Satellite Communication and Positioning System - Lecture Notes, Dispense di Ingegneria Aerospaziale

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Satellite

communication

and positioning

system

Prof. Lorenzo Luini

A.Y. 2020/

Lecture Notes by Andrea Pizzetti

Telecommunication Systems Pagina 1

Electric Field

The Coulomb Force rules the movement of charges in the space. It acts on any charge immersed in an electric field. The value of Conductivity classifies the type of material. It's expressed in Siemens per meter. The value of conductivity is large for the materials characterized by non-empty conduction band, so where we have a lot of electrons moving when an external electric field is applied. Semiconductors and insulators have empty conduction bands, but the former have a lower energy gap than the latter. If instead we consider the valence band, from quantum mechanism, we can see the electron as a particle but also as a wave. If we apply an electric field to an atom, we'll have a shift of the nucleus and of the electron, generating an Electric Dipole , which will align with the direction of the field in order to counteract it. Considering the bulk material, all the dipoles initially orient in different random directions and balance themselves. If we apply the electric field, they will align all opposite to it in order to counteract the external field. Fundamental Concepts lunedì 22 febbraio 2021 14: Introduction Pagina 2

The total charge will be the total number multiplied for the electron charge, - 1.6 10^19 Coulomb. We can define the Current as the ratio of the total charge passing in the unit time. The electrons drift with a velocity which depends on the electric field and on the Electron Mobility constant. By rearranging, we get to the conductivity again, which is obtained as the product of the electron density with the electron mobility constant and the electron charge. Considering two circular currents, we assist to the generation of a force in each point which depends on the direction of the currents, the currents absolute value and the Magnetic Permeability of the medium. Introduction Pagina 4

The same force applied on the left side is also on the right one, and this in every single piece of the circuit. We can distinguish two parts in the expression, one called Magnetic Flux Density. It's the same approach used for the electric field. What we are saying is that the current flowing in the second circuit generates a magnetic flux density which interacts with the current flowing in the first circuit, generating a force. The Magnetic Field is obtained dividing the magnetic flux density for the magnetic permeability. In the atom, the electron which orbits around the nucleus is a source of current and therefore generates a magnetic field. In this case, if we apply an external magnetic field to the material, the magnetic dipoles will align in the same direction of it, increasing the actual field value. Introduction Pagina 5

Finally, the fourth equation tells us that magnetic field originates from currents, which can be the classical conduction ones, but also induced by the electric field or by the change of electric field. What happens in an antenna? We have a time-variable current which flows inside it, so we generate a magnetic field according to the fourth of the Maxwell equation, and this magnetic field generates an electric field according to the third equation, which again generates another in the fourth equation, and so on. If there is a variation, in time, either of a signal of magnetic or electric source, we'll have an EM wave. Introduction Pagina 7

When we are far from sources of electric/magnetic field, such as in the free space, we will have zero charge density and zero conduction current. In this case we can rearrange the equations, obtaining the Wave Equation: Let's try to solve this equation, considering only one component for the electric field (in the x direction), neglecting losses (so real electric permittivity and magnetic permeability) and assuming zero conductivity. In this case we arrive to a solution: If F it's a cosine function, we get to a Monochromatic Wave. Fixing z, we get to the story of the wave in a fixed position, while fixing t, we get to a snapshot of the wave at a fixed time. From the first one, we can get the Frequency , the number of cycles at each second, while for the second one we can measure the Wavelength , distance between two peaks. What it's important is that we can define a wave only by its frequency, not its wavelength! In fact, the latter can change depending on the material and on the refractive index of that material. Waves sabato 6 marzo 2021 16: Introduction Pagina 8

Also the magnetic field will be dissipated: The attenuation constant is almost zero in rarefied medium such as the air, so often the zero losses case is ok. If there are no losses, we get the previous results, where only beta is present. Considering free space with zero conductivity, we get to an expression for beta which involves the light speed constant c. Regarding the wavelength, we already saw that its value depends on the medium and that's why a wave can be identified univocally only by the frequency. In free loss case, the wavelength has an explicit expression: Introduction Pagina 10

If we are in free space, we can compute the wavelength just as the ratio of the light velocity over the frequency. Other two important wave parameters are the Phase Velocity, which is the velocity of a point on the wave, and the Group Velocity , which is the velocity of the packet of waves. The two velocities are exactly the same for a medium with no losses: In general, the phase velocity can be higher than the group velocity, even higher than c, but this is not a problem because it's the speed of information that matters, and this is identified by the group velocity.

Dispersion

We should remember that for materials with losses the simplifications we just saw cannot be used: In this case we have to go back to the definition of gamma, compute it and take out alpha and beta, and using beta derive wavelength and phase velocity. For a lossy material, group and phase velocity do not coincide. In fact, plotting the frequency dependence on beta, called Dispersion Plot , we can see that we don't have a linear relation and the two definitions do not give us the same value. In this case we speak of a Dispersive Medium. The dispersion refers to the fact that every frequency will travel with a different speed. Introduction Pagina 11

This degree of penetration depends on the magnetic permeability, on the conductivity and on the frequency. The highest the frequency, the lower is the penetration depth. This affects the coverage of the telecommunications. When we passed from GSM to 3G in the mobile phones, we increased the frequency and the depth of penetration was lower, so with 4G we reduced the frequency again. For iron the skin depth is lower due to its higher magnetic permeability. Microwave ovens work with this principle, as a Faraday cage, made up of metallic surfaces and a small aperture where we introduce EM power. There is also a grid of metallic rods whose distances are much lower than the wavelength of the EM source, so the waves cannot escape the cage. Water is present inside any food and its molecules oscillate heating up. The key point to consider is the dependence of the wave parameters on the frequency. This force a wise choice of the bandwidth for the different service or function we want, considering that we cannot use bandwidths already in use by other services. In most of the cases, when we have lossy materials we can use two kinds of approximations according to the values of the wave parameters, which allow to avoid the full computation of gamma. Introduction Pagina 13

• Introduction Pagina

Circular Polarization

In case of a phase shift of +-pi/2 and equal electric fields, we have a Circularly Polarization: In this wave the amplitude of the electric field is constant, but it rotates around the propagation direction, drawing a circle around it. If the rotation is positive according to the right hand rule, it's called RHCP: An antenna cannot receive at the same time both RHCP and LHCP waves.

Elliptic Polarization

At last but not least, for all the other cases when the phase shift assumes any value as well as the electric fields, which are different, we have Elliptically Polarized waves. Antennas Pagina 16

An antenna emits waves in a specific type of polarization depending on its shape. The sum of 2 linearly polarized waves shifted by 90° gives a circularly or an elliptically polarized wave. The sum of 2 circularly polarized waves, one LHCP and the other RHCP, but with same amplitude, gives instead a linearly polarized wave. If the amplitude is different, we get to an elliptically polarized wave again. Antennas Pagina 17

the antenna along the direction of unitary radiation pattern over the power density would be emitted by an isotropic radiator along the same direction. An example of antenna is a linear antenna, which has a low directivity, between 1.5 and 3. By looking on the front, we'll just see a circle. If the cut is made in the mean line, we'll have a unit radiation pattern. This is a sort of isotropic radiator, it's the ideal choice for mobile phones, where we don't know the direction of the receiver. Instead, in the example of a reflector antenna, we have a much higher directivity. The radiation pattern is maximum in the direction of the main lobe. There are also secondary lobes with very low values of the radiation pattern. This is used for point-to-point applications, such as satellites. The secondary lobes cannot be avoided, but they have much lower energy than the main lobe. They are linked to the Spillover Effects , always present in reflector antennas, due to the bad efficiency of reflection, which spreads out the power. Antennas Pagina 19

Real Antennas

In the real world, the power emitted is much lower than the one incoming from the transmitter, due to conduction losses and spillover effects. The first refers to the penetration of the EM wave inside highly conductive materials, as we previously saw, while the second arise when part of radiation emitted by the feed in a reflector antenna is not received by the parabola and it's spread out. The ratio between the effective emitted power and the one incoming from the transmitter gives the Antenna Efficiency. Inserting the efficiency in the equation, we obtain the Antenna Gain as the product of its efficiency with the directivity. This is what the receiver will actually sense. The datasheet of an antenna reports the gain, the radiation pattern f, and the frequency range in between the antenna functions properly.

Receiver

An antenna can be both transmitter and receiver, we do not require to have one antenna for each one of the two functions on the same platform. The reciprocity principle tells us that the parameters of the transmitter are the same used for the receiver, once we know the radiation pattern of the former we know the best direction to point for the latter. The power measured by the receiver will depend on the radiation pattern and on the Effective Area, which gives an indication of how good is the antenna in receiving a specific wave. So, on the reception side, the efficiency is the ratio of the effective area over the geometric area. The effective area can be Antennas Pagina 20