Compute the array factor of a linear phased array in real time. Set N, d/λ, scan angle θ₀, and window function. Visualize the polar radiation pattern and dB rectangular plot.
Array Parameters
Number of elements N
Element spacing d/λ
λ
Grating lobe free: d/λ < 1/(1+|sinθ₀|)
Scan angle θ₀
°
Window Function
Sidelobe level
dB
Grating Lobe Warning: Element spacing is too large. Grating lobes are present in visible space.
Results
— °
HPBW
— dB
1st Sidelobe
— dBi
Directivity
None
Grating Lobe
Polar Beam Pattern (log scale)
Theory & Key Formulas
Array Factor (linear array):
$$AF(\theta) = \sum_{n=0}^{N-1}w_n \cdot e^{j n k d \sin\theta}$$
What exactly is "beamforming"? I hear about it with 5G and Wi-Fi routers, but what's it actually doing?
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Basically, it's making a group of small antennas act like one big, smart antenna. Instead of broadcasting a signal in all directions, you can electronically "steer" a focused beam of energy towards a specific user. In this simulator, you control that beam with the Scan Angle (θ₀) slider. Try moving it and watch the main lobe (the brightest beam) swing around the circle.
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Wait, really? So the "Number of Elements (N)" slider just adds more antennas? What's the benefit of more?
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Exactly. More elements give you a sharper, more focused beam. Think of it like using a larger lens on a camera to get a tighter zoom. A common case is a 5G base station using 64 or 256 elements. Slide N from 4 up to 16 and watch the main lobe get narrower. But there's a trade-off: more elements means a more complex and expensive system.
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Okay, and what's the deal with "Element Spacing (d/λ)"? Why is it always around 0.5 in the examples?
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Great question. Spacing is critical! If antennas are too close, they couple and become inefficient. If they're too far apart, you get unwanted duplicate beams called "grating lobes." Half-wavelength spacing (d = 0.5λ) is the gold standard—it lets you steer the beam anywhere without creating those grating lobes. Try increasing the spacing to 1.0 and then steer the beam; you'll see extra lobes pop up, which is bad for signal integrity.
Physical Model & Key Equations
The core math describing the radiation pattern of a linear array is the Array Factor (AF). It sums the contributions from each antenna element, each with a specific weight and phase shift.
$$AF(\theta) = \sum_{n=0}^{N-1}w_n \cdot e^{j n k d \sin\theta}$$
Here, $N$ is the number of elements, $w_n$ is the amplitude weighting (or "window") for the n-th element, $k = 2\pi/\lambda$ is the wavenumber, $d$ is the spacing between elements, and $\theta$ is the observation angle. The exponential term $e^{j n k d \sin\theta}$ represents the phase difference of the signal arriving at the n-th element.
When we want to steer the beam to a specific angle $\theta_0$, we introduce a progressive phase shift across the array. The pattern is often analyzed in terms of a generalized phase variable $\psi$.
$$\psi = kd(\sin\theta - \sin\theta_0)$$
For a uniform array (where all $w_n = 1$), the array factor simplifies to a closed-form expression. Its magnitude shows the classic sinc-like pattern, defining the main lobe and sidelobes.
The variable $\theta_0$ is your scan angle. Changing it shifts the entire pattern. The positions of the nulls (where the signal is zero) and the width of the main beam are determined by $N$ and $d$.
Frequently Asked Questions
If the spacing exceeds λ/2, lobes with the same intensity as the desired main beam (grating lobes) appear in other directions. This can cause a decrease in antenna gain and false detection, so operation at λ/2 or less is generally recommended.
The window function trades off sidelobe level and main beam width. For example, the Hamming window suppresses sidelobes but widens the beam, while a uniform distribution (rectangular window) gives the narrowest beam but higher sidelobes. Choose according to the application.
As the scan angle increases, the projected aperture area of the array decreases, widening the beam. Additionally, if the element spacing is close to λ/2, grating lobes are more likely to enter the visible region, causing the pattern to deform asymmetrically.
This tool calculates only the array factor assuming point sources, and does not account for the directivity of individual elements, mutual coupling, or feed network losses. In actual design, these factors must be corrected separately through additional simulations or measurements.
Real-World Applications
5G and mmWave Cellular Networks: This is the most direct application. At high frequencies like 28 GHz, signals attenuate quickly. Phased arrays in base stations and user devices form narrow, steerable beams to maintain a strong link. The simulator's parameters directly model the arrays used in these systems, where controlling sidelobe levels is crucial to avoid interfering with neighboring cells.
Radar and Military Systems: Phased array radars can electronically scan their beam across the sky at incredible speeds, without moving a heavy dish. This allows them to track multiple targets simultaneously. The "Sidelobe Level" control in the simulator relates to stealth technology—low sidelobes make a radar harder to detect from the side.
Satellite Communications: Modern satellites use phased arrays on their downlink beams to dynamically shape coverage areas on Earth. For instance, a satellite can focus more signal power over a densely populated city and less over an ocean, optimizing bandwidth. The beam steering you practice here is fundamental to that process.
Medical Imaging and Therapy: Ultrasound and certain cancer treatment systems use acoustic phased arrays. The same beamforming principles apply: by controlling the timing (phase) of signals from many transducers, they can focus ultrasonic energy precisely on a tumor deep inside the body without damaging surrounding tissue.
Common Misconceptions and Points to Note
First, do you think "the more elements, the absolutely better"? While directivity indeed becomes sharper, in reality, cost, computational load, and physical installation space skyrocket. For example, doubling the number of elements from 16 to 32 only improves the beamwidth by about 1/√2 (approximately a 30% narrowing), requiring a cost-effectiveness judgment. Furthermore, you need complex circuitry and a control system to drive all elements, which also increases the risk of failure.
Next, the misconception that "the Hann window is the best." The Hann window certainly suppresses sidelobes, but it widens the main beam width, resulting in a directivity gain loss of about 1.5 dB. For instance, if your goal is precise communication with a distant point, a Uniform (equal amplitude) window, which has a sharper main beam even with somewhat higher sidelobes, might be more advantageous. Understand the trade-off of "what you sacrifice versus what you gain" according to your application.
A common pitfall in parameter setting is ignoring the combination of "scan angle" and "element spacing". Try it with the tool. You'll see that setting d=0.7λ and increasing the scan angle to 60 degrees causes a significant drop in main beam strength (the beam "degrades"). This is because the effective aperture appears smaller. In practical design, it's a golden rule to determine the element spacing after first verifying the scannable range via simulation.