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, $v<em>s(t) = Re[V e^{j(wt + \phi</em>0)}]$ 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.