resistive_wall.py

# -*- coding: utf-8 -*-
"""
Define resistive wall elements based on the Wakefield class.

@author: Alexis Gamelin
@date: 14/01/2020
"""

import numpy as np
from scipy.constants import mu_0, epsilon_0, c
from CollectiveEffect.wakefield import Wakefield, Impedance

def skin_depth(omega, rho, mu_r = 1, epsilon_r = 1):
    """ General formula for the skin depth
    
    Parameters
    ----------
    omega: angular frequency in [Hz]
    rho: resistivity in [ohm.m]
    mu_r : relative magnetic permeability, optional
    epsilon_r : relative electric permittivity, optional
    
    Returns
    -------
    delta : skin depth in [m]
    
    """
    
    delta = np.sqrt(2*rho/(np.abs(omega)*mu_r*mu_0))*np.sqrt(np.sqrt(1 + 
                   (rho*np.abs(omega)*epsilon_r*epsilon_0)**2 ) 
                    + rho*np.abs(omega)*epsilon_r*epsilon_0)
    return delta
    

class CircularResistiveWall(Wakefield):
    """
    Resistive wall Wakefield element for a circular beam pipe, approximated
    formulas from Eq. (2.77) of Chao book [1].
    
    Parameters
    ----------
    frequency: frequency points where the impedance will be evaluated in [Hz]
    length: beam pipe length in [m]
    rho: resistivity in [ohm.m]
    radius: beam pipe radius in [m]
    mu_r : relative magnetic permeability, optional
    epsilon_r : relative electric permittivity, optional
    """

    def __init__(self, frequency, length, rho, radius, mu_r = 1, epsilon_r = 1):
        super().__init__()
        
        omega = 2*np.pi*frequency
        Z1 = length*(1 + np.sign(omega)*1j)*rho/(
                2*np.pi*radius*skin_depth(omega,rho))
        Z2 = c/omega*length*(1 + np.sign(omega)*1j)*rho/(
                np.pi*radius**3*skin_depth(omega,rho))
        
        Zlong = Impedance(variable = frequency, function = Z1, impedance_type='long')
        Zxdip = Impedance(variable = frequency, function = Z2, impedance_type='xdip')
        Zydip = Impedance(variable = frequency, function = Z2, impedance_type='ydip')
        
        super().append_to_model(Zlong)
        super().append_to_model(Zxdip)
        super().append_to_model(Zydip)