Path Integrals

Path integrals compute voltage and current from electromagnetic field data by integrating electric and magnetic fields along specified paths. They are used to characterize transmission lines and to compute characteristic impedance.

Voltage Path Integrals

Voltage path integrals compute voltage by integrating the electric field along a line path.

V = - integral(E dot dl)

Use AxisAlignedVoltageIntegral for a simple axis-aligned path.

from flex_rf import web
from flex_rf.tidy3d import AxisAlignedVoltageIntegral

voltage_integral = AxisAlignedVoltageIntegral(
    center=(0, 0, 0),
    size=(0, 0, 2.0),
    sign="+",
    extrapolate_to_endpoints=True,
    snap_path_to_grid=True,
)

mode_data = web.run(mode_solver, task_name="mode_solver")
voltage = voltage_integral.compute_voltage(mode_data)

Use Custom2DVoltageIntegral for a path defined by vertices in a 2D plane.

import numpy as np
from flex_rf.tidy3d import Custom2DVoltageIntegral

vertices = np.array([
    [0, 0],
    [0, 1],
    [1, 1],
])

custom_voltage = Custom2DVoltageIntegral(
    axis=2,
    position=0.0,
    vertices=vertices,
)

voltage = custom_voltage.compute_voltage(mode_data)

The sign parameter determines integration direction. Use "+" to integrate toward the positive axis and "-" to integrate toward the negative axis.

Current Path Integrals

Current path integrals compute current using Ampere’s circuital law by integrating the magnetic field around a closed contour.

I = contour_integral(H dot dl)

Use AxisAlignedCurrentIntegral for a rectangular loop.

from flex_rf.tidy3d import AxisAlignedCurrentIntegral

current_integral = AxisAlignedCurrentIntegral(
    center=(0, 0, 0),
    size=(3, 2, 0),
    sign="+",
    snap_contour_to_grid=True,
)

current = current_integral.compute_current(mode_data)

Use Custom2DCurrentIntegral for an arbitrary closed contour. For positive current in the positive axis direction, order vertices counterclockwise when viewed from the positive axis.

import numpy as np
from flex_rf.tidy3d import Custom2DCurrentIntegral

vertices = np.array([
    [0, 0],
    [2, 0],
    [2, 1],
    [0, 1],
    [0, 0],
])

custom_current = Custom2DCurrentIntegral(
    axis=2,
    position=0.0,
    vertices=vertices,
)

current = custom_current.compute_current(mode_data)

Use CompositeCurrentIntegral to combine multiple current paths for geometries such as differential lines.

from flex_rf.tidy3d import AxisAlignedCurrentIntegral, CompositeCurrentIntegral

path_1 = AxisAlignedCurrentIntegral(
    center=(-2, 0, 0),
    size=(1, 1, 0),
    sign="+",
)

path_2 = AxisAlignedCurrentIntegral(
    center=(2, 0, 0),
    size=(1, 1, 0),
    sign="+",
)

composite_current = CompositeCurrentIntegral(
    path_specs=(path_1, path_2),
    sum_spec="sum",
)

current = composite_current.compute_current(mode_data)

sum_spec="sum" adds all currents together. sum_spec="split" keeps phase-separated contributions, which is useful when identifying dominant current directions.

Usage With Impedance Calculator

Path integrals are commonly passed to ImpedanceCalculator to compute characteristic impedance.

from flex_rf.tidy3d import ImpedanceCalculator

z_calculator = ImpedanceCalculator(
    voltage_integral=voltage_integral,
    current_integral=current_integral,
)

z0 = z_calculator.compute_impedance(mode_data)

z0, voltage, current = z_calculator.compute_impedance(
    mode_data,
    return_voltage_and_current=True,
)

Specification Classes

The integral classes above compute directly on field data. For MicrowaveModeSpec configuration, use the corresponding specification classes:

Spec	Use
AxisAlignedVoltageIntegralSpec	Axis-aligned voltage path for mode impedance.
Custom2DVoltageIntegralSpec	Custom 2D voltage path for mode impedance.
AxisAlignedCurrentIntegralSpec	Axis-aligned current contour for mode impedance.
Custom2DCurrentIntegralSpec	Custom 2D current contour for mode impedance.
CompositeCurrentIntegralSpec	Composite current contour for mode impedance.

Best Practices

Integral	Recommendation
Voltage	Follow the electric field path between conductors.
Current	Enclose the current-carrying region.
Grid alignment	Use snap_path_to_grid=True or snap_contour_to_grid=True when available.
Endpoints	Use extrapolate_to_endpoints=True when voltage endpoints lie on conductor boundaries.
Direction	Keep sign consistent with the desired power-flow direction.

Related Pages

Page	Why open it
RF Mode Analysis	Use path integral specs with MicrowaveModeSpec.
Impedance Calculator	Convert path-integral voltage and current into impedance.