June 10, 202612 min read

I validated RayRF's PMC boundary against a 2003 IEEE paper, with openEMS as referee

RayRF field viewer showing the surface current standing wave on the dipole arms at 12.6 GHz over the PMC boundary — Surface currents on the dipole arms at 12.6 GHz, 1.5 mm above the PMC wall.

RayRF radiation pattern tab showing the 3D gain pattern of the dipole over the PMC boundary with E and H plane cuts — The same antenna's 3D gain pattern and principal-plane cuts.

Short answer

RayRF v1.0.55 adds per-face boundary conditions, including a PMC wall. I rebuilt the reference cases from Yang and Rahmat-Samii's 2003 EBG paper, a dipole 1.5 mm over PEC and PMC ground planes plus the mushroom EBG array itself, then ran the identical geometry on identical meshes in openEMS. The two solvers put the EBG match dip within 0.4 percent of each other, both consistent with the paper, and analytic image theory backs the boundary physics. RayRF needed 46 seconds on a GPU for the run openEMS finished in 2 hours 23 minutes. The replication also flushed out a real via discretization bug, now fixed. Everything is downloadable below.

A PMC is the one boundary you cannot check on a bench, because the material does not exist. When I added it to RayRF I wanted more than the internal physics tests passing. So I pulled a paper that RF engineers have cited for two decades, rebuilt its reference cases at the exact published dimensions, and then made a second, independent solver judge the same geometry cell for cell.

The paper

Fan Yang and Yahya Rahmat-Samii, "Reflection phase characterizations of the EBG ground plane for low profile wire antenna applications," IEEE Transactions on Antennas and Propagation, vol. 51, no. 10, pp. 2691-2698, October 2003. It is one of the standard references for electromagnetic band-gap ground planes. Before getting to the EBG surface, the authors set the stage with a clean question: what happens when you put a half-wavelength-class dipole extremely close to an ideal ground plane of each polarity?

Their Figure 3 answers it with FDTD. Over a PEC, the dipole's return loss never gets below -3.5 dB. The anti-phase image current 3 mm away kills the radiation, and the feed sees a near-short. Over a PMC, the in-phase image shifts the input impedance off 50 ohms instead, and the dipole manages a moderate -7.2 dB dip. Their mushroom EBG surface reaches -27 dB. I replicated all three configurations. The one thing I did not replicate is the paper's separate reflection-phase characterization of the infinite EBG surface, which needs periodic unit-cell boundaries and a plane-wave source RayRF does not have.

Figure 3 from Yang and Rahmat-Samii 2003: FDTD return loss of a dipole over PEC, PMC and EBG ground planes — Fig. 3 of the paper (copyright IEEE, reproduced for comparison): dipole return loss over PEC, PMC and EBG ground planes.

The setup

Everything is defined relative to the free-space wavelength at 12 GHz, which is 24.98 mm. The dipole is 0.40 wavelengths long with a wire radius of 0.005 wavelengths, centered 0.06 wavelengths above a one-wavelength-square ground plane, fed at the center against a 50 ohm reference. RayRF draws planar copper, so the round wire becomes a flat strip using the standard equivalence of strip width equals four times the wire radius. On the 0.25 mm reference mesh the worst dimension snap is 0.07 percent:

Paper dimensions and their grid-snapped equivalents. The paper never states its feed model, so the 0.5 mm gap is my choice, and its effect is checked against analytic theory below.
Quantity	Paper	RayRF	Cells at 0.25 mm
Dipole length (tip to tip)	9.993 mm	10.0 mm	40
Strip width (4x wire radius)	0.4997 mm	0.5 mm	2
Feed gap (center, in-plane port)	not stated	0.5 mm	2
Height above ground	1.499 mm	1.5 mm	6
Ground plane side	24.98 mm	25.0 mm	100

RayRF editor with the dipole and the 25 by 25 mm ground sheet, stackup showing the 1.5 mm air gap — The PEC-sheet project in the editor. The stackup is dipole, 1.5 mm air gap, ground.

Zoomed RayRF editor view of the two dipole arms and the in-plane port at the center gap — The center feed: two 4.75 mm arms and an in-plane area port across the 0.5 mm gap.

Two RayRF features carry this whole exercise. The first is the in-plane port direction that shipped alongside the boundary work: a printed dipole needs its excitation across a horizontal gap on a single copper layer, not the vertical probe a patch uses. The second is the per-face boundary matrix. For the ideal-ground cases there is no copper sheet at all: the bottom face of the simulation domain itself is the ground plane, set to PEC or PMC with a fixed 1.5 mm spacing below the dipole. That makes the ground mathematically infinite, which is exactly what the textbook image-theory check assumes.

RayRF Simulate tab with the boundary condition matrix: bottom face PMC at 1.5 mm fixed spacing, all other faces PML — The Simulate tab for the PMC case: -Z is a PMC wall 1.5 mm below the geometry, every other face stays PML.

RayRF mesh preview showing the domain box with PML labels on five faces and PMC on the bottom face under the dipole — Mesh preview with the boundary overlay. The staircased strip sits 6 cells above the PMC face.

All RayRF runs in this post are the CUDA backend on an RTX 5070 Ti, which runs these domains at 16 to 22 GCell/s. Every run ends on the solver's own energy ringdown criterion, with the step cap raised to one million so nothing terminates early. That matters for the EBG, which rings two orders of magnitude longer than the bare dipole cases.

What the paper says, what RayRF gets

RayRF S-parameters tab for the dipole over the finite PEC sheet: shallow dip to about -3 dB near 12.5 GHz — Over the 25 mm PEC sheet: S11 never leaves the -3 dB neighborhood. The Smith trace hugs the edge, a near-total reflection.

RayRF S-parameters tab for the dipole over the infinite PMC boundary: dip to -8.2 dB at 12.4 GHz — Over the PMC wall: a moderate dip to -8.2 dB at 12.4 GHz. Zin at the dip is 108 - 22j ohms, well above the 61 ohms the same dipole shows in free space.

RayRF S11 sweeps drawn over the axes of the paper's Figure 3, PEC and PMC curves tracking the published ones — RayRF sweeps drawn over the paper's Fig. 3 axes.

The behavior is the paper's, feature for feature. Over PEC the dipole refuses to match anywhere in the band, bottoming out at -2.9 dB against the paper's -3.5 dB. Over PMC it manages -8.2 dB against the paper's -7.2 dB, with the dip 2.5 percent below where the paper's curve reads. The ground-size checks came out the paper's way too: going from the 25 mm sheet to a 50 mm sheet moved the PEC dip by 0.09 dB, and replacing the sheet with the infinite wall changed nothing past the second decimal. At 1.5 mm height the image current dominates and the plane's edges are spectators.

RayRF field viewer showing the induced image current distribution on the finite PEC ground sheet — The image current itself: induced surface current on the 25 mm PEC sheet, directly under the dipole.

A second referee: the same cases in openEMS

Agreement with a 23-year-old plot read by eye only goes so far, so I rebuilt the same cases in openEMS, the open-source FDTD this machine already runs for benchmarks. Same coordinates, same uniform cell size, same band and port, PML on the same faces, and openEMS's own PMC boundary on the bottom face for the ideal-ground case. Both solvers ran to their own ringdown criteria. That makes openEMS an independent referee on identical input, with only each engine's internal discretization choices left to differ.

S11 curves from RayRF and openEMS at the matched 0.25 mm mesh for the PMC, PEC sheet and EBG cases — Same geometry, same 0.25 mm mesh. Solver times in the legends: RayRF on the GPU, openEMS on all 16 CPU cores.

Matched 0.25 mm mesh, dip position and depth per solver. The paper column is its stated values; its dip positions are read off the plot.
Case	RayRF	openEMS	Difference	Paper
Free-space dipole	12.46 GHz, -19.7 dB	13.25 GHz, -15.5 dB	6.1%	not shown
Dipole over PMC	12.38 GHz, -8.2 dB	12.91 GHz, -7.0 dB	4.2%	-7.2 dB
Dipole over PEC sheet	12.53 GHz, -2.9 dB	13.44 GHz, -3.5 dB	7.0%	-3.5 dB
Dipole over EBG array	12.72 GHz, -10.0 dB	12.67 GHz, -18.0 dB	0.4%	-27 dB at ~12.8 GHz

The dipole cases differ by 4 to 7 percent between the two solvers, and the free-space row explains why. The exact same strip on the exact same grid resonates at 12.46 GHz in RayRF and 13.25 GHz in openEMS, against a converged value of 13.46 GHz. The two engines rasterize a 2-cell-wide zero-thickness strip differently, RayRF running it about 7 percent electrically long at this cell size and openEMS about 2 percent. That is a property of staircased thin strips at coarse cells, not of any boundary, and the paper's own in-house FDTD carried the same class of offset. The EBG row is the remarkable one: the quantity this paper is actually about lands within 0.4 percent across two independent codes.

What different run times buy you

Validation numbers mean little without the cost attached, so here is the same physics on three uniform meshes, every run ringdown-terminated. The 0.5 mm rung matches the cell size the paper's own 2003 FDTD implies. Solver time is the stepping loop on the RTX 5070 Ti:

The mesh ladder for the two ideal-ground cases. Depth is stable from the cheapest run onward, and the dip position converges first-order: the remaining error roughly halves with each cell-size halving.
Case	Cell size	Cells	Solver time	Dip	S11
Dipole over PMC	0.5 mm	0.35 M	0.04 s	11.50 GHz	-8.6 dB
	0.25 mm	2.8 M	0.22 s	12.38 GHz	-8.2 dB
	0.125 mm	22 M	4.3 s	12.79 GHz	-7.9 dB
Dipole over PEC sheet	0.5 mm	0.66 M	0.05 s	11.59 GHz	-2.0 dB
	0.25 mm	5.3 M	0.47 s	12.53 GHz	-2.9 dB
	0.125 mm	42 M	8.4 s	13.08 GHz	-3.3 dB

The pattern worth internalizing: the physics verdict, poor match over PEC and a moderate dip over PMC, is already correct in the 0.04 second run. What refinement buys is the dip position, pinned by the free-space anchor above. For scale, openEMS needs 31 to 73 seconds on 16 cores for the middle rung of these cases. RayRF's GPU engine clears all six ideal-ground runs in under 14 seconds combined.

The independent referee: image theory

For an infinite plane, image theory is exact: the ground is replaced by a mirror dipole at twice the height, anti-phase for PEC, in-phase for PMC, and the input impedance follows from the self and mutual impedance of a two-dipole pair. I computed that with the induced-EMF method, a few dozen lines of numpy that know nothing about FDTD. Two checks fall out of having both ideal-wall runs, and they cancel everything about my modeling choices:

Superposition: the average of the PEC and PMC input impedances must equal the free-space dipole's impedance, point by point. In the 10 to 14 GHz band that holds within 4 ohms on a 77 ohm scale, about 5 percent.
The half-difference of the two runs isolates the image contribution itself. Its resistive part matches the analytic mutual impedance within about 3 percent RMS over the same band.

RayRF S11 curves compared with analytic image theory, raw and with the measured electrical length factor applied — RayRF (solid) vs image theory (dashed) for free space, PEC and PMC. Left: raw. Right: the FDTD axis scaled by the single electrical-length factor measured on the free-space case.

Those two identities are the core of the boundary validation. The feed model, the strip approximation and the staircased mesh all cancel in them, so what remains is whether the wall reflects with the right sign and the right magnitude. It does, for both polarities, and openEMS's PMC agrees through its own independent implementation.

The paper's main act: the mushroom EBG array

The reason this paper is famous is the third curve, so I built it. The EBG ground plane is a 7x7 array of 3.0 mm square patches with 0.5 mm gaps on a 1.0 mm substrate at eps 2.20, one grounding via per patch at 0.25 mm diameter, and the same dipole 0.5 mm above the patch tops. That is 49 patches, 49 vias, five stackup layers, and the same overall 1.5 mm antenna height as the PEC and PMC cases.

RayRF editor showing the 7x7 mushroom patch array on the substrate with the dipole crossing the center — The EBG project: 7x7 patches over the eps 2.20 substrate, dipole across the center row.

RayRF field viewer showing surface currents concentrated on the three center patches of the EBG array at 12.6 GHz — Patch-layer surface currents at 12.6 GHz: the three patches under the dipole carry the resonance.

RayRF S-parameters tab for the dipole over the EBG array showing the deep match dip and narrow low-band array resonances — The EBG run at 0.25 mm: the match dip near 12.7 GHz, plus the narrow low-band resonances of the lossless finite array. openEMS reproduces both, including the ripple cluster.

The EBG ladder, every run terminated by its own ringdown criterion (the lossless array rings for hundreds of thousands of steps). Deepest dip in the 11 to 15 GHz match band. The paper reports -27 dB at roughly 12.8 GHz.
Cell size	Cells	Steps run	Solver time	EBG dip	S11
0.5 mm	0.66 M	129,560	5.3 s	11.98 GHz	-20.8 dB
0.25 mm	5.3 M	163,640	46 s	12.72 GHz	-10.0 dB
0.125 mm	42 M	555,180	22 min	13.73 GHz	-19.1 dB
openEMS, 0.25 mm	5.3 M	311,208	2 h 23 min	12.67 GHz	-18.0 dB

Honest scorecard. The EBG mechanism shows up at every mesh: a deep match dip in the right band with the input resistance pulled to 40 to 44 ohms, and the narrow low-band spikes of a lossless finite array sitting where the paper's curve shows its own ripple. At the matched 0.25 mm mesh, RayRF and openEMS agree on the dip position to 0.4 percent and both sit within about 1 percent of the paper. Depth at a deep match stays hypersensitive, which is why the three sources span -10 to -27 dB while agreeing on where the dip is. Across meshes the dip position still walks upward by 6 to 8 percent per halving, the strip artifact plus sub-cell features (0.25 mm vias, 0.5 mm gaps, the dipole 2 to 4 cells above the patches), so treat the absolute frequency as mesh-class dependent in any FDTD, this paper's own included. The starkest evidence for that: at the paper's 0.5 mm cell size, openEMS's geometric rasterizer closes the 1-cell gaps entirely and produces no EBG dip at all, while RayRF's mask-based rasterizer keeps the gaps and still resolves a clean -20.8 dB dip.

S11 curves for the PMC and EBG cases at three cell sizes with solver times, openEMS overlaid at 0.25 mm — The post-fix mesh ladders with solver seconds, openEMS dashed at 0.25 mm. Left: the PMC dip converges cleanly. Right: the EBG dip agrees across solvers at the matched mesh.

Replication found a real bug

This is the part I am happiest about, because it is what validation campaigns are for. The first EBG mesh ladder refused to converge in a way no setup variable could explain. I swept the board-to-boundary air spacing, the PML thickness, the excitation bandwidth and the step caps, and none of them moved the dip. What did explain it was in the solver: the via barrel zeroed the vertical E field through the cell at its end layer as well as the cells below it, which in RayRF's Yee convention extends the metal one cell past the top plate. Every mushroom via carried a spurious PEC stub poking into the air gap above its patch: all of the 0.5 mm gap at 0.5 mm cells, half at 0.25, a quarter at 0.125. Exactly the mesh-dependent loading the ladder showed.

The convention was provably inconsistent with the rest of the code: RayRF's own lumped port integrates the plate-to-plate voltage over the half-open cell range, which pins where a conductor sheet lives on the grid. The fix is the same half-open rule for the via barrel, applied to both the CUDA and CPU solvers. Geometries without vias are bit-identical before and after, the bundled patch example among them. With the fix, the coarse EBG rung moved 6.4 percent and the finer rungs barely moved, confirming the stub was the coarse-mesh distortion. A via stub one cell tall above a patch is irrelevant on a typical PCB. Under an antenna whose entire operating gap is two cells, it was not.

Pitfalls this exercise exposed

Things that made these mistakes easy to make, now on the fix list. The via stub was invisible in every view because the patch metal drew over it. A run that ends on the step cap instead of ringdown only says so in the log, and the EBG needs about a hundred times more steps than a patch, so the first ladder quietly under-resolved its narrow resonances until the caps were lifted. And features smaller than a cell are silently absorbed: a 0.25 mm via on a 0.5 mm mesh becomes a full-cell column in RayRF and disappears into a closed gap in openEMS. The planned remedies are a visible end-on-cap warning in the app, an exporter warning when drawn features fall below the cell size, and a via'd resonator in the regression suite.

What does not line up, and why

The honest residual

Dip positions in this post sit a few percent from their targets, and the error budget is known. The staircased strip dipole runs electrically long by 7.4 percent at 0.25 mm in RayRF and about 2 percent in openEMS, converging together toward the thick-dipole theory value. That offset shifts each solver's curves as a set and cancels in the image-theory identity checks. What it does not do is favor either solver: it is the price of representing a 0.5 mm round wire as staircased cells, in 2003 or now. The EBG dip carries the extra sub-cell sensitivity quantified above, and its truly converged position likely sits above the paper's published 12.8 GHz, which was itself computed at the coarsest rung of this ladder.

Worth restating plainly: the PMC here is a domain wall rather than a finite floating sheet, and a PMC cannot be measured against hardware because no such material exists. Published results, a second independent solver, and analytic theory together are the strongest available test for this feature, which is why this post exists.

Run it yourself

The archive has all seven projects, both solvers' outputs, and the analytic referee:

projects/: the seven .rfsim files. The PEC-sheet, PMC and EBG cases ship with results embedded, so the S-Parameters, Radiation Pattern and Field Viewer tabs have data the moment they open.
results/: RayRF S11 sweeps as CSV with columns freq_hz, s11_db, zin_real, zin_imag, plus the mesh ladder table.
openems/: the driver script that reproduces the openEMS half of this post, and the four openEMS sweeps it produced.
analytic/: image_theory.py and its output. Plain numpy, no RayRF required.

Download the validation kit (zip, 3 MB). Requires RayRF v1.0.55 or newer. The EBG project's five layers are above the Hobby plan's 3-layer cap, so re-running that one takes a trial or Pro license. Open a project, hit Run Simulation, and compare against the CSVs. If you find a discrepancy, I want to know about it.

For how RayRF holds up against physical hardware rather than a paper, the VNA validation page and the measurement write-up cover that side. The matched-mesh speed methodology is the same as the RayRF vs openEMS benchmark.

Frequently asked questions

What is a PMC boundary condition?

A perfect magnetic conductor is the dual of a perfect electric conductor: it reflects with 0 degrees of phase shift instead of 180. A horizontal current sitting just above a PMC is reinforced by its image instead of cancelled, which is why low-profile antennas want PMC-like ground planes. No such material exists in nature. It matters as an idealization, and as the limit that engineered EBG surfaces approximate near their resonance.

Why does a dipole match better over PMC than over PEC?

Image currents. Over a PEC, the image of a horizontal dipole at 1.5 mm height is anti-phase, so it cancels the radiation and the input resistance collapses to about 8 ohms in this setup. The antenna is effectively shorted and reflects almost everything. Over a PMC the image is in-phase, the input impedance rises to about 108 ohms at the dip, and the antenna radiates. It is still mismatched to 50 ohms, which is why the paper and both solvers see only a moderate 7 to 8 dB dip rather than a deep match.

How was openEMS set up for the comparison?

Same coordinates, same uniform cell sizes, same 9 to 21 GHz Gaussian band, same 50 ohm in-plane lumped port across the same 0.5 mm feed gap, PML on the same five faces and a PMC boundary on the bottom face for the ideal-ground case. Both solvers ran until their own energy ringdown criterion was met, with the step cap set far out of reach. The driver script and the openEMS sweeps are in the download so the comparison is reproducible.

Can RayRF simulate the EBG mushroom surface from the paper?

The finite 7x7 mushroom array under the dipole, yes: ground plane, eps 2.20 substrate, 49 patches, 49 vias, and the dipole on top. That run is in this post, cross-checked against openEMS at the same mesh, and in the download. What RayRF does not have is periodic unit-cell boundaries with a plane-wave source, so the paper's separate reflection-phase characterization of the infinite surface was not replicated.

Where do I get the project files?

There is a zip linked in the post with all seven .rfsim projects, the solver S11 sweeps as CSV, the openEMS driver script and its sweeps, the mesh ladder data, and the analytic image-theory script. The projects need RayRF v1.0.55 or newer. The EBG project has 5 layers, which is above the Hobby plan's 3-layer cap, so re-running that one takes a trial or Pro license.

Test it on your own antenna

Per-face PML, PEC and PMC boundaries and in-plane ports are in RayRF v1.0.55. Draw the geometry, set the bottom face to PMC, and run. 30-day free trial, no card required.

Start the free trial

validationPMCopenEMSboundary conditionsantennas