Weak form of Helmholtz equation

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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

A way to memorize the boundary conditions

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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

Recognize the trigonometric functions again

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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

Dot product in cylindrical coordinates

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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

Differential operators in curvilinear coordinates

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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

AD650 Voltage Frequency Converter

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The AD650 is a charge balance voltage-to-frequency converter. The input signal is converted into an equivalent current by the input resistance RIN. This current is exactly balanced by an internal feedback current delivered in short, timed bursts from the switched 1 mA internal current source as shown in Fig.1. These bursts of current may be … Read more

Space Charge Limitation

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The charges at the cathode reduces the potential that is present in the absence of the electron so that further emission is prevented. The maxium current is said to be the space charge limitation. Child and Langmuir first studied the space charge limitation in 1910s[1,2]. It assumes that two infinite parallel plane electrodes at fixed … Read more

Maxwell’s equation

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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