Electromagnetics Principles and Transmission Line Fundamentals

Chapter 1: Electromagnetics Fundamentals

Gauss's Law

Gauss's Law for Electricity: The total electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity ($\epsilon_0$).

Gauss's Law for Magnetism (Eq. 6.3): This integral is zero because magnetic field lines always form closed loops; magnetic monopoles do not exist.

Gauss's Law for Electricity (Eq. 6.1): This integral can be non-zero since positive and negative charges can be isolated, leading to the surface integral equaling $Q$, the enclosed charge.

Wave Characteristics

The velocity with which the envelope—or equivalently the wave group—travels through the medium is called the group velocity.

A traveling wave is characterized by a spatial wavelength ($\lambda$), a time period ($T$), and a phase velocity ($v_p$).

An electromagnetic (EM) wave consists of oscillating electric and magnetic field intensities and travels in free space at the velocity of light ($c$). The EM spectrum encompasses gamma rays, X-rays, visible light, infrared waves, and radio waves.

Phasor analysis is a useful mathematical tool for solving problems involving time-periodic sources.

Introduction to Electromagnetics

Electromagnetics: This is the study of electric and magnetic phenomena and their engineering applications.

The Four Fundamental Forces of Nature: These are the nuclear force, weak interaction, electromagnetic force, and gravitational force.

Electromagnetics consists of three branches:

• Electrostatics, which pertains to stationary charges.
• Magnetostatics, which pertains to DC currents.
• Electrodynamics, which pertains to time-varying currents.

Chapter 2: Transmission Line Theory

Transmission Line Types and Parameters

Higher-order transmission lines: Waves propagating along these lines have at least one significant field component in the direction of propagation.

Transmission Line Parameters are:

• $R'$: The combined resistance of both conductors per unit length, in $\Omega/\text{m}$.
• $L'$: The combined inductance of both conductors per unit length, in $\text{H/m}$.
• $G_1$: The conductance of the insulation medium between the two conductors per unit length, in $\text{S/m}$.
• $C_1$: The capacitance of the two conductors per unit length, in $\text{F/m}$.

The pertinent constitutive parameters apply to all three lines and consist of two groups:

1. $\epsilon_c$ and $\mu_c$ are the magnetic permeability and electrical conductivity of the conductors.
2. $\epsilon, \mu, \sigma$ are the electrical permittivity, magnetic permeability, and electrical conductivity of the insulation material separating them.

Wave Behavior on Lines

The presence of two waves on the line propagating in opposite directions produces a standing wave.

If sinusoidal waves of different frequencies travel on a transmission line with the same phase velocity, the line is called nondispersive.

Specific Line Structures

An air line: This is a transmission line in which air separates the two conductors.

The microstrip line: This structure has two geometric parameters: the width of the elevated strip ($w$), and the thickness (height) of the dielectric layer ($h$).

Characterization of Transmission Lines

A transmission line is fully characterized by two fundamental parameters: its propagation constant ($\gamma$) and characteristic impedance ($Z_0$). Both are specified by the angular frequency ($\omega$) and the line parameters $R', L', G', C'$.