such waves. The transmission line structure behaves like a low‐pass filter.
2.1.3 Kelvin–Heaviside Transmission Line Equations in Time Domain
Figure (2.6b) shows the lumped element equivalent circuit model of a section Δx of the transmission line. The primary line constants R, L, C, G on the per unit length (p.u.l.) basis are related to the lumped circuit elements as Rlum = RΔx, Llum = LΔx, Clum = CΔx, and Glum = GΔx. Figure (2.6b) shows the voltage loop equation and the current node equation for a small line section Δx. These are written as follows:
The Loop Equation
Differential voltage change across line length Δx = Voltage drop across inductor + Voltage drop across resistance
The instantaneous line voltage v(t) and the instantaneous line current i(t) are functions of the space–time variables x and t. In the limiting case of Δx → 0, equation (2.1.16) is written as
(2.1.17)
The Node Equation
Differential shunt current at the node = Current through conductance + Current through capacitor
(2.1.18)
In the limiting case, Δx → 0; and the above equation is written as
(2.1.19)
The pair of coupled voltage and current transmission line equations in the time‐domain summarized below is known as “the time domain telegrapher's equations”:
The above pair of equations can also be called the Kelvin–Heaviside transmission line equations. The coupled Kelvin–Heaviside transmission line equations relate the voltage and current on a transmission line through the line parameters, RLCG. These parameters are known as the primary constants of a line. The RLCG parameters depend on the physical configuration of a line, i.e. on its physical shapes, dimensions, and the electrical properties of the medium. They could be frequency‐dependent parameters also. For simple transmission lines such as a pair of wires and coaxial lines, the closed‐form formulas are available to compute them. However, these parameters could also be obtained through measurements. The empirical expressions for the RLCG parameters of a microstrip line are also available [J.3]. The transmission line equations are the complementary parts of Maxwell's field equations that relate the time‐varying electric and magnetic fields in a physical medium through the primary constants of a medium‐conductivity (σ), permittivity (ε) and permeability (μ). It is discussed in chapter 4. The transmission line equations can be obtained from Maxwell's equations. It is discussed in section (7.3) of chapter 7.
The voltage and current coupled variables of equation (2.1.20) can be separated. The separation of the variables leads to the wave equations for the voltage and current waves on a transmission line. On differentiating equation (2.1.20a) with respect to the variable x, the voltage wave equation is obtained:
On substituting
(2.1.22)
The above partial differential equation describes the time‐domain voltage wave on a lossy transmission line. Likewise, an equation could be written to describe the current wave on a transmission line:
(2.1.23)
A lossless transmission line has, R = G = 0. The voltage and current waves on a lossless line are given by the following 1D PDEs:
On comparing the above equations with equation (2.1.11a), the velocity of propagation, for both the current and voltage waves, is
(2.1.26)
It is like the velocity of propagation of an electromagnetic wave in a dielectric medium obtained from Maxwell's equations, where the primary constant of the line L and C are replaced by the medium constants permeability μ and permittivity ε. The EM‐wave is discussed in chapter 4.
2.1.4 Kelvin–Heaviside Transmission Line Equations in Frequency‐Domain
The time‐harmonic instantaneous voltage in the frequency domain, i.e. in the phasor form, is written as
(2.1.27)
where “Re” stands for the real part of the voltage phasor
(2.1.28)
