Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 28

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, 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.