Control fusion plasma | microwave | live formula | algebra | function plot
Gyrotron is a microwave oscillators. It changes the DC energy to high power millimetre microwave. It has been successfully used in the magnetic confinement thermonuclear fusion research as the microwave source for the electron cyclotron resonance heating and current drive. A typical gyrotron cavity consists of a central section and one taper section at both … Read more
Weak form is an important tool for the analysis of electromagnetic problems. It is a bridge from the partial differential equations to the linear algebra. For example, the Helmholtz equation with wave number \(k_c\) is \(\nabla_t^2 \phi+k_c^2 \phi=0\) It is so-called “strong form”, in which the unknow is in the 2nd order differential operator. It … Read more
It is very important to specify boundary conditions at material interfaces and physical boundaries in order to get a full description of an electromagnetic problem. However the boundary conditions between two media is difficult to memorize. Here we present a way to memorize them. The maxwell’s equations are \(\nabla \times H=J+\partial_t D\) \(\nabla \times E=-\partial_t … Read more
We always learn the trigonometric functions from the geometry and use them to solve geometry problem. It it quite difficult to memorize some trigonometric identities. We re-define the sine and cosine function from the view of the solution of the ordinary differential equations in this article. It is similar with the Bessel functions. The Helmholtz … Read more
The dot product of the two vectors in cartesian coordinates is defined as: Multiply corresponding components and the add the results. Thus \( \vec a \cdot \vec b=a_1b_1+a_2b_2+a_3b_3\) If there are two vectors in cylindrical coordinates, \(\vec{A} = A_r \hat{r} + A_\theta \hat{\theta} + A_z \hat{z}\) \(\vec{B} = B_r \hat{r} + B_\theta \hat{\theta} + B_z … Read more
We have to use the cylindrical coordinates when the problem is cylindrical symmetry. For example, the eigenmode in a circular waveguide. Althrough there is a general theory for the differential operators in general curvilinear coordinates, it it very difficult to be understanded. There is also the formular for the differential operators in cylindrical coordinates. However … Read more
The electrons beam is usually confined by magnetics in the vacuum tube. The plasma is usually confined by magnetics in tokomaks. The motion is govend by the Newton’s 2nd law. However the cylinderical coordinates are used in both cases as the fields are axially symmetric. The common question is how to get the equations of … Read more
Maxwell’s equation in vacuum Name Integral equations Differential equations Gauss’s law Gauss’s law for magnetism Maxwell–Faraday equation(Faraday’s law of induction) Ampère’s circuital law (with Maxwell’s addition) Next, We obtain the electromagnetic wave equation in a vacuum. It is also called Helmholtz equation. Take the Curl of the Faraday equation \(\nabla \times \left(\nabla \times \mathbf {E} \right)=\nabla \times … Read more