Longitudinal-section mode

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Longitudinal-section modes are a set of a particular kind of electromagnetic transmission modes found in some types of transmission line. They are a subset of hybrid electromagnetic modes (HEM modes). [1] HEM modes are those modes that have both an electric field and a magnetic field component longitudinally in the direction of travel of the propagating wave. Longitudinal-section modes, additionally, have a component of either magnetic or electric field that is zero in one transverse direction. In longitudinal-section electric (LSE) modes this field component is electric. In longitudinal-section magnetic (LSM) modes the zero field component is magnetic. Hybrid modes are to be compared to transverse modes which have, at most, only one component of either electric or magnetic field in the longitudinal direction.

The mode of electromagnetic radiation describes the field pattern of the propagating waves. Electromagnetic modes are analogous to the normal modes of vibration in other systems, such as mechanical systems.

Transmission line specialized cable or other structure designed to carry alternating current of radio frequency

In radio-frequency engineering, a transmission line is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

Electric field spatial distribution of vectors representing the force applied to a charged test particle

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. Mathematically the electric field is a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strengh. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

Contents

Derivation and notation

There is an analogy between the way transverse modes (TE and TM modes) are arrived at and the definition of longitudinal section modes (LSE and LSM modes). When determining whether a structure can support a particular TE mode, one sets the electric field in the z direction (the longitudinal direction of the line) to zero and then solves Maxwell's equations for the boundary conditions set by the physical structure of the line. One can just as easily set the electric field in the x direction to zero and ask what modes that gives rise to. Such modes are designated LSE{x} modes. Similarly there can be LSE{y} modes and, analogously for the magnetic field, LSM{x} and LSM{y} modes. When dealing with longitudinal-section modes, the TE and TM modes are sometimes written as LSE{z} and LSM{z} respectively to produce a consistent set of notations and to reflect the analogous way in which they are defined. [2]

A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of the radiation in the plane perpendicular to the radiation's propagation direction. Transverse modes occur in radio waves and microwaves confined to a waveguide, and also in light waves in an optical fiber and in a laser's optical resonator.

Maxwells equations set of partial differential equations that describe how electric and magnetic fields are generated and altered by each other and by charges and currents

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at the speed of light. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.

Both LSE and LSM modes are a linear superposition of the corresponding TE and TM modes (that is, the modes with the same suffix numbers). Thus, in general, the LSE and LSM modes have a longitudinal component of both electric and magnetic field. Likewise the LSM modes are found by setting one of the transverse components of magnetic field to zero with analogous results. [2]

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Occurrence

LSE and LSM modes can occur in some types of planar transmission line with non-homogeneous transmission media. There are some structures that are unable to support a pure TE or TM mode and consequently the transmission mode must necessarily be hybrid. [3]

Planar transmission line Transmission lines with flat ribbon-like conducting or dielectric lines

Planar transmission lines are transmission lines with conductors, or in some cases dielectric (insulating) strips, that are flat, ribbon-shaped lines. They are used to interconnect components on printed circuits and integrated circuits working at microwave frequencies because the planar type fits in well with the manufacturing methods for these components. Transmission lines are more than simply interconnections. With simple interconnections, the propagation of the electromagnetic wave along the wire is fast enough to be considered instantaneous, and the voltages at each end of the wire can be considered identical. If the wire is longer than a large fraction of a wavelength, these assumptions are no longer true and transmission line theory must be used instead. With transmission lines, the geometry of the line is precisely controlled so that its electrical behaviour is highly predictable. At lower frequencies, these considerations are only necessary for the cables connecting different pieces of equipment, but at microwave frequencies the distance at which transmission line theory becomes necessary is measured in millimetres. Hence, transmission lines are needed within circuits.

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Coplanar waveguide

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References

  1. Zhang & Li, p. 188
  2. 1 2 Zhang & Li, pp. 294–299
  3. Zhang & Li, pp. 332

Bibliography

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