MHD Core: Difference between revisions
Line 296: | Line 296: | ||
== Key Constants and Physical Quantities == | == Key Constants and Physical Quantities == | ||
* '''Planck's Constant (''h''):''' \(6.62607015 \times 10^{-34}\) Js | * '''Planck's Constant (''h''):''' <math>\(6.62607015 \times 10^{-34}\) Js</math> | ||
* '''Reduced Planck's Constant (''\hbar''):''' \(\frac{h}{2\pi}\) | * '''Reduced Planck's Constant (''<math>\hbar/math>''):''' \(\frac{h}{2\pi}\)/math> | ||
* '''Speed of Light (''c''):''' \(3.0 \times 10^8\) m/s | * '''Speed of Light (''c''):''' <math>\(3.0 \times 10^8\) m/s/math> | ||
* '''Elementary Charge (''e''):''' \(1.602176634 \times 10^{-19}\) C | * '''Elementary Charge (''e''):''' <math>\(1.602176634 \times 10^{-19}\) C/math> | ||
* '''Vacuum Permittivity (''\epsilon_0''):''' \(8.854187817 \times 10^{-12}\) F/m | * '''Vacuum Permittivity (''<math>\epsilon_0/math>''):''' <math>\(8.854187817 \times 10^{-12}\) F/m/math> | ||
---- | ---- | ||
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. | 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. |
Revision as of 09:57, 10 November 2024
Magneto Hydro Dynamic Core
A Levitation Power Core
Fundimental Technology for the operation of a Star Speeder and Magneto Speeder
https://github.com/Jthora/MHD-Core
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: Zero-point energy
- : Reduced Planck's constant
- : Angular frequency
Casimir Effect Force Between Two Plates:
- FCasimir: Casimir force
- c: Speed of light
- A: Area of the plates
- L: Separation between the plates
Dynamic Casimir Effect Photon Generation Rate:
- : Photon generation rate
- cavity: Resonant frequency of the cavity
- : Modulation amplitude of cavity length
- : Original cavity length
Expectation Value of the Energy-Momentum Tensor:
- Tμν: Energy-momentum tensor
- gμν: Metric tensor of spacetime
Electromagnetic Field Specialist's Equations
Modified Wave Equation with Scalar Potential:
- : Scalar potential
- : Charge density
- : Vacuum permittivity
Magnetic Flux Quantum:
- 0: Magnetic flux quantum
- h: Planck's constant
- e: Elementary charge
Material Development
Superconducting Material Properties
Yttrium Barium Copper Oxide (YBCO):
- Critical Temperature (Tc): Approximately 92 K
- Critical Current Density (Jc): 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:
- : Magnetic flux through a superconducting loop
- n: Integer (quantum number)
- 0: Magnetic flux quantum
Energy Gap in Superconductors:
- : Energy required to break a Cooper pair
- : Energy gap parameter
Engineering Design
Levitation System Equations
Magnetic Force Equation:
- mag: Magnetic force
- : Magnetic moment
- : Magnetic field
Electrostatic Force Equation:
- elec: Electrostatic force
- Q: Electric charge
- : Electric field
Control Systems and Simulations
Levitation Control Equations
State Equations:
- : Position vector
- : Velocity vector
- m: Mass of the core
- dist: Disturbance force
Cost Function for Nonlinear Model Predictive Control (NMPC):
- J: Cost function
- Tp: Prediction horizon
- ref: Reference position
- : Control input
- Q, R: Weighting matrices
Charge Regulation Equations
Sliding Surface Definition:
- s(t): Sliding surface
- e(t) = q_{\text{ref}}(t) - q(t): Charge error
- : Positive constant
Control Law:
- 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
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
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:
- : Acceleration
- : Coil currents
State-Space Representation:
- : Orientation angles
- : Angular velocities
- mag, elec: Magnetic and electrostatic torques
- : Moment of inertia tensor
Sliding Surface for Adaptive Sliding Mode Control (ASMC):
Control Law for ASMC:
Control Algorithms Parameters
Parameters Definitions:
- m: Mass of the core
- , : Position and velocity vectors
- , : Orientation and angular velocity vectors
- mag, elec: Magnetic and electrostatic forces
- dist: Disturbance forces
- : Moment of inertia tensor
- mag, elec: Magnetic and electrostatic torques
- e(t): Error signal
- : Positive constant for sliding surface
- k: Adaptive gain for control law
- : Control input vector
Key Constants and Physical Quantities
- Planck's Constant (h): Failed to parse (syntax error): {\displaystyle \(6.62607015 \times 10^{-34}\) Js}
- Reduced Planck's Constant (<math>\hbar/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>
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.