EE-348 Electromagnetics Notes

Basics of Fields

  • Static vs. Dynamic Fields:
    • Static: Steady or constant over time (e.g., DC current).
    • Dynamic: Varies with time; can be further categorized.
    • Periodic: Time variation repeats with a period, T.
    • Sinusoidal: Common time-varying function for ease of analysis.
    • Advantage: Simple mathematical form, aligns well with Fourier analysis.

Field Quantities and Units

  • Electrostatics:

    • Condition: Stationary charges (i.e.,
      \frac{dq}{dt} = 0)
    • Electric field intensity E (V/m) and Electric flux density D (C/m²), where D = \varepsilon E.

  • Magnetostatics:

    • Condition: Steady currents (i.e.,
      \frac{dI}{dt} = 0)
    • Magnetic flux density B (T) and Magnetic field intensity H (A/m).
    • Relationship: A time-varying electric field generates a time-varying magnetic field and vice versa, characterized by B = \mu H.

EM Fields & Materials

  • Free-Space Values:

    • Electrical permittivity \varepsilon = 8.854 \times 10^{-12} F/m
    • Magnetic permeability \mu = 4\pi \times 10^{-7} H/m.
    • Conductivity: \sigma = 0 S/m.

  • Material Polarization:

    • When an electric field E is applied, material atoms align to form electric dipoles, referred to as polarization.

Wave Properties

  • Types of Waves: Found in various forms (water waves, sound, electromagnetic fields).
    • Carry energy and have velocity (e.g., EM wave in free space c = 3 \times 10^8 m/s).
    • Superposition: Waves can interact linearly.

1D Traveling Waves

  • Mathematical Model:

    • y(x,t) = A \cos(\frac{2\pi}{T}t - \frac{2\pi}{\lambda}x + \phi_0)
    • Parameters: Amplitude A, Period T, Wavelength \lambda, and Phase \phi_0.

  • Wave Characteristics:

    • Amplitude is sinusoidal; period varies spatially and temporally.
    • Understanding direction of propagation: Positive direction if wave moves right.

Phase Velocity and Direction

  • Phase Velocity: Derived as

    • u_p = \frac{dx}{dt} = \frac{\lambda}{T}
    • Also expressed in terms of frequency: u_p = f \lambda.

  • Superposition of Waves:

    • Superimposing waves can create standing wave patterns with distinct peaks and valleys, depending on constructive and destructive interference.

EM Wave in Lossy Medium

  • Model:

    • Includes attenuation characterized by an exponential decay: y(x,t) = A e^{-\alpha x} \cos(wt - \beta x + \phi_0), where \alpha indicates attenuation constant.

  • Example:

    • Propagation of a laser beam in a medium leads to amplitude decay modeled by 150 e^{-0.003 \cdot distance}.

Complex Numbers & Phasors

  • Phasor Representation:

    • z = x + jy maps to polar form z = |z| e^{j\theta}.
    • Fundamental: Euler's identity e^{j\theta} = \cos\theta + j\sin\theta forms the basis for circuit analysis and signal processing.

  • S-domain Analysis:

    • Steady-state sinusoidal solutions translate into phasor representation where time-dependence is suppressed.
    • In phasor form, vs(t) = Re[V e^{j(wt + \phi0)}] expresses sinusoidal inputs.

Waveform Analysis and Example Problem

  • Waveform Example: Propagating electromagnetic wave can be analyzed for frequency, amplitude, and phase using given expressions.
  • Check Analysis through Circuit Simulation: Using tools like LTspice to validate theoretical results.

Conclusion

  • This guide captures essential concepts of electromagnetics, focusing on waves, fields, and circuit analysis, serving as a basis for further explorations in engineering applications.