# Inset feed for 50 ohms target_Z = 50 if target_Z < R_edge: y0_m = (L_m / math.pi) * math.acos(math.sqrt(target_Z / R_edge)) y0_mm = y0_m * 1000 else: y0_mm = None # cannot match with inset; use other method
Designing a patch antenna manually requires solving several transmission-line equations. A automates these calculations, taking inputs like operating frequency, substrate permittivity, and height, and returning key dimensions: patch length (L), patch width (W), inset feed position, and input impedance. 2. Fundamental Design Equations The most common patch shape is rectangular. For a given resonant frequency ( f_r ), substrate relative permittivity ( \varepsilon_r ), and substrate height ( h ), the steps are: 2.1 Patch Width (W) [ W = \fracc2 f_r \sqrt\frac\varepsilon_r + 12 ] where ( c = 3 \times 10^8 , m/s ) (speed of light). 2.2 Effective Permittivity (( \varepsilon_reff )) Due to fringing fields: [ \varepsilon_reff = \frac\varepsilon_r + 12 + \frac\varepsilon_r - 12 \left[ 1 + 12 \frachW \right]^-1/2 ] 2.3 Fringing Extension ((\Delta L)) [ \frac\Delta Lh = 0.412 \frac(\varepsilon_reff + 0.3)(W/h + 0.264)(\varepsilon_reff - 0.258)(W/h + 0.8) ] 2.4 Actual Patch Length (L) [ L = \fracc2 f_r \sqrt\varepsilon_reff - 2\Delta L ] 2.5 Inset Feed Position (for 50Ω matching) The input resistance at the edge ((R_in)) is approximately: [ R_in(y=y_0) = R_in(0) \cos^2\left(\frac\pi y_0L\right) ] where (R_in(0)) is the edge resistance (typically 150–300 Ω). For 50Ω: [ y_0 = \fracL\pi \cos^-1\sqrt\frac50R_in(0) ] (y_0) is measured from the center toward the edge. 3. Example Calculation (Manual) Design goal: Rectangular patch antenna at ( f_r = 2.45 , GHz ) (Wi-Fi/Bluetooth).
return "Width (mm)": round(W_mm, 2), "Length (mm)": round(L_mm, 2), "Effective permittivity": round(ereff, 3), "Delta L (mm)": round(delta_L_mm, 3), "Edge resistance (ohms)": round(R_edge, 1), "Inset from center for 50Ω (mm)": round(y0_mm, 2) if y0_mm else "N/A" microstrip patch antenna calculator
# Approximate edge resistance R_edge = 90 * (er**2 / (er - 1)) * (L_m / W_m)**2 if R_edge > 300: R_edge = 300 # practical limit
# Fringing extension num = (ereff + 0.3) * (W_m / h + 0.264) den = (ereff - 0.258) * (W_m / h + 0.8) delta_L_m = 0.412 * h * (num / den) delta_L_mm = delta_L_m * 1000 # Inset feed for 50 ohms target_Z =
| Step | Parameter | Formula/Value | Result | |------|-----------|---------------|--------| | 1 | Width W | ( \frac3e82 \times 2.45e9 \times \sqrt(4.4+1)/2 ) | ≈ 37.26 mm | | 2 | ( \varepsilon_reff ) | ( \frac4.4+12 + \frac4.4-12 (1 + 12 \times 1.6/37.26)^-0.5 ) | ≈ 3.74 | | 3 | ΔL | ( 0.412 \times 1.6 \times \frac(3.74+0.3)(37.26/1.6+0.264)(3.74-0.258)(37.26/1.6+0.8) ) | ≈ 0.729 mm | | 4 | Length L | ( \frac3e82 \times 2.45e9 \times \sqrt3.74 - 2 \times 0.729 ) | ≈ 29.06 mm | | 5 | Edge resistance (R_in(0)) | Approx. formula | ≈ 220 Ω | | 6 | Inset (y_0) for 50Ω | ( \frac29.06\pi \cos^-1\sqrt50/220 ) | ≈ 8.4 mm from center |
# Effective length and physical length Leff = c / (2 * f * math.sqrt(ereff)) L_m = Leff - 2 * delta_L_m L_mm = L_m * 1000 Fundamental Design Equations The most common patch shape
FR-4 (( \varepsilon_r = 4.4 ), ( h = 1.6 , mm )).