1 Antennas: from Theory to Practice 3. Field Concepts and Radio Waves Yi HUANG Department of Electrical Engineering & Electronics The University of Liverpool Liverpool L69 3GJ

2 Objectives of This Chapter Use Maxwell’s equations to obtain wave solutions. Introduce the concepts of the plane wave, intrinsic impedance, and polarisation; Discuss radio propagation mechanisms and propagation characteristics in various media; Review basic radio propagation models; Compare the circuit concepts and field concepts; Examine the concept of skin depth from both the field and circuit points of views.

3 3.1Wave Equation and Solutions For a time harmonic case the time factor is Maxwell’s equations

4 In source free region, we obtain the wave equation A solution is

5 For the loss-free case, the velocity of an electromagnetic wave (including light) in free space is about 30,000,000 m/s

6 The power density: H  E zH  E z 3.2 Plane Wave, Intrinsic Impedance and Polarisation Plane wave:

7 Polarisation: is described by the locus of the tip of the E vector as time progresses. For a wave propagating towards z-direction E E E x x y y Linear a or b =0 Elliptical Axial ratio = a/b Circular a = b Note: there are RCP and LCP

8 Intrinsic impedance of the material is defined as the ratio of the electric and magnetic fields. In a loss-free medium: In free space: Generally speaking:

9 3.3Radiowave Propagation Mechanisms Reflection and transmission Snell’s law: Reflection and transmission coefficients: How to obtain them?

10 Two principal polarisations To obtain the reflection and transmission coefficients, we introduce equivalent transmission line model:

11 The characteristic Impedances are: Thus: From the power point of view:

12 Example 3.1 Perfect conductor: obtain the reflection and transmission coefficients between air and a perfect conductor. The conductivity of a perfect conductor is infinite the characteristic impedance of its equivalent transmission line is zero for any polarisation and incident angle, i.e. Z 2 = 0, thus

13 Example 3.2 Ground: if the relative permittivity of a ground is 9 and the conductivity is very small and negligible, plot the reflection coefficient as a function of the incident angle for both parallel and perpendicular polarisations.

14 Brewster’s angle  For parallel polarisation, the reflection coefficient vanishes (= 0) at a particular incident angle, this angle is called Brewster’s angle:  When the incident angle is greater than the Brewster’s angle for parallel polarisation,

15 The critical angle is the incident angle that gives a transmitted angle of 90 degrees when the wave is from a dense medium to a less dense medium, such as from water into air. From Snell’s law, we obtain this special angle For non-normal incidence, the reflection coefficients are different for the two principle polarisations. As a result, if an incident wave is a combination of these two orthogonal waves, the combined signal after the reflection will be changed. e.g., for a conductor: RCP wave becomes a LCP wave!

16 Radio propagation through a wall

17 Example 3.3 A brick wall has a relative permittivity of 4 and a thickness of 20 cm, the loss is negligible. –a). If the operational frequency is 2.45 GHz for wireless applications (such as bluetooth), plot the reflection coefficient as a function of the incident angle for both parallel and perpendicular polarisations. –b). If the incident angle is 45 degrees, plot the reflection coefficient as a function of the frequency for both parallel and perpendicular polarisations.

18 The reflection is minimised when the thickness of the wall is an integer of half of the effective wavelength.

19 Diffraction and Huygens’s Principle Huygens’ Principle states that each point on a primary wave front can be considered as a new source of a secondary spherical wave. The relative (to the direct ray) power density

20 Scattering Unlike the other propagation mechanisms where the size of the medium or the obstacle is much larger than the wavelength, scattering occurs when the obstacle is comparable or even small than the wavelength. –In scattering, there are no energy transformation results, only a change in the spatial distribution of the radiation.

21 3.4Radio Wave Propagation Characteristics in Media Media classification This classification is useful for evaluating the EM properties in terms of the loss tangent but is not accurate for classifying whether a medium is lossy or not!

22 A more accurate consideration should take the complex permittivity spectrum into account:

23 Propagation Through the Ionosphere The ionosphere is the region above the troposphere (where the air is), from about 80 to 400 km above the earth. It is a collection of ions, which are atoms that have some of their electrons stripped off leaving two or more electrically charged objects. The sun's rays cause the ions to form which slowly recombine. –Reflection at low frequencies (up to about 30 MHz). –Scattering, refraction and absorption when high frequency waves (above 100 MHz) pass through it. –Faraday rotation: the wave polarisation plane/line is rotated through the ionosphere.

24 Propagation in Rain The major effect of rain on radiowaves is attenuation due to absorption and scattering over a wide range of the spectrum: where R is the rain rate in mm/h, and a and b are constants that depend on frequency and temperature of the rain.

Radio Wave Propagation Models Free space model Path loss: Received power:

26 Two-ray Model/Plane Earth Model Received power: Path-loss: The 1st Fresnel zone: Path-loss: (not a function of freq) 20 dB/dec, d < D f 40 dB/dec, d > D f

27 Multipath Models No analytical equations to give an accurate prediction of the radio propagation pathloss. Empirical and statistical representations are available for various scenarios. Most of the popular outdoor pathloss prediction tools are based on Okumura and Hata’s formulation –based on measured data for 100 MHz and 3 GHz General form for pathloss: 1.5 < n < 4 0 < X < 20

28  Multipath fast fading: the received signal changes significantly (> 30 dB) over a very short distance (few wavelengths), resulted from the complex and vector summation of signals.  Delay spread: multi-copies of the original signal arrive at the destination at different time through different paths, which may cause dispersion and inter-symbol interference  The delay spread is often employed to define the channel’s coherence bandwidth B C (similar to a filter’s bandwidth in certain sense).  Doppler frequency shift is employed to define the channel’s coherence time T c

29 Fading channels Typical values of RMS delay spread is, – ~2 ms for outdoor urban cellular (Bc = 100kHz) –~100 ns for indoor environment (Bc = 2MHz) Tc Bc Pulse duration Signal BW Slow fading Fast fading Freq selective Flat Fast fading Freq selective Slow fading

30 Statistically, the power density function (PDF) of the received (short-term) signal envelope follows certain distribution –When there is a line-of-sight ray, it follows the Gaussian distribution and this channel is therefore called the Gaussian channel; –When there is a partial line-of-sight ray (the path is partially blocked by obstacles such as trees), it follows the Rician distribution and this channel is therefore called the Rician channel; –When there is no line-of-sight ray, it follows the Rayleigh distribution and this channel is therefore called the Rayleigh channel.

31 3.6Comparison of Circuit Concepts and Field Concepts

32 Correspondence of the Circuit Concepts and the Field Concepts

33 Skin Depth  Field concept: skin depth is defined as the distance  through which the amplitude of a traveling plane wave decreases by factor 1/e, or 37%, or dB over one skin depth Circuit concept: skin depth is defined as the depth below the surface of the conductor at which the current density decays to 1/e (about 37%) of the current density at the surface. The per unit resistance of a wire:

34 Mathematically Skin depth and resistance of a gold track of dimensions 7  m x 16  m x  m