8: Guided Electromagnetic Waves

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Page ID: 48166

Markus Zahn
Massachusetts Institute of Technology via MIT OpenCourseWare

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The uniform plane wave solutions developed in Chapter 7 cannot in actuality exist throughout all space, as an infinite amount of energy would be required from the sources. However, TEM waves can also propagate in the region of finite volume between electrodes. Such electrode structures, known as transmission lines, are used for electromagnetic energy flow from power (60 Hz) to microwave frequencies, as delay lines due to the finite speed c of electromagnetic waves, and in pulse forming networks due to reflections at the end of the line. Because of the electrode boundaries, more general wave solutions are also permitted where the electric and magnetic fields are no longer perpendicular. These new solutions also allow electromagnetic power flow in closed single conductor structures known as waveguides.

8.1: The Transmission Line Equations
The general properties of transmission lines are illustrated in Figure 8-1 by the parallel plate electrodes a small distance d apart enclosing linear media with permittivity \(\varepsilon \) and permeability \(\mu \). Because this spacing \( d\) is much less than the width \(w\) or length \(l\), we neglect fringing field effects and assume that the fields only depend on the \( z\) coordinate.
8.2: Transmission Line Transient Waves
The easiest way to solve for transient waves on transmission lines is through use of physical reasoning as opposed to mathematical rigor. Since the waves travel at a speed \(c\), once generated they cannot reach any position \(z\) until a time \(z/c\) later. Waves traveling in the positive \(z\) direction are described by the function \(\textrm{V}_{+}\left ( t-z/c \right )\) and waves traveling in the \(-z\) direction by \(\textrm{V}_{-}(t + z/c)\). However, at any time \(t\) and position \(z\),
8.3: Sinusoidal Time Variations
Often transmission lines are excited by sinusoidally varying sources so that the line voltage and current also vary sinusoidally with time:
8.4: Arbitrary Tions
A lossless transmission line excited at \(z = -l\) with a sinusoidal voltage source is now terminated at its other end at \(z =0\) with an arbitrary impedance \(Z_L\), which in general can be a complex number. Defining the load voltage and current at \(z =0\) as
8.5: Stub Tuning
In practice, most sources are connected to a transmission line through a series resistance matched to the line. This eliminates transient reflections when the excitation is turned on or off.
8.6: The Rectangular Waveguide
We showed in Section 8-1-2 that the electric and magnetic fields for \(\textrm{TEM}\) waves have the same form of solutions in the plane transverse to the transmission line axis as for statics. The inner conductor within a closed transmission line structure such as a coaxial cable is necessary for \(\textrm{TEM}\) waves since it carries a surface current and a surface charge distribution, which are the source for the magnetic and electric fields. A hollow conducting structure, called a waveguide
8.7: Dielectric Waveguide
We found in Section 7-10-6 for fiber optics that electromagnetic waves can also be guided by dielectric structures if the wave travels from the dielectric to free space at an angle of incidence greater than the critical angle. Waves propagating along the dielectric of thickness \(2d\) in Figure 8-30 are still described by the vector wave equations derived in Section 8-6-1.
8.8: Problems

Thumbnail: Waveguide flange UBR320 for microwaves. (Public Domain; Catslash via Wikipedia)