Rectangular Wave Guides

If the two wires are joined as in Fig. 27(b) and excited as previously explained, the voltage and current distribution remain essentially the same. If several such conductors are connected as in Fig. 27(c), the voltage and current will again be distributed as in (b). If now, many shorted quarter-wave conductors are placed side by side in perfect contact, the rectangular wave guide of Fig. 27 (d) results.

If the rectangular wave guide of Fig. 28 is excited (by a vertical "antenna" wire passing through the center of the guide) at a frequency such that the greatest cross-section dimension is λ/2 (as in Fig. 27), it follows that the voltage and current distribution is as indicated. If this is true, within the wave guide the electric-field component and the magnetic-field component of the electromagnetic wave must be as shown in Fig. 28 (a). At the center of the guide the voltage is large as indicated by the height of curve E; hence, the electric field will be

Figure 27. Diagrams for illustrating certain elementary relations between short-circuited transmission lines and wave guides.

intense as shown by the concentration of vertical solid arrows. The voltage across the "short circuit" caused by the sidewalls of the tube is zero (by analogy with a transmission line), and hence no electric field exists at these walls.

Figure 28. Approximate field distributions in rectangular wave guides for the TE1.0 mode. In (a), the solid arrows are the electric field and the broken arrows are the magnetic field, looking into the wave guide at transverse section n-nr. Energy flow is in. The magnetic lines are, of course, continuous. In (6) is shown the magnetic field distribution, looking into the top of the guide. The electric field is shown, as dots at section n-n' and crosses at m-m'. Energy flow is to the right. In (c) is shown the field distribution, looking into the side of the guide. The arrows represent the electric field, and the dots and crosses represent the magnetic field. Energy flow is to the right.

The magnitude of the current flowing in the inner surfaces of the wave guide is shown by curve I of Fig. 28(a). Large currents are flowing in the sidewalls, and in the top and bottom of the guide near the sides; hence, the magnetic field will be as shown by the broken horizontal lines. Because magnetic lines of force are continuous, the magnetic lines of Fig. 28 (a) must be arranged within the guide as shown by the top view of Fig. 28(b). A side view of the fields is shown in Fig. 28 (c).

Cutoff Frequency. A wave guide is similar to a high-pass filter in that there is a low-frequency limit beyond which it will not conduct. If the cross-sectional dimensions of a rectangular wave guide are as shown in Fig. 29, the lowest frequency f₀ (and the longest free-space wavelength λ₀) that will be transmitted down the guide is given by the equation

where f₀ is the frequency in cycles per second, λ₀ is in centimeters, c is the velocity in centimeters per second of a wave in free space, and a is the dimension in centimeters indicated in Fig. 29. From this it is noted that the lowest frequency that will be transmitted is that of a wave that will just fit into a space twice the width of the guide. For instance, if a rectangular wave guide has an a dimension of 7.5 centimeters, it will pass a wave of minimum frequency f₀ = 3 X 10¹⁰/15 = 2 x 10⁹ cycles or 2000 megacycles, and a wavelength of λ₀ = 2 X 7.5 = 15 centimeters. Attention is called to the fact that the dimensions a and b are often interchanged. The usage in this book is in accordance with the standards.³⁷ As is evident the short dimension b has no effect in determining the cutoff frequency. However, a very high voltage (Fig. 28) may exist between the top and the bottom of a guide, and, if dimension b is small, the guide may flash across.

Figure 29. Dimensions of a rectangular wave guide are specified in the standards as shown.

Mode of Operation. The distribution of electric and magnetic lines of force of Fig. 28 is one of many possible arrangements. This specific arrangement and the values given by equation 35 are for the dominant mode, which, for a given wave guide, is the mode of operation that will give the lowest cutoff frequency and the longest wavelength that the guide will transmit. In many applications a wave guide is operated in the vicinity of its dominant mode, one reason being that the attenuation is low for this condition. The mode of operation of Fig. 28 is known as the TE_1.0 mode, for the following reasons. In free space (page 443) electric energy travels away from the source (antenna) as a transverse electromagnetic or TEM wave. That is, both the electric lines of force and the magnetic lines of force composing the wave are at right angles to the direction of wave propagation. In a wave guide, in which energy is confined to the space within the metal guide, the resultant wave traveling down the guide is composed of two or more waves that are reflected back and forth within the guide. For this reason the resultant wave is not a wave composed of an electric and a magnetic field at right angles to the direction of propagation and is not a TEM wave. In wave guides, there are components of either the electric field or the magnetic field in the direction of propagation. If a wave guide is assumed to be lossless, in the transverse electric (TE) mode, there is no electric-field component in the direction of propagation of the resultant wave. Similarly, in the transverse magnetic (TM) mode, there is no magnetic-field component in the direction of propagation of the resultant wave.

As mentioned previously the mode of operation in Fig. 28 is known as the TE_1.0 mode. The meaning of the two capital letters has been explained. For rectangular wave guides, the first subscript denotes the number of half waves of electric field intensity (voltage) along the large dimension a, and the second subscript denotes the number of half waves of electric-field intensity (voltage) along the small dimension b. These designations are different from the designations often used but are in accordance with the standards.³⁷

Wave Velocities. In considering transmission over open-wire lines and cables, the phase constant, or phase shift per unit distance, was calculated from the line constants, and the wave velocity and wavelength were computed from the phase constant. This is entirely satisfactory for transmission over such circuits, and, because of the way in which the values are found, the term phase velocity is often applied. It will also be recalled that wavelength and phase velocity were determined by measuring standing wave peaks on open-wire lines.

But in a wave guide, in a sense, the energy being propagated down the guide is being reflected back and forth within the tube along a zigzag path from wall to wall.³⁴ If a probe is inserted into a wave guide³³ and if the distribution of the resultant electric field is studied, it will be found that the distance between maximum values or "peaks" of electric field is such that apparently the wavelength is longer than would be expected. Since in general it is considered that A = v/f and v = λf, it follows that, if the apparent wavelength in a guide (as measured with a probe) is greater than it would be in free space, then the apparent wave velocity or phase velocity in a guide is greater than the velocity of an electromagnetic wave in free space. This apparent phase constant β is given by the equation

where β will be the apparent phase shift in radians per centimeter, w = 2πf, c is the velocity of light in centimeters per second, and a is the long dimension of the guide in centimeters (Fig. 29). If β is known, then the apparent or phase velocity in centimeters is

For the wave guide previously considered, and at a frequency of 3 X 10⁹ cycles,

Using this value the apparent or phase velocity is

V = ω/β = 6.28 * 3 * 10⁹/0.47 = 4.02 x 10¹⁰ centimeters per second,

which is considerably above the actual velocity that an electromagnetic wave in free space may have, that is, 3 x 10¹⁰ centimeters per second. The corresponding apparent wavelength in the guide is λ₀ = V/f = 4.02 X 10¹⁰/3 X 10⁹ = 13.88 centimeters; yet the actual wavelength in free space would be V_s = 3 X 10¹⁰/3 X 10⁹ = 10 centimeters. Thus, the apparent, or phase, velocity is greater than the velocity in free space and also is greater than the group velocity, which is the velocity with which a signal, such as a speech-modulated wave, would be propagated down the guide. These phenomena are caused by the zigzag path previously mentioned.³⁴

Attenuation. Attenuation can be computed from equations or can be obtained from curves.³⁹ For copper walls and an air dielectric and at the dominant or TE_1.0 mode, the attenuation of a rectangular wave guide is given by the relation³⁹

where a will be in decibels per foot when a and b are the dimensions in inches shown in Fig. 29, λ_c is the cutoff wavelength, and X is the free-space wavelength.

Other Modes of Operation. Other methods of excitation and other modes of operation are used. Furthermore, wave guides other than the rectangular type are employed (references 33 to 36).