 dB at broadside, 8.91 dB at 45°, and 9.12 dB at 60°. The scan loss you measure on real hardware comes from the element pattern, not from the array factor, which is why the PHASER loses gain toward the edges of its scan volume. That is Lesson 21’s subject.

We can run every result so far in reverse. If we fix the beamwidth and scan volume as requirements, we can design the array using the following three-step process.

1. Aperture. Invert the half-power formula, with \(\theta_\text{HP}\) in radians:

   \[Nd = \frac{0.886\ \lambda}{\theta_\text{HP}}.\]

   If the specification has to hold across a scan volume, divide by \(\cos\theta_0\) at the widest scan angle, because that is where the beam is widest.

2. Spacing. Lesson 23 derives the condition that keeps grating lobes out of visible space:

   \[d < \frac{\lambda}{1 + \lvert\sin\theta_0\rvert_\text{max}}.\]

   At \(d = \lambda/2\) the criterion holds for any scan angle, which is why half-wavelength spacing is the default.

3. Element count. \(N = Nd/d\), rounded up. Every element is an antenna, a phase shifter, an amplifier and a control line, so this is the number that sets the cost of the array.

The L20 Lab sweeps the PHASER with all eight elements on, then with the center four, then with the center pair, and measures the beamwidth each time. Predict first, measure second, reconcile third. The eight-element row is this lesson’s worked example at 10.3 GHz, with the HPBW solved exactly from the array factor; the 0.886 rule gives 13.2°. The other two rows are yours: Part 1 of the L20 Lab packet asks for them at your kit’s Find HB100 frequency, with the methods of this lesson. Three details are worth knowing before you start.

The 0.886 rule carries a small-angle step, and the wider the beam, the worse that step gets. Before you use it for a small aperture, decide whether the beam is narrow enough; where it is not, solve the array factor’s half-power point directly. For two elements the array factor is just \(\cos(\psi/2)\), which you can solve by hand.

The null formula \(\sin\theta = m\lambda/Nd\) only gives a null where the right-hand side is at most 1. Check that for each aperture before you write a first-null beamwidth down, and decide what that means for the sidelobes too.

The first sidelobe drifts with element count. The −13 dB figure is the large-array limit; eight elements give −12.8 dB, and smaller arrays land higher. The sweep’s noise floor sits roughly 23 dB below the uniform-taper peak.

The sweep is a discrete grid. With Use Bits on, as in every lab preset, the GUI steps the element-to-element phase one phase-shifter LSB, 2.8125°, per point. One step moves \(\sin\theta\) by 2.8125°/173.2° = 0.0162, which is 0.93° of angle at broadside and 1.07° at 30°, widening as \(1/\cos\theta\). A half-power width read as the span between the outermost samples above −3 dB comes out short by up to one step: 12.1° to 13.1° for the full array, against 13.3° from the pattern. Interpolating between the two samples that straddle each −3 dB crossing recovers the width to within 0.1°. The nulls are read at their lowest sample, which puts the measured FNBW between 30.1° and 31.1°. That spread comes from the sampling grid, not from the array.

Lesson 21 addresses what the array factor leaves out. Every result in this lesson treats the elements as isotropic points, which is why the directivity came out independent of scan angle and why the pattern in the widget is the same at 60° as at boresight. Real patches have their own pattern, it rolls off toward the edges of the scan volume, and the full array pattern is the product of the two. That product is where the array’s measured gain, its usable scan range, and its cross-polarization all come from.

The L20 Lab puts your predictions in front of the hardware. You will load the Array Factor preset, sweep the full array, then drop to four elements and two, then leave only the two end elements on, and compare measured beamwidths against the numbers you calculated in Part 1 of the packet. Bring Part 1 filled in; a measurement you cannot predict within a couple of degrees is a measurement you cannot use to find a fault later. Read the L20 Lab procedure before you go to the bench.