Anand K. Verma - Introduction To Modern Planar Transmission Lines

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:

(2.1.21)

On substituting from equation (2.1.20b)in equation (2.1.21), the voltage wave equation is

(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:

(2.1.24)

(2.1.25)

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.

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 . The voltage phasor is given by the following expression:

(2.1.28)

The phasor is nothing but a polar form of a complex quantity. Likewise, the instantaneous current in the phasor form is

(2.1.29)

where current phasor is

(2.1.30)

The phasor is either a constant or a function of only the space variable. It is not a function of time t . The phasor is shown with a tilde (~) sign in this chapter. However, in the subsequent chapters, the tilde (~) sign is dropped. The phasor is used at a single frequency. Using the phasor notation, the voltage across R, L, and the current through C, G; given by equations (2.1.12)– (2.1.15)in the time domain, can be rewritten in the frequency‐domain:

(2.1.31)

In the above equations, the time derivative ∂ /∂t is replaced by jω converting the expression from the time‐domain to frequency‐domain. Following the conversion process, the time‐domain coupled voltage‐current transmission line equation (2.1.20), is rewritten in the frequency‐domain:

(2.1.32)

On separation of the voltage and current variables, the following voltage and current wave equations are obtained in the frequency‐domain:

(2.1.33)

It is noted that the second‐order partial differential wave equations in the time‐domain are converted to the second‐order ordinary differential equations in the frequency‐domain. The factors at the right‐hand side of the above equations help to define a secondary parameter γ, known as the complex propagation constant of a transmission line:

(2.1.34)

where γ = α + jβ. The real part α(Np/m) of the complex propagation constant γ is called the attenuation constant and the imaginary part β (rad/sec) is the propagation constant of a lossy transmission line. The parameter β is also known as the phase‐shift constant or phase constant . On separating the real and imaginary parts of the above equation, the following expressions are obtained:

(2.1.35)

(2.1.36)

The attenuation constant α, and propagation constant β are given in terms of the primary line constants, R, L, C, G. Normally α and β are frequency‐dependent. Thus, the phase velocity of both the current and voltage waves, given by v p= ω/β is frequency‐dependent. This kind of transmission line is known as the dispersive transmission line . A complex wave traveling on a lossy dispersive line gets distorted as each component of the complex wave travels with different phase velocity.

Using the complex propagation constant, the wave equation (2.1.33a and b)are rewritten as