of chapter 4.
2.1.2 Circuit Model of Transmission Line
A physical transmission line, supporting the voltage/current wave, can be modeled by the lumped R, L, C, G components, i.e. the resistance, inductance, capacitance, and conductance per unit length (p.u.l.), respectively. The two‐conductor transmission line can acquire many physical forms. A few of these forms are shown in Fig (2.5). The lines as shown in Fig (2.5a–c) support the wave propagation in the transverse electromagnetic mode, i.e. in the TEM‐mode; while Fig (2.5c) shows the quasi‐TEM mode‐supporting microstrip. For TEM mode wave propagation, the electric field and magnetic field are normal to each other and also normal to the direction of wave propagation. For the TEM mode, there is no field component along the direction of propagation. However, the quasi‐TEM mode also has component of weak fields along the longitudinal direction of wave propagation. The quasi‐TEM mode is a hybrid mode discussed in the subsection (7.1.4) of chapter 7.
All TEM mode supporting transmission lines can be represented by a parallel two‐wire transmission line shown in Fig (2.6a). A transmission line is a 1D wave supporting structure. Its cross‐sectional dimension is much less than λ/4; otherwise, its TEM nature is changed. The longitudinal dimension can have any value, from a fraction of a wavelength to several wavelengths. The mode theory of the electromagnetic (EM) wave propagation is further discussed in chapter 7.
Figure 2.5 Cross‐section of a few two‐conductor transmission lines.
Figure 2.6 RLCG lumped circuit model of a transmission line.
A two‐conductor transmission line, or any other line supporting the TEM mode, is modeled as a chain of discrete passive RLCG components. As a matter of fact, by cascading several sections of discrete L‐network of the series L and shunt C elements, or even discrete L‐network of the series C and shunt L elements, an artificial transmission line can also be constructed. The artificial transmission line is discussed in section (3.4) of chapter 3, and also in the chapters 19 and 22. It plays a very important role in modern microwave planar technology. The behavior of a transmission line is determined in terms of the resistor (R), inductor (L), capacitor (C), and conductance (G); all line elements are in per unit length, i.e. p.u.l. Kelvin introduced the modeling of the telegraph cable laid in the ocean using the RC circuit model. Heaviside further introduced L and G components in the circuit model to improve the modeling of the lossy transmission line [B.1, B.2, J.1, J.2]. Kelvin RC‐circuit model of the transmission line leads to the diffusion equation, not to the wave equation. Whereas using the RLGC circuit model, Heaviside finally obtained the wave equation for the voltage/current on a transmission line. Using the RLCG circuit model, shown in Fig (2.6b), the voltage, and current equations are obtained for the transmission line. The set of the coupled voltage and current equations are normally called the telegrapher's equations; as it was originally developed for the telegraph cables. However, the set of coupled transmission line equations can be called the Kelvin–Heaviside transmission line equations to recognize their contributions.
The Resistance of a Line
The electrical loss in a transmission line, known as the conductor loss, is due to the finite conductivity of the line. It is modeled as the resistance R p.u.l. It is also influenced by the skin effect phenomena at a higher frequency. The instantaneous current i(t) flowing through a lumped resistance Rlum is related to the instantaneous voltage drop v(t) by Ohm's law:
The Inductance of a Line
The current flowing in a conductor creates the magnetic field around itself. So the magnetic energy stored in the space around the transmission line, i.e. the time‐varying current supporting line section, is modeled by a series inductor L p.u.l. The inductance of a line is not lumped at one point, i.e. it is not confined at one point. It is distributed over the whole length of a line. The instantaneous voltage across a lumped inductor Llum is related to the current flowing through it by
(2.1.13)
The Capacitance of a Line
In a two‐conductor transmission line, the conductors separated by a dielectric medium form a distributed system of capacitance. The electric field energy stored in a line is modeled by the shunt capacitance C p.u.l. The instantaneous shunt current through a lumped capacitor Clum is related to the instantaneous voltage across it by
(2.1.14)
The Conductance of a Line
If the medium between two conductors of the transmission line is not a perfect dielectric, i.e. if it has finite conductivity, then a part of the line current shunts through the medium causing the dielectric loss. The dielectric loss of the line is modeled by shunt conductance G p.u.l. The instantaneous shunt current is related to the instantaneous voltage across a lumped conductance by
Figure (2.6a) shows a physical transmission line that supports the TEM mode wave propagation. Figure (2.6b) shows that this line could be modeled as a chain of the lumped RLCG structure. More numbers of RLCG sections per wavelength are needed to model a transmission line. The RLCG model is the modeling of a transmission line by the lossy series inductor and lossy shunt capacitor. The transmission line supports the wave propagation from power AC to RF and above. Likewise, the corresponding
