Circular Waveguide

Simulate the dominant TE11 mode in a circular metallic waveguide and verify the cut-off frequency against the analytical solution.

This tutorial covers:

• Cylindrical coordinate system for circular cross-sections

• Mode-matched circular waveguide port setup

• Cut-off frequency extraction and comparison with theory

Octave/Matlab Script

Simulation Parameters

Define the waveguide dimensions and the frequency range of interest. The radius rad sets the TE11 cut-off frequency (~251 MHz for 350 mm); the chosen frequency span (300–500 MHz) therefore straddles the onset of propagation and captures strongly dispersive behaviour.

physical_constants;
unit = 1e-3; %drawing unit in mm

% waveguide dimensions
length = 2000;
rad = 350;     %waveguide radius in mm

% frequency range of interest
f_start =  300e6;
f_stop  =  500e6;

mesh_res = [10 2*pi/49.999 10]; %targeted mesh resolution

FDTD Parameters and Excitation

Configure the FDTD solver for cylindrical coordinates (CoordSystem=1) and drive it with a Gaussian pulse centred in the band. EndCriteria of 1e-4 stops the time-stepping once the remaining energy falls below that fraction of the peak, keeping runtime short while ensuring the impulse has decayed completely. PML layers on both z-faces absorb outgoing energy without spurious reflections.

FDTD = InitFDTD('EndCriteria',1e-4,'CoordSystem',1);
FDTD = SetGaussExcite(FDTD,0.5*(f_start+f_stop),0.5*(f_stop-f_start));

% boundary conditions
BC = [0 0 0 0 3 3]; %pml in pos. and neg. z-direction
FDTD = SetBoundaryCond(FDTD,BC);

Cylindrical Mesh Setup

A cylindrical mesh (r, azimuth, z) is the natural coordinate system for this geometry: it avoids staircase errors on the curved conducting wall and keeps cell counts manageable. SmoothMeshLines distributes lines evenly from the axis to the wall in r, over a full 2pi in azimuth, and along the waveguide length in z.

CSX = InitCSX('CoordSystem',1); % init a cylindrical mesh
mesh.r = SmoothMeshLines([0 rad], mesh_res(1)); %mesh in radial direction
mesh.a = SmoothMeshLines([0 2*pi], mesh_res(2)); % mesh in aziumthal dir.
mesh.z = SmoothMeshLines([0 length], mesh_res(3));
CSX = DefineRectGrid(CSX, unit,mesh);

Waveguide Port Definition

Two TE11 mode ports bookend the waveguide: port 1 (excitation flag = 1) injects the dominant circular-waveguide mode; port 2 at the far end acts as a passive detector. Placing each port several cells inward from the PML boundary ensures the mode is fully formed before reaching the absorber and that the port plane samples only the travelling wave.

start=[mesh.r(1)   mesh.a(1)   mesh.z(8)];
stop =[mesh.r(end) mesh.a(end) mesh.z(15)];
[CSX, port{1}] = AddCircWaveGuidePort( CSX, 0, 1, start, stop, rad*unit, 'TE11', 0, 1);

start=[mesh.r(1)   mesh.a(1)   mesh.z(end-13)];
stop =[mesh.r(end) mesh.a(end) mesh.z(end-14)];
[CSX, port{2}] = AddCircWaveGuidePort( CSX, 0, 2, start, stop, rad*unit, 'TE11');

Field Dump Configuration

Register a volumetric electric-field dump in HDF5 format (FileType=1) spanning the entire simulation domain. SubSampling of 4 in every direction reduces the file size by a factor of 64 while still capturing the spatial field structure at a resolution sufficient for visualisation in ParaView or AppCSXCAD.

CSX = AddDump(CSX,'Et','FileType',1,'SubSampling','4,4,4');
start = [mesh.r(1)   mesh.a(1)   mesh.z(1)];
stop  = [mesh.r(end) mesh.a(end) mesh.z(end)];
CSX = AddBox(CSX,'Et',0 , start,stop);

Write XML and Run Simulation

Serialise the complete simulation setup to an openEMS XML file and hand it to the solver. CleanupSimPath removes any previous run so stale field dumps do not contaminate post-processing results.

Sim_Path = 'tmp';
Sim_CSX = 'circ_wg.xml';

CleanupSimPath(Sim_Path);

WriteOpenEMS([Sim_Path '/' Sim_CSX],FDTD,CSX);

RunOpenEMS(Sim_Path, Sim_CSX)

Post-Processing: Port Calculation

calcPort reads the time-domain probe signals, transforms them to the frequency domain, and decomposes incident and reflected wave amplitudes at each port. S-parameters follow directly from the voltage wave ratios; the complex wave impedance ZL is the ratio of total voltage to total current and reveals the dispersive character of the TE11 mode near cut-off.

freq = linspace(f_start,f_stop,201);
port = calcPort( port, Sim_Path, freq);

s11 = port{1}.uf.ref./ port{1}.uf.inc;
s21 = port{2}.uf.ref./ port{1}.uf.inc;
ZL = port{1}.uf.tot./port{1}.if.tot;

S-Parameter Plot

Plot S11 and S21 in dB across the frequency band. Near cut-off S11 rises sharply as the mode cannot propagate; well above cut-off S21 should approach 0 dB (lossless transmission) and S11 should drop, confirming that the waveguide is matched to the TE11 mode ports.

figure
plot(freq*1e-6,20*log10(abs(s11)),'k-','Linewidth',2);
xlim([freq(1) freq(end)]*1e-6);
grid on;
hold on;
plot(freq*1e-6,20*log10(abs(s21)),'r--','Linewidth',2);
l = legend('S_{11}','S_{21}','Location','Best');
set(l,'FontSize',12);
ylabel('S-Parameter (dB)','FontSize',12);
xlabel('frequency (MHz) \rightarrow','FontSize',12);

Waveguide Impedance Comparison

Overlay the numerically extracted wave impedance (real and imaginary parts of ZL) against the analytic TE11 value stored in port{1}.ZL. Close agreement between the two validates both the mode excitation and the port normalisation; the strong frequency dependence near cut-off is the hallmark of dispersive waveguide propagation.

figure
plot(freq*1e-6,real(ZL),'Linewidth',2);
hold on;
grid on;
plot(freq*1e-6,imag(ZL),'r--','Linewidth',2);
plot(freq*1e-6,port{1}.ZL,'g-.','Linewidth',2);
ylabel('ZL (\Omega)','FontSize',12);
xlabel('frequency (MHz) \rightarrow','FontSize',12);
xlim([freq(1) freq(end)]*1e-6);
l = legend('\Re(Z_L)','\Im(Z_L)','Z_L analytic','Location','Best');
set(l,'FontSize',12);

Images

S-parameters of the circular waveguide showing TE11 cut-off

Simulated wave impedance vs. analytic TE11 result

Notes

This tutorial is the cylindrical-coordinate counterpart to the Rectangular Waveguide tutorial — the port setup, post-processing, and result interpretation are nearly identical. The calcPort field variable naming (uf/if, inc/ref/tot) is explained there.

Suggested Modifications

• Try exciting and detecting different or multiple modes simultaneously.

• Add an asymmetric dielectric load inside the waveguide to observe mode conversion (requires multi-mode port detection).

• Insert a periodic dielectric grating to produce frequency-selective Bragg reflections.