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adding code for simulation of magnetic coils based on Joschka's code and magpylib

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Lennart Naeve 2025-05-28 17:33:31 +02:00
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coil_design/2025_04_29 (trap frequencies for rect coils).ipynb Normal file
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coil_design/2025_05_01 (all coils magpy).ipynb Normal file
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# -*- coding: utf-8 -*-
"""
Created on Mon Aug 16 11:51:36 2021
@author: Joschka
"""
import numpy as np
from scipy import special as sp
from src import physical_constants as cs
#HH = +1 --> Helmholtz configuration, HH = -1 --> Anti Helmholtz configuration
def B_field_ideal(N_windings,I_current,R_radius,d_distance,HH,z_arg):
""" Calculate B-field for ideal point like wires with R_radius and d_distance
HH = +1 --> Helmholtz configuration, HH = -1 --> Anti Helmholtz configuration
"""
B = cs.mu_0*N_windings*I_current/2 * R_radius**2 * (1/(R_radius**2+(z_arg-d_distance/2)**2)**(3/2) +HH* 1/(R_radius**2+(z_arg+d_distance/2)**2)**(3/2))
B *= 1e4 #conversion Gauss
return B
#Argument for elliptic integrals
def k_sq(R_loop, z_loc, r, z):
""" Argument for elliptic integrals"""
return (4*R_loop*r)/((R_loop+r)**2 + (z-z_loc)**2)
def B_z_loop(I_current, R_loop, z_loc, r, z):
"""calculate z-component of B-field at position r and z for each individual loop"""
B_z = 2e-7*I_current * 1/( (R_loop+r)**2 + (z-z_loc)**2 )**(1/2) *(sp.ellipk(k_sq(R_loop,z_loc,r,z))+ sp.ellipe(k_sq(R_loop,z_loc,r,z))*(R_loop**2 - r**2 - (z-z_loc)**2)/((R_loop-r)**2 + (z-z_loc)**2))
B_z *= 1e4 #conversion to gauss
return B_z
def B_r_loop(I_current, R_loop, z_loc, r, z):
"""calculate r-component of B-field at position r and z for each individual loop"""
B_r = 2e-7*I_current/r * (z-z_loc)/( (R_loop+r)**2 + (z-z_loc)**2 )**(1/2) *(-sp.ellipk(k_sq(R_loop,z_loc,r,z))+ sp.ellipe(k_sq(R_loop,z_loc,r,z))*(R_loop**2 + r**2 + (z-z_loc)**2)/((R_loop-r)**2 + (z-z_loc)**2))
B_r *= 1e4 #conversion to gauss
return B_r
def B_multiple(I_current, HH, R_inner, distance_coils, layers, windings, wire_width, wire_height, x, z):
"""Returns Bz along z-axis and B_r along r-axis
HH = +1 --> Helmholtz configuration, HH = -1 --> Anti Helmholtz configuration"""
z_start = (distance_coils/2 - windings * wire_height/2 + wire_height/2)*1e-3
R_start = (R_inner + wire_width/2 )*1e-3
if x[0] <=0:
x_neg = np.abs([el for el in x if el<0])
x_pos = np.array([el for el in x if el>0])
x_zero = np.array([el for el in x if el == 0])
B_z = np.zeros(len(z))
B_x_pos = np.zeros(len(x_pos))
B_x_neg = np.zeros(len(x_neg))
B_x_zero = x_zero #If there is a zero in the x-array it is assured that the x component of the field at this point is zero
for xx in range(0,layers):
for zz in range(0,windings):
z_pos = z_start + zz * wire_height*1e-3
R_pos = R_start + xx * wire_width*1e-3
B_z += B_z_loop(I_current,R_pos,z_pos,0,z) + B_z_loop(HH*I_current,R_pos,-z_pos,0,z)
B_x_pos += B_r_loop(I_current,R_pos,z_pos,x_pos,0) + B_r_loop(HH*I_current,R_pos,-z_pos,x_pos,0)
B_x_neg += B_r_loop(I_current,R_pos,z_pos,x_pos,0) + B_r_loop(HH*I_current,R_pos,-z_pos,x_pos,0)
B_x_neg *=-1 #B_x_neg is pointing in opposite x_direction
B_x = np.concatenate((B_x_neg,B_x_zero,B_x_pos))
return B_z,B_x
def B_multiple_raster(I_current, HH, R_inner, distance_coils, layers, windings, wire_width, wire_height, x, z):
"""Returns Bz along z-axis and B_r along r-axis
HH = +1 --> Helmholtz configuration, HH = -1 --> Anti Helmholtz configuration"""
z_start = (distance_coils/2 - windings * wire_height/2 + wire_height/2)*1e-3
R_start = (R_inner + wire_width/2 )*1e-3
if x[0] <=0:
x_neg = np.abs([el for el in x if el<0])
x_pos = np.array([el for el in x if el>0])
x_zero = np.array([el for el in x if el == 0])
B_z = np.zeros(len(z))
B_x_pos = np.zeros(len(x_pos))
B_x_neg = np.zeros(len(x_neg))
B_x_zero = x_zero #If there is a zero in the x-array it is assured that the x component of the field at this point is zero
#dividing each wire into 10 x 10 raster
raster = 10
I_current /=raster**2
for xx in range(0,layers):
for zz in range(0,windings):
for xx_in in range(0,raster):
for zz_in in range(0,raster):
z_pos = z_start + (zz * wire_height - wire_height/2 + zz_in * wire_height/(raster-1) )*1e-3
R_pos = R_start + (xx * wire_width - wire_width/2 + xx_in * wire_width/(raster-1) )*1e-3
B_z += B_z_loop(I_current,R_pos,z_pos,0,z) + B_z_loop(HH*I_current,R_pos,-z_pos,0,z)
B_x_pos += B_r_loop(I_current,R_pos,z_pos,x_pos,0) + B_r_loop(HH*I_current,R_pos,-z_pos,x_pos,0)
B_x_neg += B_r_loop(I_current,R_pos,z_pos,x_pos,0) + B_r_loop(HH*I_current,R_pos,-z_pos,x_pos,0)
B_x_neg *=-1 #B_x_neg is pointing in opposite x_direction
B_x = np.concatenate((B_x_neg,B_x_zero,B_x_pos))
return B_z,B_x
def B_multiple_raster_test(I_current, HH, R_inner, distance_coils, layers, windings, wire_width, wire_height, x, z):
"""Returns Bz along z-axis and B_r along r-axis
HH = +1 --> Helmholtz configuration, HH = -1 --> Anti Helmholtz configuration"""
z_start = (distance_coils/2 - windings * wire_height/2 + wire_height/2)*1e-3
R_start = (R_inner + wire_width/2 )*1e-3
if x[0] <=0:
x_neg = np.abs([el for el in x if el<0])
x_pos = np.array([el for el in x if el>0])
x_zero = np.array([el for el in x if el == 0])
B_z = np.zeros(len(z))
B_x_pos = np.zeros(len(x_pos))
B_x_neg = np.zeros(len(x_neg))
B_x_zero = x_zero #If there is a zero in the x-array it is assured that the x component of the field at this point is zero
#dividing each wire into 10 x 10 raster
raster = 10
I_current /=raster
for xx in range(0,layers):
for zz in range(0,windings):
for xx_in in range(0,1):
for zz_in in range(0,raster):
z_pos = z_start + (zz * wire_height - wire_height/2 + zz_in * wire_height/(raster-1) )*1e-3
R_pos = R_start + (xx * wire_width)*1e-3
B_z += B_z_loop(I_current,R_pos,z_pos,0,z) + B_z_loop(HH*I_current,R_pos,-z_pos,0,z)
B_x_pos += B_r_loop(I_current,R_pos,z_pos,x_pos,0) + B_r_loop(HH*I_current,R_pos,-z_pos,x_pos,0)
B_x_neg += B_r_loop(I_current,R_pos,z_pos,x_pos,0) + B_r_loop(HH*I_current,R_pos,-z_pos,x_pos,0)
B_x_neg *=-1 #B_x_neg is pointing in opposite x_direction
B_x = np.concatenate((B_x_neg,B_x_zero,B_x_pos))
return B_z,B_x

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# -*- coding: utf-8 -*-
"""
Created on Tue Aug 17 15:17:44 2021
@author: Joschka
"""
import numpy as np
import matplotlib.pyplot as plt
import logging as log
from scipy import special as sp
import matplotlib
# matplotlib.use('Qt5Agg')
# matplotlib.use('Agg')
# get_ipython().run_line_magic('matplotlib', 'qt')
# get_ipython().run_line_magic('matplotlib', 'inline')
# import time
from src import physical_constants as cs
# logger = log.getLogger('example')
# log.setLevel(log.info)
log.basicConfig(level=log.WARNING, format='%(message)s')
# TODO: Docstrings for every function
# TODO: Implement conventions
class BCoil:
def __init__(self, HH, distance, radius, layers, windings, wire_width, wire_height, insulation_thickness=0.
, is_round=False, winding_scheme=False, red_N=0):
"""
Creates object that represents a configuration of two coils, with symmetry axis = z
:param HH: HH = 1 --> current in same direction, homogeneous field, HH = -1 --> current in opposite direction, quadrupole field
:param distance: distance between center of coils
:param radius: radius to center of coil
:param layers: number of radial (horizontal) layers
:param windings: number of axial (vertical) layers
:param wire_width: width of conductive wire core
:param wire_height: height of conductive wire core
:param insulation_thickness: thickness of wire insulation (radial increase of thickness not diameter!)
:param is_round: True --> Round wire, False --> rectangular wire
:param winding_scheme: 0: standard, layer on layer, 1: with offset (for round wires primarily), 2: like 1, but alternatingly 8 --> 7 windings per layer
:param red_N: account for lower number of windings, reduces N by red_N
"""
if not is_round:
if winding_scheme == 1 or winding_scheme == 2:
log.warning('Is there a reason you want to wind a not round wire like that?')
if is_round:
if wire_width != wire_height:
log.error('Wire is round but width != height')
raise ValueError("Wire is round but width != height")
self.HH = HH
self.distance = distance * 1e-3
self.radius = radius * 1e-3
self.layers = layers
self.windings = windings
self.wire_width = wire_width * 1e-3
self.wire_height = wire_height * 1e-3
self.insulation_thickness = insulation_thickness * 1e-3
self.is_round = is_round
self.winding_scheme = winding_scheme
self.red_N = red_N
# Standard get/set functions
@property
def get_HH(self):
return self.HH
@property
def get_distance(self):
return self.distance
@property
def get_radius(self):
return self.radius
@property
def get_layers(self):
return self.layers
@property
def get_windings(self):
return self.windings
@property
def get_wire_width(self):
return self.wire_width
@property
def get_wire_height(self):
return self.wire_height
@property
def get_insulation_thickness(self):
return self.insulation_thickness
@property
def get_is_round(self):
return self.is_round
@property
def get_winding_scheme(self):
return self.winding_scheme
def get_zmin(self):
return self.distance / 2 - self.get_coil_height() / 2
def get_zmax(self):
return self.distance / 2 + self.get_coil_height() / 2
def get_R_inner(self):
return self.radius - self.get_coil_width() / 2
def get_R_outer(self):
return self.radius + self.get_coil_width() / 2
def set_R_outer(self, R_outer):
R_outer *= 1e-3
self.radius = R_outer - self.get_coil_width() / 2
def set_R_inner(self, R_inner):
R_inner *= 1e-3
self.radius = R_inner + self.get_coil_width() / 2
def set_d(self, d):
d *= 1e-3
self.distance = d
def set_d_min(self, d_min):
d_min *= 1e-3
self.distance = d_min + self.get_coil_height()
def set_d_max(self, d_max):
d_max *= 1e-3
self.distance = d_max - self.get_coil_height()
def get_wire_area(self):
"""
calculates wire area in m^2
:return: float
"""
if self.is_round:
return np.pi * (self.wire_width / 2) ** 2
return self.wire_width * self.wire_height
def get_N(self):
"""
Calculates number of windings
"""
N = self.layers * self.windings
log.debug(f"N = {N}")
if self.winding_scheme == 2:
N -= self.layers // 2
log.debug(f"N = {N}")
return N-self.red_N
def get_wire_length(self):
"""
:return: Approximate length of wire per coil (m)
"""
return self.get_N() * 2 * self.radius * np.pi
def get_tot_wire_height(self):
""" returns wire height incl. insulation"""
return self.wire_height + 2 * self.insulation_thickness
def get_tot_wire_width(self):
""" returns wire width incl. insulation"""
return self.wire_width + 2 * self.insulation_thickness
def get_coil_height(self):
if self.winding_scheme == 1:
return self.get_tot_wire_height() * (self.windings + 0.5)
return self.get_tot_wire_height() * self.windings
def get_coil_width(self):
if self.winding_scheme == 1 or self.winding_scheme == 2:
if self.is_round:
log.info("Coil width: Be aware of the fact that this is an idealized situation of coil winding (slope "
"of windings changes each layer)")
return self.layers * self.get_tot_wire_width() - (self.layers - 1) * (
2 - np.sqrt(3)) * self.get_tot_wire_width() / 2 # width is reduced due to winding offset
return self.get_tot_wire_width() * self.layers
def get_cross_section(self):
return self.get_coil_height() * self.get_coil_width()
def winding_raster(self):
"""
generates raster of flowing currents
:param raster_value: wire height/raster_value is distance between rastered points in one wire
:return: 2 dim array [[z1,R1], [z2,R2], ...]
"""
outer_raster = np.zeros((self.get_N(), 2))
it = 0
z_start = self.get_zmin() + self.get_tot_wire_height() / 2 # (distance_coils/2 - windings * wire_height/2 + wire_height/2)*1e-3
R_start = self.get_R_inner() + self.get_tot_wire_width() / 2 # (R_inner + wire_width/2 )
log.debug(f"N = {self.get_N()}")
for xx in range(0, self.layers):
for zz in range(0, self.windings):
if self.winding_scheme == 2 and xx % 2 == 1 and zz == self.windings - 1:
continue # leave out every last winding in every second layer
if self.red_N !=0:
if xx == self.layers-1 and zz >= self.windings - self.red_N - 1:
continue
z_pos = z_start + zz * self.get_tot_wire_height()
R_pos = R_start + xx * self.get_tot_wire_width()
# correct for different winding techniques and round wire
if self.winding_scheme == 1 or self.winding_scheme == 2:
if xx % 2 == 1:
z_pos += self.get_tot_wire_height() / 2
if self.is_round:
R_pos -= xx * ((2 - np.sqrt(3)) * self.get_tot_wire_width() / 2)
log.debug(f"lay = {xx}, wind = {zz}")
outer_raster[it] = [z_pos, R_pos]
# print(outer_raster[it])
it += 1
return outer_raster
def inner_raster(self, raster_value, plot_radius_offset=0.):
"""
Gives back inner raster for one wire around 0,0
Args:
raster_value (int): if N produces a N x N raster for rectangular and cut out of this for round
plot_radius_offset: reduce radius of wire for plotting
Returns: array containing raster around 0,0 [[z_pos_in_1,r_pos_in_1],...]
"""
# TODO: Less important, but rastering for round wires could be improved
if raster_value == 0:
raster_value = 1
log.info("raster value is 0 increased to 1")
if raster_value == 1:
return [0, 0]
if self.is_round & (raster_value % 2 == 0):
raster_value += 1
log.info(
f"for round wire rastering works better with uneven rastering value --> rastering value set to: {raster_value}")
inner_raster = np.zeros((raster_value ** 2, 2))
it = 0
for xx_in in range(0, raster_value):
for zz_in in range(0, raster_value):
z_pos_in = - self.wire_height / 2 + zz_in * self.wire_height / (raster_value - 1)
r_pos_in = - self.wire_width / 2 + xx_in * self.wire_width / (raster_value - 1)
inner_raster[it] = [z_pos_in, r_pos_in]
it += 1
# delete points out of round geometry
if self.is_round:
delete_list = []
for i in range(0, len(inner_raster)):
abs = np.sqrt(inner_raster[i, 0] ** 2 + inner_raster[i, 1] ** 2)
if abs > (self.wire_width / 2 - plot_radius_offset):
delete_list.append(i)
inner_raster = np.delete(inner_raster, delete_list, axis=0)
return inner_raster
def full_raster(self, raster_value, plot_radius_offset=0.):
"""
Args:
raster_value: rastering value
plot_radius_offset: reduce radius of wire for plotting
Returns: [ wire 1:[[z_in1,r_in2], [z_in2,r_in,2],...],wire 2:[ [,] , ]...]...]
"""
outer_ras = self.winding_raster()
inner_ras = self.inner_raster(raster_value, plot_radius_offset=plot_radius_offset)
full_ras = np.zeros((len(outer_ras), len(inner_ras), 2))
for i in range(0, len(full_ras)):
full_ras[i] = outer_ras[i] + inner_ras
return full_ras
def plot_raster(self, raster_value=100, plot_radius_offset=0.):
full_structure = self.full_raster(raster_value, plot_radius_offset=plot_radius_offset) * 1e3
if self.get_coil_width() > self.get_coil_height():
extension = self.get_coil_width()
else:
extension = self.get_coil_height()
extension *= 1e3
plt.figure(77, figsize=(5, 5))
plt.scatter(full_structure[:, :, 1], full_structure[:, :, 0], linewidths=0.001, s=.1)
plt.xlabel("radius [mm]")
plt.ylabel("z position [mm]")
plt.axvline(x=self.get_R_inner() * 1e3 - 0.1, lw=5, color="red")
plt.xlim(1e3 * self.get_R_inner() - 0.5, 1e3 * self.get_R_inner() + extension + 0.5)
plt.ylim(1e3 * self.get_zmin() - 0.5, 1e3 * self.get_zmin() + extension + 0.5)
plt.show()
def print_info(self):
print(" ")
# print(f"{self.get_zmin()}")
# print(self.__name__)
print(
f"HH = {self.HH}, Distance = {self.distance * 1e3} mm, z_min = {self.get_zmin() * 1e3:.3f} mm, z_max = {self.get_zmax() * 1e3:.3f} mm")
print(
f"Radius = {self.radius * 1e3:.3f} mm, Radius_inner = {self.get_R_inner() * 1e3:.3f} mm, Radius_outer = {self.get_R_outer() * 1e3:.3f} mm")
print(
f"Coil width = {self.get_coil_width() * 1e3:.3f} mm, Coil height = {self.get_coil_height() * 1e3:.3f} mm"
)
print(
f"layers = {self.layers}, windings = {self.windings}, wire_width = {self.wire_width * 1e3:.3f}, wire_height = {self.wire_height * 1e3:.3f} mm, round wire = {self.is_round} ")
print(" ")
def print_basic_info(self):
if self.HH == 1:
print("Offset coil:")
else:
print("Gradient coil:")
print(
f" layers = {self.layers}, windings = {self.windings} \n "
f" wire: core = {self.wire_height * 1e3:.2f} mm, tot = {self.get_tot_wire_height() * 1e3:.2f} mm, total length for pair = {2 * self.get_wire_length():.0f} m"
)
print(
f" Distance = {self.distance * 1e3:.2f} mm, Radius = {self.radius * 1e3:.2f} mm"
)
print(
f" Coil width = {self.get_coil_width() * 1e3:.2f} mm, Coil height = {self.get_coil_height() * 1e3:.2f} mm"
)
print("")
def B_field_ideal_z(self, I_current, z_arg):
"""
Calculate B-field for ideal point like wires with R_radius and d_distance
HH = +1 --> Helmholtz configuration, HH = -1 --> Anti Helmholtz configuration
z_arg in mm
"""
z_SI = z_arg * 1e-3
N_windings = self.layers * self.windings
B = cs.mu_0 * N_windings * I_current / 2 * self.radius ** 2 * (
1 / (self.radius ** 2 + (z_SI - self.distance / 2) ** 2) ** (3 / 2) + self.HH * 1 / (
self.radius ** 2 + (z_arg + self.distance / 2) ** 2) ** (3 / 2))
B *= 1e4 # conversion Gauss
return B
@staticmethod
def make_axis(lim,precision,two_axis=True):
"""
Gives back two arrays x and z in mm, symmetric around zero
Args:
lim: positive (and negative if lim2 = None) limit of axis
precision: nr_points per mm
two_axis: If True --> gives back two arrays with given parameter
Returns: array
"""
nr_points = (2 * lim) * precision + 1
nr_points = int(nr_points)
z = np.linspace(-lim, lim, nr_points)
if two_axis:
x = np.linspace(-lim, lim, nr_points)
return x, z
return z
@staticmethod
def mm_it(x, x_pos):
"""
Takes argument in mm returns position in x-array
Args:
x: array of interest
x_pos: position to find position of
Returns: int of position of x_pos in array x
"""
precision = len(x)//(2*np.abs(x[0]))
it = int(len(x)//2 + precision * x_pos)
return it
@staticmethod
def k_sq(R_loop, z_loc, r, z):
""" Argument for elliptic integrals"""
return (4 * R_loop * r) / ((R_loop + r) ** 2 + (z - z_loc) ** 2)
@staticmethod
def __B_z_loop(I_current, r_loop, z_loop, r_pos, z_pos):
"""
calculate z-component of B-field at position r and z for each individual loop
Args:
I_current: Current in [A]
r_loop: Radial position of current loop
z_loop: Axial position of current loop along symmetry axis
r_pos: radial position of calculation in [m]
z_pos: axial position of calculation in [m]
Returns: z component of B-field at position r, z in G
"""
B_z = 2e-7 * I_current * 1 / ((r_loop + r_pos) ** 2 + (z_pos - z_loop) ** 2) ** (1 / 2) * (
sp.ellipk(BCoil.k_sq(r_loop, z_loop, r_pos, z_pos)) + sp.ellipe(BCoil.k_sq(r_loop, z_loop, r_pos, z_pos)) * (
r_loop ** 2 - r_pos ** 2 - (z_pos - z_loop) ** 2) / ((r_loop - r_pos) ** 2 + (z_pos - z_loop) ** 2))
B_z *= 1e4 # conversion to gauss
return B_z
@staticmethod
def __B_r_loop(I_current, r_loop, z_loop, r_pos, z_pos):
"""
calculate z-component of B-field at position r and z for each individual loop
Args:
I_current: Current in [A]
r_loop: Radial position of current loop
z_loop: Axial position of current loop along symmetry axis
r_pos: radial position of calculation in [m]
z_pos: axial position of calculation in [m]
Returns: z component of B-field at position r, z in G
"""
B_r = 2e-7 * I_current / r_pos * (z_pos - z_loop) / ((r_loop + r_pos) ** 2 + (z_pos - z_loop) ** 2) ** (1 / 2) * (
-sp.ellipk(BCoil.k_sq(r_loop, z_loop, r_pos, z_pos)) + sp.ellipe(BCoil.k_sq(r_loop, z_loop, r_pos, z_pos)) * (
r_loop ** 2 + r_pos ** 2 + (z_pos - z_loop) ** 2) / ((r_loop - r_pos) ** 2 + (z_pos - z_loop) ** 2))
B_r *= 1e4 # conversion to gauss
return B_r
def B_field_z(self,I_current):
pass
def B_field(self, I_current, x, z, raster = 7):
"""
Returns Bz along z-axis and B_x along x-axis,
HH = +1 --> Helmholtz configuration, HH = -1 --> Anti Helmholtz configuration
"""
x_SI = x * 1e-3
z_SI = z * 1e-3
if x[0] <= 0: # divide array into positive and negative values
x_negative = np.abs([el for el in x_SI if el < 0])
x_positive = np.array([el for el in x_SI if el > 0])
x_zero = np.array([el for el in x_SI if el == 0])
B_z = np.zeros(len(z))
B_x_pos = np.zeros(len(x_positive))
B_x_neg = np.zeros(len(x_negative))
B_x_zero = x_zero # 0 in x-array --> field is zero at this position
calc_raster = self.full_raster(raster)
if self.get_N() != len(calc_raster):
log.error("N is not equal length of raster")
rastering_value = len(calc_raster[0])
I_current /= rastering_value # divide current into smaller currents for mapping the whole wire
# start = time.time()
for wire in range(0, self.get_N()):
for ii in range(0, rastering_value):
# extract position information out of raster
z_pos = calc_raster[wire, ii, 0]
r_pos = calc_raster[wire, ii, 1]
B_z += BCoil.__B_z_loop(I_current, r_pos, z_pos, 0, z_SI) + BCoil.__B_z_loop(self.HH * I_current, r_pos,
-z_pos, 0, z_SI)
B_x_pos += BCoil.__B_r_loop(I_current, r_pos, z_pos, x_positive, 0) + BCoil.__B_r_loop(
self.HH * I_current, r_pos, -z_pos, x_positive, 0)
B_x_neg += BCoil.__B_r_loop(I_current, r_pos, z_pos, x_negative, 0) + BCoil.__B_r_loop(
self.HH * I_current, r_pos, -z_pos, x_negative, 0)
B_x_neg *= -1 # B_x_neg is pointing in opposite x_direction
B_x = np.concatenate((B_x_neg, B_x_zero, B_x_pos))
# end = time.time()
# print(f"time = {end - start} s")
return B_z, B_x
def B_field_asym(self, I_current, x, z, raster = 7, rad_diff=0.1e-3):
"""
Returns Bz along z-axis and B_x along x-axis,
HH = +1 --> Helmholtz configuration, HH = -1 --> Anti Helmholtz configuration
"""
x_SI = x * 1e-3
z_SI = z * 1e-3
if x[0] <= 0: # divide array into positive and negative values
x_negative = np.abs([el for el in x_SI if el < 0])
x_positive = np.array([el for el in x_SI if el > 0])
x_zero = np.array([el for el in x_SI if el == 0])
B_z = np.zeros(len(z))
B_x_pos = np.zeros(len(x_positive))
B_x_neg = np.zeros(len(x_negative))
B_x_zero = x_zero # 0 in x-array --> field is zero at this position
calc_raster = self.full_raster(raster)
if self.get_N() != len(calc_raster):
log.error("N is not equal length of raster")
rastering_value = len(calc_raster[0])
I_current /= rastering_value # divide current into smaller currents for mapping the whole wire
# start = time.time()
for wire in range(0, self.get_N()):
for ii in range(0, rastering_value):
# extract position information out of raster
z_pos = calc_raster[wire, ii, 0]
r_pos = calc_raster[wire, ii, 1]
B_z += BCoil.__B_z_loop(I_current, r_pos, z_pos, 0, z_SI) + BCoil.__B_z_loop(self.HH * I_current, r_pos-rad_diff,
-z_pos, 0, z_SI)
B_x_pos += BCoil.__B_r_loop(I_current, r_pos, z_pos, x_positive, 0) + BCoil.__B_r_loop(
self.HH * I_current, r_pos-rad_diff, -z_pos, x_positive, 0)
B_x_neg += BCoil.__B_r_loop(I_current, r_pos, z_pos, x_negative, 0) + BCoil.__B_r_loop(
self.HH * I_current, r_pos-rad_diff, -z_pos, x_negative, 0)
B_x_neg *= -1 # B_x_neg is pointing in opposite x_direction
B_x = np.concatenate((B_x_neg, B_x_zero, B_x_pos))
# end = time.time()
# print(f"time = {end - start} s")
return B_z, B_x
def B_quality(self,z_pos=1, raster = 7):
z = np.array([0, z_pos])
Bz, Bx = self.B_tot_along_axis(1, z, z, raster=raster)
print(f" Diff B +/- {z_pos} mm, relative: {(Bz[1] - Bz[0]) / Bz[0]}")
def Grad_quality(self, z_pos=1, I=1, raster = 7):
"""Does not work!!!"""
z = np.arange(-1, z_pos + 1, 0.001)
x = np.array([0])
z_0 = 1000
z_pos_ind = z_0 + 1000 * z_pos
print(z[z_0])
print(z[z_pos_ind])
B_z_tot, B_x_tot = self.B_field(I, x, z, raster=raster)
print(B_z_tot[z_0])
Bz_grad = BCoil.grad(B_z_tot, z)
print(Bz_grad[z_0])
print(f"rel Diff Grad B +/- {z_pos} mm relative: {(Bz_grad[z_pos_ind] - Bz_grad[z_0]) / Bz_grad[z_0]}")
def max_field(self, I_current, raster = 7):
B_z_0 = 0
calc_raster = self.full_raster(raster)
if self.get_N() != len(calc_raster):
log.error("N is not equal length of raster")
rastering_value = len(calc_raster[0])
I_current /= rastering_value # divide current into smaller currents for mapping the whole wire
for wire in range(0, self.get_N()):
for ii in range(0, rastering_value):
# extract position information out of raster
z_pos = calc_raster[wire, ii, 0]
r_pos = calc_raster[wire, ii, 1]
B_z_0 += BCoil.__B_z_loop(I_current, r_pos, z_pos, 0, 0) + BCoil.__B_z_loop(self.HH * I_current, r_pos,
-z_pos, 0, 0)
print(f" Max Field = {B_z_0:.2f} G")
return B_z_0
def max_field_one_coil(self, I_current, raster = 7):
"""
Returns and prints max (z-) field in Gauss at center of one coil of configuration pair if second one is not powered
Args:
I_current: Current in [A]
raster: rastering value for each wire (increasing increases accuracy), 7 is fine for most cases
Returns:
Max B-field at center position of coil
"""
B_z_0 = 0
calc_raster = self.full_raster(raster)
if self.get_N() != len(calc_raster):
log.error("N is not equal length of raster")
rastering_value = len(calc_raster[0])
I_current /= rastering_value # divide current into smaller currents for mapping the whole wire
for wire in range(0, self.get_N()):
for ii in range(0, rastering_value):
# extract position information out of raster
z_pos = calc_raster[wire, ii, 0]
r_pos = calc_raster[wire, ii, 1]
B_z_0 += BCoil.__B_z_loop(I_current, r_pos, z_pos, 0, z_pos)
print(f" Max Field = {B_z_0:.2f} G")
return B_z_0
def max_gradient(self, i_current, raster = 7):
x, z = BCoil.make_axis(0.5, 1000)
Bz, Bx = self.B_field(i_current, x, z, raster = raster)
bz_grad = BCoil.grad(Bz, z)
bx_grad = BCoil.grad(Bx, x)
print(f" Max z Gradient = {bz_grad[len(z)//2]:.2f} G/cm")
print(f" Max x Gradient = {bx_grad[len(x)//2]:.2f} G/cm")
# return bx_grad[len(x)//2]
def B_tot_along_axis(self, I_current, x, z, raster=7):
"""
return B_tot_z, B_tot_x
Returns B_tot_z along z-axis and B_tot_x along x-axis,
HH = +1 --> Helmholtz configuration, HH = -1 --> Anti Helmholtz configuration
"""
# Convert axis to SI units
x_SI = x * 1e-3
z_SI = z * 1e-3
if x[0] <= 0:
x_neg = np.abs([el for el in x_SI if el < 0])
x_pos = np.array([el for el in x_SI if el > 0])
x_zero = np.array([el for el in x_SI if el == 0])
B_tot_z = np.zeros(len(z))
B_x_pos = np.zeros(len(x_pos))
B_x_neg = np.zeros(len(x_neg))
B_x_zero = x_zero # If there is a zero in the x-array it is assured that the x component of the field at this point is zero
B_z_x = np.zeros(len(x_SI))
# dividing each wire into 10 x 10 raster
calc_raster = self.full_raster(raster)
if self.get_N() != len(calc_raster):
log.error("N is not equal length of raster")
rastering_value = len(calc_raster[0])
I_current /= rastering_value # divide current into smaller currents for mapping the whole wire
# start = time.time()
for wire in range(0, self.get_N()):
for ii in range(0, rastering_value):
# extract position information out of raster
z_pos = calc_raster[wire, ii, 0]
r_pos = calc_raster[wire, ii, 1]
# z-field along z-axis (x-Field always zero)
B_tot_z += BCoil.__B_z_loop(I_current, r_pos, z_pos, 0, z_SI) + BCoil.__B_z_loop(self.HH * I_current,
r_pos, -z_pos, 0, z_SI)
# x-field along x-axis
B_x_pos += BCoil.__B_r_loop(I_current, r_pos, z_pos, x_pos, 0) + BCoil.__B_r_loop(self.HH * I_current,
r_pos, -z_pos, x_pos,
0)
B_x_neg += BCoil.__B_r_loop(I_current, r_pos, z_pos, x_neg, 0) + BCoil.__B_r_loop(self.HH * I_current,
r_pos, -z_pos, x_neg,
0)
# Bz along x-axis:
B_z_x += BCoil.__B_z_loop(I_current, r_pos, z_pos, x_SI, 0) + BCoil.__B_z_loop(self.HH * I_current,
r_pos, -z_pos, x_SI, 0)
B_x_neg *= -1 # B_x_neg is pointing in opposite x_direction
# B_x_x = np.zeros(len(x_SI))
B_x_x = np.concatenate((B_x_neg, B_x_zero, B_x_pos))
B_tot_x = np.sqrt(B_x_x ** 2 + B_z_x ** 2)
B_tot_z = np.abs(B_tot_z)
return B_tot_z, B_tot_x
def B_multiple_3d(self, I_current, x, z, raster=4):
z_start = self.get_zmin() + self.wire_height / 2
R_start = self.get_R_inner() + self.wire_width / 2
x_SI = x * 1e-3
z_SI = z * 1e-3
B = np.zeros([len(z_SI), len(x_SI), 2])
calc_raster = self.full_raster(raster)
if self.get_N() != len(calc_raster):
log.error("N is not equal length of raster")
rastering_value = len(calc_raster[0])
# TODO: why division by zero?
I_current /= rastering_value # divide current into smaller currents for mapping the whole wire
# start = time.time()
for el in range(0, len(x_SI)):
for wire in range(0, self.get_N()):
for ii in range(0, rastering_value):
# extract position information out of raster
z_pos = calc_raster[wire, ii, 0]
r_pos = calc_raster[wire, ii, 1]
# compute z-value of field
B[:, el, 1] += BCoil.__B_z_loop(I_current, r_pos, z_pos, np.abs(x_SI[el]), z_SI) + BCoil.__B_z_loop(
self.HH * I_current, r_pos, -z_pos, np.abs(x_SI[el]), z_SI)
# compute x-value
if x_SI[el] < 0:
B[:, el, 0] -= BCoil.__B_r_loop(I_current, r_pos, z_pos, -x_SI[el], z_SI) + BCoil.__B_r_loop(
self.HH * I_current, r_pos, -z_pos, -x_SI[el], z_SI)
elif x_SI[el] > 0:
B[:, el, 0] += BCoil.__B_r_loop(I_current, r_pos, z_pos, x_SI[el], z_SI) + BCoil.__B_r_loop(
self.HH * I_current, r_pos, -z_pos, x_SI[el], z_SI)
elif x_SI[el] == 0:
B[:, el, 0] = 0
return B
@staticmethod
def B_tot_3d(B):
return np.sqrt(B[:, :, 0] ** 2 + B[:, :, 1] ** 2)
def plot_3d(self, I_current, x_lim, z_lim):
x = np.arange(-x_lim, x_lim, 5)
print(x)
z = np.arange(-z_lim, z_lim, 5)
print(z)
x_m, z_m = np.meshgrid(x, z)
if self.is_round:
B = self.B_multiple_3d(I_current, x, z, raster=3)
else:
B = self.B_multiple_3d(I_current, x, z, raster=2)
B_tot = BCoil.B_tot_3d(B)
for xx in range(0, len(x)):
for zz in range(0, len(z)):
if B_tot[zz, xx] > 15:
B[zz, xx, :] /= B_tot[zz, xx] / 15
plt.figure()
plt.quiver(x_m, z_m, B[:, :, 0], B[:, :, 1])
plt.xlabel("x-axis [mm]")
plt.ylabel("z-axis [mm]")
plt.show()
@staticmethod
def curv(B_f, z):
return np.gradient(np.gradient(B_f, z), z) * 1e2
@staticmethod
def grad(B_f, z):
return np.gradient(B_f, z) * 1e1
def B_quick_plot(self, I_current, abs=True, x_lim=50, z_lim=50, nr_points=200):
x = np.linspace(-x_lim, x_lim, nr_points)
z = np.linspace(-z_lim, z_lim, nr_points)
if abs:
B_z, B_x = self.B_tot_along_axis(I_current, x, z)
else:
B_z, B_x = self.B_field(I_current, x, z)
plt.figure(11)
plt.plot(z, B_z, linestyle="solid", label=r"$B_{tot}$ along z-axis")
plt.plot(x, B_x, label=r"$B_{tot}$ along x-axis")
plt.title("B-field")
plt.ylabel(r"B-field [G]")
plt.xlabel("x-axis / z-axis [mm]")
plt.legend()
plt.show()
def B_grad_quick_plot(self, I_current, x_lim=50, z_lim=50, nr_points=200):
x = np.linspace(-x_lim, x_lim, nr_points)
z = np.linspace(-z_lim, z_lim, nr_points)
B_z, B_x = self.B_field(I_current, x, z)
# TODO: Think about Gradient calculation of tot B_field
B_z = BCoil.grad(B_z, z)
B_x = BCoil.grad(B_x, x)
plt.figure(12)
plt.plot(z, B_z, linestyle="solid", label=r"z grad of Bz along z-axis")
plt.plot(x, B_x, label=r"x Grad of B_x along x-axis")
plt.title("Gradient of B-field")
plt.ylabel(r"B-field [G/cm]")
plt.xlabel("x-axis / z-axis [mm]")
plt.legend()
plt.show()
def B_curv_quick_plot(self, I_current, x_lim=50, z_lim=50, nr_points=200):
x = np.linspace(-x_lim, x_lim, nr_points)
z = np.linspace(-z_lim, z_lim, nr_points)
B_z, B_x = self.B_field(I_current, x, z)
B_z = BCoil.curv(B_z, z)
B_x = BCoil.curv(B_x, x)
plt.figure(13)
plt.plot(z, B_z, linestyle="solid", label=r"$B_{tot}$ along z-axis")
plt.plot(x, B_x, label=r"$B_{tot}$ along x-axis")
plt.title("Curvature of B-field")
plt.ylabel(r"Curv. B-field [G/cm$^2$]")
plt.xlabel("x-axis / z-axis [mm]")
plt.legend()
plt.show()
def Bz_plot_HH_comp(self, Coil2, I_current, x, z):
B_z, B_x = self.B_field(I_current, x, z)
B_z_2, B_x_2 = Coil2.B_field(I_current, x, z)
B_z_curvature = np.gradient(np.gradient(B_z, z), z) * 1e2
B_z_curvature_2 = np.gradient(np.gradient(B_z_2, z), z) * 1e2
plt.figure(100, figsize=(13, 10))
# plt.rcParams.update({'font.size': 15})
plt.suptitle("Helmholtz coil field Bz along z-axis")
# Field plot
##########################
plt.subplot(2, 1, 1)
plt.plot(z, B_z, linestyle="solid", label=r"$Bz$")
plt.plot(z, B_z_2, linestyle="solid", label=r"$B_{z2}$")
# plt.xlim(-0.01,0.01)
plt.title("B-field")
plt.ylabel(r"$Bz$ [G]")
plt.xlabel("z-axis [mm]")
plt.legend()
plt.subplot(2, 1, 2)
plt.plot(z, B_z_curvature, linestyle="solid", label=r"$\nabla_z^2 Bz$")
plt.plot(z, B_z_curvature_2, linestyle="solid", label=r"$\nabla_z^2 B_{z2}$")
plt.ylabel(r"$\nabla_z^2 Bz [G/cm^2]$")
plt.xlabel("z-axis [mm]")
plt.xlim(-10, 10)
plt.title("Curvature of B-field")
plt.legend(loc='lower right')
plt.show()
def cooling(self, I_current, T):
"""
Print current density and power for one coil
Parameters
----------
I_current : current in A
T : temperature in degree Celsius
Returns
-------
None.
"""
j_dens = I_current / self.get_wire_area() * 1e-6
Power = self.power(I_current, T)
Voltage = self.resistance(T) * I_current
#print("")
print("Cooling:")
print(f" Current I = {I_current} A")
print(f" current density = {j_dens:.2f} A/mm^2")
print(f" Power = {Power:.2f} W")
print(f" U = {Voltage:.2f} V")
def power(self, I_current, T):
"""
Power for one coil
"""
P = self.resistance(T) * I_current ** 2
return P
def inductivity(self):
return cs.mu_0 * (self.layers * self.windings) ** 2 * self.radius ** 2 * np.pi / self.get_coil_height()
def induct_perry(self):
"""
Calculates inductance of one rectangular coil via empirical formular by perry in μH
"""
L = 4 * np.pi * (self.windings * self.layers) ** 2 * (1e2 * self.radius) ** 2 / (
0.2317 * 100 * self.radius + 0.44 * 100 * self.get_coil_height() + 0.39 * 100 * self.get_coil_width())
return L * 1e-9
@staticmethod
def resistivity_copper(T):
"""
For annealed copper
Args:
T: at temperature T
Returns: resistivity in Ohm * meter
"""
R_ref = cs.rho_copper_20
T_ref = 20 # degree celsius
alpha = 0.00393
R = R_ref * (1 + alpha * (T - T_ref))
return R
def resistance(self, tempr):
"""
Calculates resistance of one coil of configuration
Parameters
----------
tempr : double
temperature in degree Celsius
Returns
-------
double
resistance in Ohm
"""
return BCoil.resistivity_copper(tempr) * self.get_wire_length() / self.get_wire_area()
def tau(self, tempr = 22):
return self.induct_perry()/self.resistance(tempr)
def main():
AHH_Coil = BCoil(HH=-1, distance=54, radius=48, layers=8, windings=16,
wire_height=0.5, wire_width=0.5, insulation_thickness=(0.546 - 0.5) / 2,
is_round=True, winding_scheme=2, red_N=1)
AHH_Coil.B_grad_quick_plot(1)
AHH_Coil.Grad_quality()
# main()
if __name__ == "__main__":
log.info("Run main Coil class")
main()

13
coil_design/src/coil_class_add_functions.py Normal file
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# -*- coding: utf-8 -*-
"""
Created on Wed Sep 15 12:01:05 2021
@author: Joschka
"""
import numpy as np
from src import physical_constants as cs
print(resistivity_copper(30))

197
coil_design/src/coil_helpers.py Normal file
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import matplotlib.pyplot as plt
import numpy as np
from src import coil_class as BC
import magpylib as magpy
import matplotlib.ticker as ticker
def circular_HH(HH, radius, distance, layers, windings,
I = 1, #A
wire_height = .5 *1e-3,
wire_width = .5 *1e-3,
insulation_thickness = (0.568-0.5)/2 *1e-3,
print_info = True):
"""returns magpylib system for a circular HH-coil (parameters in m) and prints info on it if requested"""
#### give out coil parameter s####
HH_Coil = BC.BCoil(HH = HH, distance = distance*1e3, radius = radius*1e3, layers = layers, windings = windings,
wire_height = wire_height*1e3, wire_width = wire_width*1e3, insulation_thickness = insulation_thickness*1e3,
is_round = True, winding_scheme= 2)
if print_info:
print("Single coil parameters:")
HH_Coil.print_info()
print(f"N = {HH_Coil.get_N()}")
print(f"wire length = {HH_Coil.get_wire_length()} m")
print(f"R = {HH_Coil.resistance(25)} Ohm")
print(f"L = {HH_Coil.induct_perry()*1e3} mH")
print(f"tau = {HH_Coil.tau() *1e3} ms")
HH_Coil.plot_raster()
print("Pair in series:")
print(f"wire length = {2*HH_Coil.get_wire_length()} m")
print(f"R = {2*HH_Coil.resistance(25)} Ohm")
print(f"L = {2*HH_Coil.induct_perry()*1e3} mH")
print(f"tau = {HH_Coil.tau() *1e3} ms")
print(f"\nat {I} A:")
print(f"V = {2*HH_Coil.resistance(25)*I} V")
print(f"P = {2*HH_Coil.resistance(25)*I**2} W")
#### calculate field properties ####
#define coils
winding_offsets = HH_Coil.full_raster(raster_value=1)[:,0]
coil1 = magpy.Collection()
for coord in winding_offsets:
winding = magpy.current.Circle(
current=1,
diameter=2*coord[1],
position=(0,0,coord[0]),
)
coil1.add(winding)
coil2 = magpy.Collection()
for coord in winding_offsets:
winding = magpy.current.Circle(
current=HH*1,
diameter=2*coord[1],
position=(0,0,-coord[0]),
)
coil2.add(winding)
#move to test if slight changes in positioning have a large effect
#coil1.move([0.5e-3,0,0])
# Combine them into a collection (like multiple windings)
system = magpy.Collection(coil1, coil2)
return system, 2*HH_Coil.get_wire_length()
def rect_current(current, a,b,position, corner_radius):
"""returns rectangular mapy current with given dimensions (in m)"""
#first quadrant
angles = np.linspace(0,np.pi/2,50)
radii = np.zeros((len(angles),3))
radii[:,0] = np.cos(angles)
radii[:,1] = np.sin(angles)
radii *= corner_radius
vertices = np.array([a/2-corner_radius, b/2-corner_radius, 0]) + radii
#second quadrant
angles = np.linspace(np.pi/2,np.pi,50)
radii = np.zeros((len(angles),3))
radii[:,0] = np.cos(angles)
radii[:,1] = np.sin(angles)
radii *= corner_radius
vertices = np.append(vertices,np.array([-(a/2-corner_radius), b/2-corner_radius, 0]) + radii, axis=0)
#third quadrant
angles = np.linspace(np.pi,1.5*np.pi,50)
radii = np.zeros((len(angles),3))
radii[:,0] = np.cos(angles)
radii[:,1] = np.sin(angles)
radii *= corner_radius
vertices = np.append(vertices,np.array([-(a/2-corner_radius), -(b/2-corner_radius), 0]) + radii, axis=0)
#fourth quadrant
angles = np.linspace(1.5*np.pi, 2*np.pi,50)
radii = np.zeros((len(angles),3))
radii[:,0] = np.cos(angles)
radii[:,1] = np.sin(angles)
radii *= corner_radius
vertices = np.append(vertices,np.array([a/2-corner_radius, -(b/2-corner_radius), 0]) + radii, axis=0)
#close loop
vertices = np.append(vertices, [vertices[0]], axis=0)
# Create a rectangular current coil
coil1 = magpy.current.Polyline(position=position, vertices=vertices, current=current)
return coil1
def rectangular_HH(HH, a, b, distance, corner_radius, layers, windings,
I = 1, #A
wire_height = .5 *1e-3,
wire_width = .5 *1e-3,
insulation_thickness = (0.568-0.5)/2 *1e-3,
print_info = True):
"""returns magpylib system for a rectangular HH-coil (parameters in m) and prints info on it if requested"""
#### give out coil parameter s####
HH_Coil = BC.BCoil(HH = HH, distance = 0, radius = 0, layers = layers, windings = windings,
wire_height = wire_height*1e3, wire_width = wire_width*1e3, insulation_thickness = insulation_thickness*1e3,
is_round = True, winding_scheme= 2)
if print_info:
N = HH_Coil.get_N()
print("Single coil parameters: \n")
#HH_Coil.print_info()
print(f"coil width = {HH_Coil.get_coil_width()*1e3} mm")
print(f"coil height = {HH_Coil.get_coil_height()*1e3} mm")
print(f"N = {HH_Coil.get_N()}")
wire_length = N*(2*a - 4*corner_radius + 2*b - 4*corner_radius + 2*np.pi*corner_radius)
print(f"wire length = {wire_length} m")
R = HH_Coil.resistivity_copper(25) * wire_length / HH_Coil.get_wire_area()
print(f"R = {R} Ohm")
#use abbotts formula to get upper bound on inductivity
L = 1e-5*max(a,b)**2 * N**2 /(3*max(a,b) +
9*HH_Coil.get_coil_height() +
10*HH_Coil.get_coil_width())
print(f"L = {L*1e3} mH")
tau = L/R
print(f"tau = {tau *1e3} ms")
HH_Coil.plot_raster()
print("Pair in series:")
print(f"wire length = {2*wire_length} m")
print(f"R = {2*R} Ohm")
print(f"L = {2*L*1e3} mH")
print(f"tau = {tau *1e3} ms")
print(f"\nat {I} A:")
print(f"V = {2*R*I} V")
print(f"P = {2*R*I**2} W")
#### calculate field properties ####
#define coils
winding_offsets = HH_Coil.full_raster(raster_value=1)[:,0]
coil1 = magpy.Collection()
for coord in winding_offsets:
winding = rect_current(
current=1,
a=a+coord[1], b=b+coord[1],
position=[0,0,distance/2+coord[0]],
corner_radius=corner_radius+coord[1]/2
)
coil1.add(winding)
coil2 = magpy.Collection()
for coord in winding_offsets:
winding = rect_current(
current=HH*1,
a=a+coord[1], b=b+coord[1],
position=[0,0,-(distance/2+coord[0])],
corner_radius=corner_radius+coord[1]/2
)
coil2.add(winding)
#move to test if slight changes in positioning have a large effect
#coil1.move([0.5e-3,0,0])
# Combine them into a collection (like multiple windings)
system = magpy.Collection(coil1, coil2)
return system, 2*wire_length

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coil_design/src/physical_constants.py Normal file
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# -*- coding: utf-8 -*-
"""
Created on Mon Aug 16 11:54:40 2021
@author: Joschka
"""
import numpy as np
#once and forever
mu_0 = 1.25663706212e-6
mu_B = 9.2740100783e-24
h = 6.62607015e-34
h_bar = h/(2*np.pi)
k_B = 1.38064852e-23
sigma_B = 5.670374419e-8
#Dysprosium
m_Dy_164 = 2.72210785e-25
Gamma_626 = 8.5e5
f_626 = 478.839e12
lambda_626 = 626.082e-9
#specific constants
rho_copper_20 = 1.72e-8
density_copper = 8.940e3
# water
water_dens = 999.78 # kg/m^3
water_c_p = 4190 # J/kg*K