Editing MHD Core (section)

= MHD Core Project: Mathematical Equations and Data =

This document compiles all the mathematical equations, values, and data discussed in the MHD Core project presentations.

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== Theoretical Foundations ==

=== Quantum Field Theorist's Equations ===

'''Zero-Point Energy of a Quantum Harmonic Oscillator:'''

<math>E = \frac{1}{2} \hbar \omega</math>

* ''E'': Zero-point energy
* ''<math>\hbar</math>'': Reduced Planck's constant
* ''<math>\omega</math>'': Angular frequency

'''Casimir Effect Force Between Two Plates:'''

<math>F_{\text{Casimir}} = \frac{\pi^2 \hbar c}{240} \frac{A}{L^4}</math>

* ''F''<sub>Casimir</sub>: Casimir force
* ''c'': Speed of light
* ''A'': Area of the plates
* ''L'': Separation between the plates

'''Dynamic Casimir Effect Photon Generation Rate:'''

<math>\Gamma = \frac{\pi \omega_{\text{cavity}}^2}{3c^2} \left( \frac{\Delta L}{L} \right)^2</math>

* ''<math>\Gamma</math>'': Photon generation rate
* ''<math>\omega</math>''<sub>cavity</sub>: Resonant frequency of the cavity
* ''<math>\Delta L</math>'': Modulation amplitude of cavity length
* ''<math>L</math>'': Original cavity length

'''Expectation Value of the Energy-Momentum Tensor:'''

<math>\langle T_{\mu\nu} \rangle = -\frac{\hbar c}{720 \pi^2} \frac{1}{L^4} g_{\mu\nu}</math>

* ''T''<sub>μν</sub>: Energy-momentum tensor
* ''g''<sub>μν</sub>: Metric tensor of spacetime

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=== Electromagnetic Field Specialist's Equations ===

'''Modified Wave Equation with Scalar Potential:'''

<math>\nabla^2 \phi - \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} = -\frac{\rho}{\epsilon_0}</math>

* ''<math>\phi</math>'': Scalar potential
* ''<math>\rho</math>'': Charge density
* ''<math>\epsilon_0</math>'': Vacuum permittivity

'''Magnetic Flux Quantum:'''

<math>\Phi_0 = \frac{h}{2e}</math>

* ''<math>\Phi</math>''<sub>0</sub>: Magnetic flux quantum
* ''h'': Planck's constant
* ''e'': Elementary charge

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== Material Development ==

=== Superconducting Material Properties ===

'''Yttrium Barium Copper Oxide (YBCO):'''

* '''Critical Temperature (''T''<sub>c</sub>):''' Approximately 92 K
* '''Critical Current Density (''J''<sub>c</sub>):''' Exceeding <math>(1 \times 10^6)</math> A/cm² at 77 K

'''Barium Zirconate Nanoparticles Enhancement:'''

* '''Increase in Critical Current Density:''' 30% under high magnetic fields

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=== Quantum Behaviors in Superconducting Materials ===

'''Cooper Pair Formation:'''

* Electrons form bound pairs enabling zero electrical resistance.

'''Flux Quantization Equation:'''

<math>\Phi = n \Phi_0</math>

* ''<math>\Phi</math>'': Magnetic flux through a superconducting loop
* ''n'': Integer (quantum number)
* ''<math>\Phi</math>''<sub>0</sub>: Magnetic flux quantum <math>(\Phi_0 = \frac{h}{2e})</math>

'''Energy Gap in Superconductors:'''

<math>\Delta E = 2\Delta</math>

* ''<math>\Delta E</math>'': Energy required to break a Cooper pair
* ''<math>\Delta</math>'': Energy gap parameter

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== Engineering Design ==

=== Levitation System Equations ===

'''Magnetic Force Equation:'''

<math>\mathbf{F}_{\text{mag}} = \nabla (\mathbf{m} \cdot \mathbf{B})</math>

* ''<math>\mathbf{F}</math>''<sub>mag</sub>: Magnetic force
* ''<math>\mathbf{m}</math>'': Magnetic moment
* ''<math>\mathbf{B}</math>'': Magnetic field

'''Electrostatic Force Equation:'''

<math>\mathbf{F}_{\text{elec}} = Q \mathbf{E}</math>

* ''<math>\mathbf{F}</math>''<sub>elec</sub>: Electrostatic force
* ''Q'': Electric charge
* ''<math>\mathbf{E}</math>'': Electric field

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== Control Systems and Simulations ==

=== Levitation Control Equations ===

'''State Equations:'''

<math>
\begin{cases}
\dot{\mathbf{x}} = \mathbf{v} \\
\dot{\mathbf{v}} = \frac{1}{m} \left( \mathbf{F}_{\text{mag}} + \mathbf{F}_{\text{elec}} + \mathbf{F}_{\text{dist}} \right)
\end{cases}
</math>

* ''<math>\mathbf{x}</math>'': Position vector
* ''<math>\mathbf{v}</math>'': Velocity vector
* ''m'': Mass of the core
* ''<math>\mathbf{F}</math>''<sub>dist</sub>: Disturbance force

'''Cost Function for Nonlinear Model Predictive Control (NMPC):'''

<math>J = \int_{t}^{t+T_p} \left[ \|\mathbf{x}_{\text{ref}}(t) - \mathbf{x}(t)\|_Q^2 + \|\mathbf{u}(t)\|_R^2 \right] dt</math>

* ''J'': Cost function
* ''T''<sub>p</sub>: Prediction horizon
* ''<math>\mathbf{x}</math>''<sub>ref</sub>: Reference position
* ''<math>\mathbf{u}</math>'': Control input
* ''Q'', ''R'': Weighting matrices

=== Charge Regulation Equations ===

'''Sliding Surface Definition:'''

<math>s(t) = e(t) + \lambda \int_{0}^{t} e(\tau) d\tau</math>

* ''s(t)'': Sliding surface
* ''e(t) = q_{\text{ref}}(t) - q(t)'': Charge error
* ''<math>\lambda</math>'': Positive constant

'''Control Law:'''

<math>u(t) = -k \cdot \text{sign}(s(t)) + \dot{q}_{\text{ref}}(t)</math>

* ''u(t)'': Control input
* ''k'': Adaptive gain
* ''sign(s(t))'': Sign function

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== Acoustic Integration ==

=== Hypersound Frequencies and Phonon Interactions ===

* '''Hypersound Frequency Range:''' Above 1 GHz
* '''Phonon-Electron Coupling:''' Interaction mechanism between high-frequency phonons and electrons in materials.

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== Environmental Alignment ==

=== Schumann Resonance Frequencies ===

{| class="wikitable"
|+ Schumann Resonance Modes
|-
! Mode !! Frequency (Hz) !! Wavelength (km)
|-
| 1 || ~7.83 || ~38,300
|-
| 2 || ~14.3 || ~21,000
|-
| 3 || ~20.8 || ~14,400
|-
| 4 || ~27.3 || ~11,000
|-
| 5 || ~33.8 || ~8,900
|}

* '''Variability:''' Frequencies can shift by ±0.5 Hz due to ionospheric conditions.

=== Geomagnetic Pulsation Frequencies ===

{| class="wikitable"
|+ Geomagnetic Pulsations
|-
! Category !! Frequency Range !! Associated Phenomena
|-
| Pc1 || 0.2–5.0 Hz || Electromagnetic ion cyclotron waves
|-
| Pc2 || 5–10 mHz || Field line resonances
|-
| Pc3 || 10–45 mHz || Cavity modes in the magnetosphere
|-
| Pc4 || 45–150 mHz || Large-scale magnetospheric oscillations
|-
| Pc5 || 1–7 mHz || Solar wind coupling effects
|}

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== Mathematical Modeling ==

=== System Dynamics Equations ===

'''Core Motion Equations:'''

<math>m \ddot{\mathbf{x}} = \mathbf{F}_{\text{mag}}(\mathbf{x}, \dot{\mathbf{x}}, \mathbf{I}) + \mathbf{F}_{\text{elec}}(\mathbf{x}, \dot{\mathbf{x}}, Q) + \mathbf{F}_{\text{dist}}</math>

* ''<math>\ddot{\mathbf{x}}</math>'': Acceleration
* ''<math>\mathbf{I}</math>'': Coil currents

'''State-Space Representation:'''

<math>
\begin{cases}
\dot{\mathbf{x}} = \mathbf{v} \\
\dot{\mathbf{v}} = \frac{1}{m} \left( \mathbf{F}_{\text{mag}} + \mathbf{F}_{\text{elec}} + \mathbf{F}_{\text{dist}} \right) \\
\dot{\boldsymbol{\theta}} = \boldsymbol{\omega} \\
\dot{\boldsymbol{\omega}} = \mathbf{I}^{-1} \left( \boldsymbol{\tau}_{\text{mag}} + \boldsymbol{\tau}_{\text{elec}} + \boldsymbol{\tau}_{\text{dist}} \right)
\end{cases}
</math>

* ''<math>\boldsymbol{\theta}</math>'': Orientation angles
* ''<math>\boldsymbol{\omega}</math>'': Angular velocities
* ''<math>\boldsymbol{\tau}</math>''<sub>mag</sub>, ''<math>\boldsymbol{\tau}</math>''<sub>elec</sub>: Magnetic and electrostatic torques
* ''<math>\mathbf{I}</math>'': Moment of inertia tensor

'''Sliding Surface for Adaptive Sliding Mode Control (ASMC):'''

<math>s(t) = e(t) + \lambda \int_{0}^{t} e(\tau) d\tau</math>

'''Control Law for ASMC:'''

<math>u(t) = -k \cdot \text{sign}(s(t)) + \dot{q}_{\text{ref}}(t)</math>

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== Control Algorithms Parameters ==

'''Parameters Definitions:'''

* ''m'': Mass of the core
* ''<math>\mathbf{x}</math>'', ''<math>\mathbf{v}</math>'': Position and velocity vectors
* ''<math>\boldsymbol{\theta}</math>'', ''<math>\boldsymbol{\omega}</math>'': Orientation and angular velocity vectors
* ''<math>\mathbf{F}</math>''<sub>mag</sub>, ''<math>\mathbf{F}</math>''<sub>elec</sub>: Magnetic and electrostatic forces
* ''<math>\mathbf{F}</math>''<sub>dist</sub>: Disturbance forces
* ''<math>\mathbf{I}</math>'': Moment of inertia tensor
* ''<math>\boldsymbol{\tau}</math>''<sub>mag</sub>, ''<math>\boldsymbol{\tau}</math>''<sub>elec</sub>: Magnetic and electrostatic torques
* ''e(t)'': Error signal
* ''<math>\lambda</math>'': Positive constant for sliding surface
* ''k'': Adaptive gain for control law
* ''<math>\mathbf{u}(t)</math>'': Control input vector

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== Key Constants and Physical Quantities ==

* '''Planck's Constant (''h''):''' <math>(6.62607015 \times 10^{-34}) Js</math>
* '''Reduced Planck's Constant (''<math>\hbar </math>''):''' <math>(\frac{h}{2\pi})</math>
* '''Speed of Light (''c''):''' <math>(3.0 \times 10^8) m/s </math>
* '''Elementary Charge (''e''):''' <math>(1.602176634 \times 10^{-19}) C </math>
* '''Vacuum Permittivity (''<math>\epsilon_0 </math>''):''' <math>(8.854187817 \times 10^{-12}) F/m </math>

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This document compiles all the mathematical equations, values, and data relevant to the MHD Core project, providing a comprehensive reference for team members and stakeholders.