Antennas and Propagation…. - Lecture 16

Antennas and Propagation.Lecture 162OverviewDipole AntennaAntenna GainAntenna LengthRadiation MechanismRadiation PatternAntenna EfficiencyBeam WidthTypes of AntennasPropagation ModesNoise CategoriesDipole AntennaOne of the most widely used antenna types is the half-wave dipole.The half-wave dipole, also called a doublet, is formally known as the Hertz antenna.A dipole antenna is two pieces of wire, rod, or tubing that are one-quarter wavelength long at the operating resonant frequency.34Converting a Transmission Line into an AntennaBending at right angles produces an efficient radiatorOptimum radiation occurs when the length is 1/2 of a wavelengthMagnetic fields now support each otherThree-dimensional radiation pattern for a dipole.Dipole AntennaVertically MountedThis pattern is a donut shape with the antenna passing thru the center.There is no radiation from the end of the antenna.5Antenna gain (G)A dipole antenna gain is 1.64A half-wave dipole antenna has a power gain of 1.64 (or 2.15 dB) over an isotropic source.Antenna gain relative to a dipole antenna can be expressed in decibels as dBd.Thus, an antenna with a gain of 3 dBd would have a gain of 5.15 dBi (3 dB + 2.15 dB)6Actual Antenna LengthsA dipole resonates best when it is approx. 95% of the actual “half-wavelength length”Shortcut: Lfeet = 468/f MHz (This is in Feet)1 ft = .3048 mDipole hung vertically is closest to an isotropic radiatorBottom of dipole antenna should be at least ½ a wavelength off the groundMay make total structure height unreasonable78What to look for in AntennasFreq/WavelengthBeam PatternBandwidthGain9RADIATION MECHANISMIdeally all incident energy must be reflected back when open circuit. But practically a small portion of electromagnetic energy escapes from the system that is it gets radiated.This occurs because the line of force don’t undergo complete phase reversal and some of them escapes.G10RADIATION MECHANISM The amount of escaped energy is very small due to mismatch between transmission line and surrounding space.Also because two wires are too close to each other, radiation from one tip will cancel radiation from other tip.( as they are of opposite polarities and distance between them is too small as compared to wavelength )G11RADIATION MECHANISM .To increase amount of radiated power open circuit must be enlarged , by spreading the two wires.Due to this arrangement, coupling between transmission line and free space is improved.Also amount of cancellation has reduced.The radiation efficiency will increase further if two conductors of transmission line are bent so as to bring them in same line.12TYPES OF ANTENNASAccording to their applications and technology available, antennas generally fall in one of two categories: 1.Omnidirectional or only weakly directional antennas which receive or radiate more or less in all directions. These are employed when the relative position of the other station is unknown or arbitrary. They are also used at lower frequencies where a directional antenna would be too large, or simply to cut costs in applications where a directional antenna isn't required. 2. Directional or beam antennas which are intended to preferentially radiate or receive in a particular direction or directional pattern.13TYPES OF ANTENNAS According to length of transmission lines available, antennas generally fall in one of two categories: 1. Resonant Antennas – is a transmission line, the length of which is exactly equal to multiples of half wavelength and it is open at both ends. 2.Non-resonant Antennas – the length of these antennas is not equal to exact multiples of half wavelength. In these antennas standing waves are not present as antennas are terminated in correct impedance which avoid reflections. The waves travel only in forward direction .Non-resonant antenna is a unidirectional antenna.14RADIATION PATTERNThe radiation pattern of an antenna is a plot of the relative field strength of the radio waves emitted by the antenna at different angles. It is typically represented by a three dimensional graph, or polar plots of the horizontal and vertical cross sections. It is a plot of field strength in V/m versus the angle in degrees.The pattern of an ideal isotropic antenna , which radiates equally in all directions, would look like a sphere. Many non-directional antennas, such as dipoles, emit equal power in all horizontal directions, with the power dropping off at higher and lower angles; this is called an omni directional pattern and when plotted looks like a donut.15RADIATION PATTERNThe radiation of many antennas shows a pattern of maxima or "lobes" at various angles, separated by “nulls", angles where the radiation falls to zero. This is because the radio waves emitted by different parts of the antenna typically interfere, causing maxima at angles where the radio waves arrive at distant points in phase, and zero radiation at other angles where the radio waves arrive out of phase. In a directional antenna designed to project radio waves in a particular direction, the lobe in that direction is designed larger than the others and is called the "main lobe". The other lobes usually represent unwanted radiation and are called “sidelobes". The axis through the main lobe is called the "principle axis" or “boresight axis".16ANTENNA GAINGain is a parameter which measures the degree of directivity of the antenna's radiation pattern. A high-gain antenna will preferentially radiate in a particular direction. Specifically, the antenna gain, or power gain of an antenna is defined as the ratio of the intensity (power per unit surface) radiated by the antenna in the direction of its maximum output, at an arbitrary distance, divided by the intensity radiated at the same distance by a hypothetical isotropic antenna.17The gain of an antenna is a passive phenomenon - power is not added by the antenna, but simply redistributed to provide more radiated power in a certain direction than would be transmitted by an isotropic antenna. High-gain antennas have the advantage of longer range and better signal quality, but must be aimed carefully in a particular direction. Low-gain antennas have shorter range, but the orientation of the antenna is relatively inconsequential. ANTENNA GAIN18For example, a dish antenna on a spacecraft is a high-gain device that must be pointed at the planet to be effective, whereas a typical Wi-Fi antenna in a laptop computer is low-gain, and as long as the base station is within range, the antenna can be in any orientation in space. In practice, the half-wave dipole is taken as a reference instead of the isotropic radiator. The gain is then given in dBd (decibels over dipole)ANTENNA GAIN19ANTENNA EFFICIENCYEfficiency of a transmitting antenna is the ratio of power actually radiated (in all directions) to the power absorbed by the antenna terminals. The power supplied to the antenna terminals which is not radiated is converted into heat. This is usually through loss resistance in the antenna's conductors, but can also be due to dielectric or magnetic core losses in antennas (or antenna systems) using such components.20POLARIZATIONThe polarization of an antenna is the orientation of the electric field (E-plane) of the radio wave with respect to the Earth's surface and is determined by the physical structure of the antenna and by its orientation. A simple straight wire antenna will have one polarization when mounted vertically, and a different polarization when mounted horizontally.Reflections generally affect polarization. For radio waves the most important reflector is the ionosphere - signals which reflect from it will have their polarization changed LF,VLF and MF antennas are vertically polarized21BEAM-WIDTHBeam-width of an antenna is defined as angular separation between the two half power points on power density radiation pattern OR Angular separation between two 3dB down points on the field strength of radiation pattern It is expressed in degrees22BEAM-WIDTH23ISOTROPIC ANTENNAIsotropic antenna or isotropic radiator is a hypothetical (not physically realizable) concept, used as a useful reference to describe real antennas. Isotropic antenna radiates equally in all directions. Its radiation pattern is represented by a sphere whose center coincides with the location of the isotropic radiator.24It is considered to be a point in space with no dimensions and no mass. This antenna cannot physically exist, but is useful as a theoretical model for comparison with all other antennas. Most antennas' gains are measured with reference to an isotropic radiator, and are rated in dBi (decibels with respect to an isotropic radiator).ISOTROPIC ANTENNA25HALF WAVE DIPOLE ANTENNAThe half-wave dipole antenna is just a special case of the dipole antenna. Half-wave term means that the length of this dipole antenna is equal to a half-wavelength at the frequency of operation.The dipole antenna, is the basis for most antenna designs, is a balanced component, with equal but opposite voltages and currents applied at its two terminals through a balanced transmission line.26To make it crystal clear, if the antenna is to radiate at 600 MHz, what size should the half-wavelength dipole be?One wavelength at 600 MHz is = c / f = 0.5 meters. Hence, the half-wavelength dipole antenna's length is 0.25 meters.The half-wave dipole antenna is as you may expect, a simple half-wavelength wire fed at the center as shown in Figure HALF WAVE DIPOLE ANTENNA27DipoleDipoles have an radiation pattern, doughnut symmetrical about the axis of the dipole. The radiation is maximum at right angles to the dipole, dropping off to zero on the antenna's axis. 2829Folded antenna is a single antenna but it consists of two elements.First element is fed directly while second one is coupled inductively at its end.Radiation pattern of folded dipole is same as that of dipole antenna i.e figure of eight (8).FOLDED DIPOLEAdvantagesInput impedance of folded dipole is four times higher than that of straight dipole.Typically the input impedance of half wavelength folded dipole antenna is 288 ohm.Bandwidth of folded dipole is higher than that of straight dipole.30HERTZ ANTENNAThe Hertzian dipole is a theoretical short dipole (significantly smaller than the wavelength) with a uniform current along its length. A true Hertzian dipole cannot physically exist, since the assumed current distribution implies an infinite charge density at its ends, and significant radiation requires a very high current over its very short length.3132LOOP ANTENNAA loop antenna is a radio antenna consisting of a loop of wire with its ends connected to a balanced transmission lineIt is a single turn coil carrying RF current through it.The dimensions of coil are smaller than the wavelength hence current flowing through the coil has same phase.Small loops have a poor efficiency and are mainly used as receiving antennas at low frequencies. Except for car radios, almost every AM broadcast receiver sold has such an antenna built inside of it or directly attached to it. 33A technically small loop, also known as a magnetic loop, should have a circumference of one tenth of a wavelength or less. This is necessary to ensure a constant current distribution round the loop. As the frequency or the size are increased, a standing wave starts to develop in the current, and the antenna starts to have some of the characteristics of a folded dipole antenna or a self-resonant loop.Self-resonant loop antennas are larger. They are typically used at higher frequencies, especially VHF and UHF, where their size is manageable. They can be viewed as a form of folded dipole and have somewhat similar characteristics. The radiation efficiency is also high and similar to that of a dipole.LOOP ANTENNA3435Radiation pattern of loop antenna is a doughnut pattern.Can be circular or square loopNo radiation is received normal to the plane of loop and null is obtained in this direction.Application: Used for direction finding applicationsLOOP ANTENNA36A turnstile antenna is a set of two dipole antennas aligned at right angles to each other and fed 90 degrees out-of-phase. The name reflects that the antenna looks like a turnstile when mounted horizontally. When mounted horizontally the antenna is nearly omnidirectional on the horizontal plane. TURNSTILE ANTENNAWhen mounted vertically the antenna is directional to a right angle to its plane and is circularly polarized. The turnstile antenna is often used for communication satellites because, being circularly polarized, the polarization of the signal doesn't rotate when the satellite rotates.TURNSTILE ANTENNA37RHOMBIC ANTENNAStructure and construction4 wires are connected in rhombic shape and terminated by a resistor.Mounted horizontally and placed > ^/2 from ground.Highest development of long wire antenna is rhombic antenna.38RHOMBIC ANTENNAStructure and construction4 wires are connected in rhombic shape and terminated by a resistor.Mounted horizontally and placed > ^/2 from ground.Highest development of long wire antenna is rhombic antenna.39RHOMBIC ANTENNA40AdvantagesEasier to constructIts i/p impedance and radiation pattern are relatively constant over range of frequencies.Maximum efficiencyHigh gain can be obtained.DisadvantagesLarge site area and large side lobes.RHOMBIC ANTENNA41ApplicationLong distance communication, high frequency transmission and reception.Point to point communication.Radio communication.Short wave radio broadcasting.RHOMBIC ANTENNA42The Discone AntennaThe discone antenna is characterized by very wide bandwidth, covering a 10:1 frequency rangeIt also has an omnidirectional pattern in the horizontal plane and a gain comparable to that of a dipoleThe feedpoint resistance is typically 50 ohmsTypically, the length of the surface of the cone is about one-quarter wavelength at the lowest operating frequency43The Helical AntennaSeveral types of antennas are classified as helicalThe antenna in the sketch has its maximum radiation along its long axisA quarter-wave monopole can be shortened and wound into a helix— common in rubber ducky antenna used with many handheld transceivers44ANTENNA ARRAYSAntenna arrays is group of antennas or antenna elements arranged to provide desired directional characteristics.Generally any combination of elements can form an array.However equal elements of regualar geometry are usually used.Simple antenna elements can be combined to form arrays resulting in reinforcement in some directions and cancellations in others to give better gain and directional characteristicsArrays can be classified as broadside or end-fireExamples of arrays are:The Yagi ArrayThe Log-Periodic Dipole ArrayThe Turnstile ArrayThe Monopole Phased ArrayOther Phased Arrays45YAGI-UDA ANTENNAIt is a directional antenna consisting of a driven element (typically a dipole or folded dipole) and additional parasitic elements (usually a so-called reflector and one or more directors). All the elements are arranged collinearly and close together.The reflector element is slightly longer (typically 5% longer) than the driven dipole, whereas the so-called directors are a little bit shorter. The design achieves a very substantial increase in the antenna's directionality and gain compared to a simple dipole. 46YAGI-UDA ANTENNA47Typical spacing between elements vary from about 1/10 to 1/4 of a wavelength, depending on the specific design.The elements are usually parallel in one plane.Radiation pattern is modified figure of eightBy adjusting distance between adjacent directors it is possible to reduce back lobeImproved front to back ratioYAGI-UDA ANTENNA48YAGI-UDA ANTENNA49ANTENNA APPLICATIONSThey are used in systems such as Radio broadcastingBroadcast televisionTwo-way radio Communication receiversRadarCell phones Satellite communications.50ANTENNA CONSIDERATIONSThe space available for an antennaThe proximity to neighborsThe operating frequencies The output powerMoney5152Antenna MatchingSometimes a resonant antenna is too large to be convenientOther times, an antenna may be required to operate at several widely different frequencies and cannot be of resonant length all the timeThe problem of mismatch can be rectified by matching the antenna to the feedline using an LC matching network53ReflectorsIt is possible to construct a conductive surface that reflects antenna power in the desired directionThe surface may consist of one or more planes or may be parabolicTypical reflectors are:Plane and corner ReflectorsThe Parabolic Reflector54Cell-Site AntennaFor cellular radio systems, there is a need for omnidirectional antennas and for antennas with beamwidths of 120º, and less for sectorized cellsCellular and PCS base-station receiving antennas are usually mounted in such a way as to obtain space diversityFor an omnidirectional pattern, typically three antennas are mounted on a tower with a triangular cross section and the antennas are mounted at 120º intervals55Mobile and Portable AntennaMobile and portable antennas used with cellular and PCS systems have to be omnidirectional and smallThe simplest antenna is the quarter-wavelength monopole are these are usually the ones supplied with portable phonesFor mobile phones, and common configuration is the quarter-wave antenna with a half-wave antenna mounted collinearly above it5657Antenna GainRelationship between antenna gain and effective areaG = antenna gainAe = effective areaf = carrier frequencyc = speed of light ( 3  108 m/s) = carrier wavelength5859Propagation ModelsGround Wave (GW) Propagation: 30MHz60Ground Wave PropagationFollows contour of the earth.Can propagate considerable distances.Frequency bands: ELF, VF, VLF, LF, MF.Spectrum range: 30Hz ~ 3MHz, e.g. AM radio.61Sky Wave PropagationSignal reflected from ionized layer of upper atmosphere back down to earth, which can travel a number of hops, back and forth between ionosphere and earth’s surface. HF band with intermediate frequency range: 3MHz ~ 30MHz.e.g: International broadcast. 62Line-of-Sight PropagationTx. and Rx. antennas are in the effective ‘line of sight’ range. Includes both LOS and non-LOS (NLOS) caseFor satellite communication, signal above 30 MHz not reflected by ionosphere.For ground communication, antennas within effective LOS due to refraction. Frequency bands: VHF, UHF, SHF, EHF, Infrared, optical lightSpectrum range : 30MHz ~ 900THz.6364LOS CalculationsLine-of-Sight EquationsEffective, or radio, line of sightd = distance between antenna and horizon (km)h = antenna height (m)K = adjustment factor to account for refraction, rule of thumb K = 4/3Maximum distance between two antennas for LOS propagation:65LOS Wireless Transmission ImpairmentsAttenuation and attenuation distortionFree space lossNoiseAtmospheric absorptionMultipathRefractionThermal noise66AttenuationStrength of signal falls off with distance over transmission mediumAttenuation factors for unguided media:Received signal must have sufficient strength so that circuitry in the receiver can interpret the signalSignal must maintain a level sufficiently higher than noise to be received without errorAttenuation is greater at higher frequencies, causing distortion67Free Space LossFree space loss, ideal isotropic antenna Pt = signal power at transmitting antennaPr = signal power at receiving antenna = carrier wavelengthd = propagation distance between antennasc = speed of light ( 3  108 m/s)where d and  are in the same units (e.g., meters)68Free Space LossFree space loss equation can be recast:6970Free Space LossFree space loss accounting for gain of other antennas can be recast as71Categories of NoiseThermal NoiseIntermodulation noiseCrosstalkImpulse Noise 72Noise (1)Thermal noise due to thermal agitation of electrons.Present in all electronic devices and transmission media.As a function of temperature.Uniformly distributed across the frequency spectrum, hence often referred as white noise.Cannot be eliminated – places an upper bound on the communication system performance.Can cause erroneous to the transmitted digital data bits.73Noise (2): Noise on digital dataError in bits74Thermal NoiseThe noise power density (amount of thermal noise to be found in a bandwidth of 1Hz in any device or conductor) is:N0 = noise power density in watts per 1 Hz of bandwidthk = Boltzmann's constant = 1.3803  10-23 J/KT = temperature, in kelvins (absolute temperature) 0oC = 273 Kelvin75Thermal NoiseNoise is assumed to be independent of frequencyThermal noise present in a bandwidth of B Hertz (in watts):or, in decibel-watts (dBW),76Noise TerminologyIntermodulation noise – occurs if signals with different frequencies share the same mediumInterference caused by a signal produced at a frequency that is the sum or difference of original frequenciesCrosstalk – unwanted coupling between signal pathsImpulse noise – irregular pulses or noise spikesShort duration and of relatively high amplitudeCaused by external electromagnetic disturbances, or faults and flaws in the communications system77Signal to Noise Ratio – SNR (1)Ratio of the power in a signal to the power contained in the noise present at a particular point in the transmission. Normally measured at the receiver with the attempt to eliminate/suppressed the unwanted noise.In decibel unit, where PS = Signal Power, PN = Noise PowerHigher SNR means better quality of signal.78Signal to Noise Ratio – SNR (2)SNR is vital in digital transmission because it can be used to sets the upper bound on the achievable data rate. Shannon’s formula states the maximum channel capacity (error-free capacity) as:Given the knowledge of the receiver’s SNR and the signal bandwidth, B. C is expressed in bits/sec.In practice, however, lower data rate are achieved.For a fixed level of noise, data rate can be increased by increasing the signal strength or bandwidth.79Expression Eb/N0Ratio of signal energy per bit to noise power density per HertzThe bit error rate for digital data is a function of Eb/N0Given a value for Eb/N0 to achieve a desired error rate, parameters of this formula can be selectedAs bit rate R increases, transmitted signal power must increase to maintain required Eb/N080Other ImpairmentsAtmospheric absorption – water vapor and oxygen contribute to attenuationMultipath – obstacles reflect signals so that multiple copies with varying delays are receivedRefraction – bending of radio waves as they propagate through the atmosphere81Multipath PropagationReflection - occurs when signal encounters a surface that is large relative to the wavelength of the signalDiffraction - occurs at the edge of an impenetrable body that is large compared to wavelength of radio waveScattering – occurs when incoming signal hits an object whose size in the order of the wavelength of the signal or less82The Effects of Multipath PropagationMultiple copies of a signal may arrive at different phasesIf phases add destructively, the signal level relative to noise declines, making detection more difficultIntersymbol interference (ISI)One or more delayed copies of a pulse may arrive at the same time as the primary pulse for a subsequent bit83Types of FadingFast fadingSlow fadingFlat fadingSelective fadingRayleigh fadingRician fading84Error Compensation MechanismsForward error correctionAdaptive equalizationDiversity techniques85Forward Error CorrectionTransmitter adds error-correcting code to data blockCode is a function of the data bitsReceiver calculates error-correcting code from incoming data bitsIf calculated code matches incoming code, no error occurredIf error-correcting codes don’t match, receiver attempts to determine bits in error and correct86Adaptive EqualizationCan be applied to transmissions that carry analog or digital informationAnalog voice or videoDigital data, digitized voice or videoUsed to combat intersymbol interferenceInvolves gathering dispersed symbol energy back into its original time intervalTechniquesLumped analog circuitsSophisticated digital signal processing algorithms87Diversity TechniquesDiversity is based on the fact that individual channels experience independent fading eventsSpace diversity – techniques involving physical transmission pathFrequency diversity – techniques where the signal is spread out over a larger frequency bandwidth or carried on multiple frequency carriersTime diversity – techniques aimed at spreading the data out over time8889AntennasWLANsAntennas are most often used to increase the range of a wireless LAN systemProper antenna selection can also enhance security of a wireless LANare most sensitive to RF signals whose wavelength is an even multiple of the antenna’s length (including fractional multiples – such as ½ or ¼)90Complimentary Session for Antennas Lecture 1691Complimentary SessionQ:Find the optimum wavelength and frequency for a half-wave dipole of length 10 m.Ans: The length of a half-wave dipole is one-half the wavelength of the signal that can be transmitted most efficiently. Therefore, the optimum wavelength in this case is λ = 20 m. The optimum free space frequency is f = c/λ = (3x108)/20 = 15 MHz.9293QQ: It turns out that the depth in the ocean to which airborne electromagnetic signals can be detected grows with the wavelength. Therefore, the military got the idea of using very long wavelengths corresponding to about 30 Hz to communicate with submarines throughout the world. If we want to have an antenna that is about one-half wavelength long, how long would that be?94AnsQ: It turns out that the depth in the ocean to which airborne electromagnetic signals can be detected grows with the wavelength. Therefore, the military got the idea of using very long wavelengths corresponding to about 30 Hz to communicate with submarines throughout the world. If we want to have an antenna that is about one-half wavelength long, how long would that be?Ans:-We have λf = c; in this case λx 30 = 3 x 108 m/sec, which yields a wavelength of 10,000 km. Half of that is 5,000 km which is comparable to the east-to-west dimension of the continental U.S. While an antenna this size is impractical, the U.S. Defense Department has considered using large parts of Wisconsin and Michigan to make an antenna many kilometers in diameter.9596QQ:- The audio power of the human voice is concentrated at about 300 Hz. Antennas of the appropriate size for this frequency are impracticably large, so that to send voice by radio the voice signal must be used to modulate a higher (carrier) frequency for which the natural antenna size is smaller. What is the length of an antenna one-half wavelength long for sending radio at 300 Hz?An alternative is to use a modulation scheme, as described in one of the Lectures, for transmitting the voice signal by modulating a carrier frequency, so that the bandwidth of the signal is a narrow band centered on the carrier frequency. Suppose we would like a half-wave antenna to have a length of 1 m. What carrier frequency would we use?97AnsWhat is the length of an antenna one-half wavelength long for sending radio at 300 Hz? An alternative is to use a modulation scheme, as described in one of the Lectures, for transmitting the voice signal by modulating a carrier frequency, so that the bandwidth of the signal is a narrow band centered on the carrier frequency. Suppose we would like a half-wave antenna to have a length of 1 m. What carrier frequency would we use?9899QQ:- Stories abound of people who receive radio signals in fillings in their teeth. Suppose you have one filling that is 2.5 mm (0.0025 m) long that acts as a radio antenna. That is, it is equal in length to one-half the wavelength. What frequency do you receive?Ans:- 100101Q:- It has been studied that if a source of electromagnetic energy is placed at the focus of the paraboloid, and if the paraboloid is a reflecting surface, then the wave will bounce back in lines parallel to the axis of the paraboloid. To demonstrate this, consider the parabola y2= 2px shown in Figure. Let P(x1, y1 ) be a point on the parabola and PF be the line from P to the focus. Construct the line L through P parallel to the x-axis and the line M tangent to the parabola at P. The angle between L and M is Beta, and the angle between PF and M is Alpha. The angle Alpha is the angle at which a ray from F strikes the parabola at P. Because the angle of incidence equals the angle of reflection, the ray reflected from P must be at an angle a to M. Thus, if we can show that Alpha = Beta, we have demonstrated that rays reflected from the parabola starting at F will be parallel to the x-axis.First show that tan (Beta) = (p/y1 ). Hint: Recall from trigonometry that the slope of a line is equal to the tangent of the angle the line makes with the positive x direction. Also recall that the slope of the line tangent to a curve at a given point is equal to the derivative of the curve at that point.Now show that tan (Alpha) = (p/y1 ) , which demonstrates that Alpha = Beta. Hint: Recall from trigonometry that the formula for the tangent of the difference between two angles Alpha1 and Alpha2 is tan( Alpha2 - Alpha1 ) = (tan (Alpha2) - tan (Alpha1)/( 1 + tan Alpha2 X tan Alpha1 ) .102Complimentary Session(a)103First, take the derivative of both sides of the equation y2 = 2px:The slope of PF is (y1 – 0)/(x1 – (p/2)). Therefore:Complimentary Session(b)104