Network Analysis And Transmission Lines Textbook Pdf

[Santi Dubrova]

Thecost function can be simple: for instance, it can be defined as the length of a line. The cost function can also be more elaborate and take into account not only length of the lines but also their capacity, maximum transmission (travel) rate and other line characteristics, for instance to obtain a reasonable approximation of travel time. There can even be cases in which the nodes visited add to the cost of the path as well. These may be called turning costs, which are defined in a separate turning-cost table for each node, indicating the cost of turning at the node when entering from one line and continuing on another. This is illustrated in Figure below.

In network allocation, we have a number of target locations that function as resource centres, and the problem is which part of the network to exclusively assign to which service centre.

This may sound like a simple allocation problem, in which a service centre is assigned those line (segments) to which it is nearest, but usually the problem statement is more complicated. The additional complications stem from the requirements to take into account (a)the capacity with which a centre can produce the resources (whether they are medical operations, seats for school pupils, kilowatts or bottles of milk), and (b) the consumption of the resources, which may vary amongst lines or line segments. After all, some streets have more accidents, more children who live there, more industry in high demand of electricity or just more thirsty workers.

Computations on networks comprise a different set of analytical functions in GISs. Here, the network may consist of roads, public transport routes, high-voltage power lines, or other forms of transportation infrastructure. Analysis of networks may entail shortest path computations (in terms of distance or travel time) between two points in a network for routing purposes. Other forms are to find all points reachable within a given distance or duration from a start point for allocation purposes, or determination of the capacity of the network for transportation between an indicated source location and sink location.

Network analysis can be performed on either raster or vector data layers, but they are more commonly done on the latter, as line features can be associated with a network and hence can be assigned typical transportation characteristics, such as capacity and cost per unit.

In network partitioning, the purpose is to assign lines and/or nodes of the network in a mutually exclusive way to a number of target locations. Typically, the target locations play the role of service centres for the network. This may be any type of service, e.g. medical treatment, education, water supply. This sort of network partitioning is known as a network allocation problem. Another problem is trace analysis. Here, one wants to determine that part of a network that is upstream (or downstream) from a given target location. Such problems exist in tracing pollution along river/stream systems, but also in tracking down network failures in energy distribution networks.

Optimal-path finding techniques are used when a least-cost path between two nodes in a network must be found. The two nodes are called origin and destination. The aim is to find a sequence of connected lines to traverse from the origin to the destination at the lowest possible cost.

In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmission must be taken into account. This applies especially to radio-frequency engineering because the short wavelengths mean that wave phenomena arise over very short distances (this can be as short as millimetres depending on frequency). However, the theory of transmission lines was historically developed to explain phenomena on very long telegraph lines, especially submarine telegraph cables.

Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas (they are then called feed lines or feeders), distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses. RF engineers commonly use short pieces of transmission line, usually in the form of printed planar transmission lines, arranged in certain patterns to build circuits such as filters. These circuits, known as distributed-element circuits, are an alternative to traditional circuits using discrete capacitors and inductors.

Ordinary electrical cables suffice to carry low frequency alternating current (AC), such as mains power, which reverses direction 100 to 120 times per second, and audio signals. However, they are not generally used to carry currents in the radio frequency range,[1] above about 30 kHz, because the energy tends to radiate off the cable as radio waves, causing power losses. Radio frequency currents also tend to reflect from discontinuities in the cable such as connectors and joints, and travel back down the cable toward the source.[1][2] These reflections act as bottlenecks, preventing the signal power from reaching the destination. Transmission lines use specialized construction, and impedance matching, to carry electromagnetic signals with minimal reflections and power losses. The distinguishing feature of most transmission lines is that they have uniform cross sectional dimensions along their length, giving them a uniform impedance, called the characteristic impedance,[2][3][4] to prevent reflections. Types of transmission line include parallel line (ladder line, twisted pair), coaxial cable, and planar transmission lines such as stripline and microstrip.[5][6] The higher the frequency of electromagnetic waves moving through a given cable or medium, the shorter the wavelength of the waves. Transmission lines become necessary when the transmitted frequency's wavelength is sufficiently short that the length of the cable becomes a significant part of a wavelength.

At microwave frequencies and above, power losses in transmission lines become excessive, and waveguides are used instead,[1] which function as "pipes" to confine and guide the electromagnetic waves.[6] Some sources define waveguides as a type of transmission line;[6] however, this article will not include them.

Mathematical analysis of the behaviour of electrical transmission lines grew out of the work of James Clerk Maxwell, Lord Kelvin, and Oliver Heaviside. In 1855, Lord Kelvin formulated a diffusion model of the current in a submarine cable. The model correctly predicted the poor performance of the 1858 trans-Atlantic submarine telegraph cable. In 1885, Heaviside published the first papers that described his analysis of propagation in cables and the modern form of the telegrapher's equations.[7]

In the simplest case, the network is assumed to be linear (i.e. the complex voltage across either port is proportional to the complex current flowing into it when there are no reflections), and the two ports are assumed to be interchangeable. If the transmission line is uniform along its length, then its behaviour is largely described by a two parameters called characteristic impedance, symbol Z0 and propagation delay, symbol τ p \displaystyle \tau _p . Z0 is the ratio of the complex voltage of a given wave to the complex current of the same wave at any point on the line. Typical values of Z0 are 50 or 75 ohms for a coaxial cable, about 100 ohms for a twisted pair of wires, and about 300 ohms for a common type of untwisted pair used in radio transmission. Propagation delay is proportional to the length of the transmission line and is never less than the length divided by the speed of light. Typical delays for modern communication transmission lines vary from 3.33 ns/m to 5 ns/m.

When sending power down a transmission line, it is usually desirable that as much power as possible will be absorbed by the load and as little as possible will be reflected back to the source. This can be ensured by making the load impedance equal to Z0, in which case the transmission line is said to be matched.

Some of the power that is fed into a transmission line is lost because of its resistance. This effect is called ohmic or resistive loss (see ohmic heating). At high frequencies, another effect called dielectric loss becomes significant, adding to the losses caused by resistance. Dielectric loss is caused when the insulating material inside the transmission line absorbs energy from the alternating electric field and converts it to heat (see dielectric heating). The transmission line is modelled with a resistance (R) and inductance (L) in series with a capacitance (C) and conductance (G) in parallel. The resistance and conductance contribute to the loss in a transmission line.

The total loss of power in a transmission line is often specified in decibels per metre (dB/m), and usually depends on the frequency of the signal. The manufacturer often supplies a chart showing the loss in dB/m at a range of frequencies. A loss of 3 dB corresponds approximately to a halving of the power.

Propagation delay is often specified in units of nanoseconds per metre. While propagation delay usually depends on the frequency of the signal, transmission lines are typically operated over frequency ranges where the propagation delay is approximately constant.

The transmission line model is an example of the distributed-element model. It represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:

When the elements R \displaystyle R and G \displaystyle G are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the L \displaystyle L and C \displaystyle C elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:

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