Patch Antenna Designer

Model (Balanis ch. 14, transmission-line / two-slot): W = (c/2f₀)√(2/(εr+1)); εeff Hammerstad (1+12h/W)^−½; ΔL Hammerstad fringing (14-2); L = c/(2f₀√εeff) − 2ΔL. Slot conductance G₁ and mutual G₁₂ (J₀ weighting) by Simpson integration of (14-12)/(14-18a); edge resistance R₁ = 1/(2(G₁+G₁₂)) (odd mode, + sign, 14-17). Inset / probe depth from R₁cos²(πy₀/L) (14-20a). Directivity D₂ = D₀·2/(1+G₁₂/G₁) with D₀ = (2πW/λ₀)²/I₁ (14-57, 14-59).

Bandwidth: radiation-limited fractional BW (VSWR≤2) = 3.77·((εr−1)/εr²)·(W/L)·(h/λ₀) → Q_rad = 1/(√2·BW); Q_c = h√(πf₀μ₀σ), Q_d = 1/tanδ; 1/Q_t = 1/Q_rad+1/Q_c+1/Q_d; reported BW = 1/(√2·Q_t); η = Q_t/Q_rad (conductor + dielectric loss only — no surface-wave loss modelled). Gain = D₂ + 10logη. Feed line by Hammerstad–Jensen synthesis; inset notch gap ≈ W_f/2 (rule of thumb). Probe inductance not modelled. Suggested ground plane ≥ patch + 6h per side.