# Waveguide (electromagnetism)

A waveguide is a waveguide for electromagnetic waves, mainly in the centimeter wavelength range and below ( approximately 3 GHz to 200 GHz). Waveguides are metal tubes with mostly rectangular, circular or elliptical cross -section, in which can be in contrast to very low loss cables transmit such high frequencies.

- 4.1 rectangular waveguide
- 4.2 circular waveguide
- 4.3 waveguides with elliptical cross-section

## Physical background

Does an electromagnetic wave is perpendicular to a highly conductive interface, it is reflected into itself. With a suitable distance a parallel second interface, it may be formation of a standing wave. More walls forming a resonant cavity. However, the electromagnetic waves in the interior of which are standing waves; is a stationary electric and magnetic alternating field. The possible resonance frequencies of the standing waves depend on the distance of the walls from each other.

In a waveguide on the other hand moves the electric and magnetic field continues: Imagine a long tube with a rectangular cross-section before, in which a plane wave perpendicularly incident on a narrow side and is reflected between two walls back and forth. The wavelength is twice as large as the distance between the two walls, so that a standing wave is formed. The wavelength is then reduced in size as the wave can propagate in only one particular angle, in which again a standing wave is created between the two walls. To the wavelength along the longer wall must again be twice as large as the distance between the two walls. The resulting wavelength in the longitudinal direction results in a wave propagating along the waveguide. We therefore speak for the distinction of standing waves by a traveling wave.

The minimum width of a rectangular waveguide is approximately half the wavelength of the transmitted frequency - if and only one antinode in the transverse direction fits into it. You can connect to the lowest frequency used in the corresponding device therefore from the width of a rectangular waveguide. The corresponding wavelength is called the critical wavelength λk or the cutoff wavelength? C ( c for "cut - off"). It is calculated according to the relation λk = 2 · a (where a is the longer side of the rectangular waveguide cross-section, see sketch above).

Waveguide can also be placed under increased internal gas pressure in order to transmit higher power can without flashover or air shocks occur. The phenomenon multipaction means the constructive superposition of several different wavelengths, thus very high field strengths can arise in this context.

## Modes

The type of spread described above can be such that an integral multiple of the half wavelength fits between the narrow sides. The various possible states are the so-called vibrational modes, short modes, and are labeled with the numbers corresponding to this multiple; ie: 1,2,3, ....

At higher frequencies are joined to the horizontal transverse modes nor the vertical between the top and bottom of the tube, where different modes occur independently in turn. Therefore, each specifying two numbers is used to describe a mode in the rectangular waveguide necessary: for example, ( 2,3 ) mode. This is one of the numbers for each of the transverse modes in the direction of the electric and the magnetic field component (E- and H- directions).

The field lines of the electric field are always perpendicular to the outer conductor and extending from one side of the wall to the other. Depending on how many having extreme values of the field distribution across the entire width of the waveguide, the mode designation takes its first index. The width of a waveguide is denoted by a. At a maximum of the minimum number of the electric field distribution is thus referred to as a - or shaft.

Analogously, the number of maxima in the field distribution of the electric field over the entire height of the waveguide to the second index. The height of the waveguide is denoted by b. The field strength can remain constant over the entire height of the waveguide (ie, it must give no maximum ), then one speaks of one or shaft.

Similar modes are also available in round waveguides. Here, however, are still added modes which have a homogeneous field distribution along the pipe circumference.

The input and output coupling of the RF energy is carried by slots coupling loops, rods, funnels ( horn ) or holes - depending on whether the energy into another waveguide, in a coaxial cable or open space to reach. Location and shape of these coupling elements determine the mode and the propagation direction of the waves.

### E-/H-Moden

Electric and magnetic field are always perpendicular to each other with electromagnetic waves. Thus, the wave can propagate in one spatial direction, wave components must exist in this direction in space. Shows the electric field in the propagation direction, it is called E- modes. Displays the magnetic field in the propagation direction, it is called H- modes. The figure shows a longitudinal section through a waveguide direction (z direction ).

### Waveguide wavelength and cut-off frequency

While the distance of the peaks of the field distribution dependent on the x- or y - direction from the free space wavelength of the wave, the waveguide wavelength of the spacing of the maxima in the z- direction, that is in the propagation direction significantly.

The waveguide depends on the wavelength according to the equation of the free-space wavelength, the width of the waveguide, and the height and the respective mode. The waveguide wavelength is always greater than the free space wavelength of the same frequency. There is no difference between the wavelength of the hollow conductor, and modes of the corresponding modes.

The waveguide wavelength depends non- linearly on the free-space wavelength. There exists a wavelength for which the waveguide wavelength approaches infinity.

An infinite waveguide wavelength means that the wave is not capable of propagation. Since the waveguide wavelength depends on a mode of a certain frequency from the dimensions of the waveguide, not any modes are able to propagate in a waveguide. The higher value is a fashion, the greater is its cutoff frequency, or the smaller the cut-off wavelength.

The cutoff frequency divides the frequency range into two regions, the damping region and the propagation region. The decisive factor is the behavior of the propagation coefficient γ of the frequency.

The attenuation range the shaft is not capable of propagation. The propagation coefficient is as purely real. The wave is therefore damped aperiodically. Evanescent modes can be excited and at least temporarily bind a portion of the wave energy. Is the frequency of the wave is equal to the limit frequency, the propagation coefficient is equal to zero. The wave is reflected at a right angle between the sides of the waveguide, without any power transmission taking place.

For frequencies above the cutoff frequency, the wave can propagate. The propagation coefficient is ideally and thus purely imaginary. The shaft is thus not attenuated but spreads out in the waveguide with a frequency-dependent phase shift. In real waveguide also evanescent wave is attenuated. These bear the losses in only finitely conducting waveguide wall at ( surface currents ). The ( loss) from the surface currents on the power transport is dependent fashion, tends to decrease with higher modes and tends to increase due to the skin effect. Since waveguides are filled with air or gas in general, there are no dielectric losses. This is an essential factor for their use at very high frequencies

## Wave impedance

The characteristic impedance of the amplitudes of the combined electric and magnetic field strength of an electromagnetic wave. The waveguide is frequency dependent and different for the TM and TE mode, but has all the waveguides have the same value.

Where flimit the cut-off frequency of the respective modes and means of the free-space wave resistance.

Above the cut-off frequency ( f > flimit ) the impedance is real-valued and in the waveguide is energy propagates. Below the cutoff frequency, the impedance contrast, is imaginary and the waves penetrates with a weakening of the amplitude in the waveguide a.

## Various waveguide modes and their

Basically have all waveguide or waveguide same characteristics. This includes a cut-off frequency below which there is no wave propagation.

While in a coaxial TEM - waves propagate ( electric and magnetic fields are always perpendicular to the propagation direction), only so-called H- waves are found in a waveguide (also TE waves ) and E -wave ( TM waves ) in which magnetic or electric field components in the direction of propagation have.

Waveguides have a high-pass behavior with a cutoff frequency. Both rectangular and circular waveguide have at the bottom of said fundamental wave types. Do these fundamental waves no way spread (relative to H- and E - waves) due to the dimensions of the waveguide itself, including other types of waves to be spread out. See also cavity. Above the limit frequency, the propagation of the waves (for example, group velocity, phase velocity and wavelength) is dependent on the frequency. The wave propagation in the waveguide is thus in principle dispersive.

The following rules apply for the existence of modes:

- Electric and magnetic field lines are always perpendicular to each other
- Magnetic field lines are always closed and are unable to make walls - they can only affect walls
- Electric field lines can not occur along the walls, but take only perpendicular to it

### Rectangular waveguide

For a rectangular waveguide, as previously mentioned, the greatest dimension of decisive. That is, the width determines the waves can propagate in this guide.

For the E- wave in the propagation direction applies:

Wherein m and n represent the mode numbers (m: x-direction ( horizontal) and n Y- direction ( vertically ) with respect to spreading in the longitudinal direction z). a is the larger dimension of the waveguide. See also Maxwell's equations.

As a result, the so-called fundamental type of the e- wave is the wave, since the above equation with the values of m = 0 or n = 0 results, and thus there is no e- component in the propagation direction. Thus least - waves must be able to develop in the direction of propagation in rectangular waveguide.

Typical of rectangular waveguide, however, the H10 wave.

### Circular waveguide

For the circular waveguide, the vibrational modes arise over the Bessel function and its derivatives and zeros with which the propagatable H- and E - waves for the circular waveguide can be determined. According to calculations by the Bessel functions is obtained for the circular waveguide with radius than the fundamental mode whose cutoff wavelength is calculated on the first zero of the first derivative of the first order Bessel function, which is located on the site of 1.841:

Due to the higher wave attenuation of the shaft relative to the shaft, it is often desirable to improve the propagation of the latter wave type, or to prevent the former. For this reason, the inside of the circular waveguide are often provided with grooves, which disturbs the propagation of the wave, but not the shaft. See picture below ( waveguide with an elliptical cross section). The cutoff wavelength of the wave is calculated with:

Thus, the cutoff wavelength is smaller than that of the fundamental wave, which is why the hollow conductor no longer behaves in a single mode for the shaft.

### Waveguides with elliptical cross-section

Besides rectangular waveguides can also waveguide having a circular cross-section or elliptical cross-section used. Mathematically, circular waveguide using the Bessel functions calculate. The cutoff wavelength also corresponds with the round and elliptical waveguides roughly twice the transverse dimension ( λk ≈ 2 · a). As a rule of thumb that elliptical waveguide in its transverse dimensions are slightly larger than a rectangular hollow conductor with the same cutoff frequency.

Elliptical waveguide can be technically low as flexible cables make (Figure). Thus, greater lengths thereof kept in rolls or "Cable" drums and transported. Also let elliptical waveguide smaller bending radii as round or square.

## Ports and connections of a waveguide

In a waveguide, the electric energy with a probe ( rod antenna ) is coupled, which extends λ / 4 away from the closed end in the waveguide. The detaching from the probe of the coupler wave "sees" on three sides of the infinite resistance (see special case λ / 4) of the shorted λ/4-Leitung. The electromagnetic wave can therefore only propagate in the remaining direction.

When connecting a waveguide to other devices, the electrical resistance along the entire circumference has to be very small, since in the wall, high currents flow. A simple interference contact can not be kept low transition resistance, which is why one uses an AC λ/2-Transformationsleitung that transforms a short circuit at A in a short-circuit in C (see special case λ / 2). Choosing a waveguide mode in which a voltage maximum occurs at C (see middle panel ).

Because in the middle flow only small currents at point B, a possible transfer resistance affects there little. At this point, you can even install an insulating rubber sheet to fill the hollow conductor with protective gas can. This principle of resonance seal is also used for sealing the door of a microwave oven the high-frequency technology.

The conclusion of a waveguide can be done without reflection by means of a shaft sump.

## Waveguide frequency bands

A waveguide with specific dimensions is used in each case only makes sense in a certain frequency range with less than an octave bandwidth. Below the lower cutoff frequency no propagation is possible and the electromagnetic wave is damped blind, above the upper frequency limit unwanted higher modes can propagate in addition to the desired fundamental mode. Commercially available rectangular waveguide are available, among others, for the following frequency ranges:

This table is based on a width -to-height ratio of 2:1. The lower recommended transmission frequencies are in the middle to the 1.26 times over that resulting from the width of the critical lower limit frequencies, the frequencies are in the upper transmission means, the 1.48 times the recommended lower transmission frequencies. The factor of 1.86 (mean ) of the upper transmission frequencies for the respective critical lower limit frequency ensures single-mode propagation ( value <2).

To WRxxx designation of the waveguide: The width of the waveguide in% of an inch is expressed (1 inch (in) = 25.4 mm). A WR -28 waveguide thus 28 % of an inch = 7.11 mm wide.

## Waveguide in practice

Waveguides are used:

- In a microwave oven ( this is only a short distance between the magnetron and cavity available)
- On radio equipment and radio telescopes to the antenna feed (see horn )
- In the satellite to the antenna feed and the feeding of the receiving amplifier
- The plasma generator as a connection between the magnetron and the plasma chamber,
- Back to RADAR devices for the transmission of high transmission pulse power to the antenna and the received echoes to the receiver
- In particle accelerators to power the acceleration chambers.

A dehydrator keeps moisture away from the environment, which could distort the adaptation of the waveguide.

## Literature

- Horst Stöcker: Paperback physics. 4th edition, published by Harry German, Frankfurt am Main, 2000, ISBN 3-8171-1628-4
- Technique of message transfer part 3 Wireline communications - line art. Institut eV for the development of modern teaching methods, Bremen