be used to understand the problems related to voice transmission over telephonic channels. The telephony was coming into existence. The modern telephone system is an outcome of the efforts of several innovators. However, Graham Bell got the first patent of a telephone in the year 1874. The transmitted telephonic voice signal was distorted. Therefore, an analytical model was urgently needed to improve the quality of telephonic transmission. Heaviside in 1876 introduced the line inductance L p.u.l. and reformulated the cable theory of Kelvin using Kirchhoff circuital laws [B1, B.3]. The formulation resulted in the wave equation for both the voltage (V) and current (I) waves on the line:
(1.1.4)
In the case of line inductance L = 0, the above equation is reduced to the diffusion type cable equation (1.1.3) of Kelvin. Using the Fourier method, Heaviside solved the aforementioned time‐domain equation. Only in 1887, he could introduce the line conductance G p.u.l in his formulation to account for the leakage current in an imperfect insulating layer between two conductors. Finally, Heaviside obtained a set of coupled transmission line equations using all four line constants R, L, C, and G. Subsequently, the coupled transmission line equations were called the Telegrapher's equations. At the end, Heaviside obtained the following modified wave equation:
To solve the above time‐domain equation, Heaviside developed his own intuitive operational method approach by defining the operator ∂/∂t → p. The use of the operator reduced the above partial differential equation to the ordinary second‐order differential equation. Finally, he solved the equation under initial and final conditions at the ends of a finite length line. In the process, he obtained the expressions for the characteristic impedance and propagation constant in terms of line parameters. Heaviside could obtain results for the line under different conditions. For a lossless line, R = G = 0, the equation (1.1.5b) is obtained. Conceptually, the characteristic impedance provided a mechanism to explain the phenomenon of wave propagation on an infinite line. At each section of the line, it behaved like a secondary Huygens's source providing the forward‐moving wave motion. Heaviside also obtained the condition for the dispersionless transmission on a real lossy line, and suggested the inductive loading of a line to reduce the distortion in both the telegraph and telephone lines. Afterward, his intuitive operational method approach developed into the formal Laplace transform method, widely used to solve the differential equations [J.19, J.20, B.1–B.3, B.13].
The method of Heaviside was further extended by Pupin in 1899 and 1900. Pupin introduced the harmonic excitation in the wave equation as a real part of the source V0ejpt [J.21, J.22]. This was an indication of the use of the modern phasor solution of the wave equation. Similar analytical works, and also practical inductive loading of the line was done by Campbell at Bell Laboratory. He published the results in 1903 [J.23]. In July 1893, Steinmetz introduced the concept of phasor to solve the AC networks of RLC circuits. In 1893, Kennelly also published the use of complex notation in Ohm's law for the AC circuits [J.24]. Carson in 1921 applied the method to solve Maxwell's equations for the wave propagation on closely spaced lines, and also analyzed for the mutual impedances. Carson in 1927 developed the electromagnetic theory of the Electric Circuits, and paved the way for the modeling of the wave phenomena using the circuit models [J.25, J.26].
Peijel in 1918, and Levin in 1927 analyzed the wave propagation on the parallel lines. Levin extended the telegrapher's equations to the multiconductor transmission lines using Maxwell's equations [J.27]. In 1931 Bewley presented a set of wave equations on the coupled multiconductor lines. Subsequently, Pipes introduced the matrix method to formulate the wave propagation problem on the multiconductor lines [J.28, J.29]. Thus, the theoretical foundation was laid to deal with the complex technical problems related to transmission lines. Starting with Marconi wireless in 1895, several improvements took place in the long‐range wireless telegraphy. Also, the audio broadcasting was developed between 1905 and 1906 and commercially, around 1920–1923, in the long‐wave, medium‐wave, and short‐wave RF frequency bands [B.5]. Now, the time was ripe for microwave and mm‐wave communication.
The above discussion shows that the Telegrapher's equations have come in existence due to the contributions of both Kelvin and Heaviside. To recognize their contributions, we call in this book the Telegrapher's equations as the Kelvin‐Heaviside transmission line equations. Also, as the characteristic impedance behaves as the secondary Huygens's source, so it can also be viewed as the Huygens's load. Such Huygens's load distributed over a surface forms the modern Huygens's metasurface, discussed in the chapter 22 of this book.
1.1.4 Waveguides as Propagation Medium
Heaviside reformulated Maxwell equation in 1884. He rejected the idea of EM‐wave propagation in a hollow metallic cylinder. In his opinion, two conductors, alternatively one conductor and the earth as a ground conductor are essential for the EM‐wave propagation. However, in 1893 J.J. Thomson expressed the possibility of the EM‐wave propagation in a hollow cylinder [B.12]. Next year, Oliver Lodge verified it experimentally. In the year 1895, J.C. Bose used the waveguide and horn antenna for the mm‐wave transmission and reception. In 1897 he reported the work at Royal Institution in London [B.5]. However, it was Rayleigh who carried out a detailed solution of boundary‐value problems. He obtained the normal mode solution, showing wave propagation in the form of the distinct discrete modes, i.e. the normal modes. He obtained his solutions for both the TE and TM modes, and introduced the concept of the cutoff frequency for modes. He further examined the EM‐wave propagation on a dielectric waveguide [J.30]. In 1920 Rayleigh, Sommerfeld and Debye continued the researches in this direction.
However, only in 1930 proper experimental investigations of the wave propagation in the waveguides were undertaken by G. C. Southworth at Bell Labs, and W.L. Barrow at MIT. In 1934, microwave commercial link was established, and in 1936, Southworth and Barrow discovered the possibility of using the waveguide as a transmission medium. However, they published their works only in 1936 [J.31, J.32, B.5]. During the same time‐period, Brillouin also investigated the wave propagation in a tube [J.33]. Serious analytical work on waveguides was further undertaken by J.R Carson, S.P. Mead, and S.A. Schelkunoff around 1933 [J.34]. Almost forgotten analytical works of Rayleigh was reinvented. Chu and Barrow further investigated the EM‐waves propagation in the elliptical and rectangular hollow metallic pipes [J.35]. During 1934, Schelkunoff extended the concept of impedance to the EM‐wave propagation in the coaxial line, and obtained the transmission line equations using the electromagnetic theory [J.36]. In 1937, he further extended the theory to the TE and TM mode guided wave propagations, and obtained the circuit models of mode supporting waveguides. Finally, Schelkunoff generalized the standard telegrapher's equation, using Maxwell's EM‐theory to represent an infinite set of uncoupled and coupled modes of a waveguide by the system of uncoupled and coupled transmission line equations [J.37–J.39]. Subsequently, his method has been extended to planar lines in an inhomogeneous medium supporting the hybrid modes [B.13].
During the World War‐II period, important theoretical and practical works were done in the field of waveguide technology for the development of the waveguide‐based components and systems. The development of Radar provided the impetus for such research activities.
1.2 Planar Transmission Lines
A brief review of the development of planar transmission lines, influencing modern microwave
