1 · Aperture field
Complex aperture distribution
\[E_a(x,y)=A(x,y)e^{j\Phi(x,y)}\]
Amplitude \(A\) is constrained from 0 to 1. Phase \(\Phi\) is edited from −180° to +180°. Together they define the complex excitation across the aperture.
Reference
Equations and conventions
The calculator uses normalized wavelength units, so the numerical model sets \(\lambda=1\) and \(k=2\pi/\lambda=2\pi\). The aperture lies in the XY plane, +Z is boresight, \(\theta\) is measured from +Z, and \(\phi\) is measured from +X toward +Y.
2 · Coordinates
Direction cosines
\[u=\sin\theta\cos\phi,\qquad v=\sin\theta\sin\phi\]
These variables map a far-field direction onto the aperture's spatial-frequency coordinates. They are used in both the continuous integral and discrete summation.
3 · Continuous 2D aperture
Far-field aperture factor
\[F(\theta,\phi)=\int_0^{L_x}\int_0^{L_y} A(x,y)e^{j\Phi(x,y)}e^{jk(xu+yv)}\,dy\,dx\]
This is the spatial Fourier-transform relationship between the complex aperture field and its far-field pattern. The implementation evaluates this integral numerically on a sampled grid. The same expression reduces to a single integral for the 1D aperture.
4 · Discrete array
Array-factor summation
\[F(\theta,\phi)=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1} A_{m,n}e^{j\Phi_{m,n}}e^{jk(x_m u+y_n v)}\]
For discrete mode, element locations are generated from the requested aperture dimensions and spacing \(d_x/\lambda\) and \(d_y/\lambda\). In 1D only \(d/\lambda\) is needed. This tool models the array factor only; no individual element pattern is multiplied into the result.
5 · Beam steering
Linear steering phase
\[\Phi_{steer}(x,y)=-k\left(xu_0+yv_0\right)\]
Here \(u_0=\sin\theta_0\cos\phi_0\) and \(v_0=\sin\theta_0\sin\phi_0\). With the sign convention used above, this phase ramp places the coherent maximum at the requested steering direction.
6 · Normalization
Normalized pattern in dB
\[F_N=\frac{|F|}{\max|F|},\qquad P_{dB}=20\log_{10}(F_N)\]
The displayed pattern is normalized so its peak is 0 dB. A user-selected floor such as −40 dB limits the visual range and avoids displaying numerical values near negative infinity.
7 · Physical aperture limit
Maximum theoretical directivity
\[D_{max}=\frac{4\pi A_{phys}}{\lambda^2},\qquad A_{phys}=L_xL_y\]
For a 10λ × 10λ rectangular aperture, \(A_{phys}=100\lambda^2\), so \(D_{max}\approx1256.6\), or about 31.0 dBi. This is a directivity limit, not gain, because conductor, dielectric, feed, mismatch, and other losses are not modeled.
8 · Directivity estimates
2D aperture and 1D line-array directivity
\[D_{est,2D}=\eta_{ap}D_{max}\]
\[D_{1D}=\frac{4\pi U_{max}}{\int_{4\pi}U(\theta,\phi)d\Omega}\]
For a 2D aperture, the calculator uses the physical-aperture limit multiplied by the sampled illumination/phase efficiency. For a 1D line aperture or linear array, physical area is undefined, so the calculator instead evaluates the full-sphere array-factor directivity under an isotropic-element/line-source assumption. The displayed 1D “max” value is the uniform broadside reference for the same line geometry.
9 · 2D cuts
Signed-angle convention
\[\alpha\in[-90^\circ,90^\circ],\quad u=\sin\alpha\cos\phi_c,\quad v=\sin\alpha\sin\phi_c\]
For Cartesian and polar cuts, a signed angle is used. Positive and negative angles represent opposite sides of boresight in the selected cut plane. XZ corresponds to \(\phi_c=0^\circ\), YZ to \(\phi_c=90^\circ\), and Custom uses the user-entered cut azimuth.
10 · Design synthesis
First-order aperture and spacing rules
\[\mathrm{HPBW}_{deg}\approx\frac{50.8\lambda}{L}\]
\[\frac{d}{\lambda}\leq\frac{1}{1+\sin\theta_{max}}\]
The Design page uses the broadside uniform-aperture HPBW approximation to estimate required electrical length. For a scanning discrete array, the spacing inequality is used as a visible-region grating-lobe limit; Auto spacing adds a user-selectable safety margin in Advanced mode.
11 · Scan loss
Projected planar aperture
\[\Delta G_{scan}\approx-10\log_{10}(\cos\theta)\]
For the 2D planar-aperture design estimate, scan loss includes the first-order projected-area reduction. The actual element pattern can create additional scan loss and is not inferred from S-parameters alone.
12 · Periodic unit cell
Active mismatch from scan-dependent S11
\[\eta_{mismatch}=1-|\Gamma_{active}|^2,\qquad L_m=-10\log_{10}(\eta_{mismatch})\]
When a periodic-unit-cell CSV is imported, the Design page evaluates active S11 over the requested scan region. If several spacing candidates are present, Auto spacing favors a scan-safe candidate with stronger worst-case active return loss.
13 · Finite array network
Active reflection from an S-matrix
\[\mathbf b=\mathbf S\mathbf a,\qquad \Gamma_{active,m}=\frac{b_m}{a_m}\]
\[\eta_{accepted}=1-\frac{\|\mathbf b\|^2}{\|\mathbf a\|^2}\]
For an uploaded finite-array Touchstone matrix, ApertureLab constructs the steering excitation vector \(\mathbf a\), computes the reflected wave vector \(\mathbf b\), and estimates total accepted power. This captures network-level mutual-coupling and mismatch behavior for the supplied finite array, but it still does not provide an embedded element radiation pattern.
14 · EM import workflow
Recommended simulation data
For a periodic unit cell, sweep scan angle and export frequency_ghz, dx_lambda, dy_lambda, theta_deg, phi_deg, s11_db. Include multiple spacing candidates if you want the Design page to compare spacing using EM data. For a finite array, export a full Touchstone S-matrix and specify the port grid. Radiation/embedded-element-pattern data is a separate future refinement needed for a fully realized-gain pattern prediction.
16 · Multi-port matching
Loaded-port S-matrix reduction
\[\Gamma_{in}=S_{11}+\frac{S_{12}S_{21}\Gamma_L}{1-S_{22}\Gamma_L}\]
For a two-port antenna with Port 1 driven and Port 2 terminated by a load reflection coefficient \(\Gamma_L\), the secondary load changes the effective reflection seen at Port 1 through the measured coupling terms. For three or more ports, ApertureLab uses the matrix form below rather than reducing each port independently.
\[\mathbf a_L=(\mathbf I-\mathbf \Gamma_L\mathbf S_{LL})^{-1}\mathbf \Gamma_L\mathbf S_{Lm}a_m\]
\[\Gamma_{eff}=S_{mm}+\mathbf S_{mL}\frac{\mathbf a_L}{a_m}\]
17 · Component ports
Reactive tuning loads
\[\Gamma_L(f)=\frac{Z_L(f)-Z_0}{Z_L(f)+Z_0}\]
A component port is not independently driven in the current matching workflow. It is loaded with an ideal reactive L/C termination to ground. A port marked 50 Ω terminated uses a matched 50 Ω load. After the secondary-port loads are applied, the selected main port is reduced to an effective one-port impedance and the conventional series/shunt matching network is optimized against 50 Ω.
18 · Multi-band matching
Band-priority weighting
\[J_{avg}=\frac{\sum_b w_b\,J_b}{\sum_b w_b},\qquad 0\le w_b\le1\]
Each selected frequency band is evaluated independently first, then the band-level results are combined with the user-defined priority weights. This prevents a wider or more densely sampled band from automatically dominating a narrower band. A weight of 1 gives full priority; a weight of 0 excludes that band from the optimization while leaving it configured in the UI. Worst-case optimization applies the weight to each band’s worst reflected-power result; average, center-frequency, center-weighted, and −10 dB bandwidth objectives use weighted band-level aggregation.
19 · Maximum components
Neutral matching positions
\[Z_{series,neutral}=0,\qquad Z_{shunt,neutral}\to\infty\]
“Maximum components” means the optimizer may leave some physical positions electrically unused. A neutral series position is a bypass/short (0 Ω), while a neutral shunt position is an open circuit (∞ Ω). This is topology-aware: a shunt 0 nH inductor would short the RF node to ground and is therefore not an unused component; similarly a series 0 pF capacitor would open the signal path. ApertureLab explicitly compares lower-order matches when a larger maximum is requested, so increasing the allowed component count cannot discard a better lower-order solution.
20 · Total antenna efficiency
Radiation efficiency, mismatch, and total efficiency
\[\eta_m(f)=1-|S_{11}(f)|^2\]
\[\eta_{total}(f)=\eta_{rad}(f)\,\eta_m(f)=\eta_{rad}(f)\,[1-|\Gamma_{in}(f)|^2]\]
For one-port antenna matching with ideal lossless matching components, ApertureLab can optimize the network for total antenna efficiency rather than S11 alone. If radiation efficiency is supplied, total efficiency is computed directly. If total efficiency is supplied, radiation efficiency is derived from the original imported S1P:
\[\eta_{rad}(f)=\frac{\eta_{total,original}(f)}{1-|S_{11,original}(f)|^2}\]
If both curves are supplied, the tool cross-checks them. During hypothetical S11 editing, the radiation-efficiency curve is held fixed because the tool has no full-wave information telling it how the radiation efficiency itself would change.
21 · Important scope
What is not included
The Main pattern simulator remains an aperture/array-factor model and does not include element pattern, polarization, conductor/dielectric loss, feed-network loss, platform scattering, or radome effects. The Design page can optionally use periodic active-S11 or a finite-array S-matrix to add mismatch/coupling information, but S-parameters alone do not create a full-wave radiation pattern or realized-gain model.