Module in Radio Propagation

Transcription

1 1 This chapter is a brief overview of the subject of radio waves: what the are, how the can be characterised, and how their properties are related to those of the media in which the travel. It discusses electromagnetic waves as solutions of Mawell s equations, and introduces and derives epressions for the velocit, wavelength, period, attenuation, power densit and energ flow of these waves. A lot of the mathematical details of the derivations are not included in this chapter: the interested reader is referred to an good tetbook on the subject, for eample Antennas and Propagation for Wireless Communication Sstems b Simon R. Saunders and Alejandro Aragon-Zavala, or Radio Wave Propagation and Antennas b John Griffiths. 1.1 Introduction to In the nineteenth centur, phsicists such as Michael Farada and André-Marie Ampere were discovering that electric fields and magnetic fields were related. Specificall, that a changing magnetic field would produce a changing electric field around it, and that a changing electric field would produce a changing magnetic field around it, which would then produce a changing electric field around it, which would then produce a changing magnetic field, which would then produce and so on, with a pattern of electric and magnetic fields oscillating awa into space. James Clerk Mawell formulated the well-known Mawell s equations to describe the mathematical relationships between the electric and magnetic fields, and predicted that these electromagnetic waves should travel at 186, miles per second: eactl the speed that light had previousl been measured to travel at. Mawell suggested that light was an electromagnetic wave, but did not live to see this prediction proved. In a uniform medium without boundaries, Mawell s equations can be written in the form: dh = µ H= J+ ε dt t ρ. H=. = ε (1.1) where is the electric field strength, H is the magnetic field strength 1, ε is the permittivit of the medium, µ is the permeabilit of the medium, J is the current densit and ρ the charge densit. If we restrict the analsis to mediums in which there are no charged particles in the medium, and no currents flowing other than those caused b the electric fields in the waves, then we can write: J = σ (1.) where σ is the conductivit of the medium. Solving these equations gives: 1 More correctl, H is the magnetising field strength, and a related qualit B is known as the magnetic field, but I ll stick with H here for simplicit; this is common notation in electronic engineering. 7 Dave Pearce Page 1 14/1/8

2 = µε t + µσ t (1.3) which is known as a three-dimensional wave-equation. 1. Plane Waves This three-dimensional wave-equation has several solutions, but the simplest is the plane wave: a wave in which the electric field at all points in a plane is equal. (This is quite a good approimation when ou re a long wa from the transmitter in free space.) For eample, consider a wave in which the electric field is given b: = ( ) cos( = ep α ωt β ) (1.4) = where, and are the components of the electric field along the, and aes respectivel, and, α, β and ω are constants. In this case, the electric field is alwas parallel to the -ais, and has the same value everwhere in ever plane parallel to the -plane. The electric field varies in phase and amplitude along the -ais onl. Figure 1-1 A Wave with lectric Field Varing Along the -ais B substituting equation (1.4) into equation (1.3) we can derive that: α = ω 1 µε 1 + ( σ / ωε) 1 (1.5) and β = ω 1 µε 1 + ( σ / ωε) + 1 (1.6) 7 Dave Pearce Page 14/1/8

3 1..1 Attenuation The constant α determines the attenuation of the wave as it passes through the medium. At an location, the electric field is entirel in the direction of the -ais, and is given b: ( α ) ( ω β ) = ep cos t (1.7) and this is an oscillation with amplitude ep( α). A plot of the amplitude against distance of this wave would look like a decaing eponential, as shown below. Over a distance 1/α meters, the amplitude of a wave decas b a factor of e.7188, since: ( α ) ( α) ( α + ) ep( α ) ( α ) ( ) 1 ep = = = ep() 1 = ep 1/ ep ep 1 (1.8) This is known as one neper of attenuation: a neper is a factor of 1/e in amplitude. Since the attenuation over a distance of 1/α meters is one neper, in one metre the wave eperiences α nepers of attenuation, and α is quoted in terms of nepers per metre. (One db is a factor of 1.1 = 1.59 in power, and since power is proportional to the square of amplitude, an attenuation of one neper is equivalent to an attenuation of db.) Figure 1- Amplitude Attenuation with Distance It s interesting to note that the formula for the attenuation gives a value of α = when the conductivit of the medium σ =. Radio waves do not normall eperience no power loss during propagation through a material that does not conduct electricit, such as the lower part of the arth s atmosphere. When the attenuation is small or the frequenc is ver high, we can approimate: At least provided it s not raining, snowing, hailing, or otherwise has liquid water present. Also, this statement isn t true at ver high frequencies, where energ can be lost from the radio waves due to molecular resonance in the atmosphere. 7 Dave Pearce Page 3 14/1/8

4 and hence derive the approimate result: σ 1 ( σ / ωε) (1.9) ω ε σ µ α = (1.1) ε and in this case the attenuation is not a function of frequenc. On the other hand, if the frequenc is ver low, so that: which gives the approimate result: 1 ( / ) σ + σ ωε (1.11) ωε µ σω α = (1.1) ε which shows that at ver low frequencies, the attenuation tends to increase with frequenc. 1.. Wavelength and Wave Number The wavelength λ is the distance between two successive maimum values of electric (or magnetic) field. The formula for electric field of the plane wave we have been considering: ( α ) ( ω β ) = ep cos t (1.13) at an given time t has maimum values where cos(ω t β ) = 1. This implies that for these minima: ω t β= πn (1.14) and for two consecutive maimum values at distances 1 and at the same time: ( t ) ( t ) ω β ω β = π 1 β ( ) = π 1 π 1 = = λ β (1.15) the quantit β is known as the wavenumber, although it s much more common to deal with the wavelength λ Period and Frequenc The period T is the time between two successive maimum values of electric (or magnetic) field passing the same place. Again, we can determine this from the formula for electric field of the plane wave we have been considering: ( α ) ( ω β ) = ep cos t (1.16) 7 Dave Pearce Page 4 14/1/8

5 at an given place has maimum values where cos(ω t β ) = 1. Again, this implies that: ω t β= πn (1.17) and for two consecutive maimum values at times t 1 and t at the same place: ( t ) ( t ) ω β ω β = π 1 ω ( t t ) = π 1 (1.18) π T = t1 t = ω the quantit ω is known as the angular frequenc of the wave, and is measured terms of radians per second (rad/s). The frequenc of the wave is the number of maimum values that go past a defined spot ever second. If the time between these maima is π / ω, then the frequenc is: f 1 ω = = (1.19) T π The frequenc is measured in Hert (H) Polarisation The plane wave described above is not the onl solution of Mawell s equations that results in a plane wave travelling along the -ais. There is another solution in which the electric field is entirel parallel to the -ais everwhere, and there is no component parallel to the -ais. When the -ais is vertical, this is known as a verticall polarised wave, and the wave described above with the electric field parallel to the horiontal ais is known as a horiontall-polarised wave. = = = ep( α ) cos( ωt β) (1.) Figure 1-3 A Plane Wave Polarised Along the -ais 7 Dave Pearce Page 5 14/1/8

6 Since Mawell s equations are linear, the sum of an two solutions of Mawell s equations is another solution, and adding horiontall and verticall polarised waves together can result in plane waves in which the electric field is at an fied angle to the horiontal. Antennas are designed to transmit or receive just one polarisation. This can allow two separate transmissions in the same locations to work without interfering with each other (provided there is nothing that rotates the polarisation: for eample reflecting some verticall polarised waves when a horiontall polarised wave is incident). There is another interesting form of polarisation that occurs when a horiontall polarised wave and a verticall polarised wave are combined 9-degrees out of phase: circular polarisation. The wave in this case can be described b: = ( ) sin ( = ep α ωt β ( ) cos( = ep α ωt β ) ) (1.1) so that the amplitude of the electric field is now constant at all times, since: = + ( α ) ( ω β ) ( ω β) = ep sin t + cos t = ep ( α ) (1.) It is just the direction of the electric field that changes. This is notoriousl difficult to draw, but it looks something like this: Figure 1-4 A Circularl-Polarised Plane Wave Just as there are two tpes of linear polarisation (horiontal and vertical), there are two tpes of circular polarisation: clockwise and anticlockwise. An antenna designed to receive clockwise circularl-polarised waves will not receive an power from an anticlockwise circularl polarised wave, and vice versa. 7 Dave Pearce Page 6 14/1/8

7 However, since a circularl polarised wave is the linear combination of a horiontall polarised wave and a verticall polarised wave, an antenna designed to receive circularl-polarised wave will receive half the power in an incident linearl polarised wave, no matter what the orientation of the linear polarisation. Similarl, a receive antenna designed to receive horiontall-polarised wave will receive half the power in a circularl-polarised incident wave Phase and Group Velocit In general, wave packets travel through space with two characteristic speeds. The first, and easiest to understand, is the phase velocit. This is the speed of the wavefronts of a continuous single-frequenc wave, so that for a user travelling at this speed, all the time. This can onl happen if: ( ωt β) cos = 1 (1.3) ω t = β u p ω = = t β (1.4) Therefore the phase velocit is just u p = ω / β. Using equation (1.6), this can be written as: u p ω = = β ( / ) µε + σ ωε + 1 (1.5) and when there is no conductivit in the medium, this results in the simple formula: u p 1 = (1.6) µε An important related quantit is the refractive inde n, defined as the ratio of the phase velocit in free space c to the phase velocit in the medium, u p : c n = = ε rµ r (1.7) u p where ε r is the relative permittivit and µ r the relative permeabilit respectivel: ε = εε µ = µµ (1.8) r r where ε is the permittivit and µ the permeabilit of free space. When sending a pulse of energ rather than one continuous frequenc, there is a range of different frequencies being transmitted 3, and something more interesting happens in certain tpes of media 3 Given b the Fourier transform of the shape of the transmitted pulse. 7 Dave Pearce Page 7 14/1/8

8 (known as dispersive media) in which the phase velocit is a function of frequenc. In this case the pulse of energ travels at a different speed from the wavefronts 4. Usuall the speed with which the energ travels is more interesting, and this is known as the group velocit. It can be determined b considering what happens to the pulse in terms of the frequenc components that make up the pulse: the pulse occurs where all the frequenc components of the pulse add up in phase. In other words, the pulse travels at such a speed that the phase of each component is not a function of frequenc. Mathematicall: ( t ) independent of φ = ω β = ω dφ d β = t = dω d ω u g dω = = t dβ (1.9) (at least for the simple case of isotropic media where the wave-number β is not a function of time or position). Now since the refractive inde u p = c / n, and u p = ω / β, we can derive that: cβ = ω n dβ dn c = n+ ω dω dω dω c ug = = dβ n + ω. dn/ dω (1.3) and it's evident that when the refractive inde is not a function of frequenc, the group velocit is just equal to the phase velocit u p. 1.3 Wave Impedance and nerg Flow The wave impedance Z w is defined as the ratio of the magnitude of the and H fields at an point in space. The magnitude of the H field can be derived from the field using Mawell s equation: which gives for the plane wave discussed here 5 : = µ H (1.31) t Z w = = H jωµ σ + jωε (1.3) 4 Go and drop a pebble in a lake and look at the ripples sometime: ou'll notice that the peaks of the waves travel at a different speed from the ripples themselves. 5 The factor j indicates that this impedance is comple is related to the fact that when the medium is conductive, the electric and magnetic fields are no longer in phase. In a non-conductive medium, σ =, and the electric and magnetic fields are in phase with each other, and the factor in j cancels out. 7 Dave Pearce Page 8 14/1/8

9 In free space σ =, and the permittivit and permeabilit of the medium are just the permittivit and permeabilit of free space, and we can simplif this to: Z µ = = = 1π = 377ohms H ε (1.33) nerg Flow and the Ponting Vector nerg is stored in electric and magnetic fields: this is the energ that can be converted into kinetic energ when like-charged particles accelerate awa from each other when allowed to move freel. The energ stored in an electric field b unit volume is: G ε = (1.34) and in a magnetic field is: G H µ H = (1.35) Again, in free space, there is an equal energ stored in the electric and magnetic components of an electromagnetic wave, and therefore the total energ stored per unit volume of space in the wave is: G W ε µ H = + = ε (1.36) As an electromagnetic field passes through space, this energ is moved in the direction of propagation of the radio wave. A power densit can be defined for this wave in terms of a power per unit area: this is the energ that moves through a unit area perpendicular to the direction of propagation in unit time. This energ is moving at the speed of light, so the volume of energ that passes through a unit area in one second is just the energ in a clinder (or other shape), meters long, and one meter in cross-sectional area. In free space, this implies that: d 1 ε ε ε µ µ Z 1 P = ε c= = = = π (1.37) Note that this result is epressed in terms of the mean square electric field, and for a sinusoidall oscillating electric field, this is related to the peak amplitude electric field b: = (1.38) This result is often quoted in terms of the Ponting vector, which in free space is given b: S= H (1.39) This vector points in the direction of the flow of energ, and has a magnitude equal to the energ flow in the radio wave at ever point. 7 Dave Pearce Page 9 14/1/8

10 1.4 Ke Points At this point ou should: Know that one solution of Mawell's equations in isotropic media is plane waves, travelling in one direction; Know how to relate the wavelength and attenuation of these waves to the phsical parameters of the media; Understand the difference between phase and group velocit, and how to calculate both; Understand the concept of wave impedance and know how to calculate the wave impedance from the phsical parameters of a medium; Know the difference between horiontall and verticall polarised waves, and circularl polarised waves. 1.5 Tutorial Questions (No need to attempt all of them - just make sure ou know how to attack at least all those not marked with a *.) 1) A radio wave in free space has a frequenc of GH. What s the wavelength? ) A Watt is a unit of power, equivalent to a rate of using energ of one joule per second. In similar terms, what is a neper? A medium has an attenuation co-efficient of.1 nepers/meter. A beam of light of power 1W is shone in a laer of this medium meters wide. How much power gets to the other side? What is the loss of this medium in db? And in db/m? (Neglect an reflections at the surfaces of this medium). *3) Under what conditions does a medium have a phase co-efficient (defined as π divided b the wavelength in meters) approimatel equal to the attenuation in nepers? Give an eample of a material in which these conditions eist. 7 Dave Pearce Page 1 14/1/8