02

 

2

Properties of Electromagnetic Waves ‘I have no picture of this electromagnetic field that is in any sense accurate . . .  I see some kind of vague, shadowy, wiggl wiggling ing lines. . .

So if you have some difficulty in making such a picture, you should not be worried that your  difficulty is unusual’. Richard P. Feynman, Nobel Prize Laureate in Physics

2.1 INTR INTRODUCT ODUCTION ION Some of the key properties of electromagnetic waves travelling in free space and in other unifor uni form m med media ia are int introd roduce uced d in thi thiss cha chapte pterr. They estab establis lish h the bas basic ic par paramet ameters ers and relationships which are used as standard background when considering problems in antennas and propagation in later chapters. This chapter does not aim to provide rigorous or complete derivations of these relationships; for a fuller treatment see books such as, [Kraus, 98] or [Hay [H ayt, t, 01 01]. ]. It do does es,, ho howe weve verr, ai aim m to show show that that an any y un unif ifor orm m me medi dium um ca can n be spec specifi ified ed by a sm smal alll set of descriptive parameters and that the behaviour of waves in such media may easily be calculated.

2.2 MAXWELL MAXWELL’S ’S EQUA EQUATIONS The ex The exis iste tenc ncee of prop propaga agati ting ng el elec ectro troma magn gneti eticc wa wave vess ca can n be pr pred edic icte ted d as a di direc rectt co cons nseq eque uence nce of Maxwell’s equations [Maxwell, 1865]. These equations specify the relationships between the variations of the vector electric field E field  E and  and the vector magnetic field H field  H in  in time and space within a medium. The   E  field strength is measured in volts per metre and is generated by either a time-varying magnetic field or a free charge. The H The  H field  field is measured in amperes per metre and is generated by either a time-varying electric field or a current. Maxwell’s four equations can be summarised in words as  An electric field is produced by a time-varying magnetic field   A magnetic field is produced by a time-varying electric field or by a current  Electric field lines may either start and end on charges ;   or are continuous

ð2:1Þ

 Magnetic field lines are continuous

The firs firstt two equa equatio tions ns,, Maxw Maxwell ell’’s curl equ equati ations, ons, con contai tain n con consta stants nts of prop proport ortiona ionalit lity y which which dictate the strengths of the fields. These are the permeability of the medium    in henrys per

Second d Edi Edition tion  Antennas and Propagation Propagation for Wireless Wireless Communication Systems   Secon ´ Alejandro Aragon-Zavala  2007 John Wiley & Sons, Ltd

Simon Simon R. Saund Saunders ers aand nd

 

26

 

Antenna Antennas s and Propagation Propagation for Wireless Communic Communication ation Systems

metre and the permittivity of the medium  "  in farads per metre. They are normally expressed relative to the values in free space:



¼  "¼" " 0

ð2:2Þ ð2:3Þ

   

r 

0 r 

where   0   and  " 0  are the values in free space, given by

 ¼ 4  10 H m 10  "  ¼ 8:854  10  36   F m 1

7

0

ð2:4Þ ð2:5Þ

9

1

12

0

and r ; "r  are the rel relati ative ve va values lues (i. (i.e. e. r  "r  1 in free free spa space) ce).. Fre Freee spa space ce str strictl ictly y ind indica icates tes a vacuum, but the same values can be used as good approximations in dry air at typical temperatures temper atures and pressu pressures. res.

 ¼  ¼

2.3 PLANE WAVE PROPER PROPERTIES TIES Many solutions to Maxwell’s equations exist and all of these solutions represent fields which co coul uld d ac actu tual ally ly be pr prod oduc uced ed in pr prac acti tice ce.. Ho Howe weve verr, th they ey ca can n al alll be repr repres esen ente ted d as a sum sum of  plane  plane waves, which represent the simplest possible time varying solution. Figure 2.1 shows a plane wave, propagating parallel to the  z -axis at time  t  0.

¼

 x  l 

 E 0

 E   x 

Motion (Propagation/Poynting (Propagation/Poyn ting vector)

 z

 H 0  y

 H  y

Wave front Figure 2.1:  A plane wave propagating through space at a single moment in time

The elec The electr tric ic and and ma magn gnet etic ic field fieldss ar aree pe perp rpend endic icul ular ar to each each ot othe herr and and to th thee di dire rect ction ion of  propagation of the wave; the direction of propagation is along the   z  axis; the vector in this direction is the propa propagation gation vector   or or Poynting  Poynting vector vector.. The two fields are in phase phase at any point in time or in space. Their magnitude is constant in the  xy  plane, and a surface of constant phase (a wavefront ) forms a plane parallel to the  xy  plane, hence the term  plane wave.  The oscillating electric field produces a magnetic field, which itself oscillates to recreate an electric field and so on, in accordance with Maxwell’s curl equations. This interplay between the two fields stores energy and hence carries power along the Poynting vector. Variation, or  modulation, of the properties of the wave (amplitude, frequency or phase) then allows information to be carried in the wave between its source and destination, which is the central aim of a wireless communication system.

 

Properties of Electromagnetic Electromagnetic Waves

 

27

2.3.1 Field Rel Relations ationships hips The electric field can be written as

  ð2:6Þ ¼ E   cosð!t   kzÞ^x where   E   is the field amplitude [V m  ],   ! ¼ 2 f  is the angular frequency in radians for a  E

0

1

0

frequency  f  [Hz],   [Hz],  t  is  is the elapsed time [s],  k  is   is the wavenumber [m

1

],  z  is distance along the

unit it ve vect ctor or in the the po posi siti tive ve x dir direct ection ion.. The wav wavenu enumbe mberr rep repres resent entss the rat ratee  z-axi -axiss (m (m)) an and d^ x is a un of change of the phase of the field with distance; that is, the phase of the wave changes by  kr  radians over a distance of  r   r  metres.  metres. The distance distance ov over er which the phas phasee of the wave changes by 2  radians is the wavelength   l. Thus k 

¼ 2l

ð2:7Þ

 

Similarly Simila rly,, the magneti magneticc field vector vector H  can be written as  H can H

¼  H   cosð!t   kzÞ^y

ð2:8Þ

 

0

^  is a unit vector in the positive  y  direction. where  H 0  is the magnetic field amplitude and   y

In both both Eqs Eqs.. (2. (2.6) 6) aand nd ((2.8 2.8), ), it has bee been n ass assume umed d tha thatt the m medi edium um in in whic which h the wa wave ve tra trave vels ls is lossless, so the wave amplitude stays constant with distance. Notice that the wave varies sinusoidally in both time and distance. It is often convenient to represent the phase and amplitude of the wave using complex quantities, so Eqs. (2.6) and (2.8) become E

¼ E   exp½ j jð!t   kzÞ^x

 

ð2:9Þ

H

¼  H   exp½ j jð!t   kzÞ^y

 

ð2:10Þ

0

and 0

The real quantities may then be retrieved by taking the real parts of Eqs. (2.9) and (2.10). Complex notation will be applied throughout this book.

2.3.2 Wave Imped Impedance ance Equatio Equa tions ns (2.6 (2.6)) an and d (2 (2.8 .8)) sa sati tisf sfy y Ma Maxwe xwell ll’’s eq equa uatio tions ns,, pro provi vide ded d th thee rati ratio o of th thee fie field ld amplitudes is a constant for a given medium,

r  ffiffi

jEj ¼   E  ¼   E  ¼   ¼  Z  jHj  H   H  "  x

0

 y

0

ð2:11Þ

 

where  Z  is   is called the  wave impedance  and has units of ohms. In free space,   r  the wave impeda impedance nce becom becomes es

 ¼ "  ¼ 1 and

 ¼  Z   ¼

 Z 

0

r  ffiffi ffi ffi ffi ffi ffi ffi ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi r   ffi ffi ffi  ¼        ¼ 0 "0

4

10

7

  36

109

120

 377

r 

 

ð2:12Þ

 

28

 

Antenna Antennas s and Propagation Propagation for Wireless Communic Communication ation Systems

Thus, in free space or any uniform medium, it is sufficient to specify a single field quantity together togeth er with Z  in order to specify the total field for a plane wave.

2.3.3 Poyn Poynting ting V Vector ector The Poynting vector   S, measured in watts per square metre, describes the magnitude and direction of the power flow carried by the wave per square metre of area parallel to the   xy plane, i.e. the  power density  of the wave. Its instantaneous value is given by

¼ E  H

S

 

ð2:13Þ

Usually, only the time average of the power flow over one period is of concern,

 ¼ 12 E  H  ^z

Sav

0

 

0

ð2:14Þ

The direction vector in Eq. (2.14) emphasises that E that  E,, H  and S i.e.  Sav  H and  Sav  form a right-hand set, i.e. S is in the the direc directi tion on of mo move veme ment nt of a ri righ ght-h t-han ande ded d co cork rksc scre rew w, tu turn rned ed from from th thee E di dire recti ction on to th thee H  direction.

2.3.4 Phase V Velocit elocity y The velocity of a point of constant phase on the wave, the  phase velocity v  at which wave fronts advance in the S the  S  direction, is given by v

  1 ¼ !k  ¼ p  "

 

 ffiffi ffi

Hence the wavelength   l  is given by l

ð2:15Þ

v

2:16

¼  f 

 

ð

Þ

This book is concerned entirely with frequencies from around 30 MHz to 300 GHz, i.e. free space wavelengths from 10 m to 1 mm. In free space the phase velocity becomes v

¼ c ¼ p   1  "  3  10

8

 ffiffi ffi ffi ffi 0 0

m s1

ð2:17Þ

Note that light is an example of an electromagnetic wave, so   c  is the speed of light in free space.

2.3.5 2.3 .5 Los Lossy sy Medi Media a So far only lossless media have been considered. When the medium has significant conductivity ducti vity,, the amplitude of the wave diminishes with dista distance nce trav travelled elled through the medium

 

Properties of Electromagnetic Electromagnetic Waves

 

29

as energy is removed from the wave and converted to heat, so Eqs. (2.9) and (2.10) are then replaced by E

¼ E   exp½ j jð!t   kzÞ  a z^x

H

¼  H   exp½ j jð!t   kzÞ  a z^y

0

 

ð2:18Þ

 

ð2:19Þ

and 0

The con constan stantt   a   is kn kno own as the the   attenuation attenuation const constant  ant , wi with th un unit itss of per per me metr tree [m1], whic wh ich h de depe pend ndss on the the pe perm rmeab eabil ilit ity y an and d pe perm rmitt ittiv ivit ity y of th thee me mediu dium, m, th thee frequ frequen ency cy of  the wave and the   conductivity  of the medium,    , measured in siemens per metre or perohm-met ohm -metre re [m]1. Togethe ogetherr    ,      and   "   are are kn kno own as th thee   constitutive constitutive parame parameters ters   of  the medium. In co cons nsequ equen ence ce,, the the fie field ld st stre reng ngth th (b (both oth el elect ectri ricc an and d ma magn gneti etic) c) di dimi mini nish shes es ex expo pone nent ntia iall lly y as the wave travels through the medium as shown in Figure 2.2. The distance through which the 0:368 36:8% of its initial value is its wave travels before its field strength reduces to  e 1 skin depth   , which is given by

¼

 

¼ 1a

¼

ð2:20Þ

 

| E| 1.2

E0

1

0.8

0.6

E0 / e

0.4

0.2

0 0 .5 0   0.

1 1.5 2 2. 5 3 3. 5 4 4.5 Distance along propagation vector [skin depths]

Figure 2.2:  Field attenuation in lossy medium

5

 

 

30

Antenna Antennas s and Propagation Propagation for Wireless Communic Communication ation Systems

¼ 0 is

Thus the amp Thus amplit litude ude o off th thee ele electr ctric ic fiel field d str streng ength th aatt a poi point nt z comp  compare ared d wit with h its v valu aluee at z given by

ð Þ ¼ E ð0Þe =   z

E   z

ð2:21Þ

Table 2.1 gives expressions for  a  and  k  which   which apply in both lossless and lossy media. Note that the expressions may be simplified depending on the relative values of      and   !". If     domina domi nates tes,, the the ma mate teria riall is a goo good d con conduct ductor  or ; if  is very very sm smal all, l, th thee ma mate teri rial al is a goo good d ins insula ulator  tor  or  good dielectric. Table 2.1:  Attenuation constant, wave number, wave impedance, wavelength and phase velocity for plane waves in lossy media (after [Balanis, 89])

¼

n ck =! =! in all cases

  Exact expression

Attenuation constant 1 a  [m ]

ffi  ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi #ffi v  u ffiffi ffi ffi"ffi r  þ     utv   ffiu ffi ffi ffi"ffi r  ffi  ffiffi ffi ffi ffi ffi ffiffi ffi ffi ffiffi ffi ffi ffi ffi ffi #ffi t þ    þ r  ffiffi ffi ffi ffi ffi ffi ffi  

Wave number k  [m  [m 1]

!

 

!

Wave impedance  Z  [  [ ]

" 2

" 2

1

    !"

2

2

1

    !"

Good dielectric (insulator) =!" 2 1

 

ð

r  ffiffi

1

1

 !p "

þ

 ffiffi ffi ffi r   ffi ffi 

2 k 

2  !  p  "

!

  1  p  "

 "

   j!"

Phase velocity v  [m s 1]

 

 ffiffi ffi ffi

k 

ð

Þ 

 

r   ffi ffi ffi ffi ffi 

 

r   ffi ffi ffi ffi ffi   ffi ffi ffi ffi r    ð þÞ s  ffiffi ffi ffi ffi  s  ffiffi ffi ffi

   2 "



 j!

Wavelength l  [m]

Þ

Good conductor =!" 2 1

     

!  2

!  2

! 1 2 

2



 j

2

! 

2!

 

 ffiffi ffi

Examplee 2.1 Exampl

A linearly polarised plane wave at 900 MHz travels in the positive   z  direction in a medium with constitutive parameters   r  1; "r  3 and   0:01 S m1. The electric field magnitude at   z 0 is 1 V m1. Calculate:

 ¼

¼

 ¼

¼

(a) the wav wavee imp impedan edance; ce; (b) the m magn agnitu itude de of th thee mag magneti neticc fiel field d at  z 0; 2 (c) the av avera erage ge power power aava vaila ilable ble in a 0. 0.5 5 m area perpendicular to the direction of 

¼

propaga at  zfor th 0;e wav (d) propagation the ti time metion tak taken en the wavee to trav travel el thr throug ough h 10 ccm; m; (e) the dist distance ance tra travell velled ed by the w wave ave bef before ore its fi field eld stre strength ngth dro drops ps to one ten tenth th of  its value at  z 0.

¼

¼

 

Properties of Electromagnetic Electromagnetic Waves

 

31

Solution Referring to Table 2.1,

  !"

¼ 2   900 0:1001  3  6

109 36

 0:07  1

so the material can clearly be regarded as a good insulator. (a) For an in insula sulator tor,, the wa wave ve im imped pedance ance is gi given ven by

 ffi ffi r  r   ffi ffi ffi ffi ffi     ¼   ¼ p     ffiffi ffi  p p   ffiffi   p  

 Z 

 "

0 "r "0

377

  Z 0

"r 

3

218 

(b) Fro From m Eq. (2.11 (2.11)) the magn magneti eticc field amp amplitu litude de is given given by  1   ¼¼  Z E  ¼ 218  4:6 m A m

 H 

1

(c) The ava availa ilable ble aver average age pow power er is the magn magnitu itude de of the time time-av -avera erage ge Poynti Poynting ng vector multiplied by the collection area, i.e.

P

1  0:005  0:5 ¼ 1:25mW ¼ SA ¼ EH   A ¼ 2 2

(d) The time ta taken ken to tra travel vel a gi given ven dis distance tance is ssimply imply the d distance istance d divid ivided ed by the phase velocity d p    0:1 p  p  p  t  ¼ ¼ d  "   ¼ d   "  " ¼ c "   3  10 3   0:6 ns v d 

0 r  0

r 

8

 ffiffi ffi ffi  ffiffi ffi ffi ffi ffi ffi  ffiffi ffi

 ffiffi

(e) Rea Rearran rrangin ging g Eq Eq.. (2 (2.21 .21)) yi yield eldss

 z

¼  ln E E ðð z0ÞÞ

where

 

1

 2

" 

r  ffiffi  ¼

¼ a   

 2

  Z 

and from Table 2.1 the approximation holds for good insulators. Thus

 z

  2  1 ¼    2 Z  ln  E E ðð z0ÞÞ ¼  0:01 ln  218 10 ¼ 2:11 m

 

32

 

Antenna Antennas s and Propagation Propagation for Wireless Communic Communication ation Systems

2.4 POLARISA POLARISATION TION 2.4.1 Pola Polarisat risation ion States The alig The alignme nment nt of the the el elec ectri tricc fie field ld ve vect ctor or of a pl plan anee wa wave ve rela relati tive ve to th thee di dire rect ctio ion n of  propagation defines the  polarisation  of the wave. In Figure 2.1 the electric field is parallel to the   x  axis, so this wave is   x  polarised. This wave could be generated by a straight wire antennaa paralle antenn parallell to the x  axis. An entire entirely ly distinct y -polarised plane wave could be generated with same direction of propaga propagation tion and recov recovered ered independe independently ntly of the This otherprinciple wave using pairsthe of transmit and receive antennas with perpendicular polarisation. is sometimes used in satellite communications to provide two independent communication ch chan annel nelss on the the sa same me eart earth h sa sate tell llit itee link link.. If th thee wa wave ve is ge gene nera rate ted d by a ve vert rtic ical al wi wire re antenna (H (H  field horizontal), then the wave is said to be  vertically polarised ; a wire antenna parallel to the ground (E (E   field horizontal) primarily generates waves that are   horizontally  polarised. The waves described so far have been  linearly polarised , since the electric field vector has a single direction along the whole of the propagation axis. If two plane waves of equal ampl am plit itude ude an and d or orth thog ogona onall po pola lari risa satio tion n ar aree co comb mbin ined ed wi with th a 90 pha  phase se dif differ ferenc ence, e, the res result ulting ing wave will be   circularly polarised  (CP), in that the motion of the electric field vector will describe a circle centred on the propagation vector. The field vector will rotate by 360    for every ever y wavel wavelength engthsince travell travelled. ed. can Circul Circularly polarised areusing mostantennas commonly usedare in oriented satellite satellite communications, they be arly generated andwaves received which in any direction around their axis without loss of power. They may be generated as either right-hand righthand circularly polarised (RHCP) or leftleft-hand hand circu circularly larly polarised (LHCP); RHCP descri des cribes bes a wav wavee wit with h the ele electr ctric ic fiel field d vec vector tor rot rotati ating ng clo clockwi ckwise se whe when n loo lookin king g in the direction of propagation. In the most general case, the component waves could be of unequal amplitudes or at a phase angle other than 90 . The result is an   elliptically polarised  wave,   wave, where the electric field vector still rotates at the same rate but varies in amplitude with time, thereby describing an ellipse. In this case, the wave is characterised by the ratio between the maximum and minimum values of the instantaneous electric field, known as the axial ratio,  AR, E maj maj  AR

ð2:22Þ

¼ E 

min min

 AR is defi defined ned to be pos positi itive ve for lef left-h t-hand and pola polaris risati ation on and neg negati ative ve for rig rightht-han hand d polari polarisat sation ion.. These various polarisation  states  are illustrated in Figure 2.3.

2.4.2 Mathem Mathematical atical Represe Representatio ntation n of Pola Polarisat risation ion All of the polarisation states illustrated in Figure 2.3 can be represented by a compound electric field vector E vector  E composed  composed of  x polarised plane waves with amplit amplitudes udes E  x  x  and  y  linearly polarised and  E   yy , E  x ^ x

E

¼

E  y ^ y

þ

2:23

 

ð

Þ

The relative values of  E   E  x x   and   E  y  for the six polarisation states in Figure 2.3 are as shown in Table 2.2, assuming that the peak amplitude of the wave is   E 0  in all cases and where the

 

Properties of Electromagnetic Electromagnetic Waves

 

33

 Direction of propa gation (z-axis) is out of th e page

 y

 y

Linear  y-polarised

 x 

 x 

 x -polarised  y

Right hand

 y

Left hand

 x 

Right hand

 y

Circular

 x   y

Left hand

 x 

Elliptical

 x 

Minor Major Figure 2.3  Possible polarisation states for a  z -directed plane wave

complex comple x constan constantt a depends upon upon the axial rati ratio. o. The axial rat ratio io is given in terms of   E  E  x  and  E  y as follows [Siwiak, 1998]:

2  þ 6 ¼ 64  

  E  y

2

cos arg E  y E  x E  y sin arg E  y E  x

1

 AR

3   ½ ð Þ  ð Þ 7  7 ½ ð Þ  ð Þ  5

The exponent in Eq. (2.24) is chosen such that  AR

1

arg E  x

ð2:24Þ

arg E  x

 1:

2.4.3 Random Po Polaris larisation ation The polarisation states considered in the previous section involved the sum of two linearly polarised waves whose amplitudes were constant. These waves are said to be   completely  polarised , in that they have a definite polarisation state which is fixed for all time. In some cases, however, the values of  E   E  x x   and  E  y y  may vary with time in a random fashion. This could

Table 2.2:  Relative electric field values for the polarisation states illustrated in Figure 2.3  

Polarisation State

Linear   x Linear   y Right-hand circular Left-hand circular Right-hand elliptical Left-hand elliptical

 

      

E  x

p  p   p p   p 

E 0 = 2 0 E 0 = 2 E 0 = 2 aE 0 = 2 aE 0 = 2

 

E  y

0          

p 2 p 2 p 2 p 22

E 0 = jE 0 = jE 0 = jE 0 = jE 0 =

 

 

34

Antenna Antennas s and Propagation Propagation for Wireless Communic Communication ation Systems

happen if the fields were created by modulating random noise onto a carrier wave of a given frequency. If the resultant fields are completely uncorrelated, then the wave is said to be completely unpolarised , and the following condition holds: E  E  x E  y

½

¼0

 

ð2:25Þ

where E [.] [ .] indi indica cate tess the the time time-a -ave vera rage ged d valu valuee of th thee qu quan anti tity ty in br brac acke kets, ts, or it itss expectation; see Appendix A for a definition of  correlation.   correlation.  In the most general case, when   E  x   and   E  y   are partially correlated, the wave can be expressed as the sum of an unpolarised wave and a completely polarised wave. It is then said to be  partially polarised.

2.5 CONCL CONCLUSION USION Propagation of waves in uniform media can conveniently be described by considering the properties of plane waves, whose interactions with the medium are entirely specified by their frequency and polarisation and by the constitutive parameters of the medium. Not all waves are are plan plane, e, bu butt all all wa wav ves ca can n be de desc scri ribe bed d by a sum sum of pl plan anee wa wave vess wi with th ap appr prop opri riat atee am ampl plit itud ude, e, phase, polarisation and Poynting vector. Later chapters will show how the characteristics of  propagation and antennas in a wireless communication system can be described in terms of  the behaviour of plane waves in random media.

REFERENCES [Balanis, 89] C. A. Balanis,   Advanced engineering electromagnetics, John Wiley & Sons, Inc., New York, ISBN 0-471-62194-3, 1989. [Hayt, 01] W. H. Hayt Jr and J. Buck,  Engineering electromagnetics, 6th edn, McGraw-Hill, New York, ISBN 0-07-230424-3, 2001. [Kraus, 98] J. D. Kraus and K. Carver,  Electromagnetics, McGraw-Hill, New York, ISBN 0-07-289969-7, 1998. [Max [M axwel well, l, 18 1865 65]] J. Cler Clerk k Ma Maxwe xwell ll,, A dynamical theory of the electromagnetic field , Sci Scienti entific fic Papers, 1865, reprinted by Dover, New York, 1952. [Siwiak, 98] K. Siwiak,  Radiowave propagation and antennas for personal communications , 2nd edn, Artech House, Norwood MA, ISBN 0–89006-975-1, 1998.

PROBLEMS 2.1 Prove Eq. (2.14). 2.2 2.2 H Ho ow far far mu must st a plan planee wa wave ve of fr freq eque uenc ncy y 60 GHz GHz tr trav avel el in orde orderr fo forr th thee ph phas asee of th thee wa wav ve  to be retarded by 180  in a lossless medium with   r  1 (a  non-magnetic  medium) and "r  3:5? 2.3 What is the aver average age power densit density y carried in a plane wave with electr electric ic field amplitude

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1

10 V m ? What electric and magnetic field strengths are produced when the same power density is carried by a plane wave in a lossless non-magnetic medium with "r  4:0?

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Properties of Electromagnetic Electromagnetic Waves

 

35

2.4 By what proportion is a 400 MHz plane wave reduced after travelling 1.5 m through a non-magnetic non-ma gnetic material with consta constants nts   1000,  " r  10? 2.5 2.5 A plane plane wa wave ve tr trav avel elss throu through gh fr free ee spac spacee and and ha hass an av aver erage age Poy Poynt ntin ing g ve vect ctor or of  2 magnitude magnitu de 10 W m . What is the peak electric field strength? 2.6 Calcula Calculate te the distance required for the electric field of a 5 GHz propagatin propagating g plane wave 2 to diminish to 13.5% (e ) given  " r  3,   r  2 and    100? 2.7 Rep Repeat eat Ex Examp ample le 2. 2.1(e) 1(e) for tthe he ca case se when   10. Compare Compare your answer with tthe he 2.11 m

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found in requency Example thisgold-plated to explain.why the surfaces of copper conductors in high-frequenc high-f y 2.1(e), circu circuits itsand areuse often gold-plated. 2.8 2.8 Co Comp mpare are the the at atte tenu nuati ation on of a pla plane ne wa wave ve tr trav avel elli ling ng 1 m th thro roug ugh h a no non-m n-mag agne neti ticc me mediu dium m 1  with    104 S m and  " r  3 at 100 MHz, 1 GHz and 10 GHz. 2.9 Describe, in your own words, the physical meaning of Maxwell’s equations and why they are important for wireless communications. 2.10 A vertically polarised plane wave at 1900 MHz travels in the positive   z  direction in a medium with constitutive parameters   r  1,   "r  3 and     1 0 S m1 . The electric field fiel d mag magnit nitude ude at   z 0 is 1.5 V m1. Calculate: (a) the wave impedance; (b) the magnitude of the magnetic field at   z 0; (c) the average power available in a 1.3 m 2 area perpendicular to the direction of propagation at   z 0; (d) the time taken for the wave to travel through 15 cm; (e) the distance travelled by the wave before its field

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strength drops to one fifth of its value at  z

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¼ 0.