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ChristopherMayesChristopherMayes
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Add resistive wall impedance
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‎docs/examples/wakefields/resistive_wall_impedance.ipynb‎

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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "04477050",
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"\n",
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"from pmd_beamphysics.wakefields import FlatResistiveWallImpedance"
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]
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},
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{
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"cell_type": "markdown",
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"id": "0b806816",
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"metadata": {},
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"source": [
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"## Basic Usage\n",
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"\n",
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"Create an impedance object by specifying the geometry and material properties:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "b22ca02e",
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"metadata": {},
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"outputs": [],
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"source": [
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"imp = FlatResistiveWallImpedance(\n",
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" half_gap=4.5e-3, # m (half-gap between plates)\n",
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" conductivity=2.4e7, # S/m (DC conductivity)\n",
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" relaxation_time=8e-15, # s (Drude relaxation time)\n",
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")\n",
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"\n",
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"imp"
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]
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},
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{
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"cell_type": "markdown",
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"id": "e545b33a",
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"metadata": {},
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"source": [
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"The characteristic length scale $s_0$ is computed automatically:\n",
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"\n",
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"$$s_0 = \\left( \\frac{2a^2}{Z_0 \\sigma_0} \\right)^{1/3}$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "d75ad878",
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"metadata": {},
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"outputs": [],
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"source": [
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"print(f\"Characteristic length s₀ = {imp.s0*1e6:.2f} µm\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "418a1551",
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"metadata": {},
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"source": [
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"## Computing Impedance $Z(k)$\n",
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"\n",
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"The longitudinal impedance is computed by integrating the surface impedance kernel over all transverse modes."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "4f8884e5",
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"metadata": {},
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"outputs": [],
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"source": [
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"ks = np.linspace(0, 3e5, 100)\n",
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"Zk = imp.impedance(ks)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "fd741ae8",
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"metadata": {},
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"outputs": [],
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"source": [
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"fig, ax = plt.subplots(figsize=(8, 5))\n",
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"ax.plot(ks * 1e-3, np.real(Zk), label=r\"Re[$Z(k)$]\")\n",
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"ax.plot(ks * 1e-3, np.imag(Zk), label=r\"Im[$Z(k)$]\")\n",
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"ax.set_xlabel(r\"$k$ (1/mm)\")\n",
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"ax.set_ylabel(r\"$Z(k)$ (Ω/m)\")\n",
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"ax.legend()\n",
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"ax.set_title(\"Longitudinal Impedance\")\n",
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"plt.tight_layout()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "40091658",
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"metadata": {},
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"source": [
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"The built-in plot method provides a quick visualization:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "15d074d0",
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"metadata": {},
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"outputs": [],
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"source": [
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"imp.plot_impedance(k_max=5e5, n_points=200)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "74a4b289",
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"metadata": {},
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"source": [
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"## Computing Wakefield $W(z)$\n",
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"\n",
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"The wakefield is computed from the real part of the impedance using a cosine transform:\n",
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"\n",
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"$$W(z) = \\frac{2}{\\pi} \\int_0^{k_{\\max}} \\text{Re}[Z(k)] \\cos(kz) \\, dk$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "6adce166",
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"metadata": {},
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"outputs": [],
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"source": [
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"zs = np.linspace(0, 200e-6, 30)\n",
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"Wz = imp.wakefield(zs, k_max=1e6)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "50a34a22",
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"metadata": {},
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"outputs": [],
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"source": [
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"fig, ax = plt.subplots(figsize=(8, 5))\n",
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"ax.plot(zs * 1e6, Wz * 1e-12)\n",
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"ax.set_xlabel(r\"$z$ (µm)\")\n",
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"ax.set_ylabel(r\"$W(z)$ (V/pC/m)\")\n",
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"ax.set_title(\"Longitudinal Wakefield\")\n",
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"plt.tight_layout()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "cc872afe",
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"metadata": {},
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"source": [
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"The built-in plot method:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "23b4135d",
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"metadata": {},
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"outputs": [],
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"source": [
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"imp.plot_wakefield(z_max=200e-6, n_points=40, k_max=1e6)"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fe70c4f2",
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"metadata": {},
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"source": [
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"## Low-Level Functions\n",
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"\n",
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"The module also provides low-level functions for more control:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "61e275b9",
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"metadata": {},
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"outputs": [],
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"source": [
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"from pmd_beamphysics.wakefields.resistive_wall_impedance import (\n",
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" ac_conductivity,\n",
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" longitudinal_impedance,\n",
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" wakefield_from_impedance,\n",
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")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "6e8572a6",
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"metadata": {},
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"source": [
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"### AC Conductivity\n",
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"\n",
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"The Drude model for frequency-dependent conductivity:\n",
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"\n",
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"$$\\sigma(k) = \\frac{\\sigma_0}{1 - i k c \\tau}$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "05c898c3",
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"metadata": {},
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"outputs": [],
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"source": [
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"sigma0 = 2.4e7 # S/m\n",
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"ctau = 2.4e-6 # m\n",
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"\n",
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"ks = np.linspace(0, 1e6, 200)\n",
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"sigma_ac = ac_conductivity(ks, sigma0, ctau)\n",
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"\n",
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"fig, ax = plt.subplots(figsize=(8, 5))\n",
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"ax.plot(ks * 1e-3, np.real(sigma_ac) / sigma0, label=r\"Re[$\\sigma$]/$\\sigma_0$\")\n",
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"ax.plot(ks * 1e-3, np.imag(sigma_ac) / sigma0, label=r\"Im[$\\sigma$]/$\\sigma_0$\")\n",
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"ax.set_xlabel(r\"$k$ (1/mm)\")\n",
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"ax.set_ylabel(r\"Normalized conductivity\")\n",
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"ax.legend()\n",
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"ax.set_title(\"AC Conductivity (Drude Model)\")\n",
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"plt.tight_layout()"
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]
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},
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{
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"cell_type": "markdown",
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"id": "1573edcb",
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"metadata": {},
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"source": [
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"### Direct Impedance Calculation\n",
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"\n",
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"Use `longitudinal_impedance` directly with explicit parameters:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "c6d40f1c",
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"metadata": {},
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"outputs": [],
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"source": [
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"a = 4.5e-3 # half-gap [m]\n",
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"sigma0 = 2.4e7 # conductivity [S/m]\n",
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"ctau = 2.4e-6 # relaxation distance [m]\n",
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"\n",
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"k_test = 1e5 # 1/m\n",
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"Zk_test = longitudinal_impedance(k_test, a, sigma0, ctau)\n",
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"print(f\"Z({k_test:.0e} 1/m) = {Zk_test:.4f} Ω/m\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "1715a4f6",
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"metadata": {},
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"source": [
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"### Wakefield from Custom Impedance Function\n",
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"\n",
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"The `wakefield_from_impedance` function accepts any callable that returns $Z(k)$:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "3a99efcb",
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"metadata": {},
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"outputs": [],
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"source": [
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"from functools import partial\n",
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"\n",
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"# Create a custom impedance function\n",
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"my_Zk = partial(longitudinal_impedance, a=4.5e-3, sigma0=2.4e7, ctau=2.4e-6)\n",
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"\n",
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"# Compute wakefield at a single point\n",
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"z_test = 50e-6 # m\n",
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"Wz_test = wakefield_from_impedance(z_test, my_Zk, k_max=1e6)\n",
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"print(f\"W({z_test*1e6:.0f} µm) = {Wz_test:.2e} V/C/m = {Wz_test*1e-12:.2f} V/pC/m\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fe65f33a",
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"metadata": {},
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"source": [
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"## Performance Notes\n",
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"\n",
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"The impedance calculation involves nested numerical integration (over k and a transverse integration variable), which can be slow for large arrays. For production use with many particles, consider:\n",
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"\n",
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"1. Pre-computing Z(k) on a grid and interpolating\n",
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"2. Using the `ResistiveWallWakefield` class with pseudomode fits for faster evaluation\n",
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"\n",
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"See the [resistive_wall_wakefield](resistive_wall_wakefield.ipynb) notebook for the pseudomode approach."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "f76779c2",
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"metadata": {},
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"outputs": [],
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"source": [
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"%%timeit -n 1 -r 3\n",
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"# Time a typical impedance calculation\n",
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"ks = np.linspace(0, 3e5, 50)\n",
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"Zk = imp.impedance(ks)"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "pmd_beamphysics-dev",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.13.9"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}

‎mkdocs.yml‎

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- examples/fields/solenoid_modeling.ipynb
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- Wakefields:
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- examples/wakefields/resistive_wall_wakefield.ipynb
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- examples/wakefields/resistive_wall_impedance.ipynb
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- Utilities:
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- examples/units.ipynb
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- examples/labels.ipynb

‎pmd_beamphysics/wakefields/__init__.py‎

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from .resistive_wall import ResistiveWallWakefield
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from .resistive_wall_impedance import (
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FlatResistiveWallImpedance,
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ac_conductivity,
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characteristic_length,
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longitudinal_impedance,
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sinhc,
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surface_impedance,
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wakefield_from_impedance,
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)
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__all__ = ["ResistiveWallWakefield"]
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__all__ = [
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"ResistiveWallWakefield",
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"FlatResistiveWallImpedance",
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"ac_conductivity",
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"characteristic_length",
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"longitudinal_impedance",
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"sinhc",
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"surface_impedance",
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"wakefield_from_impedance",
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]

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