Introduction To Modern Planar Transmission Lines. Anand K. Verma

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potential with momentum. The magnetic vector potential could be considered as the potential momentum per unit charge, just as the electric scalar potential ϕ is the potential energy per unit charge [J.16]. The potential momentum is obtained as follows:

      (1.1.2)

In the above equation, is the force acting on the charge q, and is given by equation (1.1.1b).

Lebedev in 1900 experimentally demonstrated the radiation pressure, demonstrating momentum carried by the EM‐wave. The energy and momentum carried by the EM‐wave indicate that the light radiation could be viewed as some kind of particle, not a wave phenomenon. A particle is characterized by energy and momentum. Such a dual nature of light is a quantum mechanical duality phenomenon. Einstein introduced the concept of the light particle, called “photon” to explain the interaction of light with matter, i.e. the photoelectric effect. However, Lorentz retained the classical wave model to explain the interaction between radiation and matter via polarization of dipoles in a material creating its frequency‐dependent permittivity.

      It is to be noted that at a location in the space, even for zero and fields, the potentials and ϕ could exist [B.11]. Aharonov–Bohm predicted that the potential fields and ϕ, in the absence of and , could influence a charged particle. Tonomura and collaborators experimentally confirmed the validity of Aharonov–Bohm prediction. The Aharonov–Bohm effect demonstrates that and fields only partly describe the EM‐fields in quantum mechanics. The vector potential also has to be retained for a complete description of the EM‐field quantum mechanically [J.3, J.16, J.17]. However, to solve the classical electromagnetic problems, such as guided wave propagation and radiation from antenna, Heaviside formulation of Maxwell equations and potential functions as additional tools is adequate.

      EM‐Modeling of Medium

      The above brief review omitted developments in the electromagnetic properties of the material medium. A few important developments could be summarized. In 1837, Faraday introduced the concept of the dielectric constant of a material. In 1838, he introduced the concept of electric polarization in dielectrics under the influence of the external electric field. Soon after the discovery of the electron in 1897 by J.J. Thomson, the models around electrons were developed to describe the electromagnetic properties of a material. Around 1898, John Gaston Leathem obtained an important relation , connecting the displacement of charges in a material with polarization. Kelvin, in the year 1850, developed the concept of magnetic permeability and susceptibility with separate concepts of , , and to characterize a magnetic material. In 1900, Drude developed the electrical conduction model, now known as the Drude model, after electron theory. Subsequently, the model was extended to the dielectric medium by Lorentz in 1905. The model called the Drude‐Lorentz model explains the dispersive property of dielectrics. In the year 1912, Debye developed the concept of dipole moment and obtained equations relating it to the dielectric constant. These models laid the foundation to study of the electric and magnetic properties of natural and engineered materials under the influence of external fields [B.4, B.6, B.7, B.12]

      1.1.3 Development of the Transmission Line Equations

      Kelvin's Cable Theory

      During the period 1840–1850, several persons conceived the idea of telegraph across the Atlantic Ocean. Finally, in the year 1850, the first under‐sea telegraphy, between Dover (Kent, England) and Calais (France), was made operational. However, no cable theory was available at that time to understand the electrical behavior of signal transmission over the undersea cable.

      In 1854, Kelvin modeled the under‐sea cable as a coaxial cable with an inner conductor of wire surrounded by an insulating dielectric layer, followed by the saline sea‐water acting as the outer conductor [J.18, B.1]. The coaxial cable was modeled by him as a distributed RC circuit with the series resistance R per unit length (p.u.l.) and shunt capacitance C p.u.l. It was the time of the fluid model of electricity. Kelvin further conceived the flow of electricity similar to the flow of heat in a conductor. Fourier analysis of 1D heat flow was in existence since 1822. Following the analogy of heat equation of Fourier, Kelvin obtained the diffusion type equation for the transmitted voltage signal over the under‐sea coaxial cable:

(1.1.3)

This is the first Cable Theory; Kelvin called the above equation the equation of electric excitation in a submarine telegraph wire. Kelvin's model did not account for the inductance L p.u.l. and the conductance G p.u.l. of the cable. The cable inductance L is due to the magnetic effect of current, and G is due to the leakage current between the inner and outer conductors. However, cable theory was a great success. Following the method of Fourier, he solved the equation for both the voltage and current signals. At any distance x on the cable, a definite time‐interval was needed to get the maximum current of the received signal. The galvanometer was used to detect the received current. This time‐interval called the retardation time of the received current signal also depends on the square of the distance. Moreover, the telegraph signals constituted of several waves of different frequencies, and their propagation velocities were frequency‐dependent. It limited the speed of signal transmission for long‐distance telegraphy. The conclusions of Kelvin's analysis were ignored, and 1858 transatlantic cable worked only for three weeks. It failed due to the application of 2000 V potential pulse on the cable. The speed of transmission was just 0.1 words per minute. Finally, following Kelvin's advice and using a very sensitive mirror galvanometer invented by him, the transatlantic telegraph was successfully completed in 1865 with eight words per minute transmission speed [B.1–B.3].

      Heaviside Transmission Line Equation

      The limitation of the speed of telegraph signals was not understood at that time. The RC model of the cable, leading to the diffusion equation, and use of the time‐domain pulse could not explain it. Moreover,

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