rizer.electric_circuit.castera_circuit#

Mimic the electric circuit from Philippe Castera’s thesis [Castera2015].

Classes#

CasteraCircuit

Solve the following circuit, inspired by Philippe Castera's thesis [Castera2015].

Module Contents#

class rizer.electric_circuit.castera_circuit.CasteraCircuit(C: float | int, L: float | int, L_p: float | int, R_b: float | int, R_sg: float | int, R_0: float | int, I_0: float | int, U_c_0: float | int, R_model: rizer.electric_circuit.variable_resistance_models.VariableResistorModel)#

Solve the following circuit, inspired by Philippe Castera’s thesis [Castera2015].

 ┌-------------------(L-L_p)------┐
 │       ↑  │                     │    ↑
 │      u_c C                    L_p   │
R_b      │  │                     │  u_mes
 │         R_sg                  R_p   │
 │          │                     │    │
 ┖--------------------------------┘

The circuit is composed of:

  • a capacitor C,

  • an inductor L,

  • a second inductor L_p (plasma inductor),

  • a resistor R_p (plasma resistance),

  • a resistor R_b (ballast resistor),

  • a resistor R_sg (spark gap resistance, assumed to be constant).

Models require the initial values of:

  • the plasma resistance R_0,

  • the initial current trough the plasma I_0,

  • the initial voltage across the capacitor U_c_0.

This class provides the following methods:

Parameters:

  • C (float | int) – Capacitance value in Farads.

  • L (float | int) – Inductance value in Henry.

  • L_p (float | int) – Plasma inductance value in Henry.

  • R_b (float | int) – Ballast resistance in Ohms.

  • R_sg (float | int) – Spark gap resistance in Ohms.

  • R_0 (float | int) – Initial value of the plasma resistance in Ohms.

  • I_0 (float | int) – Initial current through the plasma in the circuit in Amperes.

  • U_c_0 (float | int) – Initial voltage across the capacitor in Volts.

  • R_model (VariableResistorModel) – The resistance model used in the circuit.

See also

VariableResistorModel

time: numpy.ndarray | None = None#

Time at which the circuit is solved.

i: numpy.ndarray | None = None#

Current in the circuit.

di: numpy.ndarray | None = None#

Derivative of the current.

u_c: numpy.ndarray | None = None#

Voltage across the capacitor.

R_p: numpy.ndarray | None = None#

Resistance of the plasma.

u_mes: numpy.ndarray | None = None#

Voltage across R_p and L_p.

solve(time: numpy.ndarray, method: str = 'LSODA', **kwargs) → tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray, numpy.ndarray]#

Solve the RLLC circuit numerically, given a time varying resistor.

Internally, the method uses the scipy.integrate.solve_ivp function to solve the ODE system.

Parameters:

  • time (np.ndarray) – Time at which the circuit is solved.

  • method (str) – The numerical method used to solve the ODE. Default is “LSODA”.

  • **kwargs – Additional arguments to pass to the ODE solver. See the documentation of scipy.integrate.solve_ivp.

Returns:

Tuple containing the following arrays:

  • i (np.ndarray): Current in the circuit.

  • didt (np.ndarray): Derivative of the current.

  • u_mes (np.ndarray): Voltage across R_p and L_p.

  • R_p (np.ndarray): Resistance of the plasma.

Return type:

tuple[np.ndarray,np.ndarray,np.ndarray,np.ndarray]

Raises:

ValueError – If the numerical solution did not converge.

See also

scipy.integrate.solve_ivp, CasteraCircuit.compute_derivatives()

compute_derivatives(t: float, y: numpy.ndarray) → numpy.ndarray#

Right-hand side of the system: \(\frac{dy}{dt} = f(t, y)\).

This method computes the derivatives of the system of ODEs that describe the Castera circuit. It is used by the CasteraCircuit.solve() method to solve the ODE system.

Parameters:

  • t (float) – Time at which the circuit is solved.

  • y (np.ndarray[float,float,float]) –

    Array containing the present values of the system:

    • y[0]=i_p (float): The current through the plasma.

    • y[1]=u_c (float): The voltage across the capacitor.

    • y[2]=R_p (float): The plasma resistance.

Returns:

Array containing the derivatives of the system:

  • dy[0]=di_p (float): The derivative of the current through the plasma.

  • dy[1]=du_c (float): The derivative of the voltage across the capacitor.

  • dy[2]=dR_p (float): The derivative of the plasma resistance.

Return type:

np.ndarray[float,float,float]

Notes

The system of ODEs is given by:

\[\begin{split}\begin{cases} \frac{di_p}{dt} &= \frac{R_b}{(R_b + R_{sg}) L} u_c - \left(\frac{R_{sg} R_b}{R_b + R_{sg}} + R_p\right) \frac{i_p}{L} \\ \frac{du_c}{dt} &= -\frac{u_c}{(R_b + R_{sg}) C} - \frac{R_b}{R_b + R_{sg}} \frac{i_p}{C} \\ \frac{dR_p}{dt} &= \text{compute_resistance_derivative}(t, [i_p, R_p]) \end{cases}\end{split}\]

See also

CasteraCircuit.solve(), rizer.electric_circuit.variable_resistance_models.VariableResistorModel.compute_resistance_derivative()