Phased Array Antenna Calculators

More elements buy three things at once. Gain rises as 10·log₁₀(N) — 3 dB per doubling. On transmit, EIRP rises twice as fast, 20·log₁₀(N), because doubling N doubles both the radiated power and the directivity; that N² behaviour is how 256 modest 0 dBm channels reach a 53 dBm-class EIRP. And the beam narrows: HPBW ≈ 0.886·λ/(N·d), about 1.6° for 64 elements at λ/2.

Steering off broadside gives some back, for two separate reasons. Seen from an angle the array looks narrower, so the projected aperture shrinks as cos θ₀ and the beam broadens by 1/cos θ₀ — twice as wide at 60°. Separately, each element radiates less to the side; that element pattern is modelled as cos^kθ with k of 1 to 1.5, costing roughly 4 dB at 60°. The plot shows both: the combined pattern can never rise above the dashed element envelope. A link budget quoting only broadside EIRP overstates edge-of-coverage by exactly this amount.

These formulas assume every element is driven equally. Real arrays taper the amplitudes to push sidelobes below the −13.2 dB uniform weighting gives, paying a slightly wider beam and a few tenths of a dB.

A phase shifter adds the same phase at every frequency, but the phase needed to steer to a given angle depends on frequency. So the beam only lands where you asked at the design frequency f₀; elsewhere in the band it drifts, following sin θ(f) = (f₀/f)·sin θ₀. That drift is beam squint.

Scanning further out and using more bandwidth both make it worse, as Δθ ≈ tan θ₀ · (Δf/f₀) shows. At broadside there is none. At 60° scan with 800 MHz at 28 GHz the band edges sit about ±1.4° off — negligible for a wide-beam 8-element array, but close to half a beamwidth for a 64-element array whose scanned beam is only ~3.2° wide, so the band edges arrive noticeably down in gain. That is why this tab reports squint as a fraction of HPBW: the absolute angle alone tells you nothing.

The fix is true time delay — a real delay rather than a fixed phase, so phase scales with frequency automatically. TTD costs area, loss and money, so most arrays use it per subarray with ordinary phase shifters inside each one, and the squint budget decides how large a subarray can be.

A patch radiates from its two open edges, which behave like a pair of slots. At resonance the reactance cancels and what remains at the edge is a pure resistance, typically 150–300 Ω — far too high to feed directly from a 50 Ω line. That is the problem this tab solves.

The voltage under the patch falls cosinusoidally from edge to centre, so the resistance at a point recessed y₀ from the edge follows R(y₀) = R(0)·cos²(πy₀/L). The inset feed cuts a notch and taps the patch where that resistance has already dropped to 50 Ω — the same trick as feeding a dipole off-centre. For a 200–300 Ω edge the tap sits about a third of the way in. Watch how steeply the curve moves: at 28 GHz on thin substrate y₀ is under a millimetre, so a few percent of etch error shifts the input resistance visibly. That is why mmWave patches are fussy to manufacture.

Treat these numbers as a starting layout, not a sign-off — the transmission-line model predicts resonance to about ±5%, and the cos² law describes an ideal tap point rather than a real notch of finite width. Their value is arriving at the full-wave solver already close, knowing which dimension moves which parameter.

A microstrip line's characteristic impedance is set almost entirely by the ratio of trace width to substrate height, W/h. The synthesis equations here invert that relationship to about 1% — close enough to draw a first-pass feed network straight from the numbers. On 0.254 mm RO4350B a 50 Ω line is around 0.55 mm wide; on 1.6 mm FR-4 it is the familiar ~3 mm.

The quarter-wave transformer is the other way to match a patch: a λg/4 section of impedance Z_t = √(Z₀·R_L) between the 50 Ω line and the high-impedance patch edge, so a 228 Ω edge needs about 107 Ω. It leaves the patch edge undisturbed, avoiding the cross-polarisation a notch can introduce — but 107 Ω is a narrow trace, and at mmWave narrow traces run into etch tolerance. That is the trade against the inset feed.

Both matches are inherently narrowband, since a λg/4 section is exact at one frequency only — though the patch itself usually limits bandwidth first. Note too that each trace width has its own ε_eff, so the transformer's physical quarter-wave length differs slightly from a quarter wave on the 50 Ω line. The calculator computes each at its own width; hand calculations often miss this.

Gain and pattern are only defined in the far field, where the wave across the aperture is effectively flat. The accepted criterion is R ≥ 2D²/λ, the distance at which phase error across the aperture is no worse than λ/16 (22.5°). Two traps: D is the largest dimension, which for a rectangular array is the diagonal, not the side; and the requirement grows with the square of size and linearly with frequency. A 10 × 10 cm array (14.1 cm diagonal) needs about 3.7 m at 28 GHz and over 10 m at 77 GHz.

Measure closer and the errors are systematic, not random: residual phase curvature fills in the nulls, raises the apparent sidelobes and reads peak gain low. Closer still, below 0.62·√(D³/λ), the probe couples to the antenna's stored energy and changes the thing being measured.

This is the calculation that sizes a direct far-field chamber: the range must clear 2D²/λ for the largest DUT at the highest frequency, with margin, and the quiet zone must cover the aperture. The two gain estimates bracket a measurement from opposite directions — the aperture form G = 4πAη/λ² is the ceiling the physical size permits, while Kraus, 41253/(θ_E·θ_H), turns measured beamwidths into an approximate directivity. A measured gain far from both usually means calibration, alignment — or a range that was not actually far field.