# Transmission lines and wave propagation pdf

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## 3.8: Wave Propagation on a TEM Transmission Line

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage , the current in a circuit , or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length , but it is otherwise dimensionless.

In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next. The propagation constant's value is expressed logarithmically , almost universally to the base e , rather than the more usual base 10 that is used in telecommunications in other situations.

The quantity measured, such as voltage, is expressed as a sinusoidal phasor. The phase of the sinusoid varies with distance which results in the propagation constant being a complex number , the imaginary part being caused by the phase change. It is probably the most widely used term but there are a large variety of alternative names used by various authors for this quantity.

These include transmission parameter , transmission function , propagation parameter , propagation coefficient and transmission constant. The primary coefficients are the physical properties of the line, namely R,C,L and G, from which the secondary coefficients may be derived using the telegrapher's equation. Note that in the field of transmission lines, the term transmission coefficient has a different meaning despite the similarity of name: it is the companion of the reflection coefficient.

The reason for the use of base e is also now made clear. Angles measured in radians require base e , so the attenuation is likewise in base e. The propagation constant for copper or any other conductor lines can be calculated from the primary line coefficients by means of the relationship.

The sign convention is chosen for consistency with propagation in lossy media. If the attenuation constant is positive, then the wave amplitude decreases as the wave propagates in the x direction. Wavelength , phase velocity , and skin depth have simple relationships to the components of the propagation constant:.

In telecommunications , the term attenuation constant , also called attenuation parameter or attenuation coefficient , is the attenuation of an electromagnetic wave propagating through a medium per unit distance from the source. It is the real part of the propagation constant and is measured in nepers per metre.

A neper is approximately 8. Attenuation constant can be defined by the amplitude ratio. The propagation constant per unit length is defined as the natural logarithm of the ratio of the sending end current or voltage to the receiving end current or voltage.

The attenuation constant for copper lines or ones made of any other conductor can be calculated from the primary line coefficients as shown above. For a line meeting the distortionless condition , with a conductance G in the insulator, the attenuation constant is given by. There are two main components to these losses, the metal loss and the dielectric loss.

The loss of most transmission lines are dominated by the metal loss, which causes a frequency dependency due to finite conductivity of metals, and the skin effect inside a conductor. The skin effect causes R along the conductor to be approximately dependent on frequency according to. Thus they are directly proportional to the frequency. The attenuation constant for a particular propagation mode in an optical fiber is the real part of the axial propagation constant. In electromagnetic theory , the phase constant , also called phase change constant , parameter or coefficient is the imaginary component of the propagation constant for a plane wave.

It represents the change in phase per unit length along the path travelled by the wave at any instant and is equal to the real part of the angular wavenumber of the wave.

For a transmission line , the Heaviside condition of the telegrapher's equation tells us that the wavenumber must be proportional to frequency for the transmission of the wave to be undistorted in the time domain. This includes, but is not limited to, the ideal case of a lossless line. The reason for this condition can be seen by considering that a useful signal is composed of many different wavelengths in the frequency domain. For there to be no distortion of the waveform , all these waves must travel at the same velocity so that they arrive at the far end of the line at the same time as a group.

Since wave phase velocity is given by. In terms of primary coefficients of the line, this yields from the telegrapher's equation for a distortionless line the condition. However, practical lines can only be expected to approximately meet this condition over a limited frequency band.

Generally speaking, the following relation. Nevertheless, it is invalid to the TE wave transverse electric wave and TM wave transverse magnetic wave. In a rectangular waveguide, the cutoff frequency is. The phase velocity equals.

The term propagation constant or propagation function is applied to filters and other two-port networks used for signal processing. In these cases, however, the attenuation and phase coefficients are expressed in terms of nepers and radians per network section rather than per unit length. Some authors [5] make a distinction between per unit length measures for which "constant" is used and per section measures for which "function" is used.

The propagation constant is a useful concept in filter design which invariably uses a cascaded section topology. In a cascaded topology, the propagation constant, attenuation constant and phase constant of individual sections may be simply added to find the total propagation constant etc. The ratio of output to input voltage for each network is given by [6]. Thus for n cascaded sections all having matching impedances facing each other, the overall propagation constant is given by.

The concept of penetration depth is one of many ways to describe the absorption of electromagnetic waves. For the others, and their interrelationships, see the article: Mathematical descriptions of opacity. From Wikipedia, the free encyclopedia. Optical and Quantum Electronics.

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## Transmission Lines, Antenna and Wave Propagation Notes PDF

Consider electromagnetic wave propagation in a source-free, lossless, unbounded medium. A simple propagating wave, which adequately represents wave propagation often encountered in real situations, is a uniform plane wave. This section introduces the fundamentals of a uniform plane wave and analyzes its propagation characteristics. Unable to display preview. Download preview PDF.

Request PDF | Guided Wave Propagation and Transmission Lines | At higher frequencies where wavelength becomes small with respect to feature size, it is.

## Plane Wave Propagation

In Section 3. In this section, we demonstrate that these expressions represent sinusoidal waves, and point out some important features. Before attempting this section, the reader should be familiar with the contents of Sections 3.

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction.

#### COMMENT 4

• Wave Propagation on Transmission Lines. A. General For an infinite uniform transmission line, ∃ no reflection waves (BTW). Then a) Propagation constant. Marissa G. - 28.03.2021 at 20:22
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• Teaching transmission lines and wave propagation is a challenging task because it involves quantities not easily observable and also because the underlying mathematical equations—functions of time, distance and using complex numbers—are not prone to an easy physical interpretation in a frequent framework of a superposition of traveling waves in distinct directions. Arunovrec - 03.04.2021 at 09:15