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NETWORK AND TRANSMISSION LINES Group A 1) Coaxial cable Coaxial lines confine virtually all of the electromagnetic wave to the area inside the cable. Coaxial lines can therefore be bent and twisted (subject to limits) without negative effects, and they can be strapped to conductive supports without inducing unwanted currents in them. In radio-frequency applications up to a few gigahertz, the wave propagates in the transverse electric and magnetic mode (TEM) only, which means that the electric and magnetic fields are both perpendicular to the direction of propagation (the electric field is radial, and the magnetic field is circumferential). However, at frequencies for which the wavelength (in the dielectric) is significantly shorter than the circumference of the cable, transverse electric (TE) and transverse magnetic (TM) waveguide modes can also propagate. When more than one mode can exist, bends and other irregularities in the cable geometry can cause power to be transferred from one mode to another. The most common use for coaxial cables is for television and other signals with bandwidth of multiple megahertz. In the middle 20th century they carried long distance telephone connections. A type of transmission line called a cage line, used for high power, low frequency applications. It functions similarly to a large coaxial cable. This example is the antenna feedline for a longwave radio transmitter in Poland, which operates at a frequency of 225 kHz and a power of 1200 kW. Microstrip A microstrip circuit uses a thin flat conductor which is parallel to a ground plane. Microstrip can be made by having a strip of copper on one side of a printed circuit board (PCB) or ceramic substrate while the other side is a continuous ground plane. The width of the strip, the thickness of the insulating layer (PCB or ceramic) and the dielectric constant of the insulating layer determine the characteristic impedance. Microstrip is an open structure whereas coaxial cable is a closed structure. 2) Signal transfer Electrical transmission lines are very widely used to transmit high frequency signals over long or short distances with minimum power loss. One familiar example is the down lead from a TV or radio aerial to the receiver. Pulse generation Transmission lines are also used as pulse generators. By charging the transmission line and then discharging it into a resistive load, a rectangular pulse equal in length to twice the electrical length of the line can be obtained, although with half the voltage. A Blumlein transmission line is a related pulse forming device that overcomes this limitation. These are sometimes used as the pulsed energy sources for radar transmitters and other devices. Stub filters If a short-circuited or open-circuited transmission line is wired in parallel with a line used to transfer signals from point A to point B, then it will function as a filter. The method for making stubs is similar to the method for using Lecher lines for crude frequency measurement, but it is 'working backwards'. One method recommended in the RSGB's radiocommunication handbook is to take an open-circuited length of transmission line wired in parallel with the feeder delivering signals from an aerial. By cutting the free end of the transmission line, a minimum in the strength of the signal observed at a receiver can be found. At this stage the stub filter will reject this frequency and the odd harmonics, but if the free end of the stub is shorted then the stub will become a filter rejecting the even harmonics. Acoustic transmission lines An acoustic transmission line is the acoustic analog of the electrical transmission line, typically thought of as a rigid-walled tube that is long and thin relative to the wavelength of sound present in it 3) Distributed element model Fig.1 Transmission line. The distributed element model applied to a transmission line. This article is an example from the domain of electrical systems, which is a special case of the more general distributed parameter systems. In electrical engineering, the distributed element model or transmission line model of electrical circuits assumes that the attributes of the circuit (resistance, capacitance, and inductance) are distributed continuously throughout the material of the circuit. This is in contrast to the more common lumped element model, which assumes that these values are lumped into electrical components that are joined by perfectly conducting wires. In the distributed element model, each circuit element is infinitesimally small, and the wires connecting elements are not assumed to be perfect conductors; that is, they have impedance. Unlike the lumped element model, it assumes non-uniform current along each branch and non-uniform voltage along each node. The distributed model is used at high frequencies where the wavelength approaches the physical dimensions of the circuit, making the lumped model inaccurate. Group B 1) Telegrapher’s equations The Telegrapher's Equations (or just Telegraph Equations) are a pair of linear differential equations which describe the voltage and current on an electrical transmission line with distance and time. They were developed by Oliver Heaviside who created the transmission line model, and are based on Maxwell's Equations. Schematic representation of the elementary component of a transmission line. The transmission line model represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line: The distributed resistance of the conductors is represented by a series resistor (expressed in ohms per unit length). The distributed inductance (due to the magnetic field around the wires, selfinductance, etc.) is represented by a series inductor (henries per unit length). The capacitance between the two conductors is represented by a shunt capacitor C (farads per unit length). The conductance of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire (siemens per unit length). The model consists of an infinite series of the elements shown in the figure, and that the values of the components are specified per unit length so the picture of the component can be misleading. , , , and may also be functions of frequency. An alternative notation is to use , , and to emphasize that the values are derivatives with respect to length. These quantities can also be known as the primary line constants to distinguish from the secondary line constants derived from them, these being the propagation constant, attenuation constant and phase constant. The line voltage and the current can be expressed in the frequency domain as When the elements and are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the and elements which greatly simplifies the analysis. For a lossless transmission line, the second order steadystate Telegrapher's equations are: These are wave equations which have plane waves with equal propagation speed in the forward and reverse directions as solutions. The physical significance of this is that electromagnetic waves propagate down transmission lines and in general, there is a reflected component that interferes with the original signal. These equations are fundamental to transmission line theory. If and where are not neglected, the Telegrapher's equations become: and the characteristic impedance is: The solutions for The constants and and , starting at at position are: must be determined from boundary conditions. For a voltage pulse and moving in the positive -direction, then the transmitted pulse can be obtained by computing the Fourier Transform, , attenuating each frequency component by , of , advancing its phase by , and taking the inverse Fourier Transform. The real and imaginary parts of computed as where atan2 is the two-parameter arctangent, and For small losses and high frequencies, to first order in and one obtains can be Noting that an advance in phase by simply computed as is equivalent to a time delay by , can be 2) Input impedance of lossless transmission line The characteristic impedance of a transmission line is the ratio of the amplitude of a single voltage wave to its current wave. Since most transmission lines also have a reflected wave, the characteristic impedance is generally not the impedance that is measured on the line. For a lossless transmission line, it can be shown that the impedance measured at a given position from the load impedance is where is the wavenumber. In calculating , the wavelength is generally different inside the transmission line to what it would be in free-space and the velocity constant of the material the transmission line is made of needs to be taken into account when doing such a calculation. Special cases [edit] Half wave length For the special case where where n is an integer (meaning that the length of the line is a multiple of half a wavelength), the expression reduces to the load impedance so that for all . This includes the case when , meaning that the length of the transmission line is negligibly small compared to the wavelength. The physical significance of this is that the transmission line can be ignored (i.e. treated as a wire) in either case. Quarter wave length Main article: quarter wave impedance transformer For the case where the length of the line is one quarter wavelength long, or an odd multiple of a quarter wavelength long, the input impedance becomes Matched load Another special case is when the load impedance is equal to the characteristic impedance of the line (i.e. the line is matched), in which case the impedance reduces to the characteristic impedance of the line so that for all and all . Short Main article: stub For the case of a shorted load (i.e. ), the input impedance is purely imaginary and a periodic function of position and wavelength (frequency) Open Main article: stub For the case of an open load (i.e. periodic ), the input impedance is once again imaginary and 3) Stepped transmission line A simple example of stepped transmission line consisting of three segments. A stepped transmission line[2] is used for broad range impedance matching. It can be considered as multiple transmission line segments connected in series, with the characteristic impedance of each individual element to be Z0,i. The input impedance can be obtained from the successive application of the chain relation where is the wave number of the ith transmission line segment and li is the length of this segment, and Zi is the front-end impedance that loads the ith segment. The impedance transformation circle along a transmission line whose characteristic impedance Z0,i is smaller than that of the input cable Z0. And as a result, the impedance curve is offcentered towards the -x axis. Conversely, if Z0,i > Z0, the impedance curve should be offcentered towards the +x axis. Because the characteristic impedance of each transmission line segment Z0,i is often different from that of the input cable Z0, the impedance transformation circle is off centered along the x axis of the Smith Chart whose impedance representation is usually normalized against Z0. 4) Transmission lines Transmission lines are a common example of the use of the distributed model. Its use is dictated because the length of the line will usually be many wavelengths of the circuit's operating frequency. Even for the low frequencies used on power transmission lines, one tenth of a wavelength is still only about 500 kilometres at 60 Hz. Transmission lines are usually represented in terms of the primary line constants as shown in figure 1. From this model the behaviour of the circuit is described by the secondary line constants which can be calculated from the primary ones. The primary line constants are normally taken to be constant with position along the line leading to a particularly simple analysis and model. However, this is not always the case, variations in physical dimensions along the line will cause variations in the primary constants, that is, they have now to be described as functions of distance. Most often, such a situation represents an unwanted deviation from the ideal, such as a manufacturing error, however, there are a number of components where such longitudinal variations are deliberately introduced as part of the function of the component. A well known example of this is the horn antenna. Where reflections are present on the line, quite short lengths of line can exhibit effects that are simply not predicted by the lumped element model. A quarter wavelength line, for instance, will transform the terminating impedance into its dual. This can be a wildly different impedance. High frequency transistors Fig.2. The base region of a bipolar junction transistor can be modelled as a simplified transmission line. Another example of the use of distributed elements is in the modelling of the base region of a bipolar junction transistor at high frequencies. The analysis of charge carriers crossing the base region is not accurate when the base region is simply treated as a lumped element. A more successful model is a simplified transmission line model which includes distributed bulk resistance of the base material and distributed capacitance to the substrate. This model is represented in figure 2. Resistivity measurements Fig. 3. Simplified arrangement for measuring resistivity of a bulk material with surface probes. In many situations it is desired to measure resistivity of a bulk material by applying an electrode array at the surface. Amongst the fields that use this technique are geophysics (because it avoids having to dig into the substrate) and the semiconductor industry also use it (for the similar reason that it is non-intrusive) for testing bulk silicon wafers.[2] The basic arrangement is shown in figure 3, although normally more electrodes would be used. To form a relationship between the voltage and current measured on the one hand, and the resistivity of the material on the other, it is necessary to apply the distributed element model by considering the material to be an array of infinitesimal resistor elements. Unlike the transmission line example, the need to apply the distributed element model arises from the geometry of the setup, and not from any wave propagation considerations.[3] The model used here needs to be truly 3-dimensional (transmission line models are usually described by elements of a one-dimensional line). It is also possible that the resistances of the elements will be functions of the co-ordinates, indeed, in the geophysical application it may well be that regions of changed resistivity are the very things that it is desired to detect.[4] Inductor windings Fig. 4. A possible distributed element model of an inductor. A more accurate model will also require series resistance elements with the inductance elements. Another example where a simple one-dimensional model will not suffice is the windings of an inductor. Coils of wire have capacitance between adjacent turns (and also more remote turns as well, but the effect progressively diminishes). For a single layer solenoid, the distributed capacitance will mostly lie between adjacent turns as shown in figure 4 between turns T1 and T2, but for multiple layer windings and more accurate models distributed capacitance to other turns must also be considered. This model is fairly difficult to deal with in simple calculations and for the most part is avoided. The most common approach is to roll up all the distributed capacitance into one lumped element in parallel with the inductance and resistance of the coil. This lumped model works successfully at low frequencies but falls apart at high frequencies where the usual practice is to simply measure (or specify) an overall for the inductor without associating a specific equivalent circuit.