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, wiggling 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 INTRODUCT INTRODUCTION ION Some of the key properties of electromagnetic waves travelling in free space and in other unifor uniform m media media are introd introduce uced d in this this chapte chapterr. They establis establish h the basic basic paramet parameters 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 [Hayt, t, 01]. 01]. It does does,, howe howeve verr, aim aim to show show that that any any unif unifor orm m me medi dium um can can be spec specifi ified ed by a smal smalll set of descriptive parameters and that the behaviour of waves in such media may easily be calculated.
2.2 MAXWELL’S MAXWELL’S EQUA EQUATIONS The The exis existe tenc ncee of prop propaga agati ting ng elec electro troma magn gneti eticc wave wavess can can be pred predic icte ted d as a direc directt cons conseq eque uence nce of Maxwell’s equations [Maxwell, 1865]. These equations specify the relationships between the variations of the vector electric field E and the vector magnetic magnetic field H 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 field is measured in amperes 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 first first two equatio equations ns,, Maxwell Maxwell’’s curl equati equations, ons, contai contain n consta constants nts of proport proportiona ionalit lity y which which dictate the strengths of the fields. These are the permeability of the medium in henrys per
Second Editi Edition on Antennas and Propagation for Wireless Communication Systems Second Alejandro Arago´ n-Zavala ß 2007 John Wiley & Sons, Ltd
Simon Simon R. Saunders Saunders and
Antennas and Propagation for Wireless Communication Systems
26
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
0 "0
¼ 4 Â 10À H mÀ 10À À ¼ 8:854 Â 10 % 36 F mÀ 7
1
9
12
ð2:4Þ ð2:5Þ
1
and r ; "r are the relative values (i.e. r "r 1 in free space). Free space strictly indicates a vacuum, but the same values can be used as good approximations in dry air at typical temperatures and pressures.
¼ ¼
2.3 PLANE WAVE PROPERTIES Many solutions to Maxwell’s equations exist and all of these solutions represent fields which could actually be produced in practice. However, they can all be represented as a sum of 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 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 electric and magnetic fields are perpendicular to each other and to the direction of propagation of the wave; the direction of propagation is along the z axis; the vector in this direction is the propagation vector or Poynting vector. The two fields are in 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 Waves
27
2.3.1 Field Relationships The electric field can be written as
¼ E cosð!t À kzÞ^x ð2:6Þ where E is the field amplitude [V mÀ ], ! ¼ 2 f is the angular frequency in radians for a À E
0
1
0
frequency f [Hz], t is the elapsed time [s], k is the wavenumber [m 1], z is distance along the z-axis (m) and ^ x is a unit vector in the positive x direction. The wavenumber represents the rate 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 metres. The distance over which the phase of the wave changes by 2 radians is the wavelength l. Thus k
¼ 2l
ð2:7Þ
Similarly, the magnetic field vector H can be written as H
¼ H cosð!t À kzÞ^y
ð2:8Þ
0
where H 0 is the magnetic field amplitude and ^ y is a unit vector in the positive y direction. In both Eqs. (2.6) and (2.8), it has been assumed that the medium in which the wave travels 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 jð!t À kzÞ^x
ð2:9Þ
H
¼ H exp½ jð!t À 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 Impedance Equations (2.6) and (2.8) satisfy Maxwell’s equations, provided the ratio of the field amplitudes is a constant for a given medium,
jEj ¼ E ¼ E ¼ jHj H H x
0
y
0
r ffiffi ¼ "
ð2:11Þ
Z
where Z is called the wave impedance and has units of ohms. In free space, r the wave impedance becomes
Z
¼ Z
0
r ffiffi ffi ffi ffi ffi ffi ffi ffiÀffi ffi ffi ffi ffi ffi ffi ffi ffi ffi r ffi ffi ffi ¼ % Â Â ¼ 0 "0
4
10
7
36
10À9
120
% 377
¼ " ¼ 1 and r
ð2:12Þ
Antennas and Propagation for Wireless Communication Systems
28
Thus, in free space or any uniform medium, it is sufficient to specify a single field quantity together with Z in order to specify the total field for a plane wave.
2.3.3 Poynting Vector 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 S
¼ E Â HÃ
ð2:13Þ
Usually, only the time average of the power flow over one period is of concern, Sav
¼ 12 E H ^z 0
ð2:14Þ
0
The direction vector in Eq. (2.14) emphasises that E, H and Sav form a right-hand set, i.e. Sav is in the direction of movement of a right-handed corkscrew, turned from the E direction to the H direction.
2.3.4 Phase Velocity The velocity of a point of constant phase on the wave, the phase velocity v at which wave fronts advance in the S direction, is given by v
¼ !k ¼ p 1"
ð2:15Þ
¼ f v
ð2:16Þ
ffiffi ffi ffi
Hence the wavelength l is given by l
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 Â 10
ffiffi ffi ffi ffi 0 0
8
m sÀ1
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 Lossy Media So far only lossless media have been considered. When the medium has significant conductivity, the amplitude of the wave diminishes with distance travelled through the medium
Properties of 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 jð!t À kzÞ À a z^x
ð2:18Þ
H
¼ H exp½ jð!t À kzÞ À a z^y
ð2:19Þ
0
and 0
The constant a is known as the attenuation constant , with units of per metre [mÀ1], which depends on the permeability and permittivity of the medium, the frequency of the wave and the conductivity of the medium, , measured in siemens per metre or perohm-metre [m]À1. Together , and " are known as the constitutive parameters of the medium. In consequence, the field strength (both electric and magnetic) diminishes exponentially as the wave travels through the medium as shown in Figure 2.2. The distance through which the wave travels before its field strength reduces to eÀ1 0:368 36:8% of its initial value is its skin depth , which is given by
¼
¼
¼ 1a
ð2:20Þ
| E| 1.2
E0
1
0.8
0.6
E0 / e
0.4
0.2
0
0
0.5
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
Antennas and Propagation for Wireless Communication Systems
30
¼ 0 is
Thus the amplitude of the electric field strength at a point z compared with its value at z given by
ð Þ ¼ E ð0ÞeÀ = z
E z
ð2:21Þ
Table 2.1 gives expressions for a and k which apply in both lossless and lossy media. Note that the expressions may be simplified depending on the relative values of and !". If dominates, the material is a good conductor ; if is very small, the material is a good insulator 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]) n ck =! in all cases
¼
Good dielectric (insulator) =!" 2 1
ð
Exact expression
Attenuation constant À1 a [m ] Wave number k [mÀ1]
v ffi ffiffi ffi ffi ffi ffi ffiffi ffi ffi ffiffi ffi ffi ffi ffi ffi ffi#ffi ut ffiffi ffi ffi"ffi r þ À v ffi ffiffi ffi ffi ffi ffi ffiffi ffi ffi ffiffi ffi ffi ffi ffi ffi ffi#ffi ut ffiffi ffi ffi"ffi r þ þ r ffiffi ffi ffi ffi ffi ffi ffi ffi
!
!
Wave impedance Z []
" 2
" 2
2
1
!"
2
1
!"
1
1
% !p "
þ
ffiffiffi r ffi ffi %
2 k
2 % !p "
!
1
Þ )
! 2
r ffi ffi ffi ffi ffi % r ffi ffi ffi ffi % ð þÞ s ffiffi ffi ffi ffi % s ffiffi ffi ffi ! 2
! 1 2
2
ffiffiffi % p ffiffi ffi ffi
%
"
k
ð
r ffi ffi ffi ffi ffi %
"
j!"
Phase velocity v [m sÀ1]
r ffiffi
2 "
%
j!
Wavelength l [m]
Þ(
Good conductor =!" 2 1
j
2
!
2!
Example 2.1 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 mÀ1. The electric field magnitude at z 0 is 1 V mÀ1. Calculate:
¼
¼
¼
¼
(a) the wave impedance; (b) the magnitude of the magnetic field at z 0; (c) the average power available in a 0.5 m2 area perpendicular to the direction of propagation at z 0; (d) the time taken for the wave to travel through 10 cm; (e) the distance travelled by the wave before its field strength drops to one tenth of its value at z 0.
¼
¼
¼
Properties of Electromagnetic Waves
31
Solution Referring to Table 2.1,
!"
¼ 2 Â 900 Â0:1001 Â 3 Â 6
10À9 36
% 0:07 ( 1
so the material can clearly be regarded as a good insulator. (a) For an insulator, the wave impedance is given by
Z
r ffi ffi r ffi ffi ffi ffi ffi % ¼ ¼ p ffiffi ffi % p ffiffi % "
0 "r "0
377
Z 0
"r
3
218
(b) From Eq. (2.11) the magnetic field amplitude is given by
H
1 ¼ Z E ¼ 218 % 4:6 m A mÀ
1
(c) The available average power is the magnitude of the time-average Poynting vector multiplied by the collection area, i.e.
P
1 Â 0:005 ¼ SA ¼ EH Â 0:5 ¼ 1:25mW A ¼ 2 2
(d) The time taken to travel a given distance is simply the distance divided by the phase velocity
t
p ¼ d v ¼ d p " ¼ d p " " ¼ d cp " % 3 Â0:110 3 % 0:6 ns
ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi ffi 0 r 0
r
8
ffiffi
(e) Rearranging Eq. (2.21) yields z
¼ À 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
¼ À 2 Z ln E E ðð z0ÞÞ ¼ À 0:01 2Â 218 ln 101 ¼ 2:11 m
Antennas and Propagation for Wireless Communication Systems
32
2.4 POLARISATION 2.4.1 Polarisation States The alignment of the electric field vector of a plane wave relative to the direction 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 antenna parallel to the x axis. An entirely distinct y-polarised plane wave could be generated with the same direction of propagation and recovered independently of the other wave using pairs of transmit and receive antennas with perpendicular polarisation. This principle is sometimes used in satellite communications to provide two independent communication channels on the same earth satellite link. If the wave is generated by a vertical wire antenna (H field horizontal), then the wave is said to be vertically polarised ; a wire antenna parallel to the ground (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 amplitude and orthogonal polarisation are combined with a 90 phase difference, the resulting 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 wavelength travelled. Circularly polarised waves are most commonly used in satellite communications, since they can be generated and received using antennas which are oriented in any direction around their axis without loss of power. They may be generated as either right-hand circularly polarised (RHCP) or left-hand circularly polarised (LHCP); RHCP describes a wave with the electric field vector rotating clockwise when looking 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, 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, AR
¼ E E
maj
min
ð2:22Þ
AR is defined to be positive for left-hand polarisation and negative for right-hand polarisation. These various polarisation states are illustrated in Figure 2.3.
2.4.2 Mathematical Representation of Polarisation All of the polarisation states illustrated in Figure 2.3 can be represented by a compound electric field vector E composed of x and y linearly polarised plane waves with amplitudes E x and E y , E
¼ E ^x þ E ^y x
y
ð2:23Þ
The relative values of E 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 Waves
33 Direction of propagation (z-axis) is out of the page
y
y
Linear y-polarised
x
x
x-polarised y
Right hand
y
Left hand
x
Right hand
y
x
y
Circular
Left hand
x
x
Elliptical
Minor Major Figure 2.3 Possible polarisation states for a z-directed plane wave
complex constant a depends upon the axial ratio. The axial ratio is given in terms of E x and E y as follows [Siwiak, 1998]:
2 þ 6 ¼ 64 1
AR
E y
3 Æ ½ ð Þ À ð Þ 7 7 ½ ð Þ À ð Þ 5 2
cos arg E y E x E y sin arg E y E x
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 Polarisation 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 x and E 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 À p p À 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 2 p 2
E 0 = jE 0 = jE 0 = jE 0 = jE 0 =
Antennas and Propagation for Wireless Communication Systems
34
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 yÃ
½
ð2:25Þ
¼0
where E [ .] indicates the time-averaged value of the quantity in brackets, or its expectation; see Appendix A for a definition of 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 CONCLUSION 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 plane, but all waves can be described by a sum of plane waves with appropriate amplitude, 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. [Maxwell, 1865] J. Clerk Maxwell, A dynamical theory of the electromagnetic field , Scientific 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 How far must a plane wave of frequency 60 GHz travel in order for the phase of the wave 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 average power density carried in a plane wave with electric field amplitude 10 V mÀ1? 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?
¼ ¼
¼
Properties of Electromagnetic Waves
35
2.4 By what proportion is a 400 MHz plane wave reduced after travelling 1.5 m through a non-magnetic material with constants 1000, "r 10? 2.5 A plane wave travels through free space and has an average Poynting vector of magnitude 10 W mÀ2. What is the peak electric field strength? 2.6 Calculate the distance required for the electric field of a 5 GHz propagating plane wave to diminish to 13.5% (eÀ2) given "r 3, r 2 and 100? 2.7 Repeat Example 2.1(e) for the case when 10. Compare your answer with the 2.11 m found in Example 2.1(e), and use this to explain why the surfaces of copper conductors in high-frequency circuits are often gold-plated. 2.8 Compare the attenuation of a plane wave travelling 1 m through a non-magnetic medium with 10À4 S mÀ1 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 mÀ1 . The electric field magnitude at z 0 is 1.5 V mÀ1. 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 strength drops to one fifth of its value at z 0.
¼
¼
¼
¼
¼ ¼
¼
¼
¼
¼
¼
¼
¼
¼
¼
