Transmission Line Theory Reflection from Resistive loads St in addition to ing Waves Effect of Lossy line on voltage in addition to current waves
Morgenstein, Joe, Film Reviewer has reference to this Academic Journal, PHwiki organized this Journal Transmission Line Theory Introduction: In an electronic system, the delivery of power requires the connection of two wires between the source in addition to the load. At low frequencies, power is considered to be delivered to the load through the wire. In the microwave frequency region, power is considered to be in electric in addition to magnetic fields that are guided from lace to place by some physical structure. Any physical structure that will guide an electromagnetic wave place to place is called a Transmission Line. Types of Transmission Lines Two wire line Coaxial cable Waveguide Rectangular Circular Planar Transmission Lines Strip line Microstrip line Slot line Fin line Coplanar Waveguide Coplanar slot line Analysis of differences between Low in addition to High Frequency At low frequencies, the circuit elements are lumped since voltage in addition to current waves affect the entire circuit at the same time. At microwave frequencies, such treatment of circuit elements is not possible since voltag in addition to current waves do not affect the entire circuit at the same time. The circuit must be broken down into unit sections within which the circuit elements are considered to be lumped. This is because the dimensions of the circuit are comparable to the wavelength of the waves according to the as long as mula: l = c/f where, c = velocity of light f = frequency of voltage/current
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Transmission Line Concepts The transmission line is divided into small units where the circuit elements can be lumped. Assuming the resistance of the lines is zero, then the transmission line can be modeled as an LC ladder network with inductors in the series arms in addition to the capacitors in the shunt arms. The value of inductance in addition to capacitance of each part determines the velocity of propagation of energy down the line. Time taken as long as a wave to travel one unit length is equal to T(s) = (LC)0.5 Velocity of the wave is equal to v (m/s) = 1/T Impedance at any point is equal to Z = V (at any point)/I (at any point) Z = (L/C)0.5 Line terminated in its characteristic impedance: If the end of the transmission line is terminated in a resistor equal in value to the characteristic impedance of the line as calculated by the as long as mula Z=(L/C)0.5 , then the voltage in addition to current are compatible in addition to no reflections occur. Line terminated in a short: When the end of the transmission line is terminated in a short (RL = 0), the voltage at the short must be equal to the product of the current in addition to the resistance. Line terminated in an open: When the line is terminated in an open, the resistance between the open ends of the line must be infinite. Thus the current at the open end is zero. Reflection from Resistive loads When the resistive load termination is not equal to the characteristic impedance, part of the power is reflected back in addition to the remainder is absorbed by the load. The amount of voltage reflected back is called voltage reflection coefficient. G = Vr/Vi where Vr = incident voltage Vi = reflected voltage The reflection coefficient is also given by G = (ZL – ZO)/(ZL + ZO)
St in addition to ing Waves A st in addition to ing wave is as long as med by the addition of incident in addition to reflected waves in addition to has nodal points that remain stationary with time. Voltage St in addition to ing Wave Ratio: VSWR = Vmax/Vmin Voltage st in addition to ing wave ratio expressed in decibels is called the St in addition to ing Wave Ratio: SWR (dB) = 20log10VSWR The maximum impedance of the line is given by: Zmax = Vmax/Imin The minimum impedance of the line is given by: Zmin = Vmin/Imax or alternatively: Zmin = Zo/VSWR Relationship between VSWR in addition to Reflection Coefficient: VSWR = (1 + G)/(1 – G) G = (VSWR 1)/(VSWR + 1) General Input Impedance Equation Input impedance of a transmission line at a distance L from the load impedance ZL with a characteristic Zo is Zinput = Zo [(ZL + j Zo BL)/(Zo + j ZL BL)] where B is called phase constant or wavelength constant in addition to is defined by the equation B = 2p/l Half in addition to Quarter wave transmission lines The relationship of the input impedance at the input of the half-wave transmission line with its terminating impedance is got by letting L = l/2 in the impedance equation. Zinput = ZL W The relationship of the input impedance at the input of the quarter-wave transmission line with its terminating impedance is got by letting L = l/2 in the impedance equation. Zinput = (Zinput Zoutput)0.5 W
Effect of Lossy line on voltage in addition to current waves The effect of resistance in a transmission line is to continuously reduce the amplitude of both incident in addition to reflected voltage in addition to current waves. Skin Effect: As frequency increases, depth of penetration into adjacent conductive surfaces decreases as long as boundary currents associated with electromagnetic waves. This results in the confinement of the voltage in addition to current waves at the boundary of the transmission line, thus making the transmission more lossy. The skin depth is given by: skin depth (m) = 1/(pmgf)0.5 where f = frequency, Hz m = permeability, H/m g = conductivity, S/m Smith chart For complex transmission line problems, the use of the as long as mulae becomes increasingly difficult in addition to inconvenient. An indispensable graphical method of solution is the use of Smith Chart. Components of a Smith Chart Horizontal line: The horizontal line running through the center of the Smith chart represents either the resistive ir the conductive component. Zero resistance or conductance is located on the left end in addition to infinite resistance or conductance is located on the right end of the line. Circles of constant resistance in addition to conductance: Circles of constant resistance are drawn on the Smith chart tangent to the right-h in addition to side of the chart in addition to its intersection with the centerline. These circles of constant resistance are used to locate complex impedances in addition to to assist in obtaining solutions to problems involving the Smith chart. Lines of constant reactance: Lines of constant reactance are shown on the Smith chart with curves that start from a given reactance value on the outer circle in addition to end at the right-h in addition to side of the center line.
Solutions to Microwave problems using Smith chart The types of problems as long as which Smith charts are used include the following: Plotting a complex impedance on a Smith chart Finding VSWR as long as a given load Finding the admittance as long as a given impedance Finding the input impedance of a transmission line terminated in a short or open. Finding the input impedance at any distance from a load ZL. Locating the first maximum in addition to minimum from any load Matching a transmission line to a load with a single series stub. Matching a transmission line with a single parallel stub Matching a transmission line to a load with two parallel stubs. Plotting a Complex Impedance on a Smith Chart To locate a complex impedance, Z = R+-jX or admittance Y = G +- jB on a Smith chart, normalize the real in addition to imaginary part of the complex impedance. Locating the value of the normalized real term on the horizontal line scale locates the resistance circle. Locating the normalized value of the imaginary term on the outer circle locates the curve of constant reactance. The intersection of the circle in addition to the curve locates the complex impedance on the Smith chart. Finding the VSWR as long as a given load Normalize the load in addition to plot its location on the Smith chart. Draw a circle with a radius equal to the distance between the 1.0 point in addition to the location of the normalized load in addition to the center of the Smith chart as the center. The intersection of the right-h in addition to side of the circle with the horizontal resistance line locates the value of the VSWR.
Finding the Input Impedance at any Distance from the Load The load impedance is first normalized in addition to is located on the Smith chart. The VSWR circle is drawn as long as the load. A line is drawn from the 1.0 point through the load to the outer wavelength scale. To locate the input impedance on a Smith chart of the transmission line at any given distance from the load, advance in clockwise direction from the located point, a distance in wavelength equal to the distance to the new location on the transmission line. Power Loss Return Power Loss: When an electromagnetic wave travels down a transmission line in addition to encounters a mismatched load or a discontinuity in the line, part of the incident power is reflected back down the line. The return loss is defined as: Preturn = 10 log10 Pi/Pr Preturn = 20 log10 1/G Mismatch Power Loss: The term mismatch loss is used to describe the loss caused by the reflection due to a mismatched line. It is defined as Pmismatch = 10 log10 Pi/(Pi – Pr) Microwave Components Microwave components do the following functions: Terminate the wave Split the wave into paths Control the direction of the wave Switch power Reduce power Sample fixed amounts of power Transmit or absorb fixed frequencies Transmit power in one direction Shift the phase of the wave Detect in addition to mix waves
Coaxial components Connectors: Microwave coaxial connectors required to connect two coaxial lines are als called connector pairs (male in addition to female). They must match the characteristic impedance of the attached lines in addition to be designed to have minimum reflection coefficients in addition to not radiate power through the connector. E.g. APC-3.5, BNC, SMA, SMC, Type N Coaxial sections: Coaxial line sections slip inside each other while still making electrical contact. These sections are useful as long as matching loads in addition to making slotted line measurements. Double in addition to triple stub tuning configurations are available as coaxial stub tuning sections. Attenuators: The function of an attenuator is to reduce the power of the signal through it by a fixed or adjustable amount. The different types of attenuators are: Fixed attenuators Step attenuators Variable attenuators Coaxial components (contd.) Coaxial cavities: Coaxial cavities are concentric lines or coaxial lines with an air dielectric in addition to closed ends. Propagation of EM waves is in TEM mode. Coaxial wave meters: Wave meters use a cavity to allow the transmission or absorption of a wave at a frequency equal to the resonant frequency of the cavity. Coaxial cavities are used as wave meters. Waveguide components The waveguide components generally encountered are: Directional couplers Tee junctions Attenuators Impedance changing devices Waveguide terminating devices Slotted sections Ferrite devices Isolator switches Circulators Cavities Wavemeters Filters Detectors Mixers
Tees Hybrid Tee junction: Tee junctions are used to split waves from one waveguide to two other waveguides. There are two ways of connecting the third arm to the waveguide along the long dimension, called E=plane Tee. along the narrow dimension, called H-Plane Tee Hybrid Tee junction: the E-plane in addition to H-plane tees can be combined to as long as m a hybrid tee junction called Magic Tee Attenuators Attenuators are components that reduce the amount of power a fixed amount, a variable amount or in a series of fixed steps from the input to the output of the device. They operate on the principle of interfering with the electric field or magnetic field or both. Slide vane attenuators: They work on the principle that a resistive material placed in parallel with the E-lines of a field current will induce a current in the material that will result in I2R power loss. Flap attenuator: A flap attenuator has a vane that is dropped into the waveguide through a slot in the top of the guide. The further the vane is inserted into the waveguide, the greater the attenuation. Rotary vane attenuator: It is a precision waveguide attenuator in which attenuation follows a mathematical law. In this device, attenuation is independent on frequency. Isolators Mismatch or discontinuities cause energy to be reflected back down the line. Reflected energy is undesirable. Thus, to prevent reflected energy from reaching the source, isolators are used. Faraday Rotational Isolator: It combines ferrite material to shift the phase of an electromagnetic wave in its vicinity in addition to attenuation vanes to attenuate an electric field that is parallel to the resistive plane. Resonant absorption isolator: A device that can be used as long as higher powers. It consists of a section of rectangular waveguide with ferrite material placed half way to the center of the waveguide, along the axis of the guide.
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