Antennas and Propagation Lecture 16 Overview n n

Antennas and Propagation…. Lecture 16

Overview n n n n n Dipole Antenna Gain Antenna Length Radiation Mechanism n Radiation Pattern Antenna Efficiency Beam Width Types of Antennas Propagation Modes Noise Categories 2

Dipole Antenna n n n One 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. 3

Converting a Transmission Line into an Antenna Bending at right angles produces an efficient radiator Magnetic fields now support each other Optimum radiation occurs when the length is 1/2 of a wavelength 4

Three-dimensional radiation pattern for a dipole. Vertically Mounted Dipole Antenna This pattern is a donut shape with the antenna passing thru the center. There is no radiation from the end of the antenna. 5

Antenna gain (G) n n A dipole antenna gain is 1. 64 A half-wave dipole antenna has a power gain of 1. 64 (or 2. 15 d. B) over an isotropic source. Antenna gain relative to a dipole antenna can be expressed in decibels as d. Bd. Thus, an antenna with a gain of 3 d. Bd would have a gain of 5. 15 d. Bi (3 d. B + 2. 15 d. B) 6

Actual Antenna Lengths n n n A dipole resonates best when it is approx. 95% of the actual “half-wavelength” Shortcut: Lfeet = 468/f MHz (This is in Feet) 1 ft =. 3048 m Dipole hung vertically is closest to an isotropic radiator Bottom of dipole antenna should be at least ½ a wavelength off the ground n May make total structure height unreasonable 7

8

RADIATION MECHANISM n n Ideally 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. G 10

RADIATION MECHANISM … n n 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 ) G 11

RADIATION MECHANISM …. n n 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. 12

TYPES OF ANTENNAS n n n According 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. 13

TYPES OF ANTENNAS … n n n 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. 14

RADIATION PATTERN n n The 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. 15

RADIATION PATTERN n n The 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". 16

ANTENNA GAIN n n Gain 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. 17

ANTENNA GAIN n n n The 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. 18

ANTENNA GAIN n n For example, a dish antenna on a spacecraft is a highgain 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 d. Bd (decibels over dipole) 19

ANTENNA EFFICIENCY n n Efficiency 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. 20

POLARIZATION n n The 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 polarized 21

BEAM-WIDTH n n n Beam-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 3 d. B down points on the field strength of radiation pattern It is expressed in degrees 22

BEAM-WIDTH 23

ISOTROPIC ANTENNA n n Isotropic 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. n Its radiation pattern is represented by a sphere whose center coincides with the location of the isotropic radiator. 24

ISOTROPIC ANTENNA n n It 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 d. Bi (decibels with respect to an isotropic radiator). 25

HALF WAVE DIPOLE ANTENNA n n n The 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. 26

HALF WAVE DIPOLE ANTENNA n n n To 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 27

Dipole n Dipoles 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. 28

FOLDED DIPOLE n n n Folded 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). 29

Advantages n n n Input 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. 30

HERTZ ANTENNA n n The 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. 31

32

LOOP ANTENNA n n A loop antenna is a radio antenna consisting of a loop of wire with its ends connected to a balanced transmission line It 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. 33

LOOP ANTENNA n n n A 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 selfresonant 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 34

LOOP ANTENNA • • Radiation pattern of loop antenna is a doughnut pattern. Can be circular or square loop No radiation is received normal to the plane of loop and null is obtained in this direction. Application: Used for direction finding applications 35

TURNSTILE ANTENNA n n n A 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. 36

TURNSTILE ANTENNA n n When 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. 37

RHOMBIC ANTENNA n Structure and construction n 4 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. 38

RHOMBIC ANTENNA n Structure and construction n 4 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. 39

RHOMBIC ANTENNA 40

RHOMBIC ANTENNA n Advantages n n n Easier to construct Its i/p impedance and radiation pattern are relatively constant over range of frequencies. Maximum efficiency High gain can be obtained. Disadvantages n Large site area and large side lobes. 41

RHOMBIC ANTENNA n Application n n Long distance communication, high frequency transmission and reception. Point to point communication. Radio communication. Short wave radio broadcasting. 42

The Discone Antenna n n The discone antenna is characterized by very wide bandwidth, covering a 10: 1 frequency range It also has an omnidirectional pattern in the horizontal plane and a gain comparable to that of a dipole The feedpoint resistance is typically 50 ohms Typically, the length of the surface of the cone is about one-quarter wavelength at the lowest operating frequency 43

The Helical Antenna n n n Several types of antennas are classified as helical The antenna in the sketch has its maximum radiation along its long axis A quarter-wave monopole can be shortened and wound into a helix— common in rubber ducky antenna used with many handheld transceivers 44

ANTENNA ARRAYS n n n Antenna 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 characteristics Arrays can be classified as broadside or end-fire Examples of arrays are: n The Yagi Array n The Log-Periodic Dipole Array n The Turnstile Array n The Monopole Phased Array 45 n Other Phased Arrays

YAGI-UDA ANTENNA n n It 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. 46

YAGI-UDA ANTENNA 47

YAGI-UDA ANTENNA n n n Typical 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 eight By adjusting distance between adjacent directors it is possible to reduce back lobe Improved front to back ratio 48

YAGI-UDA ANTENNA 49

ANTENNA APPLICATIONS n n n n They are used in systems such as Radio broadcasting Broadcast television Two-way radio Communication receivers Radar Cell phones Satellite communications. 50

ANTENNA CONSIDERATIONS n n n The space available for an antenna The proximity to neighbors The operating frequencies The output power Money 51

52

Antenna Matching n n n Sometimes a resonant antenna is too large to be convenient Other times, an antenna may be required to operate at several widely different frequencies and cannot be of resonant length all the time The problem of mismatch can be rectified by matching the antenna to the feedline using an LC matching network 53

Reflectors n n n It is possible to construct a conductive surface that reflects antenna power in the desired direction The surface may consist of one or more planes or may be parabolic Typical reflectors are: n n Plane and corner Reflectors The Parabolic Reflector 54

Cell-Site Antenna n n n For cellular radio systems, there is a need for omnidirectional antennas and for antennas with beamwidths of 120º, and less for sectorized cells Cellular and PCS base-station receiving antennas are usually mounted in such a way as to obtain space diversity For an omnidirectional pattern, typically three antennas are mounted on a tower with a triangular cross section and the antennas are mounted at 120º intervals 55

Mobile and Portable Antenna n n n Mobile and portable antennas used with cellular and PCS systems have to be omnidirectional and small The simplest antenna is the quarter-wavelength monopole are these are usually the ones supplied with portable phones For mobile phones, and common configuration is the quarterwave antenna with a half-wave antenna mounted collinearly above it 56

57

Antenna Gain n Relationship between antenna gain and effective area n n n G = antenna gain Ae = effective area f = carrier frequency c = speed of light ( 3 108 m/s) = carrier wavelength 58

59

Propagation Models n n n Ground Wave (GW) Propagation: < 3 MHz Sky Wave (SW) Propagation: 3 MHz to 30 MHz Effective Line-of-Sight (LOS) Propagation: > 30 MHz 60

Ground Wave Propagation – – Follows contour of the earth. Can propagate considerable distances. Frequency bands: ELF, VLF, MF. Spectrum range: 30 Hz ~ 3 MHz, e. g. AM radio. 61

Sky Wave Propagation – – – Signal 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: 3 MHz ~ 30 MHz. e. g: International broadcast. 62

Line-of-Sight Propagation Tx. and Rx. antennas are in the effective ‘line of sight’ range. Includes both LOS and non-LOS (NLOS) case For 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 light Spectrum range : 30 MHz ~ 900 THz. 63

LOS Calculations 64

Line-of-Sight Equations n Effective, or radio, line of sight n n n d = distance between antenna and horizon (km) h = antenna height (m) K = adjustment factor to account for refraction, rule of thumb K = 4/3 Maximum distance between two antennas for LOS propagation: n 65

LOS Wireless Transmission Impairments n n n n Attenuation and attenuation distortion Free space loss Noise Atmospheric absorption Multipath Refraction Thermal noise 66

Attenuation n n Strength of signal falls off with distance over transmission medium Attenuation factors for unguided media: n n n Received signal must have sufficient strength so that circuitry in the receiver can interpret the signal Signal must maintain a level sufficiently higher than noise to be received without error Attenuation is greater at higher frequencies, causing distortion 67

Free Space Loss n Free space loss, ideal isotropic antenna n n n Pt = signal power at transmitting antenna Pr = signal power at receiving antenna = carrier wavelength d = propagation distance between antennas c = speed of light ( 3 108 m/s) where d and are in the same units (e. g. , meters) 68

Free Space Loss n Free space loss equation can be recast: 69

70

Free Space Loss n Free space loss accounting for gain of other antennas can be recast as 71

Categories of Noise n n Thermal Noise Intermodulation noise Crosstalk Impulse Noise 72

Noise (1) n n n 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. 73

Noise (2): Noise on digital data Error in bits 74

Thermal Noise n The noise power density (amount of thermal noise to be found in a bandwidth of 1 Hz in any device or conductor) is: N 0 = noise power density in watts per 1 Hz of bandwidth k = Boltzmann's constant = 1. 3803 10 -23 J/K T = temperature, in kelvins (absolute temperature) 0 o. C = 273 Kelvin 75

Thermal Noise n n Noise is assumed to be independent of frequency Thermal noise present in a bandwidth of B Hertz (in watts): n or, in decibel-watts (d. BW), 76

Noise Terminology n Intermodulation noise – occurs if signals with different frequencies share the same medium n n n Interference caused by a signal produced at a frequency that is the sum or difference of original frequencies Crosstalk – unwanted coupling between signal paths Impulse noise – irregular pulses or noise spikes n n Short duration and of relatively high amplitude Caused by external electromagnetic disturbances, or faults and flaws in the communications system 77

Signal to Noise Ratio – SNR (1) n n n 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, n n where PS = Signal Power, PN = Noise Power Higher SNR means better quality of signal. 78

Signal to Noise Ratio – SNR (2) n n 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. n n n 79

Expression Eb/N 0 n n Ratio of signal energy per bit to noise power density per Hertz The bit error rate for digital data is a function of Eb/N 0 n n Given a value for Eb/N 0 to achieve a desired error rate, parameters of this formula can be selected As bit rate R increases, transmitted signal power must increase to maintain required Eb/N 0 80

Other Impairments n n n Atmospheric absorption – water vapor and oxygen contribute to attenuation Multipath – obstacles reflect signals so that multiple copies with varying delays are received Refraction – bending of radio waves as they propagate through the atmosphere 81

Multipath Propagation n Reflection - occurs when signal encounters a surface that is large relative to the wavelength of the signal Diffraction - occurs at the edge of an impenetrable body that is large compared to wavelength of radio wave Scattering – occurs when incoming signal hits an object whose size in the order of the wavelength of the signal or less 82

The Effects of Multipath Propagation n Multiple copies of a signal may arrive at different phases n n If phases add destructively, the signal level relative to noise declines, making detection more difficult Intersymbol interference (ISI) n One or more delayed copies of a pulse may arrive at the same time as the primary pulse for a subsequent bit 83

Types of Fading n n n Fast fading Slow fading Flat fading Selective fading Rayleigh fading Rician fading 84

Error Compensation Mechanisms n n n Forward error correction Adaptive equalization Diversity techniques 85

Forward Error Correction n Transmitter adds error-correcting code to data block n n Code is a function of the data bits Receiver calculates error-correcting code from incoming data bits n n If calculated code matches incoming code, no error occurred If error-correcting codes don’t match, receiver attempts to determine bits in error and correct 86

Adaptive Equalization n Can be applied to transmissions that carry analog or digital information n n Analog voice or video Digital data, digitized voice or video Used to combat intersymbol interference Involves gathering dispersed symbol energy back into its original time interval Techniques n n Lumped analog circuits Sophisticated digital signal processing algorithms 87

Diversity Techniques n n Diversity is based on the fact that individual channels experience independent fading events Space diversity – techniques involving physical transmission path Frequency diversity – techniques where the signal is spread out over a larger frequency bandwidth or carried on multiple frequency carriers Time diversity – techniques aimed at spreading the data out over time 88

89

Antennas…WLANs n n n Antennas are most often used to increase the range of a wireless LAN system Proper antenna selection can also enhance security of a wireless LAN …are most sensitive to RF signals whose wavelength is an even multiple of the antenna’s length (including fractional multiples – such as ½ or ¼) 90

Complimentary Session for Antennas Lecture 16 91

Complimentary Session Q: 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 = (3 x 108)/20 = 15 MHz. n 92

93

Q Q: 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 onehalf wavelength long, how long would that be? 94

Ans Q: 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 95 Michigan to make an antenna many kilometers in diameter.

96

Q Q: - 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. a. What is the length of an antenna one-half wavelength long for sending radio at 300 Hz? b. 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 97 m. What carrier frequency would we use?

Ans 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? 98

99

Q Q: - 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: - 100

101

Q: - 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 y 2= 2 px shown in Figure. Let P(x 1, y 1 ) 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. a)First show that tan (Beta) = (p/y 1 ). 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. b)Now show that tan (Alpha) = (p/y 1 ) , which demonstrates that Alpha = Beta. Hint: Recall from trigonometry that the formula for the tangent of the difference between two angles Alpha 1 and Alpha 2 is tan( Alpha 2 - Alpha 1 ) = (tan (Alpha 2) - tan (Alpha 1)/( 1 + tan Alpha 2 X tan Alpha 1 ). 102

Complimentary Session (a) First, take the derivative of both sides of the equation y 2 = 2 px: The slope of PF is (y 1 – 0)/(x 1 – (p/2)). Therefore: 103

Complimentary Session (b) 104