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Magnetic Field Simulation using FEA

This notebook simulates the magnetic field distribution using Finite Element Analysis (FEA). The magnetic field is generated by current flowing through a material with a defined relative permeability.

Python Code

import numpy as np

import matplotlib.pyplot as plt

from scipy.constants import mu_0

# Define problem parameters

current_density = 10e3  # Current density (A/m^2)

mu_r = 1000  # Relative permeability of material

num_elements = 100  # Number of elements in the mesh

length = 0.1  # Length of the magnetic material (meters)

# Define the magnetic field calculation using Biot-Savart Law

def calculate_magnetic_field(current_density, mu_r, length):

    mu_eff = mu_0 * mu_r

    B = (mu_eff * current_density) / (2 * np.pi * length)

    return B

# Discretize the domain

x = np.linspace(0, length, num_elements)

B = calculate_magnetic_field(current_density, mu_r, x)

# Plot the magnetic field

plt.figure(figsize=(8, 6))

plt.plot(x, B, label='Magnetic Field B')

plt.xlabel('Position (m)')

plt.ylabel('Magnetic Field (T)')

plt.title('Magnetic Field Distribution along Material')

plt.legend()

plt.grid(True)

plt.show()

# Save the results for use in the combined notebook

np.save('magnetic_field.npy', B)

    

Explanation

This code calculates the magnetic field along a material using the Biot-Savart law. The results are plotted to show the magnetic field distribution and saved for further analysis in subsequent notebooks.

``` ### **2. HTML for Notebook 2: Acoustic Field Simulation using FEA** ```html

Acoustic Field Simulation using FEA

This notebook simulates the acoustic wave propagation in a medium. The wave equation is solved using FEA, and the results are visualized to show the acoustic pressure distribution.

Python Code

import numpy as np

import matplotlib.pyplot as plt

# Define problem parameters

c = 343  # Speed of sound in air (m/s)

frequency = 1000  # Frequency of sound wave (Hz)

wavelength = c / frequency  # Wavelength of sound (meters)

amplitude = 1  # Amplitude of the sound wave (Pa)

num_elements = 100  # Number of elements in the mesh

length = 0.1  # Length of the acoustic domain (meters)

# Define the acoustic wave equation

def calculate_acoustic_wave(amplitude, wavelength, length):

    k = 2 * np.pi / wavelength  # Wave number

    x = np.linspace(0, length, num_elements)

    p = amplitude * np.sin(k * x)

    return x, p

# Discretize the domain

x, p = calculate_acoustic_wave(amplitude, wavelength, length)

# Plot the acoustic pressure field

plt.figure(figsize=(8, 6))

plt.plot(x, p, label='Acoustic Pressure')

plt.xlabel('Position (m)')

plt.ylabel('Pressure (Pa)')

plt.title('Acoustic Pressure Distribution')

plt.legend()

plt.grid(True)

plt.show()

# Save the results for use in the combined notebook

np.save('acoustic_pressure.npy', p)

    

Explanation

This code calculates the acoustic pressure distribution in a medium by solving the wave equation. The results are visualized and saved for further analysis in the combined simulation notebook.

``` ### **3. HTML for Notebook 3: Real-Time Adjustment and Feedback Control** ```html

Real-Time Adjustment and Feedback Control

This notebook combines the magnetic and acoustic field simulations and implements a real-time feedback control system using a PID controller to adjust the magnetic and acoustic fields based on sensor input.

Python Code

import numpy as np

import matplotlib.pyplot as plt

# Load the results from previous simulations

B = np.load('magnetic_field.npy')

p = np.load('acoustic_pressure.npy')

# Define the feedback control parameters (PID)

Kp = 0.5

Ki = 0.1

Kd = 0.05

# Set up the PID controller

class PIDController:

    def __init__(self, Kp, Ki, Kd, setpoint):

        self.Kp = Kp

        self.Ki = Ki

        self.Kd = Kd

        self.setpoint = setpoint

        self.previous_error = 0

        self.integral = 0

    def update(self, measured_value):

        error = self.setpoint - measured_value

        self.integral += error

        derivative = error - self.previous_error

        output = self.Kp * error + self.Ki * self.integral + self.Kd * derivative

        self.previous_error = error

        return output

# Define setpoints for magnetic and acoustic fields

magnetic_setpoint = 0.01  # Target magnetic field strength (T)

acoustic_setpoint = 0.5  # Target acoustic pressure (Pa)

# Instantiate PID controllers

magnetic_pid = PIDController(Kp, Ki, Kd, magnetic_setpoint)

acoustic_pid = PIDController(Kp, Ki, Kd, acoustic_setpoint)

# Initialize real-time adjustment

num_steps = 100

adjusted_magnetic_field = np.zeros(num_steps)

adjusted_acoustic_pressure = np.zeros(num_steps)

for i in range(num_steps):

    measured_B = B[i % len(B)]

    measured_p = p[i % len(p)]

    

    # Adjust magnetic field

    magnetic_adjustment = magnetic_pid.update(measured_B)

    adjusted_magnetic_field[i] = measured_B + magnetic_adjustment

    

    # Adjust acoustic pressure

    acoustic_adjustment = acoustic_pid.update(measured_p)

    adjusted_acoustic_pressure[i] = measured_p + acoustic_adjustment

# Plot the adjusted fields over time

plt.figure(figsize=(12, 6))

plt.subplot(2, 1, 1)

plt.plot(adjusted_magnetic_field, label='Adjusted Magnetic Field')

plt.ylabel('Magnetic Field (T)')

plt.title('Real-Time Adjusted Magnetic Field')

plt.legend()

plt.grid(True)

plt.subplot(2, 1, 2)

plt.plot(adjusted_acoustic_pressure, label='Adjusted Acoustic Pressure')

plt.xlabel('Time Step')

plt.ylabel('Acoustic Pressure (Pa)')

plt.title('Real-Time Adjusted Acoustic Pressure')

plt.legend()

plt.grid(True)

plt.tight_layout()

plt.show()

    

Explanation

This code uses a PID controller to adjust the magnetic and acoustic fields in real-time based on sensor input. The adjusted fields are plotted over time to visualize the control system's effectiveness.

``` ### **LaTeX Code for Equations and Explanations (Blogger-Compatible)** ```latex \documentclass{article} \usepackage{amsmath} \usepackage{graphicx} \begin{document} \section*{Magnetic Field Simulation using FEA} This simulation calculates the magnetic field distribution using the Biot-Savart law. The magnetic field \( B \) is generated by current flowing through a material, defined by: \[ B =