""" Considering parasitic capacitances/inductances in the system. ============================================================= Before a plasma breakdown, the impedance of the gas can be represented as a very high resistance (no ionization) in parallel to a constant capacitance (representing the electrode and the surrounding environment). The electrodes are generally placed in a reactor, which can have a small parasitic resistance and inductance, corresponding to the wires and the connections. The equivalent circuit of the system is the following :: ----- R_wire --- L_wire -----┐-----┐ ↑ │ │ ↑ │ │ │ │ V(x=L, t) C_el R_p │ u_p(t) │ │ │ │ │ i_reactor │ │ │ ----------<------------------┘-----┘ Let us observe what are the current and voltage in this case. """ # noqa: D205 # This sets the following figure as the thumbnail for the example gallery. # sphinx_gallery_thumbnail_number = 5 # %% # First, we import the required libraries. # ----------------------------------------- # # We start by importing the modules we need: # # - matplotlib for drawing graphs, # - numpy for array functions, # - pyresiflex for the generator, load and transmission line. import matplotlib.pyplot as plt import numpy as np from pyresiflex.cable.cable import PerfectCable from pyresiflex.generator.generator_complex_impedance import GaussianGenerator from pyresiflex.load.base_load import ComplexImpedanceBaseLoad from pyresiflex.misc.units import c_0 from pyresiflex.misc.utils import get_path_to_data from pyresiflex.solver.steady_impedance_solution import SteadyImpedanceSolution plt.style.use(get_path_to_data("article_two_columns_figure.mplstyle")) # %% # Create a trapezoidal generator. # ------------------------------- generator = GaussianGenerator( height=4e3, # [V] mean=5e-9, # [s] FWHM=10e-9, # [s] R_g=1.0, # [Ohm] C_g=0, # [F] ) # %% # Transmission line parameters. # ----------------------------- L = 4 # Length of the transmission line [m] c = 0.67 * c_0 # Wave velocity [m/s] Z_c = 75 # Characteristic impedance [Ohm] cable = PerfectCable( L=L, Z_c=Z_c, c=c, ) # %% # Create a load class for no breakdown plasma. # -------------------------------------------- class ReactorImpedance(ComplexImpedanceBaseLoad): def __init__(self, R_wire, L_wire, C_electrode, R_plasma): super().__init__(purely_resistive=False) self.R_wire = R_wire self.L_wire = L_wire self.C_electrode = C_electrode self.R_plasma = R_plasma def load_impedance(self, frequency): # Load impedance is a capacitor in parallel with a resistor. Z_wire = self.R_wire + 1j * 2 * np.pi * frequency * self.L_wire # Combine the electrode capacitance and plasma resistance in parallel. # Z_electrode = 1 / (1j * 2 * np.pi * frequency * self.C_electrode) # Z_plasma = self.R_plasma # Z_el_plasma = 1 / (1 / Z_electrode + 1 / Z_plasma) Z_el_plasma = self.R_plasma / ( 1 + 1j * 2 * np.pi * frequency * self.C_electrode * self.R_plasma ) return Z_wire + Z_el_plasma no_breakdown_plasma = ReactorImpedance( R_wire=1e-3, # Wire resistance [Ohm] L_wire=100e-9, # Wire inductance [H] C_electrode=50e-12, # Electrode capacitance [F] R_plasma=1e6, # Plasma resistance [Ohm] ) # %% # Solve the system. # ----------------- # Create the solution object with the generator, load and cable. solution = SteadyImpedanceSolution( generator=generator, load=no_breakdown_plasma, cable=cable, nb_points_ft=50_000, # Number of points for Fourier transform. max_time_ft=50_000e-9, # Maximum time for Fourier transform [s] ) x_meas_voltage = L / 2 # Position of the measurement in meters. x_meas_current = L / 2 # Position of the measurement in meters. t_end = 300e-9 # End time for the solution [s] N = 1000 # Number of points for time vector. times = np.linspace(0, t_end, N, dtype=float) # Solve for voltage, current, and energy. solution.solve(x_meas_voltage, t=times) voltage = solution.voltage current = solution.current energy = solution.energy # %% # Plot voltage, current, and energy at remote configuration. # ---------------------------------------------------------- print( f"Plotting voltage at x = {x_meas_voltage:.2f} m" f"= {x_meas_voltage / L:.2%} of the line)" ) print( f"Plotting current at x = {x_meas_current:.2f} m" f"= {x_meas_current / L:.2%} of the line)" ) fig, ax_v = plt.subplots() # Plot voltage. plot_line_v = ax_v.plot( times * 1e9, voltage * 1e-3, color="k", ) # .. Plot options for voltage. ax_v.set_xlabel(r"$\mathregular{t [ns]}$") ax_v.set_ylabel(r"$\mathregular{V \, [kV]}$") ax_v.set_ylim(-4.5, 4.5) ax_v.spines["left"].set_color("k") ax_v.set_xlim(times[0] * 1e9, times[-1] * 1e9) # Plot current. ax_i = ax_v.twinx() ax_i.plot( times * 1e9, current, ls="-", color="r", lw=4, ) # .. Plot options for current. ax_i.set_ylabel(r"$\mathregular{I \, [A]}$", color="r") # ax_i.set_ylim(-max_abs_current, max_abs_current) ax_i.set_ylim(-60, 60) ax_i.grid(visible=False) # Change color of the right y-axis to red. ax_i.spines["right"].set_color("r") # Also change the color of the ticks. ax_i.tick_params(axis="y", colors="r") # Move x-position of the y-label. ax_i.yaxis.set_label_coords(1.05, 0.5) # Set y-ticks for current. ax_i.set_yticks([-60, -40, -20, 0, 20, 40, 60]) # Plot energy. ax_e = ax_v.twinx() ax_e.plot( times * 1e9, energy * 1e3, color="b", ls="--", label="Energy", lw=2, ) # .. Plot options for energy. ax_e.set_ylabel(r"$\mathregular{E \, [mJ]}$", color="b") # Move the y-axis of ax_e to the right, by 100 points ax_e.spines["right"].set_position(("outward", 100)) ax_e.grid(visible=False) ax_e.set_ylim(0, 1.5) ax_e.set_yticks([0, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5]) # Change color of the right y-axis to blue. ax_e.spines["right"].set_color("b") # Also change the color of the ticks. ax_e.tick_params(axis="y", colors="b") plt.show() # %%