Introduction To Modern Planar Transmission Lines. Anand K. Verma

line. It is to be noted that even the voltaic pile, i.e. the chemical battery of Volta was nonexistent at that time. The Leiden jar, a capacitor to store the static electrical charges, was invented just two years before in 1745. However, much earlier in the year 1663, Otto Von Guericke studied the phenomenon of static electricity and designed a machine to produce it. Thus, the charged Leiden jar became a source of electricity, and single‐wire transmission was the communication medium. These were two important ingredients to establish the telegraph link. The third ingredient of telegraphy, the electroscope, invented by William Gilbert around 1600, acted as the receiver for the coded signals. It is interesting to note that the telegraph was conceived without any theoretical investigations on electricity. Even Coulomb's law was discovered later in 1785. However, the proper telegraphy could be developed only after the invention of voltaic pile, i.e. a chemical battery by Volta in 1799. Further development of the telegraph has an involved history. In 1837, Morse patented his telegraph in the United States, and on January 6, 1838, the first telegram was sent over 3 km distance. Cooke commercialized the telegraph in England and established the 21 km link on April 9, 1839. Thus, an era of electrical communication heralded. In 1844, long‐distance Morse's telegraph between Washington, DC, and Baltimore, Maryland was established. Before further reviewing the growth of the classical transmission lines, it is useful to have a quick look at the theoretical developments related to electricity [B.5].

      

      1.1.2 Development of Theoretical Concepts in EM‐Theory

      The ancients unfolded our story of Electromagnetism through careful observations of the phenomena of static electricity and natural magnetism. The developments of basic concepts, analytical modeling, and theoretical formulations used in the EM‐Theory are emphasized in the review. The theoretical concepts of the electric and magnetic fields followed the mathematical models developed for the gravitational force by Newton in 1687, and subsequently refined by other investigators. The development of the theoretical models of transmission lines inherited the modeling process, and mathematical method of Fourier developed for the transmission of heat in a rod.

      Electrostatics and Scalar Potential

      Newton published his theory of gravitation in his monograph Philosophiae Naturalis Principia Mathematica. Newton viewed the gravitational interaction between two masses through force. The effect of static electricity was known for a long time, at least since 600 BC. However, only in 1600, Gilbert carried out systematic studies of both magnetism and static electricity. The static electricity was generated by the rubbing of two specific objects. He suggested the word electricus for electricity, and the English word “electricity” was suggested by Thomas Browne in 1646. Gilbert also suggested that the electrical effect is due to the flow of a small stream of weightless particles called effluvium. This concept helped the formulation of one‐ and two‐fluid model of electricity. He also invented the first electrical measuring instrument, the electroscope, which helped further experimental investigations on electricity.

      In 1733, Fay proposed that electricity comes in two forms – vitreous and resinous, and on combination, they cancel each other. The flow of the two forms of electricity was explained by the two‐fluid model. During this time interval, around up to 1745, the electrical attraction and repulsion were explained using the flow of Gilbert's particle effluvium. In 1750, Benjamin Franklin proposed the one‐fluid model of electricity. The matter containing a very small quantity of electric fluid was treated as negatively charged, and the matter with excess electric fluid was treated positively charged. Thus, the negative charge was resinous electricity, and the positive charge was vitreous electricity. Now, the stage was ready for further theoretical and experimental investigations on electricity.

      In the year 1773, Lagrange introduced the concept of the gravitational field, now called the scalar potential field, created by a mass. The gravitational force of Newton was conceived as working through the gravitational field. The scalar potential field has appeared as a mechanism to explain the gravitational force interaction between two masses. Thus, a mass located in the potential field, described by a function called the potential function, experiences the gravitational force. In 1777, Lagrange also introduced the divergence theorem for the gravitational field. The nomenclature potential field was introduced by Green in 1828. Subsequently, Gauss in 1840 called it “potential.” Laplace in 1782 showed that the potential function ϕ (x,y,z) satisfies the equation ∇²ϕ = 0. Now the equation is called Laplace's equation.

Following the Law of Gravitation, Coulomb postulated similar inverse square law, now called Coulomb's law, for the electrically charged bodies. He experimentally demonstrated the inverse square law for the charged bodies in 1785. Thus, the mathematical foundation of the EM‐theory was laid by Coulomb. The law was also applicable to magnetic objects. The interaction between charged bodies was described by the electric force. In the year 1812, Poisson extended the concept of potential function from the gravitation to electrostatics. Incorporating the charge distribution function ρ, he obtained the modified Laplace's equation, written in modern terminology, ∇²ϕ = −ρ/ε₀. This equation now called Poisson's equation is the key equation to describe the potential field due to the charge distribution. In the same year, Gauss rediscovered the divergence theorem originally discovered by Lagrange for the gravitational field.

      In the year 1828, Green coined the nomenclature – the potential function, for the function of Lagrange and modern concept of the scalar potential field came into existence. Green also showed an important relation between the surface and volume integrals, now known as Green's Theorem. Green applied his method to the static magnetic field also. Green also introduced a method to solve the 3D inhomogeneous Poisson partial differential equation where the considered source is a point charge. The point charge is described by the Dirac's delta function. The solution of the Poisson's partial differential equation, using Dirac's delta function, is now called Green's function. Neumann (1832–1925) extended the Green's function method to solve the 2D potential problem and obtained the eigenfunction expansion of 2D Green's function [J.1–J.5, B.2, B.7].

      Magnetic Effect of Current

      So far, we have paid attention to the electrostatics. At this stage, Leiden jar was the only source of static electricity. A source for the continuous electric current was not available. Volta in 1799 invented the voltaic pile, i.e. a chemical battery, and the first time a continuous source of electric current came into existence. On April 21, 1820, it led to the discovery of the magnetic effect of current flowing in a wire. The electric current became the source of the magnetic field, encircling the current‐carrying wire. The magnetic effect of current was discovered by Orsted (Oersted). In the same year, Ampere showed that the co‐directional parallel currents flowing in two wires attract each other, and the counter‐currents repel each other. It was a very significant discovery, i.e. creation of the attractive and repulsive magnetic forces without any physical magnet. It firmly established the relation between electricity and magnetism. Ampere further developed an equation, presently called Ampere's Circuital Law, to connect the current flowing in a wire to the magnetic field around it and developed the right‐hand rule. He called the new field of electricity Electrodynamics and Maxwell recognized him as the Father of Electrodynamics. Ampere further modeled the natural magnetic materials as the materials composed of perpetual tiny circulating electric currents. He demonstrated the validity of his concept using the current‐carrying conductor in the helical form called a solenoid. The solenoid worked like an artificial bar magnet. In the year 1820 itself, Biot–Savart obtained the equation using the line integral to compute the magnetic field at a position in the space due to the current flowing in a wire [J.2, B.6, B.7].

      Ohm's Law

      The voltaic pile helped the discovery of the magnetic effect of current; however, surprisingly the relation between the current flowing in resistance and voltage across it, known as the Ohm's law,