Bent Patch Antenna

Simulate a microstrip patch antenna wrapped around a curved substrate and extract its S-parameters and radiation pattern. This is the cylindrical counterpart to the Simple Patch Antenna — the geometry and meshing approach are analogous, but everything is expressed in cylindrical coordinates (r, azimuth, z).

This tutorial covers:

• Cylindrical coordinate system (CoordSystem=1) for conformal antenna modelling

• Using AddBox in cylindrical coordinates — start/stop are (radius, azimuth, z)

• C-shaped simulation domain (±135° in azimuth) to reduce cell count

• Surface current dump for mode visualization at resonance

• Far-field calculation with CalcNF2FF() in cylindrical coordinates

Octave/Matlab Script

Simulation Parameters

This tutorial demonstrates a bent (conformal) microstrip patch antenna modelled in a cylindrical coordinate system. All geometric dimensions are given in millimetres; patch.radius sets the curvature of the ground plane and feed.pos controls the impedance-matching offset of the coaxial feed from the patch centre.

physical_constants;
unit = 1e-3; % all length in mm

% patch width in alpha-direction
patch.width  = 32; % resonant length in alpha-direction
patch.radius = 50; % radius
patch.length = 40; % patch length in z-direction

% substrate setup
substrate.epsR   = 3.38;
substrate.kappa  = 1e-3 * 2*pi*2.45e9 * EPS0*substrate.epsR;
substrate.width  = 80;
substrate.length = 90;
substrate.thickness = 1.524;
substrate.cells = 4;

% setup feeding
feed.pos = -5.5; %feeding position in x-direction
feed.width = 2;  %feeding port width
feed.R = 50;     %feed resistance

% size of the simulation box
SimBox.rad    = 2*100;
SimBox.height = 1.5*200;

FDTD Solver Initialization

openEMS is initialized with a cylindrical coordinate system (CoordSystem=1) because the antenna geometry is naturally described in (r, alpha, z). A Gaussian pulse centred at 2 GHz with a 1 GHz bandwidth excites the resonance while keeping the simulation short; Mur absorbing boundaries terminate all six faces of the domain.

FDTD = InitFDTD('CoordSystem', 1); % init a cylindrical FDTD
f0 = 2e9; % center frequency
fc = 1e9; % 20 dB corner frequency
FDTD = SetGaussExcite( FDTD, f0, fc );
BC = {'MUR' 'MUR' 'MUR' 'MUR' 'MUR' 'MUR'}; % boundary conditions
FDTD = SetBoundaryCond( FDTD, BC );

Geometry Initialization

CSXCAD is initialized in the same cylindrical coordinate system so that AddBox primitives are interpreted as (radius, azimuth, z) extents. Angular widths in radians are pre-computed from the physical arc lengths and the respective bending radii before any geometry is added.

CSX = InitCSX('CoordSystem',1);

% calculate some width as an angle in radiant
patch_ang_width = patch.width/(patch.radius+substrate.thickness);
substr_ang_width = substrate.width/patch.radius;
feed_angle = feed.pos/patch.radius;

Radiating Patch Element

The radiating element is a zero-thickness perfect electric conductor (PEC) placed on the outer face of the substrate at radius patch.radius + substrate.thickness. Its angular extent is derived from the desired arc length divided by the outer radius; the resonant frequency is primarily set by this arc length and the substrate permittivity.

CSX = AddMetal( CSX, 'patch' ); % create a perfect electric conductor (PEC)
start = [patch.radius+substrate.thickness -patch_ang_width/2 -patch.length/2 ];
stop  = [patch.radius+substrate.thickness  patch_ang_width/2  patch.length/2 ];
CSX = AddBox(CSX,'patch',10,start,stop); % add a box-primitive to the metal property 'patch'

Dielectric Substrate

The Rogers RO4003C-like substrate (epsilon_r = 3.38) fills the volume between the ground plane and the patch surface. A small loss term (kappa) derived from tan_delta = 1e-3 at 2.45 GHz accounts for dielectric loss that reduces radiation efficiency. The substrate.cells parameter forces at least four mesh layers through the thin dielectric so the field variation across it is adequately resolved.

CSX = AddMaterial( CSX, 'substrate' );
CSX = SetMaterialProperty( CSX, 'substrate', 'Epsilon', substrate.epsR, 'Kappa', substrate.kappa );
start = [patch.radius                     -substr_ang_width/2 -substrate.length/2];
stop  = [patch.radius+substrate.thickness  substr_ang_width/2  substrate.length/2];
CSX = AddBox( CSX, 'substrate', 0, start, stop);

Current Density Probe

A surface-current dump box (DumpType=3) records the tangential current density on the outer face of the substrate at each time step in HDF5 format. At the resonant frequency the current pattern reveals the dominant half-wave mode and can be visualized in Paraview; this diagnostic confirms correct TE10-mode operation.

CSX = AddDump(CSX, 'Jt_patch','DumpType',3,'FileType',1);
start = [patch.radius+substrate.thickness -substr_ang_width/2 -substrate.length/2];
stop  = [patch.radius+substrate.thickness +substr_ang_width/2  substrate.length/2];
CSX = AddBox( CSX, 'Jt_patch', 0, start, stop );

Ground Plane

The PEC ground plane closes the back side of the substrate at patch.radius, completing the transmission-line cross-section between patch and ground. Although it is not strictly required for simulation correctness, including it improves the geometry preview and ensures unambiguous boundary conditions on the inner substrate face.

CSX = AddMetal( CSX, 'gnd' ); % create a perfect electric conductor (PEC)
start = [patch.radius -substr_ang_width/2 -substrate.length/2];
stop  = [patch.radius +substr_ang_width/2 +substrate.length/2];
CSX = AddBox(CSX,'gnd',10,start,stop);

Feed Port

A lumped port spans the substrate in the radial direction, injecting the excitation signal and measuring the reflected wave simultaneously. The 50-ohm source resistance matches the standard coaxial reference impedance so that S11 and Zin are both referred to that reference. The port is offset from centre by feed.pos to achieve good impedance matching at the target frequency.

start = [patch.radius                      feed_angle 0];
stop  = [patch.radius+substrate.thickness  feed_angle 0];
[CSX port] = AddLumpedPort(CSX, 50 ,1 ,feed.R, start, stop, [1 0 0], true);

Mesh Generation

DetectEdges seeds mesh lines at every geometry boundary so that Yee cells align with material interfaces without manual placement. Additional radial lines are forced through the thin substrate (substrate.cells) to resolve the electric field across it, and SmoothMesh fills the remaining domain with a 1.4 growth ratio keeping the maximum cell size at lambda_min/20.

% detect all edges
mesh = DetectEdges(CSX);

% add the simulation domain size
mesh.r = [mesh.r patch.radius+[-20 SimBox.rad]];
mesh.a = [mesh.a -0.75*pi 0.75*pi];
mesh.z = [mesh.z -SimBox.height/2 SimBox.height/2];

% add some lines for the substrate
mesh.r = [mesh.r patch.radius+linspace(0,substrate.thickness,substrate.cells)];

% generate a smooth mesh with max. cell size: lambda_min / 20
max_res = c0 / (f0+fc) / unit / 20;
max_ang = max_res/(SimBox.rad+patch.radius); % max res in radiant
mesh = SmoothMesh(mesh, [max_res max_ang max_res], 1.4);

disp(['Num of cells: ' num2str(numel(mesh.r)*numel(mesh.a)*numel(mesh.z))]);
CSX = DefineRectGrid( CSX, unit, mesh );

Near-Field to Far-Field Box

A closed Huygens surface is placed at least 8 cells inside each absorbing boundary to avoid sampling the evanescent near-field region. During the FDTD run, tangential E and H fields on all six cylindrical faces are stored in HDF5 files; CalcNF2FF reads these in post-processing to transform them into the far-field radiation pattern.

start = [mesh.r(4)     mesh.a(8)     mesh.z(8)];
stop  = [mesh.r(end-9) mesh.a(end-9) mesh.z(end-9)];
[CSX nf2ff] = CreateNF2FFBox(CSX, 'nf2ff', start, stop, 'Directions',[1 1 1 1 1 1]);

Write and Run Simulation

The CSXCAD structure and FDTD settings are serialized to an XML file that openEMS reads at runtime. CSXGeomPlot opens the structure in AppCSXCAD for a visual sanity-check before the FDTD run begins. RunOpenEMS then launches the solver; wall-clock time scales with the cell count printed above.

Sim_Path = ['tmp_' mfilename];
Sim_CSX  = [mfilename '.xml'];

CleanupSimPath(Sim_Path);

% write openEMS compatible xml-file
WriteOpenEMS( [Sim_Path '/' Sim_CSX], FDTD, CSX );

% show the structure
CSXGeomPlot( [Sim_Path '/' Sim_CSX] );

% run openEMS
RunOpenEMS( Sim_Path, Sim_CSX);

S-Parameter and Impedance Post-Processing

calcPort reads the time-domain voltage and current waveforms and computes their Fourier transforms over the requested frequency vector. Input impedance Zin and reflection coefficient S11 are derived from those spectra; a deep S11 minimum identifies the resonant frequency and confirms efficient radiation into the surrounding medium.

freq = linspace( max([1e9,f0-fc]), f0+fc, 501 );
port = calcPort(port, Sim_Path, freq);

Zin = port.uf.tot ./ port.if.tot;
s11 = port.uf.ref ./ port.uf.inc;
P_in = 0.5*real(port.uf.tot .* conj(port.if.tot)); % antenna feed power

% plot feed point impedance
figure
plot( freq/1e6, real(Zin), 'k-', 'Linewidth', 2 );
hold on
grid on
plot( freq/1e6, imag(Zin), 'r--', 'Linewidth', 2 );
title( 'feed point impedance' );
xlabel( 'frequency f / MHz' );
ylabel( 'impedance Z_{in} / Ohm' );
legend( 'real', 'imag' );

% plot reflection coefficient S11
figure
plot( freq/1e6, 20*log10(abs(s11)), 'k-', 'Linewidth', 2 );
grid on
title( 'reflection coefficient S_{11}' );
xlabel( 'frequency f / MHz' );
ylabel( 'reflection coefficient |S_{11}|' );

drawnow

%find resonance frequency from s11
f_res_ind = find(s11==min(s11));
f_res = freq(f_res_ind);

Current Distribution Export

At the resonant frequency the stored time-domain surface-current data are Fourier-transformed to a single-frequency snapshot and written to a VTK file. Loading this file in Paraview allows visual inspection of the current mode shape on the patch surface; the dominant half-wave distribution confirms correct TE10-mode operation.

disp('dumping resonant current distribution to vtk file, use Paraview to visualize');
ConvertHDF5_VTK([Sim_Path '/Jt_patch.h5'],[Sim_Path '/Jf_patch'],'Frequency',f_res,'FieldName','J-Field');

Far-Field Pattern Calculation

CalcNF2FF applies the near-field to far-field transformation at the resonant frequency for two principal cuts: the elevation plane (phi = 0) sweeping theta, and the azimuth plane (theta = 90 deg) sweeping phi. The phase centre is taken at the outer patch surface so that the pattern cuts are well-defined despite the cylindrical geometry.

% calculate the far field at phi=0 degree
nf2ff = CalcNF2FF(nf2ff, Sim_Path, f_res, [-180:2:180]*pi/180, 0,'Center',[patch.radius+substrate.thickness 0 0]*unit, 'Outfile','pattern_phi_0.h5');
% normalized directivity as polar plot
figure
polarFF(nf2ff,'xaxis','theta','param',1,'normalize',1)

% calculate the far field at phi=0 degree
nf2ff = CalcNF2FF(nf2ff, Sim_Path, f_res, pi/2, (-180:2:180)*pi/180,'Center',[patch.radius+substrate.thickness 0 0]*unit, 'Outfile','pattern_theta_90.h5');
% normalized directivity as polar plot
figure
polarFF(nf2ff,'xaxis','phi','param',1,'normalize',1)

% display power and directivity
disp( ['radiated power: Prad = ' num2str(nf2ff.Prad) ' Watt']);
disp( ['directivity: Dmax = ' num2str(nf2ff.Dmax) ' (' num2str(10*log10(nf2ff.Dmax)) ' dBi)'] );
disp( ['efficiency: nu_rad = ' num2str(100*nf2ff.Prad./real(P_in(f_res_ind))) ' %']);

drawnow

Three-Dimensional Far-Field Pattern

The full 3D far-field pattern is computed on a dense (theta, phi) sphere and exported to a VTK file for volumetric visualization in Paraview. The logscale=-20 display option suppresses low-level sidelobes and highlights the main-beam structure; total radiated power and maximum directivity are printed to the console for a quick figure of merit.

disp( 'calculating 3D far field pattern and dumping to vtk (use Paraview to visualize)...' );
thetaRange = (0:2:180);
phiRange = (0:2:360) - 180;
nf2ff = CalcNF2FF(nf2ff, Sim_Path, f_res, thetaRange*pi/180, phiRange*pi/180,'Verbose',1,'Outfile','3D_Pattern.h5','Center',[patch.radius+substrate.thickness 0 0]*unit);

figure
plotFF3D(nf2ff,'logscale',-20);

Images

3D view of the bent patch antenna on curved substrate (AppCSXCAD)

Cylindrical mesh — C-shaped domain spanning ±135° in azimuth

Feed-point impedance (real and imaginary parts)

Reflection coefficient S11 over frequency

Far-field radiation pattern at phi = 0°

Far-field radiation pattern at theta = 90°

3D farfield radiation pattern