Study Notes on Fundamentals of Applied Electromagnetics

1. Pedagogical Overview and Resources

• Authors: Fawwaz T. Ulaby and Umberto Ravaioli (8th Edition).

• Core Objective: Bridging the gap between circuit analysis and electromagnetic field theory.

• Interactive Learning: Inclusion of 52 web-based simulation modules (em8e.eecs.umich.edu) for visualizing wave propagation, transmission lines, and Smith charts.

• Prerequisites: Proficiency in circuit analysis (Ohm’s Law, KCL, KVL) and multivariable calculus is required.

2. Historical Milestones in Electromagnetics

• 1800: Alessandro Volta invents the first electric battery.

• 1820: Hans Christian Ørsted discovers that electric current affects a magnetic compass needle.

• 1830s: Michael Faraday demonstrates electromagnetic induction ($V = -N \frac{d\Phi}{dt}$).

• 1873: James Clerk Maxwell publishes "A Treatise on Electricity and Magnetism," unifying the field.

• 1888: Heinrich Hertz experimentally verifies the existence of electromagnetic waves.

3. Standards, Units, and Notation

• SI Units (International System of Units):

  • Length: meter ($m$)

  • Mass: kilogram ($kg$)

  • Time: second ($s$)

  • Electric Current: ampere ($A$)

  • Note: Charge is measured in Coulombs ($C$), where $1 C = 1 A \cdot s$.

• Notation Conventions:

  • Scalars: Written in italics (e.g., $V, I, q$).

  • Vectors: Written in boldface (e.g., $\mathbf{E}, \mathbf{B}, \mathbf{D}, \mathbf{H}$).

  • Phasors: Indicated by a tilde (e.g., $\tilde{V}, \tilde{\mathbf{E}}$).

4. The Nature of Electromagnetism and Fundamental Forces

• Lorentz Force Law: Describes the total force acting on a charge $q$ moving with velocity $\mathbf{u}$ in the presence of electric ($\mathbf{E}$) and magnetic ($\mathbf{B}$) fields:
  $\mathbf{F} = q(\mathbf{E} + \mathbf{u} \times \mathbf{B})$

• Coulomb’s Law: The electrical force between two point charges $q<em>1$ and $q</em>2$: $F = \frac{1}{4\pi\epsilon<em>0} \frac{q</em>1 q_2}{R^2}$

  • Where Permittivity of free space $\epsilon_0 \approx 8.854 \times 10^{-12} \text{ F/m}$.

• Field Categories:

  • Electrostatics: Fields produced by stationary charges (constant over time).

  • Magnetostatics: Fields produced by steady currents ($dc$).

  • Electrodynamics: Time-varying fields that produce electromagnetic waves.

5. Traveling Waves

• General Wave Equation: A wave traveling in the $+z$ direction is expressed as:
  $y(z, t) = A \cos\left(\frac{2\pi t}{T} - \frac{2<br>π z}{\lambda} + \phi_0\right)$

• Key Parameters:

  • Amplitude ($A$): Maximum value of the wave.

  • Phase Velocity ($u<em>p$): The speed of wave propagation: $u</em>p = f\lambda = \frac{\omega}{k}$.

  • Angular Frequency ($\omega$): $\omega = 2\pi f = \frac{2<br>π}{T}$.

  • Wavenumber ($k$): $k = \frac{2<br>π}{\lambda}$.

  • Phase ($\phi$): The argument of the cosine function: $(\omega t - kz + \phi_0)$.

6. The Electromagnetic Spectrum

• Speed of Light ($c$): In vacuum, $c \approx 3 \times 10^8 \text{ m/s}$.

• Frequency-Wavelength Relationship: $c = f \lambda$.

• Spectrum Segments: Ranges from Gamma rays (high frequency, short wavelength) to Radio waves (low frequency, long wavelength).

7. Complex Number Review and Phasor Analysis

• Representations:

  • Rectangular: $z = x + jy$

  • Polar: $z = |z|e^{j\theta} = |z|(\cos \theta + j \sin \theta)$

  • Identity: $j = \sqrt{-1}, j^2 = -1, \frac{1}{j} = -j$

• Phasor Transformation:

  • To solve time-harmonic problems, time-domain functions are converted to phasors:
    $v(t) = V<em>m \cos(\omega t + \phi) \leftrightarrow \tilde{V} = V</em>m e^{j\phi}$

  • Linearity: Phasor analysis allows differential equations describing the system to be converted into much simpler linear algebraic equations.