Magneto Hydro Dynamic Core

A Levitation Power Core

Fundimental Technology for the operation of a Star Speeder and Magneto Speeder

MHD Core Project: Mathematical Equations and Data

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

Theoretical Foundations

Quantum Field Theorist's Equations

Zero-Point Energy of a Quantum Harmonic Oscillator:

$E={\frac {1}{2}}\hbar \omega$

E: Zero-point energy
$\hbar$ : Reduced Planck's constant
$\omega$ : Angular frequency

Casimir Effect Force Between Two Plates:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{\text{Casimir}} = \frac{\pi^2 \hbar c}{240} \frac{A}{L^4}}

F_Casimir: Casimir force
c: Speed of light
A: Area of the plates
L: Separation between the plates

Dynamic Casimir Effect Photon Generation Rate:

$\Gamma ={\frac {\pi \omega _{\text{cavity}}^{2}}{3c^{2}}}\left({\frac {\Delta L}{L}}\right)^{2}$

$\Gamma$ : Photon generation rate
$\omega$ _cavity: Resonant frequency of the cavity
$\Delta L$ : Modulation amplitude of cavity length
$L$ : Original cavity length

Expectation Value of the Energy-Momentum Tensor:

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

T_μν: Energy-momentum tensor
g_μν: Metric tensor of spacetime

Electromagnetic Field Specialist's Equations

Modified Wave Equation with Scalar Potential:

$\nabla ^{2}\phi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\phi }{\partial t^{2}}}=-{\frac {\rho }{\epsilon _{0}}}$

$\phi$ : Scalar potential
$\rho$ : Charge density
$\epsilon _{0}$ : Vacuum permittivity

Magnetic Flux Quantum:

$\Phi _{0}={\frac {h}{2e}}$

$\Phi$ ₀: Magnetic flux quantum
h: Planck's constant
e: Elementary charge

Material Development

Superconducting Material Properties

Yttrium Barium Copper Oxide (YBCO):

Critical Temperature (T_c): Approximately 92 K
Critical Current Density (J_c): Exceeding $1 \times 10^6$ A/cm² at 77 K

Barium Zirconate Nanoparticles Enhancement:

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

Quantum Behaviors in Superconducting Materials

Cooper Pair Formation:

Electrons form bound pairs enabling zero electrical resistance.

Flux Quantization Equation:

$\Phi =n\Phi _{0}$

$\Phi$ : Magnetic flux through a superconducting loop
n: Integer (quantum number)
$\Phi$ ₀: Magnetic flux quantum $(\Phi _{0}={\frac {h}{2e}})$

Energy Gap in Superconductors:

$\Delta E=2\Delta$

$\Delta E$ : Energy required to break a Cooper pair
$\Delta$ : Energy gap parameter

Engineering Design

Levitation System Equations

Magnetic Force Equation:

$\mathbf {F} _{\text{mag}}=\nabla (\mathbf {m} \cdot \mathbf {B} )$

$\mathbf {F}$ _mag: Magnetic force
$\mathbf {m}$ : Magnetic moment
$\mathbf {B}$ : Magnetic field

Electrostatic Force Equation:

$\mathbf {F} _{\text{elec}}=Q\mathbf {E}$

$\mathbf {F}$ _elec: Electrostatic force
Q: Electric charge
$\mathbf {E}$ : Electric field

Control Systems and Simulations

Levitation Control Equations

State Equations:

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

$\mathbf {x}$ : Position vector
$\mathbf {v}$ : Velocity vector
m: Mass of the core
$\mathbf {F}$ _dist: Disturbance force

Cost Function for Nonlinear Model Predictive Control (NMPC):

$J=\int _{t}^{t+T_{p}}\left[\|\mathbf {x} _{\text{ref}}(t)-\mathbf {x} (t)\|_{Q}^{2}+\|\mathbf {u} (t)\|_{R}^{2}\right]dt$

J: Cost function
T_p: Prediction horizon
$\mathbf {x}$ _ref: Reference position
$\mathbf {u}$ : Control input
Q, R: Weighting matrices

Charge Regulation Equations

Sliding Surface Definition:

$s(t)=e(t)+\lambda \int _{0}^{t}e(\tau )d\tau$

s(t): Sliding surface
e(t) = q_{\text{ref}}(t) - q(t): Charge error
$\lambda$ : Positive constant

Control Law:

$u(t)=-k\cdot {\text{sign}}(s(t))+{\dot {q}}_{\text{ref}}(t)$

u(t): Control input
k: Adaptive gain
sign(s(t)): Sign function

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.

Environmental Alignment

Schumann Resonance Frequencies

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

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

Mathematical Modeling

System Dynamics Equations

Core Motion Equations:

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

${\ddot {\mathbf {x} }}$ : Acceleration
$\mathbf {I}$ : Coil currents

State-Space Representation:

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

${\boldsymbol {\theta }}$ : Orientation angles
${\boldsymbol {\omega }}$ : Angular velocities
${\boldsymbol {\tau }}$ _mag, ${\boldsymbol {\tau }}$ _elec: Magnetic and electrostatic torques
$\mathbf {I}$ : Moment of inertia tensor

Sliding Surface for Adaptive Sliding Mode Control (ASMC):

$s(t)=e(t)+\lambda \int _{0}^{t}e(\tau )d\tau$

Control Law for ASMC:

$u(t)=-k\cdot {\text{sign}}(s(t))+{\dot {q}}_{\text{ref}}(t)$

Control Algorithms Parameters

Parameters Definitions:

m: Mass of the core
$\mathbf {x}$ , $\mathbf {v}$ : Position and velocity vectors
${\boldsymbol {\theta }}$ , ${\boldsymbol {\omega }}$ : Orientation and angular velocity vectors
$\mathbf {F}$ _mag, $\mathbf {F}$ _elec: Magnetic and electrostatic forces
$\mathbf {F}$ _dist: Disturbance forces
$\mathbf {I}$ : Moment of inertia tensor
${\boldsymbol {\tau }}$ _mag, ${\boldsymbol {\tau }}$ _elec: Magnetic and electrostatic torques
e(t): Error signal
$\lambda$ : Positive constant for sliding surface
k: Adaptive gain for control law
$\mathbf {u} (t)$ : Control input vector

Key Constants and Physical Quantities

Planck's Constant (h): $(6.62607015\times 10^{-34})Js$
Reduced Planck's Constant ( $\hbar$ ): $({\frac {h}{2\pi }})$
Speed of Light (c): $(3.0\times 10^{8})m/s$
Elementary Charge (e): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $1.602176634 \times 10^{-19}$ C }
Vacuum Permittivity ( $\epsilon _{0}$ ): $(8.854187817\times 10^{-12})F/m$

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.

MHD Core

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