Electromagnetic and Signal Processing, Appunti di Teoria dei Segnali

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ELECTROMAGNETICS AND SIGNAL

PROCESSING

Academic year 2024/

Matilde Formiconi

Contents

• 1 Introduction
  • 1.1 Electric field
  • 1.2 Electric field in conductive materials
    • 1.2.1 Anechoic chamber
  • 1.3 Polarization
  • 1.4 Current density
  • 1.5 The electric permittivity
  • 1.6 Magnetic field
  • 1.7 Magnetic Dipoles
  • 1.8 Magnetic Permeability
  • 1.9 Maxwell’s equation
  • 1.10 Antennas
  • 1.11 Helmholtz Equation
    • 1.11.1 Function Representing a Wave
  • 1.12 Phase velocity and Group velocity
  • 1.13 Phasor domain
• 2 From the Electric to the Magnetic Field
  • 2.1 Lossless Materials
  • 2.2 Conductor and Dielectric
  • 2.3 The Power of a Wave
  • 2.4 Polarization
    • 2.4.1 Linear Polarization – ϕ = 0, ∀E 1 , ∀E
    • 2.4.2 Circular Polarization – ϕ = ±π 2 , E 1 = E
    • 2.4.3 Elliptical Polarization - ∀ϕ, ∀E 1 , ∀E
  • 2.5 Normal Incidence between two media
  • 2.6 Reflection and Refraction
    • 2.6.1 Transverse Electric Case
    • 2.6.2 Transverse Magnetic Case
  • 2.7 Brewster Angle or Polarization Angle
  • 2.8 Evanescent Wave
  • 2.9 Power density between two media
• 3 Antenna
  • 3.1 Main features
    • 3.1.1 Receiver Power
  • 3.2 Main antenna type
• 4 Transmission Lines
  • 4.1 Reflection coefficient at the load section
  • 4.2 Reflection Coefficient Along the Transmission Line
  • 4.3 Input Impedance
  • 4.4 Power absorbed by the load
• 5 Signal Processing
  • 5.1 Signal
  • 5.2 Scalar and Vector signal
  • 5.3 Continuous and Discrete Signals
  • 5.4 Deterministic and Random Signal
  • 5.5 Delta
    • 5.5.1 Discrete-time Delta or Kronecker Delta
    • 5.5.2 Continuous-time Delta or Dirac’s Delta
    • 5.5.3 Things to know about Dirac’s Delta
    • 5.5.4 Scaling of a Dirac Delta
  • 5.6 Time Invariant Linear System
  • 5.7 Pulse Response
• 6 Convolution
  • 6.1 Convolution: a graphical interpretation
  • 6.2 Properties of Convoluntion
  • 6.3 Additional Properties of Convolution
  • 6.4 Convolution with Dirac Delta Functions
• 7 The Fourier Transform (FT)
  • 7.1 Typical Paradigm
  • 7.2 FT of a Rectangular Pulse
  • 7.3 FT of a Constant Function
  • 7.4 FT of a Complex Sinusoid
  • 7.5 Inverse FT
  • 7.6 Properties of the FT
  • 7.7 Frequency Resolution
  • 7.8 Autocorrelation and Cross-Correlation
  • 7.9 Matched Filter and Symbol Timing in Digital Communications
  • 7.10 Radio-Frequency Signal
• 8 Shannon Sampling Theorem
  • 8.1 Resampling
  • 8.2 Temporal Aliasing
  • 8.3 Fast FT
• 9 Wave Equation
  • 9.1 Far-Field Approximation
  • 9.2 Transmitting Antenna
  • 9.3 Transmitting Antenna Array
  • 9.4 Receiving Antenna Array

Acronyms

ADC analog-to-digital converter.

DTFT Discrete-Time Fourier Transform.

EM Electro-Magnetic.

ESD Energy Spectral Density.

FFT Fast Fourier Transform.

FT Fourier Transform.

LHCP Left-Hand Circular Polarization.

LTI Linear Time Invariant.

PEC Perfect Electric Conductor.

PSD Power Spectral Density.

RF Radio Frequency.

RHCP Right-Hand Circular Polarization.

TE Transverse Electric.

TEM Transverse Electro-Magnetic.

TM Transverse Magnetic.

2. If Q is negative, the field lines converge inward toward the charge.

Figure 2: Electric field generated by a positive and a negative charge

These field lines visually represent the direction in which a positive test charge would move if placed in the field. The density of the lines indicates the field’s strength: the closer they are, the stronger the electric field. In this way, an electron subjected to an Electric field generated by a positive charge will experience an attractive motion. If the sign of the source and the probe is the same, the motion will be repulsive.

1.2 Electric field in conductive materials

Let’s consider the simplest atom’s model, the Borh’s atom:

Figure 3: Schematic of the hydrogen atom (Z = 1) or a hydrogen-like ion (Z > 1)

According to the Bohr model, often referred to as the planetary model, electrons move around the nucleus of an atom in specific allowable paths called orbits. When an electron is in one of these orbits, its energy remains fixed and quantized, meaning it cannot have arbitrary energy values. The ground state of the hydrogen atom, where its energy is at its lowest, corresponds to the electron occupying the orbit closest to the nucleus. Orbits further from the nucleus correspond to succes- sively higher energy levels, and the electron is not allowed to exist in between these discrete orbits.

Extending this concept to atoms with a higher atomic number (Z), electrons in higher energy levels are located further away from the nucleus compared to those in lower energy states. In metallic materials, the outermost electrons, which are the least tightly bound to the nucleus, occupy a re- gion known as the conduction band. Within this band, electrons require only a small amount of energy to become free and move through the material, facilitating electrical conductivity. This is a fundamental property of conductors, as the availability of free electrons in the conduction band allows metals to efficiently transport electric charge. Now, let’s consider a metallic material in a neutral state and apply an external Electric field to it.

What we observe is that, inside the material, charge carriers (free electrons) begin to move almost instantaneously, leading to the formation of an induced Electric field that opposes the applied external field. This phenomenon is known as electrostatic induction. The total Electric field within the conductor can be expressed as:

Etot = Eext +

X^ N

i=

Ei int = 0 (2)

Principle 1.2. In electrostatics, the net Electric field inside a conducting material in equi- librium is always zero.

Therefore, there is no way for a wave to penetrate a metal. The wave will be completely reflected. It is possible to talk about Perfect Electric Conductor (PEC) when:

1. the material has an infinite number of electrons
2. the electrons have perfect mobility

The total reflection of waves inside the material mean isolation. A particular example of the application of this property is the anechoic chamber.

1.3 Polarization

Let’s now revisit the Bohr atom and examine what happens when an external Electric field is applied:

Figure 5: Consequences of an external Electric field on an atom

Due to the presence of the electric field, the electrons in the valence band form an electron cloud that tends to shift slightly in the direction opposite to that of the applied field. This displacement generates an electric dipole within the atom. When observing the material as a whole, you can notice a rearrangement of the charges, through the phenomenon called polarization. Each single dipole inside the material will generate an electric field. Therefore we can say that:

Etot = Eext +

X^ N

i=

Ei int < Eext (3)

The main conseguences of polarization are:

1. lower propagation speed of the current
2. change in the wavelength

1.4 Current density

Definition 1.3. The current density is defined as:

J = qN μqE

Where N in the charge density, μq is the charge mobility. SI unit: [A/m^2 ]

It’s possible to demonstrate that: I =

Z

S

JdS (4)

The current density is a vector whose flux through a surface S of a conductive material corresponds to the current passing through that section. It’s possible to find out that in conductors there is a relationship between the current density and the Electric field such that J = f (E). In particular:

J = σE (5)

where σ is the electric conductivity (IS unit: [S/m]).

Conductivity σ Material EM / thermal behaviour

∞ PEC no ohmic loss, perfect current flow, total reflection of EM waves, no heat generation

5. 7 × 107 Non Perfectconductor small ohmic loss; strong currents, quasi-total reflection; moder- ate heat dissipation 10 −^2 Lossy dielectric limited conduction, partial wave transmission with attenuation, some heating

0 Perfect dielectric(insulator) no conduction, full wave transmission, no Joule heating

Table 1: Conductivity values and physical properties of different materials.

The most important characteristics that we have to point out when we talk about not perfect conductor are:

1. generation of heating due to dissipation;
2. Etot ̸= 0 (at least not everywhere)

1.5 The electric permittivity

The absolute electric permittivity, often simply called electric permittivity and denoted by the Greek letter ε, is a measure of the electric polarizability of a dielectric material. A material with high permittivity polarizes more in response to an applied Electric field than a material with low permittivity, thereby storing more energy in the material. In vaccum, the permittivity of free space is symbolized as ε 0 and its value is approximately equal to 8. 85 × 10 −^12 (IS unit: [F/m]). In other materials, the permittivity constant can have a different value and is often substantially greater

Where it is possible to define the Magnetic flux density (IS unit: [W bm 2 ]=[T]):

B =

I

l 2

μ 0 4 π

I 2 dl 2 × u 12 r^2

Figure 6: Force on the first wire generated by the magnetic field of the second

The Force (8) is:

1. attractive if dl 1 dl 2 > 0
2. repulsive if dl 1 dl 2 < 0

1.7 Magnetic Dipoles

Considering Bohr’s atom again, we can interpret the movement of its electron in orbit as an “atomic current”, and consequently, the entire atom as a magnetic dipole. Now, let’s consider a material in a neutral state and apply an external magnetic field to it.

Initially, the magnetic dipoles (represented by the atoms) are randomly oriented and distributed in space, so the internal magnetic field can be considered zero. However, once the external magnetic

field is applied, all the magnetic dipoles tend to align themselves with it, generating an internal magnetic field. The total magnetic field will be:

Btot = Bext +

X^ N

i=

Bi int > Bext (9)

Theorem 1.5. A magnetic dipole is equivalent to a closed loop of electric current in terms of the generated magnetic field and of the mechanical effects of an external magnetic field on the dipole.

1.8 Magnetic Permeability

The magnetic permeability is the measure of magnetization produced in a material in response to an applied magnetic field. It is the ratio of the magnetic induction B to the magnetizing field H in a material: B = μH

The permeability constant μ 0 , also known as the magnetic constant or the permeability of free space, is the proportionality between magnetic induction and magnetizing force when forming a magnetic field in a classical vacuum. Its value is approximately equal to 4π × 10 −^7 (IS unit: [H/m]). In other material the magnetic permeability can have different value and is often greater then the free space value. In fact, the follow relationship is valid:

μ = μrμ 0 (10)

Where μr is the relative magnetic permeability:

• if μ = 1: free space
• if μ ≃ 1: diamagnetic or paramagnetic (ex. alluminium)

In general, iron can become a magnetic temporary or permanently (with high values of current).

1.9 Maxwell’s equation

In the last equation the term JC refers to the current that generates the magnetic field, σE = JI is the induced current and ε∂ ∂tE = JD is the displacement current.

1.10 Antennas

Let’s consider the simplest model of an antenna:

and apply the curl operator to both sides:

∇ × (∇ × E) = ∇ ×

−∂(μ ∂tH)

Recalling the vector identity:

∇ × (∇ × F) = ∇(∇ · F) − ∇^2 F,

rewrite the left-hand side as:

∇(∇ · E) − ∇^2 E = ∇ ×

−∂(μ ∂tH)

Assuming the material where the wave propagates is isotropic (μ = constant) and considering the wave to be far away from the source (i.e., JC = 0 and ρ = 0), the first Maxwell’s equation simplifies to ∇ · E = 0, leading to:

−∇^2 E = −μ ∂t∂(∇ × H).

Now, inserting the fourth Maxwell’s equation:

∇ × H = σE + ε∂ ∂t E,

we obtain: −∇^2 E = −μ ∂t∂

σE + ε∂ ∂tE

Rearranging:

∇^2 E = μσ ∂E ∂t

• με∂

2 E

∂t^2

This equation, known as the Helmholtz equation, describes the evolution of the Electric field as a wave propagating far from the source within a conducting medium. The equation contains second-order derivatives in both time and space, confirming that it governs the behavior of a wave function.

Example. Consider an Electric field expressed as E = Exux + Eyuy + Ez uz. In this general form, the Helmholtz equation (11) comprises three scalar equations, one for each Cartesian component. To simplify the analysis, the Electric field is assumed to be one-dimensional (E = Exux). Additionally, a monochromatic wave (i.e., single-frequency) is considered. Under these assumptions, the Helmholtz equation reduces to:

∇^2 Ex = μσ ∂t∂Ex + εμ ∂

2 ∂t^2 Ex

The corresponding solution is:

E(z, t) = E 0 e−αz^ cos(ωt − βz)ux (12)

In this expression it is possible to define:

• E 0 is the amplitude:
• e−αz^ is the attenuation constant;
• β is the phase constant [1/m] or the velocity of the wavelength

Figure 8: Plane uniform wave coming from the Helmholtz equation

In Figure 8 it can be seen that the resulting wave propagates in the z-direction and is attenuating over time.

1.11.1 Function Representing a Wave

A function f (x, t) of position x and time t represents a wave with constant amplitude propagating along the x-axis if its dependence on space and time appears only through the combination ξ = x ± ut, where u is a constant velocity:

f (x, t) = f (ξ) = f (x ± ut)

The wave is referred to as progressive when the minus sign is used, and regressive when the plus sign is used in the expression for ξ. This functional form represents a wave because, when f is regarded as a function of the variable ξ, it defines a fixed profile f (ξ). Observing this profile while moving along the x-axis at speed u, the same value ξ¯ = ¯x ± u¯t remains constant over time. At a later time ¯t + ∆t, the same value of ξ

The (14) express the phase velocity, the speed at which a point of constant phase travels as the wave propagates. This is the velocity at which the phase of any one frequency component of the wave travels and depends only on the constitutive property of the material inside which the wave is propagated (μ and ε). The group velocity of a collection of waves is defined as:

vgroup = ∂ω ∂β

This velocity fix a point (ex. a pick) and look how this point is moving.

1.13 Phasor domain

With a monochromatic wave is possible to move the previous expression in the phasor domain:

E(z, t) = E 0 e−αz^ cos(ωt − βz)ux → Eˆ = E 0 e−αz^ e−jβz^ ux

In the phasor domain we have an expansion of the domain into the complex plane. It is possible to show that the two expressions represent the same thing:

E(z, t) = ℜ( Eˆejωt) = ℜ(E 0 e−αz^ ej(ωt−βz)) =

ℜ(E 0 e−αz^ (cos(ωt − βz) + j sin(ωt − βz)) = E 0 e−αz^ cos(ωt − βz)

In this procedure, the Euler’s identity has been used.

2 From the Electric to the Magnetic Field

The Electric field in the phasor domain is given by:

E(z) = E 0 e−αz^ e−jβz^ ux = E 0 e−γz^ ux

where the propagation constant γ is defined as:

γ = α + jβ =

p jωμ(σ + jωε).

By considering Maxwell’s second equation:

∇ × E = −μ ∂ ∂t

H

and applying the phasor transformation (noting that E = E(z) ), the curl of the Electric field is computed as:

∇ × E =

ux uy uz ∂ ∂x

∂ ∂y

∂ ∂z Ex 0 0

= uy^ ∂ ∂t

Ex = uy^ ∂ ∂t

(E 0 e−γz^ ) = uyE 0 (−γe−γz^ ).

Similarly, transforming the right-hand side into the phasor domain yields:

−μ ∂ ∂t

H = −μjωH.

Equating both expressions, the magnetic field is obtained as:

−μjωH = uyE 0 (−γe−γz^ ),

which leads to:

H = E η^0 e−γz^ uy. (16)

Here, η represents the intrinsic impedance of the medium (IS unit: [Ω]) and is given by:

η = jωμγ = p jωμ jωμ(σ + jωε)

s jωμ σ + jωε.