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 q1 and q2: F = \frac{1}{4\pi\epsilon0} \frac{q1 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
    π z}{\lambda} + \phi_0\right)

  • Key Parameters:

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

    • Phase Velocity (up): The speed of wave propagation: up = f\lambda = \frac{\omega}{k}.

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

    • Wavenumber (k): k = \frac{2
      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) = Vm \cos(\omega t + \phi) \leftrightarrow \tilde{V} = Vm e^{j\phi}

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