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Your First Patch Antenna

This tutorial designs a 2.4 GHz rectangular microstrip patch on FR4, runs the FDTD simulation, and extracts the resonant frequency with Harminv. Estimated time: 20 minutes.

1. Analytical patch design

Section titled “1. Analytical patch design”

The standard rectangular patch formulas (Bahl & Trivedi 1977) give the physical dimensions:

W     = c / (2·f₀) · √(2 / (εᵣ+1))
εₑff  = (εᵣ+1)/2 + (εᵣ−1)/2 · (1 + 12h/W)^−½
ΔL    = 0.412·h · [(εₑff+0.3)(W/h+0.264)] / [(εₑff−0.258)(W/h+0.8)]
L     = c / (2·f₀·√εₑff) − 2·ΔL

In Python:

import numpy as np
C0 = 299_792_458.0  # m/s


f0    = 2.4e9        # target resonance
eps_r = 4.4          # FR4
h     = 1.6e-3       # substrate thickness (1.6 mm)
tan_d = 0.02         # loss tangent at 2.4 GHz


W       = C0 / (2*f0) * np.sqrt(2 / (eps_r + 1))
eps_eff = (eps_r+1)/2 + (eps_r-1)/2 * (1 + 12*h/W)**(-0.5)
dL      = 0.412*h * ((eps_eff+0.3)*(W/h+0.264) / ((eps_eff-0.258)*(W/h+0.8)))
L       = C0 / (2*f0*np.sqrt(eps_eff)) - 2*dL


# FR4 conductivity from loss tangent
sigma_fr4 = 2*np.pi*f0 * 8.854e-12 * eps_r * tan_d


print(f"L = {L*1e3:.2f} mm,  W = {W*1e3:.2f} mm")
print(f"eps_eff = {eps_eff:.3f},  dL = {dL*1e3:.3f} mm")
# Typical output: L = 29.38 mm,  W = 38.01 mm

2. Set up the Simulation

Section titled “2. Set up the Simulation”

Use a quarter-wavelength margin around the patch and a non-uniform z-profile to resolve the thin substrate without inflating the cell count.

from rfx import Simulation, Box, GaussianPulse
import numpy as np


# Domain: patch + λ/4 margin on all sides
lam0   = C0 / f0            # free-space wavelength ~125 mm
margin = lam0 / 4           # 31 mm absorber margin


dom_x = L + 2*margin
dom_y = W + 2*margin
dom_z = h + margin          # substrate + air above


# Non-uniform z profile: fine cells inside substrate, coarse in air
dz_sub = h / 6              # 6 cells through 1.6 mm substrate
dz_air = 1.5e-3             # 1.5 mm cells in air
n_air  = max(1, int(round(margin / dz_air)))


dz_profile = np.concatenate([
    np.full(6, dz_sub),
    np.full(n_air, dz_air),
])


sim = Simulation(
    freq_max=4e9,
    domain=(dom_x, dom_y, dom_z),
    boundary="cpml",
    cpml_layers=12,
    dx=5e-4,              # 0.5 mm lateral cell
    dz_profile=dz_profile,
)

!!! tip auto_configure() can derive dz_profile automatically. Use it when you prefer a one-shot configuration:

from rfx import auto_configure
geom = [(Box((0,0,0),(dom_x,dom_y,h)), "fr4")]
cfg  = auto_configure(geom, freq_range=(1e9, 4e9), accuracy="standard")
sim  = Simulation(**cfg.to_sim_kwargs())

3. Define materials and geometry

Section titled “3. Define materials and geometry”
# Register FR4 with frequency-accurate loss tangent at 2.4 GHz
sim.add_material("fr4_24ghz", eps_r=4.4, sigma=sigma_fr4)


# Ground plane (z = 0, one cell thick)
sim.add(Box((0, 0, 0), (dom_x, dom_y, dz_sub)), material="pec")


# Substrate slab
sim.add(Box((0, 0, 0), (dom_x, dom_y, h)), material="fr4_24ghz")


# Patch (sitting on top of substrate; one cell thick in z)
px0, py0 = margin, margin
sim.add(Box((px0, py0, h), (px0+L, py0+W, h+dz_sub)), material="pec")

!!! note rfx rasterises geometry onto the Yee grid. A patch of physical thickness dz_sub is only one cell thick in z, which is standard practice for printed conductors.

4. Add source and probe

Section titled “4. Add source and probe”

The probe-feed is placed at one-third of the patch length from the radiating edge:

feed_x = px0 + L/3
feed_y = py0 + W/2
feed_z = h / 2           # midpoint of substrate


sim.add_source(
    (feed_x, feed_y, feed_z),
    component="ez",
    waveform=GaussianPulse(f0=f0, bandwidth=0.8),
)
sim.add_probe((feed_x, feed_y, feed_z), "ez")

add_source injects a soft (no impedance loading) current source — ideal for resonance characterisation. Use add_port if you want S-parameter extraction against a 50 Ω reference.

5. Run the simulation

Section titled “5. Run the simulation”
# Build grid to get dt, then set n_steps for ~10 ns simulation time
grid    = sim._build_grid()
n_steps = int(np.ceil(10e-9 / grid.dt))
print(f"Grid: {grid.shape}, dt={grid.dt*1e12:.2f} ps, n_steps={n_steps}")


result = sim.run(n_steps=n_steps)

6. Extract resonance with Harminv

Section titled “6. Extract resonance with Harminv”
modes = result.find_resonances(freq_range=(1.5e9, 3.5e9))


if modes:
    best = min(modes, key=lambda m: abs(m.freq - f0))
    err  = abs(best.freq - f0) / f0 * 100
    print(f"FDTD resonance : {best.freq/1e9:.4f} GHz")
    print(f"Analytical     : {f0/1e9:.4f} GHz")
    print(f"Error          : {err:.2f} %")
    print(f"Q              : {best.Q:.0f}")

!!! note The ~3 % difference between FDTD and the analytical formula is expected and is a known limitation of the Hammerstad-Jensen fringe-correction formula, not an rfx accuracy problem. See Cross-Validation.

7. Optional: exploratory lumped-port run

Section titled “7. Optional: exploratory lumped-port run”

If you want a quick diagnostic S-parameter run, you can re-run with a lumped port instead of a soft source:

sim2 = Simulation(
    freq_max=4e9,
    domain=(dom_x, dom_y, dom_z),
    boundary="cpml",
    cpml_layers=12,
    dx=5e-4,
    dz_profile=dz_profile,
)
sim2.add_material("fr4_24ghz", eps_r=4.4, sigma=sigma_fr4)
sim2.add(Box((0, 0, 0), (dom_x, dom_y, dz_sub)), material="pec")
sim2.add(Box((0, 0, 0), (dom_x, dom_y, h)), material="fr4_24ghz")
sim2.add(Box((px0, py0, h), (px0+L, py0+W, h+dz_sub)), material="pec")
sim2.add_port(
    (feed_x, feed_y, feed_z), "ez",
    impedance=50,
    waveform=GaussianPulse(f0=f0, bandwidth=0.8),
)


result2 = sim2.run(n_steps=n_steps, compute_s_params=True)


import matplotlib.pyplot as plt
freqs  = result2.freqs
s11_db = 20 * np.log10(np.abs(result2.s_params[0, 0, :]) + 1e-12)


plt.figure()
plt.plot(freqs / 1e9, s11_db)
plt.xlabel("Frequency (GHz)")
plt.ylabel("|S11| (dB)")
plt.title("Patch antenna S11 — FR4 2.4 GHz")
plt.ylim(-30, 2)
plt.grid(True)
plt.tight_layout()
plt.savefig("patch_s11.png", dpi=150)
plt.show()

!!! warning For patch and microstrip feeds, treat this lumped-port S11 result as a diagnostic workflow, not yet as the canonical public validation path. The currently best-validated path for this tutorial is the current-source + Harminv resonance workflow above.

Expected results

Section titled “Expected results”
QuantityValue
FDTD resonance~2.33–2.38 GHz
Analytical prediction2.40 GHz
Discrepancy~1–3 % (formula limitation)
Q factor20–60 (lossy FR4)
Diagnostic lumped-port S11depends strongly on feed position / port model

Full script

Section titled “Full script”

The complete ready-to-run version of this tutorial is examples/04_patch_antenna.py.