Introduction To Modern Planar Transmission Lines. Anand K. Verma

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, it became obvious that the RC model couldn't 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:

(1.1.5)

      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 V₀e^jpt [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,