Introduction To Modern Planar Transmission Lines. Anand K. Verma

At the cut‐off frequency, ω = ω_c, β is zero. The cut‐off frequency is given by

      (2.3.25)

      where phase velocity of the voltage and current waves on the line at x = 0 is

(2.3.26)

      The complex propagation constants can be rewritten as follows:

      (2.3.27)

      The propagation constants β₁ and β₂ are imaginary quantities for the signal below the cut‐off frequency ω < ω_c. Under such conditions, no wave propagates on the nonuniform line. The initial signal only gets attenuated. It is called the evanescent mode. The wave propagation takes place only for ω > ω_c. Therefore, a nonuniform transmission line behaves like a high‐pass filter (HPF). However, real parts of the complex propagation constants γ₁ and γ₂ are nonzero. For p > 0, the voltage wave gets attenuated while the current wave is increased in the positive direction of wave propagation. In the backward direction, the reflected voltage and current waves have opposite behavior. The attenuation in the signal is not due to any ohmic loss of a line. It is due to the continuous reflection of the wave as it progresses on the line. The opposite behavior of the voltage and current waves maintains the constant flow of power (P) at any location on a line:

      (2.3.28)

      where . Unlike a lossless uniform transmission line, the phase velocity of the voltage and current waves on a lossless nonuniform transmission line is dispersive as given below:

      (2.3.29)

The phase velocity shows singularity at the cut‐off frequency. After the cut‐off frequency, i.e. for ω > > ω_c, it decreases, with an increase in frequency, to a value given by expression (2.3.26).

References

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3 Waves on Transmission Lines – II: (Network Parameters, Wave Velocities, Loaded Lines)

      Introduction

      The transmission line sections are used to develop various passive components. These are characterized by several kinds of matrix parameters. This chapter discusses the matrix parameters and their conversion among themselves. It also discusses various kinds of dispersion and wave propagation encountered on transmission lines. The transmission lines could be loaded by the reactive elements and resonating circuits to modify the nature of the wave propagation on the lines. Such loaded lines are important in modern planar microwave technology. Such loaded lines are introduced in this chapter. The primary purpose of this chapter is to review in detail the matrix description