Introduction To Modern Planar Transmission Lines. Anand K. Verma
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Objectives
To review the EM‐field quantities and medium parameters.
To review the basic electrical properties of media.
To obtain elementary circuit models of media.
To review Maxwell’s equations.
To present the wave equation in the unbounded lossless and lossy isotropic dielectric medium.
To review wave polarizations.
Jones matrix description of polarization states.
To present the wave equation in the unbounded lossless anisotropic dielectric medium.
4.1 Basic Electrical Quantities and Parameters
The electrical charge and the electric current are the primary electrical sources for the creation of the electric field and the magnetic field, respectively. The charge, also current (displacement current), is described by the flux field, i.e. the flux density (
4.1.1 Flux Field and Force Field
The electric and magnetic fields are visualized through the line of flux. Thus, the electric charge (Qe) is described by the electric flux (Ψe). The total charge (QT) on a physical body and the corresponding electric flux are related by Gauss’s law:
(4.1)
If the charge is distributed throughout the volume of a body, in the form of volume charge density ρe, the elemental charge in the volume element dv is ρedv. The flux is further expressed as the electric flux density (D), i.e. the flux per unit area. The flux through the elemental surface area is
Gauss’s Law for Electric Flux
Total electric flux coming out of a closed surface = Total charge enclosed inside the volume of a closed surface, i.e.
(4.1.2)
The above expression is the integral form of Gauss’s law. It can be converted to the differential form by using Gauss’s vector integral identity,
(4.1.3)
Gauss’s Law for Magnetic Flux
Similar to the electric charge distribution, the magnetic charge distribution can be assumed in a volume of the body. The magnetic charge density is expressed as ρm. The magnetic charge creates a magnetic flux Ψm. Similar to the case of the electric charge, the elemental magnetic charge in the volume dv is ρmdv, and the elemental magnetic flux coming out of the surface is
(4.1.4)
Figure 4.1 The unit vector
Again, by using Gauss’s vector integral identity, the above expression is written below in the differential form:
(4.1.5)
However, the magnetic charges are not found in nature, i.e. ρm = 0. Therefore,
(4.1.6)
The amount of charge, or current, is an absolute quantity. It does not dependent on the material medium. Thus, the corresponding flux or the flux density is also not dependent on the surrounding medium. In brief, the charge and current create the electric and magnetic flux field, i.e. the flux densities
Experiments demonstrate that the electrically charged body, or a current‐carrying conductor, interacts with other charged body, or another current‐carrying conductor. Such interaction, i.e. the mutual force, is influenced by the medium surrounding these bodies. Therefore, medium‐independent flux densities
4.1.2 Constitutive Relations