6 Lectures 3 Main Sections ~2 lectures per subject

Transcription

1 P5-Electromagnetic ields and Waves Prof. Andrea C. errari ectures 3 Main Sections ~ lectures per subject Transmission ines. The wave equation.1 Telegrapher s Equations. Characteristic mpedance.3 Reflection

2 Electromagnetic Waves in ree Space.1 Electromagnetic ields. Electromagnetic Waves.3 Reflection and Refraction of Waves Antennae and Radio Transmission.1 Antennae. Radio 3 3 OJECTES As the frequency of electronic circuits rises, one can no longer assume that voltages and currents are instantly transmitted by a wire. The objectives of this course are: Appreciate when a wave theory is needed Derive and solve simple transmission line problems Understand the importance of matching to the characteristic impedance of a transmission cable Understand basic principles of EM wave propagation in free space, across interfaces and the use of antennae 4 4

3 This course deals with transmission of electromagnetic waves 1) along a cable (i.e. a transmission line) ) through free space (the ether ). n the first half of these lectures, we will derive the differential equations which describe the propagation of a wave along a transmission line. Then we will use these equations to demonstrate that these waves ehibit reflection, have impedance, and transmit power. n the second half of these lectures we will look at the behaviour of waves in free space. We will also consider different types of antennae for transmission and reception of electromagnetic waves. Reference: OER A.D. Microwave and Optical Transmission John Wiley & Sons, 199, 1997 Shelf Mark: N Handouts The handouts have some gaps for you to fill NOTE: 1) DO NOT PANC YOU DO NOT MANAGE TO WRTE DOWN N REA TME ) Prefer to just sit back and rela? You will be able to Download a PD of the complete slides from 6 6

4 . The Wave Equation Aims To recall basic phasors concepts To introduce the generalised form of the wave equation Objectives At the end of this section you should be able to recognise the generalized form of the wave equation, its general solution, the propagation direction and velocity ntroduction An ideal transmission line is defined as: a link between two points in which the signal at any point equals the initiating signal i.e. transmission takes place instantaneously and there is no attenuation Real world transmission lines are not ideal, there is attenuation and there are delays in transmission 8 8

5 A transmission line can be seen as a device for propagating energy from one point to another The propagation of energy is for one of two general reasons: 1. Power transfer (e.g. for lighting, heating, performing work) - eamples are mains electricity, microwave guides in a microwave oven, a fibre-optic illuminator.. nformation transfer. eamples are telephone, radio, and fibre-optic links (in each case the energy propagating down the transmission line is modulated in some way). 9 9 Eamples Power Plant Consumer Home 1 1

6 Antenna Optical ibre ink Pair of wires Co-a cable PC tracks C 1 interconnects

7 Waveguides Mircostrip Dielectric of thickness T, with a conductor deposited on the bottom surface, and a strip of conductor of width W on the top surface Can be fabricated using Printed Circuit oard (PC) technology, and is used to convey microwave frequency signals 14 14

8 Microwave Oven Optical ibres 16 16

9 Phasor Notation A means A is comple m j A Re{ A} + m{ A} j A e A m{ A} A A A A Re{ A} Re j Ae β j ( t ) is short-hand for R e{ Ae β + ω } which equals: A cos( β + ωt + A) e j ( ± θ ) Proof cos( θ ) ± j sin( θ ) then Ae Ae e Ae j ( β + ω t ) j A j ( β + ω t ) j ( β + ω t + A) j( β + ωt + A) Ae A cos( β + ωt + A) + jasin( β + ωt + A) 18 18

10 ..1 The Wave Equation The generalised form of the wave equation is: t A v A Where the aplacian of a scalar A is: A + + A A A y z We will be looking at plane waves for which the wave equation is one-dimensional and appears as follows: A t A v or A 1 A v t Where A could be: Either the oltage () or the Current () as in waves in a transmission line Or the Electric ield (E) or Magnetic ield (H) as in electromagnetic waves in free space

11 There are many other cases where the wave equation is used or eample 1) Waves on a string. These are planar waves where A represents the amplitude of the wave ) Waves in a membrane, where there is variation in both and y, and the equation is of the form A A A v + t y 1 1 The constant v is called the wave speed. This comes from the fact that the general solution to the wave equation (D Alembert solution,~1747) is A f ( ± vt) Note A f ( vt) orward moving A f ( + vt) ackward moving

12 Direction of travel f ( vt) f ( + vt) (t+/v) P t t+ t t+ t t t Consider a fied point, P, on the moving waveform, i.e. a point with constant f f(-vt) will be constant if -vt is constant f t increases (t t+ t), must also increase if -vt is to be constant An increase implies that the wave is moving to the right (orward) 3 Similarly, for +vt wave is moving to left (ackward) 3 erify that A f ( ± vt) is general solution A ± vf '( ± vt) t A f '( ± vt) A v f vt t ''( ± ) A f ''( ± vt) A t v A 4 4

13 .1 Electrical Waves Aims To derive the telegrapher s equations To account for losses in transmission lines Objectives At the end of this section you should be able to recognise when the wave theory is relevant; to master the concepts of wavelenght, wave velocity, period and phase; to describe the propagation of waves in loss-less and lossy transmission lines Telegrapher s Equations et us consider a short length, δ, of a wire pair δ This could, for eample, represent a coaial cable or a small δ, any function A() can be written as A( ) A( + δ ) A( ) + δ n our case A can be oltage () or Current () 6 6

14 et us define series/loop inductance per unit length [H/m] δ δ δ t 7 7 C parallel/shunt capacitance per unit length [/m] C C Cδ δ C Cδ t C 8 8

15 δ t + δ + C C C δ C + δ δ C C Cδ CAMRDGE Cδ ( UNERSTY + δ ) Cδ EECTRONC NANOMATERAS + C ( DECES δ ) AND Cδ t DEPARTMENT t O ENGNEERNG tand SPECTROSCOPY MATERAS t GROUP t 9 9 C + δ δ t C t C t + δ Cδ t (1.1) (1.) Eqs. (1.1),(1.) are known as the telegrapher s equations They were derived in 1885 by Oliver Heaviside, and were crucial in the early development of long distance telegraphy (hence the name) 3 3

16 .1. Travelling Wave Equations et us differentiate both (1.1) and (1.) with respect to t C t (1.1a) C t C t (1.a) Then in (1.1a) substitute using (1.) Then in (1.a) substitute using (1.1) A 1 A v t C t t C (1.1a) (1.a) Same functional form as wave equation: We try a solution for in (1.1a) of the form Ae e jβ jω t v 1 C 3 3

17 β β ± ω Ae e ω C Ae e jβ jωt jβ jωt C Hence Phase Constant (1.3) Since β can be positive or negative, we obtain epressions for voltage and current waves moving forward (subscript ) and backward (subscript ) along the transmission line {( j β j β ) j ω t } Re e + e e (1.4) {( j β j β ) j ω t } Re e + e e (1.5) ossy Transmission ines Thus far we considered a lossless transmission line. Therefore we did not include any resistance along the line, nor any conductance across the line. f we now define R series resistance per unit length [Ω/m] G shunt conductance per unit length [S/m] To derive the relevant epressions for a lossy transmission line our equivalent circuit would become : 34 34

18 Rδ δ + δ R Gδ Cδ C + δ δ R C R δ δ ( + ) t δ R + t or simplicity we assume f ( ) e jωt j f ( ) e j ω ω t jω t Compare with (1.1) Then ( R + jω) jω t 36 36

19 Similarly, using Kirchoff s current law to sum currents: Gδ jωcδ ( + δ ) ( G + jωc) Compare with (1.) C t jωc Thus, we can write the epression for a lossy line starting from that of a lossless line, if we substitute in a lossless line with: ' C in a lossless line with: Then C ' ( R + jω) jω in a lossy line ( G + jωc) jω in a lossy line β ω C n a lossless line corresponds to: 1 β ' ( R + jω)( G + jωc) j in a lossy line 38 38

20 Substituting β β into (1.4) and (1.5) and defining We get γ ( R + jω)( G + jωc) α + jβ {( ( α + j β ) ( α + j β ) ) j t } ω (1.6) ( ( α + j β ) ( α + j β ) ) j ω t Re e + e e { } (1.7) Re e + e e γ is called propagation constant β is the phase constant The real term α corresponds to the attenuation along the line and is known as the attenuation constant or a forward travelling wave: e jωt e -γ e -α e j(ωt-β) oltage amplitude factor phase factor time variation e α 4 4

21 Note: At high frequencies: ω >> R and ω C >> G : γ ( + ω )( + ω ) ω ω R j G j C C j C Thus α β ω C The epressions approimate back to those for lossless lines Wave velocity: v Our epressions for voltage and current contain eponentials The one in terms of : j e ± β gives the spatial dependence of the wave, hence the wavelength: λ π β j t The other: e ω gives the temporal dependence of the wave, hence its frequency: f ω π 4 4

22 or a wave velocity v, wavelength λ and frequency f: v then f λ ω π v π β since β ω C v 1 C (1.8) Eample: Wavelength An Ethernet cable has. µhm -1 and C 86 pm -1. What is the wavelength at 1 MHz? rom λ λ π β and β ω C π λ ω C π Then π metres 44 44

23 .1.6 When must distances be accounted for in AC circuits? λ λ 4 λ arge ship is in serious trouble (as you can see) and we cannot ignore the effect of the waves A much smaller vessel caught in the same storm fares much better f a circuit is one quarter of a wavelength across, then one end is at zero, the other at a maimum f a circuit is an eighth of a wavelength across, then the difference is of the amplitude n general, if the wavelength is long in comparison to our electrical circuit, then we can use standard circuit analysis without considering transmission line effects. A good rule of thumb is for the wavelength to be a factor of 16 longer 46 46

24 λ 16 Wave Relevant λ 16 Wave Not Relevant Eample: When is wave theory relevant? A designer is creating a circuit which has a clock rate of 5MHz and has mm long tracks for which the inductance () and capacitance (C) per unit length are:.5µhm -1 C6pm -1 rom Then λ π And β ω C β π λ 36.5m π π λ ω C 36.5 m is much greater than mm (the size of the circuit board), so that wave theory is irrelevant. Note: The problem is even less relevant for mains frequencies i.e. 5 Hz. This gives λ~365 km 48 48

25 . Characteristic mpedance Aims To define and derive the characteristic impedance for lossless and lossy lines Objectives At the end of this section you should be able to describe the forward and backward waves in a transmission line and calculate the characteristic impedance ossless ines Recalling the solutions for & (equations 1.4&1.5): {( j β j β ) j ω t } Re e + e e (1.4) {( j j ) j t } β β ω Re e + e e (1.5) Differentiating (1.5) with respect to (1.4) with respect to t and multiplying by -C { jβ jβ jωt Re ( jβ ) } e + jβ e e (.1) ( Cjω e ω ) jβ C CAMRDGE Re UNERSTY Cj e e t { } NANOMATERAS jβ j EECTRONC DECES AND ωt AND SPECTROSCOPY MATERAS GROUP (.) 5 5

26 According to the second Telegrapher s equation: C t We can then equate (.1) and (.): (1.) j {( ) } jβ jωt j ( j β jβ + jβ e R jω e jωe ) j t { } Re e e e C C e Since β β ω ( β ) j( t + β ) j t e ω and e ω represent waves travelling in opposite directions, they can be treated separately. This leads to two independent epressions in and jβ e Cjω e jβ jβ β Cω jβ e Cjω e jβ jβ β Cω Note: f we consider and to have the same sign then, due to the differentiation with respect to, and have opposite signs 5 5

27 The characteristic impedance, is defined as the ratio between the voltage and the current of a unidirectional forward wave on a transmission line at any point, with no reflection: Since β Cω is always positive β Cω rom (1.3) β ω C C (.3) Units 1 1 [ β ] m [ ω] s 1 A s s 1 [ C] [ ] m m Ω m [ ] Ω C H s Ω s m m A m is the total impedance of a line of any length if there are no reflections and in phase everywhere. is real f there are reflections, the current and voltage of the advancing wave are again in phase, but not necessarily with the current and voltage of the retreating wave The lossless line has no resistors. Yet has units of Ω. The characteristic impedance does not dissipate power. t stores it 54 54

28 .. ossy ines ( α + β ) ( ( α + β ) ) ( α + β ) ( ( α + β ) ) { j j } jωt (1.6) Re e + e e { j j } jωt (1.7) Re e + e e γ α + jβ ( R + jω )( G + jωc ) Remembering that we can write the epressions for a lossy line starting from those of a lossless line, if we substitute in a lossless line with: ' C ' ( R + jω) jω ( G + jωc) jω in a lossy line C in a lossless line with: EECTRONC NANOMATERAS AND in a lossy DECES line Thus corresponds to C in a lossless line ' C ' R + jω G + jωc in a lossy line Note: at high frequencies ω >> R and ω C >> G, we recover the epression for lossless lines C 56 56

29 ..3 Summary 1) or a unidirectional wave: at all points ) or any wave: Hence and and and are in phase are in antiphase 3) or a lossless line is real with units of ohms. 4) or a lossy line is comple Characteristic mpedance Eample 1 Q: We wish to eamine a circuit using an oscilloscope. The oscilloscope probe is on an infinitely long cable and has a characteristic impedance of 5 Ω. What load does the probe add to the circuit? A: 1) Since the cable is infinitely long there are no reflections )or a unidirectional wave with no reflections / at all points, hence the probe behaves like a load of 5Ω 58 58

30 ..5 Characteristic mpedance Eample Q: A wave of 5 volts with a wavelength λ metres has a reflected wave of 1 volt f 75Ω, what are the voltage and current 3 metres from the end of the cable? A: rom Equation 1.4: π π β π λ m - 3m therefore: Since 1 [ m ] e + e e + e and jβ jβ + j3π j3π 5 1 [volts] Then 5 1 e e CAMRDGE + UNERSTY j3π j3π [amps] Reflection Aims To introduce the concept of voltage reflection coefficient and its relation to the reflected power at the load Objectives At the end of this section you should be able to calculate the voltage reflection coefficient, the incident and reflected power on the load, the conditions for ringing and quarter wave matching 6 6

31 .3.1 oltage reflection coefficient Consider a load added to the end of a transmission line rom Equation 1.4: rom Equation 1.5: e + e jβ jβ e + e jβ jβ At the load, thus + DEPARTMENT O ENGNEERNG ( ) ut: + rom our derivation of characteristic impedance: and have opposite signs relative to and Hence: ( )

32 The oltage Reflection Coefficient, ρ, is defined as the comple amplitude of the reverse voltage wave divided by the comple amplitude of the forward voltage wave at the load: ρ (3.1a) ρ + (3.1b) At the load.3. Power Reflection j { ω t} ( t) Re e cos( ωt + ) { j ω t} ( t) Re e cos( ωt + ) nstantaneous power dissipated at the load: P( t) ( t) ( t) cos( ωt + ) cos( ωt + ) Remembering the identity: 1 cos( A)cos( ) [ cos( A + ) + cos( A ) ] we get: 1 P( t) cos( ωt + + ) + cos( ) DEPARTMENT O ENGNEERNG AND SPECTROSCOPY MATERAS GROUP 64 64

33 Thus Mean power dissipated at the load: T 1 1 PAv P( t) dt cos( ) T Where At the load: + ut, from (3.1a): * is the comple conjugate of j Re{ } + m{ } j e ρ ρ 1 Re { * } DEPARTMENT O ENGNEERNG (1 + ) * j Re{ } m{ } j e Similarly: At the load: 1 + ( ) 1 Hence: (1 ρ ) ( )( ρ ) ρ * * * + ( ) 1 ρ ρ ρ CAMRDGE UNERSTY 66 66

34 * ut ρ is the comple conjugate of ρ ρ ρ 1 so: Re { } ( 1 ) ρ * * is imaginary power dissipated in the load Therefore: ncident power Reflected power The fraction of power reflected from the load is: ρ ρ.3.3 Standing Waves Reflections result in standing waves being set up in the transmission line. The oltage Standing Wave Ratio (SWR) is a measurement of the ratio of the maimum voltage to the minimum voltage. Maimum voltage SWR Minimum voltage + The SWR can be stated in terms of the reflection coefficient ρ SWR 1 ρ 1 CAMRDGE 1 UNERSTY DEPARTMENT O ENGNEERNG 68 ρ 68

35 Or alternatively (and more usefully) the reflection coefficient can be stated in terms of the SWR (which can be measured) ρ SWR SWR (3.) ρ f there is total reflection then ρ 1 and SWR is infinite. ero reflection leads to SWR Summary ρ ull power transfer requires When ρ a load is said to be matched The advantages of matching are that: 1) We get all the power to the load ) There are no echoes The simplest way to match a line to a load is to set i.e. so that the load equals the characteristic impedance Since, from (3.1b): ρ + 7 7

36 raction of power reflected ρ Reflections will set up standing waves. The oltage Standing Wave Ratio (SWR) is given by: SWR 1+ 1 ρ ρ 71 71