def setUp(self):
self.general_params = Ring(np.ones(self.n_sections) * self.C/self.n_sections,
np.tile(self.momentum_compaction,
(1, self.n_sections)).T,
np.tile(self.sync_momentum,
(self.n_sections, self.n_turns+1)),
self.particle_type, self.n_turns, n_sections=self.n_sections)
self.RF_sct_par = []
self.RF_sct_par_cpp = []
for i in np.arange(self.n_sections)+1:
self.RF_sct_par.append(RFStation(self.general_params,
[self.harmonic_numbers], [
self.voltage_program/self.n_sections],
[self.phi_offset], self.n_rf_systems, section_index=i))
self.RF_sct_par_cpp.append(RFStation(self.general_params,
[self.harmonic_numbers], [
self.voltage_program/self.n_sections],
[self.phi_offset], self.n_rf_systems, section_index=i))
# DEFINE BEAM------------------------------------------------------------------
self.beam = Beam(self.general_params,
self.n_macroparticles, self.n_particles)
self.beam_cpp = Beam(self.general_params,
self.n_macroparticles, self.n_particles)
# DEFINE SLICES----------------------------------------------------------------
number_slices = 500
cut_options = CutOptions(
cut_left=0., cut_right=self.bucket_length, n_slices=number_slices)
self.slice_beam = Profile(self.beam, CutOptions=cut_options)
self.slice_beam_cpp = Profile(self.beam_cpp, CutOptions=cut_options)
# DEFINE TRACKER---------------------------------------------------------------
self.longitudinal_tracker = []
self.longitudinal_tracker_cpp = []
for i in range(self.n_sections):
self.longitudinal_tracker.append(RingAndRFTracker(
self.RF_sct_par[i], self.beam, Profile=self.slice_beam))
self.longitudinal_tracker_cpp.append(RingAndRFTracker(
self.RF_sct_par_cpp[i], self.beam_cpp, Profile=self.slice_beam_cpp))
full_tracker = FullRingAndRF(self.longitudinal_tracker)
full_tracker_cpp = FullRingAndRF(self.longitudinal_tracker_cpp)
# BEAM GENERATION--------------------------------------------------------------
matched_from_distribution_function(self.beam, full_tracker, emittance=self.emittance,
distribution_type=self.distribution_type,
distribution_variable=self.distribution_variable, seed=1000)
matched_from_distribution_function(self.beam_cpp, full_tracker_cpp, emittance=self.emittance,
distribution_type=self.distribution_type,
distribution_variable=self.distribution_variable, seed=1000)
self.slice_beam.track()
self.slice_beam_cpp.track()
def test_losses_energy_cut(self):
longitudinal_tracker = RingAndRFTracker(self.rf_params, self.beam)
full_tracker = FullRingAndRF([longitudinal_tracker])
try:
matched_from_distribution_function(self.beam,
full_tracker,
distribution_exponent=1.5,
distribution_type='binomial',
bunch_length=1.65e-9,
bunch_length_fit='fwhm',
distribution_variable='Hamiltonian')
except TypeError as te:
self.skipTest("Skipped because of known bug in deepcopy. Exception message %s"
% str(te))
self.beam.losses_energy_cut(-3e8, 3e8)
self.assertEqual(len(self.beam.id[self.beam.id == 0]), 0,
msg='Beam: Failed losses_energy_cut, first')
self.beam.dE += 10e8
self.beam.losses_energy_cut(-3e8, 3e8)
self.assertEqual(len(self.beam.id[self.beam.id == 0]),
self.beam.n_macroparticles,
msg='Beam: Failed losses_energy_cut, second')
def setUp(self):
initial_time = 0
final_time = 1E-3
# Machine and RF parameters
radius = 25
gamma_transition = 4.4 # [1]
C = 2 * np.pi * radius # [m]
momentum_compaction = 1 / gamma_transition**2 # [1]
particle_type = 'proton'
self.ring = Ring(C, momentum_compaction, \
([0, 1E-3], [3.13E8, 3.13E8]), Proton())
self.rf_params = RFStation(self.ring, [1], [1E3], [np.pi], 1)
self.beam = Beam(self.ring, 1, 0)
self.profile = Profile(self.beam)
self.long_tracker = RingAndRFTracker(self.rf_params, self.beam)
self.full_ring = FullRingAndRF([self.long_tracker])
self.n_turns = self.ring.n_turns
self.map_ = [self.full_ring.track, self.profile.track]
self.trackIt = TrackIteration(self.map_)
def test_SPS_RL(self):
PL_gain = 1000 # gain of phase loop
rtol = 1e-4 # relative tolerance
atol = 0 # absolute tolerance
# Phase loop setup
phase_loop = BeamFeedback(self.ring, self.rf_station, self.profile,
{'machine': 'SPS_RL', 'PL_gain': PL_gain})
# Tracker setup
section_tracker = RingAndRFTracker(
self.rf_station, self.beam, Profile=self.profile,
BeamFeedback=phase_loop, interpolation=False)
tracker = FullRingAndRF([section_tracker])
# average beam position
beamAvgPos = np.zeros(self.ring.n_turns)
n_turns = self.ring.n_turns
for turn in range(n_turns):
beamAvgPos[turn] = np.mean(self.beam.dt)
self.profile.track()
tracker.track()
# difference between beam position and synchronuous position
# (assuming no beam loading)
delta_tau = beamAvgPos - (np.pi - self.rf_station.phi_rf[0, :-1])\
/ self.rf_station.omega_rf[0, :-1]
# initial position for analytic solution
init_pos = self.time_offset
omega_eff = cmath.sqrt(-PL_gain**2 + 4*self.rf_station.omega_s0[0]**2)
time = np.arange(n_turns) * self.ring.t_rev[0]
# initial derivative for analytic solution;
# defined such that analytical solution at turn 1 agrees with numerical
# solution
init_slope = 0.5 * (delta_tau[1] * omega_eff * np.exp(0.5*PL_gain*time[1])
/ np.sin(0.5*omega_eff*time[1]) - delta_tau[0]
* (PL_gain+omega_eff/np.tan(0.5*omega_eff*time[1]))).real
delta_tau_analytic = init_pos * np.exp(-0.5*PL_gain*time)
delta_tau_analytic *= np.cos(0.5*time*omega_eff).real\
+ (PL_gain+2*init_slope/init_pos)\
* (np.sin(0.5*time*omega_eff)/omega_eff).real
difference = delta_tau - delta_tau_analytic
# normalize result
difference = difference / np.max(difference)
# expected difference
difference_exp = np.array([
-1.56306635e-05, -1.55605315e-05, -2.10224435e-05, -3.18525050e-05,
-4.74014489e-05, -6.70584402e-05, -9.01307422e-05, -1.15823959e-04,
-1.43290487e-04, -1.71572162e-04, -1.99820151e-04, -2.27071730e-04,
-2.52331681e-04, -2.74668126e-04, -2.93165304e-04, -3.06972913e-04,
-3.15442474e-04, -3.17857324e-04, -3.13794970e-04, -3.02786089e-04,
-2.84680298e-04, -2.59322215e-04, -2.26874004e-04, -1.87452375e-04,
-1.41293604e-04, -8.89863575e-05, -3.08865701e-05, 3.22411495e-05,
9.97408029e-05, 1.70914181e-04, 2.44766912e-04, 3.20596833e-04,
3.97403451e-04, 4.74233283e-04, 5.50189125e-04, 6.24368453e-04,
6.95836553e-04, 7.63737143e-04, 8.27069057e-04, 8.84995559e-04,
9.36770723e-04, 9.81561780e-04, 1.01869959e-03, 1.04738842e-03,
1.06711062e-03, 1.07736961e-03, 1.07778386e-03, 1.06805613e-03,
1.04797776e-03, 1.01747638e-03, 9.76519221e-04, 9.25420191e-04,
8.64415092e-04, 7.93844624e-04, 7.14396030e-04, 6.26549187e-04,
5.31154439e-04, 4.28985322e-04, 3.21198916e-04, 2.08550190e-04,
9.21607082e-05, -2.68249728e-05, -1.47278123e-04, -2.67890543e-04,
-3.87642210e-04, -5.05244473e-04, -6.19660328e-04, -7.29670300e-04,
-8.34272846e-04, -9.32388033e-04, -1.02301036e-03, -1.10520861e-03,
-1.17824066e-03, -1.24119243e-03, -1.29350096e-03, -1.33458128e-03,
-1.36388379e-03, -1.38105465e-03, -1.38595634e-03, -1.37832214e-03,
-1.35829791e-03, -1.32588558e-03, -1.28146000e-03, -1.22518721e-03,
-1.15769141e-03, -1.07943574e-03, -9.91143310e-04, -8.93671637e-04,
-7.87961546e-04, -6.74866999e-04, -5.55444011e-04, -4.30919368e-04,
-3.02270469e-04, -1.70824836e-04, -3.77396109e-05, 9.56816273e-05,
2.28299979e-04, 3.58842001e-04, 4.86074690e-04, 6.08875045e-04,
7.26090501e-04, 8.36677390e-04, 9.39639556e-04, 1.03407702e-03,
1.11906014e-03, 1.19386315e-03, 1.25779004e-03, 1.31037519e-03,
1.35108872e-03, 1.37958003e-03, 1.39570542e-03, 1.39927441e-03,
1.39033118e-03, 1.36892681e-03, 1.33533475e-03, 1.28987173e-03,
1.23311389e-03, 1.16551418e-03, 1.08773037e-03, 1.00059786e-03,
9.04879918e-04, 8.01551710e-04, 6.91575582e-04, 5.75952750e-04,
4.55756793e-04, 3.32302985e-04, 2.06487043e-04, 7.95882588e-05,
-4.72208138e-05, -1.72823958e-04, -2.96101535e-04, -4.15925168e-04,
-5.31250383e-04, -6.41017819e-04, -7.44349685e-04, -8.40276057e-04,
-9.28032591e-04, -1.00688055e-03, -1.07610640e-03, -1.13518206e-03,
-1.18370702e-03, -1.22129557e-03, -1.24764964e-03, -1.26264035e-03,
-1.26627364e-03, -1.25857717e-03, -1.23964021e-03,
-1.20980891e-03, -1.16944284e-03, -1.11887385e-03,
-1.05870668e-03, -9.89617769e-04, -9.12311681e-04,
-8.27560752e-04, -7.36170045e-04, -6.39042153e-04,
-5.37114997e-04, -4.31247869e-04, -3.22637131e-04,
-2.12194968e-04, -1.00869243e-04, 1.03136916e-05,
1.20241207e-04, 2.27979265e-04, 3.32675130e-04,
4.33323979e-04, 5.29105666e-04, 6.19142445e-04,
7.02728753e-04, 7.79114404e-04, 8.47787258e-04,
9.08216047e-04, 9.59821724e-04, 1.00228122e-03,
1.03538392e-03, 1.05880009e-03, 1.07260841e-03,
1.07662550e-03, 1.07093155e-03, 1.05577003e-03,
1.03129797e-03, 9.97904596e-04, 9.55975595e-04,
9.05955028e-04, 8.48396342e-04, 7.83925297e-04,
7.13242537e-04, 6.36896396e-04, 5.55809454e-04,
4.70697276e-04, 3.82464668e-04, 2.91766220e-04,
1.99564879e-04, 1.06707654e-04, 1.40463177e-05,
-7.76333806e-05, -1.67470574e-04, -2.54708122e-04,
-3.38623857e-04, -4.18484684e-04])
difference_exp = difference_exp/np.max(difference_exp)
np.testing.assert_allclose(difference_exp, difference,
rtol=rtol, atol=atol,
err_msg='In TestBeamFeedback test_SPS_RL: difference between simulated and analytic result different than expected')
# DEFINE RING------------------------------------------------------------------
general_params = Ring(C, momentum_compaction, sync_momentum, Electron(),
n_turns)
RF_sct_par = RFStation(general_params, harmonic_numbers, voltage_program,
phi_offset, n_rf_systems)
# DEFINE BEAM------------------------------------------------------------------
beam = Beam(general_params, n_macroparticles, n_particles)
# DEFINE TRACKER---------------------------------------------------------------
longitudinal_tracker = RingAndRFTracker(RF_sct_par, beam)
full_tracker = FullRingAndRF([longitudinal_tracker])
# DEFINE SLICES----------------------------------------------------------------
n_slices = 500
n_bunches = 80
bunch_spacing = 1600 # buckets
filling_pattern = np.zeros(bunch_spacing * n_bunches)
filling_pattern[::bunch_spacing] = 1
# BEAM GENERATION--------------------------------------------------------------
matched_from_distribution_function(beam,
full_tracker,
emittance=emittance,
voltage_program = 16e6
phi_offset = 0
# DEFINE RING------------------------------------------------------------------
general_params = Ring(C, momentum_compaction,
sync_momentum, Proton(), n_turns)
RF_sct_par = RFStation(general_params, [harmonic_numbers], [voltage_program],
[phi_offset], n_rf_systems)
beam = Beam(general_params, n_macroparticles, n_particles)
ring_RF_section = RingAndRFTracker(RF_sct_par, beam)
full_tracker = FullRingAndRF([ring_RF_section])
fs = RF_sct_par.omega_s0[0]/2/np.pi
bucket_length = 2.0 * np.pi / RF_sct_par.omega_rf[0,0]
# DEFINE SLICES ---------------------------------------------------------------
number_slices = 3000
slice_beam = Profile(beam, CutOptions(cut_left=0,
cut_right=21*bucket_length, n_slices=number_slices))
# LOAD IMPEDANCE TABLES -------------------------------------------------------
R_S = 5e8
frequency_R = 2*RF_sct_par.omega_rf[0,0] / 2.0 / np.pi
#Phase loop
#configuration = {'machine': 'PSB', 'PL_gain': 0., 'RL_gain': [34.8,16391],
# 'PL_period': 10.e-6, 'RL_period': 7}
configuration = {
'machine': 'PSB',
'PL_gain': 0,
'RL_gain': [1.e7, 1.e11],
'period': 10.e-6
}
phase_loop = BeamFeedback(general_params, rf_params, slices_ring,
configuration)
#Long tracker
long_tracker = RingAndRFTracker(rf_params, my_beam, BeamFeedback=phase_loop)
full_ring = FullRingAndRF([long_tracker])
distribution_type = 'gaussian'
bunch_length = 200.0e-9
distribution_variable = 'Action'
matched_from_distribution_function(my_beam,
full_ring,
bunch_length=bunch_length,
distribution_type=distribution_type,
distribution_variable=distribution_variable,
seed=1223)
slices_ring.track()
# Accelerator map
elif PL_2ndLoop == 'F_Loop':
gain2nd = 0.9e-1
PLdict['machine'] = 'SPS_F'
PLdict['FL_gain'] = gain2nd
phaseLoop = BeamFeedback(ring, rf_station, profile, PLdict)
beamPosPrev = t_batch_begin + 0.5 * t_rf
# SPS --- Tracker Setup ----------------------------------------
mpiprint('Setting up tracker')
tracker = RingAndRFTracker(rf_station,
beam,
Profile=profile,
TotalInducedVoltage=inducedVoltage,
interpolation=True)
fulltracker = FullRingAndRF([tracker])
mpiprint('Creating SPS bunch from PS bunch')
# create 72 bunches from PS bunch
beginIndex = 0
endIndex = 0
PS_beam = Beam(ring, n_particles, 0.0)
PS_n_bunches = 1
for copy in range(n_bunches):
# create binomial distribution;
# use different seed for different bunches to avoid cloned bunches
matched_from_distribution_function(PS_beam,
fulltracker,
seed=seed + copy,
distribution_type='binomial',
# Define machine impedance from http://impedance.web.cern.ch/impedance/
ZTot = np.loadtxt(os.path.join(inputDir, 'Zlong_Allthemachine_450GeV_B1_LHC_inj_450GeV_B1.dat'),
skiprows=1)
ZTable = InputTable(ZTot[:, 0], ZTot[:, 1], ZTot[:, 2])
indVoltage = InducedVoltageFreq(
beam, profile, [ZTable], frequency_resolution=freq_res)
totVoltage = TotalInducedVoltage(beam, profile, [indVoltage])
# TODO add the noiseFB
tracker = RingAndRFTracker(rf, beam, BeamFeedback=PL, Profile=profile,
interpolation=True, TotalInducedVoltage=totVoltage,
solver='simple')
# interpolation=True, TotalInducedVoltage=None)
mpiprint("PL, SL, and tracker set...")
# Fill beam distribution
fullring = FullRingAndRF([tracker])
# Juan's fit to LHC profiles: binomial w/ exponent 1.5
# matched_from_distribution_function(beam, fullring,
# main_harmonic_option = 'lowest_freq',
# distribution_exponent = 1.5, distribution_type='binomial',
# bunch_length = 1.1e-9, bunch_length_fit = 'fwhm',
# distribution_variable = 'Action')
# Initial losses, slicing, statistics
beam.losses_separatrix(ring, rf)
mpiprint('dE mean: ', np.mean(beam.dE))
mpiprint('dE std: ', np.std(beam.dE))
mpiprint('dt mean: ', np.mean(beam.dE))
mpiprint('dt std: ', np.std(beam.dE))
# DEFINE SLICES----------------------------------------------------------------
number_slices = 500
cut_options = CutOptions(cut_left=0.,
cut_right=bucket_length,
n_slices=number_slices)
slice_beam = Profile(beam, CutOptions=cut_options)
# DEFINE TRACKER---------------------------------------------------------------
longitudinal_tracker = []
for i in range(n_sections):
longitudinal_tracker.append(
RingAndRFTracker(RF_sct_par[i], beam, Profile=slice_beam))
full_tracker = FullRingAndRF(longitudinal_tracker)
# BEAM GENERATION--------------------------------------------------------------
matched_from_distribution_function(beam,
full_tracker,
emittance=emittance,
distribution_type=distribution_type,
distribution_variable=distribution_variable,
seed=1000)
slice_beam.track()
# Synchrotron radiation objects without quantum excitation
rho = 11e3
SR = []
def test_SPS_RL(self):
PL_gain = 1000 # gain of phase loop
round_digit = 5 # to how many digits round the result
# Phase loop setup
phase_loop = BeamFeedback(self.ring, self.rf_station, self.profile,
{'machine':'SPS_RL', 'PL_gain':PL_gain})
# Tracker setup
section_tracker = RingAndRFTracker(
self.rf_station, self.beam, Profile=self.profile,
BeamFeedback=phase_loop, interpolation=False)
tracker = FullRingAndRF([section_tracker])
# average beam position
beamAvgPos = np.zeros(self.ring.n_turns)
n_turns = self.ring.n_turns
for turn in range(n_turns):
beamAvgPos[turn] = np.mean(self.beam.dt)
self.profile.track()
tracker.track()
# difference between beam position and synchronuous position
# (assuming no beam loading)
delta_tau = beamAvgPos - (np.pi - self.rf_station.phi_rf[0,:-1])\
/ self.rf_station.omega_rf[0,:-1]
# initial position for analytic solution
init_pos = self.time_offset
omega_eff = cmath.sqrt(-PL_gain**2 + 4*self.rf_station.omega_s0[0]**2)
time = np.arange(n_turns) * self.ring.t_rev[0]
# initial derivative for analytic solution;
# defined such that analytical solution at turn 1 agrees with numerical
# solution
init_slope = 0.5 * (delta_tau[1] * omega_eff * np.exp(0.5*PL_gain*time[1])\
/ np.sin(0.5*omega_eff*time[1]) - delta_tau[0]\
*(PL_gain+omega_eff/np.tan(0.5*omega_eff*time[1])) ).real
delta_tau_analytic = init_pos * np.exp(-0.5*PL_gain*time)
delta_tau_analytic *= np.cos(0.5*time*omega_eff).real\
+ (PL_gain+2*init_slope/init_pos)\
* (np.sin(0.5*time*omega_eff)/omega_eff).real
difference = delta_tau - delta_tau_analytic
# normalize result
difference = np.around(difference/np.max(difference), round_digit)
# expected difference
difference_exp = np.array([
-1.56306635e-05, -1.55605315e-05, -2.10224435e-05, -3.18525050e-05,
-4.74014489e-05, -6.70584402e-05, -9.01307422e-05, -1.15823959e-04,
-1.43290487e-04, -1.71572162e-04, -1.99820151e-04, -2.27071730e-04,
-2.52331681e-04, -2.74668126e-04, -2.93165304e-04, -3.06972913e-04,
-3.15442474e-04, -3.17857324e-04, -3.13794970e-04, -3.02786089e-04,
-2.84680298e-04, -2.59322215e-04, -2.26874004e-04, -1.87452375e-04,
-1.41293604e-04, -8.89863575e-05, -3.08865701e-05, 3.22411495e-05,
9.97408029e-05, 1.70914181e-04, 2.44766912e-04, 3.20596833e-04,
3.97403451e-04, 4.74233283e-04, 5.50189125e-04, 6.24368453e-04,
6.95836553e-04, 7.63737143e-04, 8.27069057e-04, 8.84995559e-04,
9.36770723e-04, 9.81561780e-04, 1.01869959e-03, 1.04738842e-03,
1.06711062e-03, 1.07736961e-03, 1.07778386e-03, 1.06805613e-03,
1.04797776e-03, 1.01747638e-03, 9.76519221e-04, 9.25420191e-04,
8.64415092e-04, 7.93844624e-04, 7.14396030e-04, 6.26549187e-04,
5.31154439e-04, 4.28985322e-04, 3.21198916e-04, 2.08550190e-04,
9.21607082e-05, -2.68249728e-05, -1.47278123e-04, -2.67890543e-04,
-3.87642210e-04, -5.05244473e-04, -6.19660328e-04, -7.29670300e-04,
-8.34272846e-04, -9.32388033e-04, -1.02301036e-03, -1.10520861e-03,
-1.17824066e-03, -1.24119243e-03, -1.29350096e-03, -1.33458128e-03,
-1.36388379e-03, -1.38105465e-03, -1.38595634e-03, -1.37832214e-03,
-1.35829791e-03, -1.32588558e-03, -1.28146000e-03, -1.22518721e-03,
-1.15769141e-03, -1.07943574e-03, -9.91143310e-04, -8.93671637e-04,
-7.87961546e-04, -6.74866999e-04, -5.55444011e-04, -4.30919368e-04,
-3.02270469e-04, -1.70824836e-04, -3.77396109e-05, 9.56816273e-05,
2.28299979e-04, 3.58842001e-04, 4.86074690e-04, 6.08875045e-04,
7.26090501e-04, 8.36677390e-04, 9.39639556e-04, 1.03407702e-03,
1.11906014e-03, 1.19386315e-03, 1.25779004e-03, 1.31037519e-03,
1.35108872e-03, 1.37958003e-03, 1.39570542e-03, 1.39927441e-03,
1.39033118e-03, 1.36892681e-03, 1.33533475e-03, 1.28987173e-03,
1.23311389e-03, 1.16551418e-03, 1.08773037e-03, 1.00059786e-03,
9.04879918e-04, 8.01551710e-04, 6.91575582e-04, 5.75952750e-04,
4.55756793e-04, 3.32302985e-04, 2.06487043e-04, 7.95882588e-05,
-4.72208138e-05, -1.72823958e-04, -2.96101535e-04, -4.15925168e-04,
-5.31250383e-04, -6.41017819e-04, -7.44349685e-04, -8.40276057e-04,
-9.28032591e-04, -1.00688055e-03, -1.07610640e-03, -1.13518206e-03,
-1.18370702e-03, -1.22129557e-03, -1.24764964e-03, -1.26264035e-03,
-1.26627364e-03, -1.25857717e-03, -1.23964021e-03,
-1.20980891e-03, -1.16944284e-03, -1.11887385e-03,
-1.05870668e-03, -9.89617769e-04, -9.12311681e-04,
-8.27560752e-04, -7.36170045e-04, -6.39042153e-04,
-5.37114997e-04, -4.31247869e-04, -3.22637131e-04,
-2.12194968e-04, -1.00869243e-04, 1.03136916e-05,
1.20241207e-04, 2.27979265e-04, 3.32675130e-04,
4.33323979e-04, 5.29105666e-04, 6.19142445e-04,
7.02728753e-04, 7.79114404e-04, 8.47787258e-04,
9.08216047e-04, 9.59821724e-04, 1.00228122e-03,
1.03538392e-03, 1.05880009e-03, 1.07260841e-03,
1.07662550e-03, 1.07093155e-03, 1.05577003e-03,
1.03129797e-03, 9.97904596e-04, 9.55975595e-04,
9.05955028e-04, 8.48396342e-04, 7.83925297e-04,
7.13242537e-04, 6.36896396e-04, 5.55809454e-04,
4.70697276e-04, 3.82464668e-04, 2.91766220e-04,
1.99564879e-04, 1.06707654e-04, 1.40463177e-05,
-7.76333806e-05, -1.67470574e-04, -2.54708122e-04,
-3.38623857e-04, -4.18484684e-04])
difference_exp = np.around(difference_exp/np.max(difference_exp),
round_digit)
self.assertListEqual(difference.tolist(), difference_exp.tolist(),
msg='In TestBeamFeedback test_SPS_RL: '
+'difference between simulated and analytic result different than'
+'expected')
