Lumped port tuning: coaxial line

A pre-flight check for a lumped port terminating a coaxial line: fill in the given quantities of your target model, run, and the scoreboard tells you how good the termination is — worst reflection, usable band, phase error. Edit the three knobs and re-run until the numbers meet your spec, then carry the settings over.

The measurement: a waveguide port launches the exact grid mode onto the lumped port under test (|S11| is the termination’s self-reflection), and a reference run of the same line with waveguide ports at both ends provides the grid-exact phase ruler. The page Lumped ports: investigations explains the setup and shows which knob moves which error; every number here is a property of your grid, so re-run whenever cross-section, resolution or band change.

import matplotlib.pyplot as plt
import numpy as np

import magnelio as mio
from magnelio import geo, plots, ports
from magnelio.constants import ETA0

Given quantities

What the target simulation dictates. cell is the cross-section cell size the test mesh may not fall below — copy the size your production mesh will actually have at the port.

r_i = 0.405e-3  # inner conductor radius [m]
r_o = 1.475e-3  # shield (dielectric outer) radius [m]
eps_r = 2.25  # solid polyethylene
f_max = 15e9  # upper band edge [Hz]
cell = 0.5 * r_i  # production cross-section cell size [m]

The knobs

The compromise you will carry into the target simulation:

• gap — end gap between inner conductor and shorted end plate; sets the broadband reflection level.

• gap_position — where the gap starts relative to the reference plane (negative = before it); sets the phase error.

• z0_port — None uses the line impedance of the grid, as measured by the waveguide-port solver; a number (e.g. 50.0) uses that instead. Sets the low-frequency reflection floor.

gap = 0.4 * (r_o - r_i)
gap_position = 0.0
z0_port = None

Derived quantities

L = 5.0 * r_o  # waveguide port to reference plane [m]
z_formula = ETA0 / (2.0 * np.pi * np.sqrt(eps_r)) * np.log(r_o / r_i)
print(f"closed-form line impedance: {z_formula:.2f} Ohm")

closed-form line impedance: 51.67 Ohm

Measurement machinery — a uniform test grid (max = min cell size), the plain line as phase reference, and the candidate termination: inner conductor to the gap start, gap, shorted end plate, lumped port bridging the gap on the axis.

def _model(length, pin_length):
    model = mio.GeometryModel(background="pec")
    dielectric = geo.Cylinder(
        origin=(0.0, 0.0, 0.0),
        radius=r_o,
        height=length,
        axis="z",
        material=mio.Material.from_isotropic(name="polyethylene", epsilon=eps_r),
    )
    inner = geo.Cylinder(
        origin=(0.0, 0.0, 0.0), radius=r_i, height=pin_length, axis="z", material="pec"
    )
    model.add(geo.Difference(dielectric, inner))
    model.add(inner)
    model.add_port(ports.PortWaveguide(name="wg", plane="zmin"))
    return model


def _mesh(model):
    return mio.Mesh.from_geometry(
        model,
        mio.MeshControl(min_cell_size=cell, max_cell_size=cell),
        f_max=f_max,
    )


ref_model = _model(L, pin_length=L)
ref_model.add_port(ports.PortWaveguide(name="far", plane="zmax"))
ref = mio.AnalysisScatteringTD(mesh=_mesh(ref_model), verbose=False).run(excited=[("wg", 0)])

z_pin = L + gap_position
z_end = z_pin + gap
model = _model(z_end, pin_length=z_pin)
mesh = _mesh(model)
z_line = mio.AnalysisScatteringTD(mesh=mesh, verbose=False).solve_ports()["wg"].z_line_num
print(f"line impedance on this grid: {z_line:.2f} Ohm ({100 * (z_line / z_formula - 1):+.1f} %)")

model.add_port(
    ports.PortLumped(
        name="dut",
        start=(0.0, 0.0, z_end),
        end=(0.0, 0.0, z_pin),
        Z0=float(z0_port if z0_port is not None else z_line),
    )
)
result = mio.AnalysisScatteringTD(mesh=mesh, ports=list(model.ports), verbose=False).run(
    excited=[("wg", 0)]
)

  mesh | feature planes
  mesh | grid lines
  mesh | materials
  mesh | conformal cells
  mesh | PEC masks
  mesh | 14 x 14 x 36 cells
  mesh | feature planes
  mesh | grid lines
  mesh | materials
  mesh | conformal cells
  mesh | PEC masks
  mesh | 14 x 14 x 38 cells
line impedance on this grid: 47.31 Ohm (-8.4 %)

The test fixture

A cut along the propagation direction: the waveguide port on the left launches down the uniform line, the inner conductor stops at gap_position relative to the reference plane, and the lumped port under test bridges the gap to the shorted end plate.

fig, ax = plots.plot_cross_section(model, "y", 0.0, flip=True, title="Coax test fixture")
ax.set_xlim(-0.3, z_end * 1e3 + 1.0)

(-0.3, 8.802999999999999)

The scoreboard

The phase error is polarity-normalised: the sign of a mode profile is a convention, so the error is referenced to the nearest multiple of 180° at the low end of the band.

f = np.asarray(result.f_axis)
band = f <= f_max
s11_db = result.db("wg", "wg")
err = result.phase("dut", "wg") - ref.phase("far", "wg")
err -= 180.0 * np.round(err[int(np.argmax(f >= f_max / 15.0))] / 180.0)

good = s11_db[band] < -20.0
f_edge = f[band][np.argmin(good)] if not good.all() else f[band][-1]
print("--- current settings — tune until this meets your spec ---")
print(f"worst |S11| in band : {s11_db[band].max():6.1f} dB")
print(f"|S11| < -20 dB up to: {f_edge / 1e9:6.2f} GHz")
print(f"max |phase error|   : {np.abs(err[band]).max():6.2f} deg")

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(9.0, 3.6))
ax1.plot(f[band] / 1e9, s11_db[band])
ax1.set_xlabel("frequency [GHz]")
ax1.set_ylabel("|S11| [dB]")
ax1.set_title("Self-reflection")
ax1.grid(True, alpha=0.3)
ax2.plot(f[band] / 1e9, err[band])
ax2.set_xlabel("frequency [GHz]")
ax2.set_ylabel("phase error [deg]")
ax2.set_title("Phase error at the reference plane")
ax2.grid(True, alpha=0.3)
fig.tight_layout()

--- current settings — tune until this meets your spec ---
worst |S11| in band :  -19.5 dB
|S11| < -20 dB up to:  14.18 GHz
max |phase error|   :  28.97 deg

The cross-section with the mesh it is actually solved on — check that the gap region resolves the way your production model does.

fig, ax = plots.plot_cross_section(model, "z", L / 2, mesh=mesh, title="Test coax cross-section")

Carry it over

Once the three numbers meet your spec, transfer gap, gap_position (relative to where your reference plane is) and the port impedance into the target model. Which knob moves which number — and why — is shown in Lumped ports: investigations; when the cross-section, resolution or band changes, run this page again.