📡Electromagnetic Interference Unit 1 – Electromagnetic Theory Fundamentals

Key Concepts and Definitions

• Electromagnetic fields are a combination of electric and magnetic fields that interact with each other and with charged particles
• Electric fields ($\vec{E}$) are created by electric charges and exert forces on other charged particles
• Magnetic fields ($\vec{B}$) are created by moving electric charges or changing electric fields and exert forces on other moving charges or magnetic dipoles
• Electromagnetic waves are self-propagating oscillations of electric and magnetic fields that travel through space at the speed of light ($c$)
  • Consist of oscillating electric and magnetic fields perpendicular to each other and to the direction of wave propagation
  • Can be characterized by their frequency ($f$), wavelength ($\lambda$), and amplitude
• Electromagnetic spectrum includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays, differentiated by their frequency and wavelength
• Permittivity ($\varepsilon$) is a measure of how much resistance is encountered when forming an electric field in a medium
• Permeability ($\mu$) is a measure of the ability of a material to support the formation of a magnetic field within itself
• Conductivity ($\sigma$) is a measure of a material's ability to conduct electric current

Maxwell's Equations Explained

• Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields and their interactions with matter
• Gauss's law for electric fields: The total electric flux through any closed surface is equal to the total electric charge enclosed within that surface ($\oint \vec{E} \cdot d\vec{A} = \frac{Q}{\varepsilon_0}$)
  • Relates the electric field to the distribution of electric charges
• Gauss's law for magnetic fields: The total magnetic flux through any closed surface is always zero ($\oint \vec{B} \cdot d\vec{A} = 0$)
  • Implies that magnetic monopoles do not exist and magnetic field lines always form closed loops
• Faraday's law of induction: A changing magnetic field induces an electric field ($\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$)
  • Describes how a time-varying magnetic field creates an electric field
• Ampère's circuital law (with Maxwell's correction): A changing electric field and electric current together produce a magnetic field ($\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$)
  • Relates the magnetic field to electric currents and changing electric fields
• Together, Maxwell's equations provide a complete description of the behavior of electromagnetic fields and waves

Electromagnetic Waves and Propagation

• Electromagnetic waves are created by accelerating electric charges and consist of oscillating electric and magnetic fields that propagate through space
• The speed of electromagnetic waves in vacuum is the speed of light ($c \approx 3 \times 10^8 \text{ m/s}$)
• In a medium, the speed of electromagnetic waves ($v$) is determined by the permittivity ($\varepsilon$) and permeability ($\mu$) of the medium: $v = \frac{1}{\sqrt{\varepsilon \mu}}$
• The wavelength ($\lambda$) and frequency ($f$) of an electromagnetic wave are related by the speed of light: $c = \lambda f$
• Electromagnetic waves can be polarized, meaning the orientation of the electric and magnetic fields can be fixed (linear polarization) or rotating (circular or elliptical polarization)
• Electromagnetic waves can interfere with each other constructively (increasing amplitude) or destructively (decreasing amplitude) when they overlap
• Reflection occurs when electromagnetic waves bounce off a surface, with the angle of incidence equal to the angle of reflection
• Refraction occurs when electromagnetic waves change direction as they pass through a boundary between two media with different refractive indices

Fields and Potentials

• Electric potential ($V$) is the potential energy per unit charge at a point in an electric field
  • Related to the electric field by the gradient operator: $\vec{E} = -\nabla V$
• Magnetic vector potential ($\vec{A}$) is a vector field whose curl gives the magnetic field: $\vec{B} = \nabla \times \vec{A}$
• Electric and magnetic fields can be derived from the electric scalar potential and magnetic vector potential using the relationships:
  • $\vec{E} = -\nabla V - \frac{\partial \vec{A}}{\partial t}$
  • $\vec{B} = \nabla \times \vec{A}$
• Gauge transformations can be applied to the potentials without changing the physical fields, allowing for mathematical simplification
• Coulomb's law describes the electric field due to a point charge: $\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r^2} \hat{r}$
• Biot-Savart law describes the magnetic field due to a current-carrying wire: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$
• Lorentz force describes the force on a charged particle in the presence of electric and magnetic fields: $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$

Time-Varying Fields

• Faraday's law of induction states that a time-varying magnetic field induces an electric field: $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$
  • This is the basis for the operation of transformers and generators
• The induced electric field is non-conservative and cannot be described by a scalar potential alone
• Lenz's law states that the induced electric field creates a current that opposes the change in magnetic flux, leading to the minus sign in Faraday's law
• Displacement current, introduced by Maxwell, is a term added to Ampère's law to maintain conservation of charge in the presence of time-varying electric fields: $\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$
  • The displacement current term $\mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$ allows for the propagation of electromagnetic waves in the absence of free charges
• Poynting's theorem describes the conservation of electromagnetic energy, with the Poynting vector ($\vec{S} = \vec{E} \times \vec{H}$) representing the directional energy flux of an electromagnetic field
• Skin effect is the tendency of high-frequency currents to flow near the surface of a conductor, reducing the effective cross-sectional area and increasing resistance

Boundary Conditions and Interfaces

• Boundary conditions specify the behavior of electromagnetic fields at the interface between two different media
• For the electric field, the normal component of the electric flux density is discontinuous by the amount of surface charge density ($\rho_s$) at the interface: $\hat{n} \cdot (\vec{D}_2 - \vec{D}_1) = \rho_s$
• The tangential component of the electric field is continuous across the interface: $\hat{n} \times (\vec{E}_2 - \vec{E}_1) = 0$
• For the magnetic field, the normal component of the magnetic flux density is continuous across the interface: $\hat{n} \cdot (\vec{B}_2 - \vec{B}_1) = 0$
• The tangential component of the magnetic field is discontinuous by the amount of surface current density ($\vec{K}$) at the interface: $\hat{n} \times (\vec{H}_2 - \vec{H}_1) = \vec{K}$
• Snell's law describes the relationship between the angles of incidence and refraction for electromagnetic waves passing through an interface: $\frac{\sin \theta_1}{\sin \theta_2} = \frac{n_2}{n_1}$
  • Where $n_1$ and $n_2$ are the refractive indices of the two media
• Fresnel equations describe the reflection and transmission coefficients for electromagnetic waves at an interface, depending on the polarization and angle of incidence

Electromagnetic Energy and Power

• Electromagnetic energy density is the sum of the electric and magnetic field energy densities: $u = \frac{1}{2} (\varepsilon \vec{E}^2 + \frac{1}{\mu} \vec{B}^2)$
• Poynting vector ($\vec{S}$) represents the directional energy flux of an electromagnetic field: $\vec{S} = \vec{E} \times \vec{H}$
  • The magnitude of the Poynting vector gives the power density (W/m²) of the electromagnetic wave
• Complex Poynting vector ($\vec{S}_c$) is used for time-harmonic fields, with the time-average power density given by the real part of $\vec{S}_c$: $\langle \vec{S} \rangle = \frac{1}{2} \text{Re}(\vec{E} \times \vec{H}^*)$
• Ohmic losses occur when electromagnetic energy is dissipated as heat in a conductor due to its finite conductivity
• Radiation resistance is a measure of the power radiated by an antenna, defined as the ratio of the power radiated to the square of the input current: $R_r = \frac{P_r}{I^2}$
• Friis transmission equation relates the power received by one antenna to the power transmitted by another, considering factors such as antenna gain, wavelength, and distance: $\frac{P_r}{P_t} = G_t G_r (\frac{\lambda}{4\pi R})^2$

Applications in EMI/EMC

• Electromagnetic interference (EMI) is the disruption of the operation of an electronic device due to electromagnetic fields from another device
• Electromagnetic compatibility (EMC) is the ability of a device to function satisfactorily in its electromagnetic environment without introducing intolerable EMI to other devices
• Common EMI sources include power lines, motors, switches, digital circuits, and radio transmitters
• EMI can be conducted (through wires or other conductors) or radiated (through electromagnetic waves)
• Shielding is the use of conductive or magnetic materials to reduce the transmission of electromagnetic fields, protecting devices from EMI
  • Shielding effectiveness depends on factors such as material properties, thickness, and frequency of the electromagnetic waves
• Grounding and bonding are important techniques for minimizing EMI by providing a low-impedance path for currents and reducing potential differences between devices
• Filtering is the use of passive or active components (such as capacitors, inductors, and ferrites) to attenuate unwanted frequencies and reduce EMI
• Proper circuit board layout and wiring practices (such as minimizing loop areas, using twisted pairs, and separating sensitive circuits) can help reduce EMI
• Electromagnetic compatibility standards (such as FCC Part 15 and CISPR) specify limits on EMI emissions and susceptibility for electronic devices to ensure their compatibility in various environments