Patch Antenna Calculator — Theory, Design Equations & Reference

Patch Geometry

A rectangular microstrip patch antenna consists of a conducting patch of width W and length L on a grounded dielectric substrate of thickness h and relative permittivity εr. The patch radiates from two radiating slots at its edges along the length dimension.

Step 1 — Patch Width W

The patch width sets the radiation resistance and influences bandwidth. It is chosen to be close to half a wavelength in the substrate but wider than the length to ensure the dominant TM₁₀ mode radiates efficiently:

Patch width

W = c 2f₀ · √ 2 εr + 1

c = speed of light · f₀ = design frequency · εr = substrate relative permittivity

Step 2 — Effective Dielectric Constant εeff

Because the fields extend both inside and outside the substrate (fringing), the patch sees an effective permittivity intermediate between εr and 1:

Effective relative permittivity

εeff = εr + 1 2 + εr − 1 2 · √ 1 1 + 12h/W

Valid for W/h > 1 · h = substrate thickness · W = patch width

Step 3 — Length Extension ΔL

Fringing fields at each radiating edge extend the effective electrical length of the patch beyond its physical dimension. Each edge contributes ΔL (Hammerstad formula):

Fringing length extension (each edge)

ΔL = 0.412 · h · (εeff + 0.3)(W/h + 0.264) (εeff − 0.258)(W/h + 0.8)

Both edges contribute → total effective length = L + 2ΔL

Step 4 — Patch Length L

The physical patch length is set so that the effective electrical length L + 2ΔL equals one half-wavelength in the effective medium:

Patch length

L = c 2f₀ · √εeff − 2ΔL

Typically L ≈ 0.45 – 0.49 × λ₀/√εr · smaller εr → longer patch

Resonant Frequency

Given a patch of known physical length L and substrate properties, the resonant frequency of the dominant TM₁₀ mode is:

TM₁₀ resonant frequency

f₀ = c 2(L + 2ΔL) · √εeff

This is the analysis equation — use it to find f₀ of an existing patch of known L

Inset Feed — Impedance Transformation

An inset feed notch cut into the patch from the non-radiating edge allows the 50 Ω feed line to connect directly to the patch at a point where the radiation resistance equals 50 Ω. The input impedance varies cosinusoidally with the inset depth y₀:

Input impedance vs inset depth

Zin(y₀) = Zin(0) · cos² ( π · y₀ L )

y₀ = L π · cos⁻¹ √ Z₀ Zin(0)

Zin(0) = edge impedance at the radiating edge (200–400 Ω typical) · Z₀ = 50 Ω feed line · y₀ = inset depth from radiating edge

Edge impedance Zin(0) is not constant — it depends on W, L, h, εr, and f₀. The closed-form estimate is Zin(0) ≈ 90 · εr²/(εr − 1) · (L/W)², but this is approximate. Use the calculator below for accurate values derived from the conductance model.

Notch Gap Width g

The notch gap on each side of the inset feed prevents the feed line from coupling directly to the patch edges. A gap of 1–2 mm (or >λg/10) is typically sufficient:

Recommended minimum notch gap

g ≥ λg 10 = c 10 · f₀ · √εeff

Typical values: 1–2 mm for 2.4 GHz on FR-4 · wider gaps reduce coupling but narrow the patch

Bandwidth

The fractional bandwidth (FBW) of a single-layer patch is proportional to substrate thickness and inversely proportional to permittivity:

Fractional bandwidth (impedance bandwidth, VSWR < 2)

FBW ≈ 3.77 · (εr − 1) εr² · W L · h λ₀

λ₀ = free-space wavelength · Thicker substrates (h > 0.05λ₀) and lower εr increase bandwidth at the cost of surface waves

Gain & Radiation Pattern

The radiation pattern of a patch is broadside (maximum normal to the patch surface). The E-plane pattern resembles a cosine function; the H-plane is broader and nearly omnidirectional above the substrate:

Directivity estimate (Balanis)

D ≈ 2W² λ₀² G_rad

G_rad = radiation conductance of one slot · Practical gain = D × radiation efficiency (typically 0.7–0.9)

Common Substrate Materials

Substrate	εr	tan δ	h (typical)	Notes
FR-4	4.2 – 4.6	0.018 – 0.025	0.8 – 1.6 mm	Low cost, lossy — avoid above 5 GHz
Rogers 4003C	3.55	0.0027	0.508 – 1.524 mm	Low loss, RF/microwave standard
Rogers 5880	2.20	0.0009	0.508 – 3.175 mm	Very low loss, wide bandwidth
PTFE (Teflon)	2.1	0.0004	0.5 – 3.0 mm	Excellent high-frequency performance
Alumina (Al₂O₃)	9.8	0.0003	0.254 – 1.0 mm	Compact size, MMIC compatible
Rogers 3010	10.2	0.0022	0.635 – 3.175 mm	High εr for compact antennas

Substrate selection rule of thumb: Use lower εr (2–4) for wide bandwidth and higher efficiency. Use higher εr (8–10) for compact size. Avoid FR-4 above 5 GHz — its high tan δ introduces significant conductor and dielectric loss, reducing gain by 1–3 dB vs a low-loss substrate.

📐 Apply the theory Proceed to Patch Antenna Calculator Enter your substrate and frequency — get W, L, y₀, and notch gap g instantly. →

Patch Antenna Calculator — Theory & Equations