🎓 Master Course: Applied Electromagnetics

This repository contains analytical formulations, simulations, and computational experiments developed for the Applied Electromagnetics course of the Master's program.

The work spans boundary value problems, wave reflection in dielectric interfaces, waveguide propagation, and transmission line analysis — integrating both analytical methods and computational approaches.

👨‍🏫 Professor: Prof. Dr. João Batista Rosa Silva

🚀 Note: All notebooks in this repository are fully compatible with Google Colab.
Simply open any .ipynb file via Colab to run the simulations and visualizations in the cloud without local setup.

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🧠 Dependencies & Libraries

The following Python libraries were used across the notebooks in this repository:

✅ Main Usage per Notebook:

Notebook	Libraries
notebooks/electrostatic_potential.ipynb	NumPy 1.26.4, SymPy 1.12, Plotly 5.20.0
notebooks/dielectric_reflection.ipynb	NumPy 1.26.4, Plotly 5.20.0, Math
notebooks/waveguide_simulation.ipynb	NumPy 1.26.4, Matplotlib 3.8.4, Math
notebooks/transmission_line_analysis.ipynb	NumPy 1.26.4, Plotly 5.20.0

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⚙️ Electrostatic Potential and Electric Field Visualization

📌 Applications

Understanding electrostatic potential distributions is essential in designing capacitive sensors, MEMS devices, insulation systems, and electrostatic actuators used in various industries.

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Objective

(a) Plot the electrostatic potential distribution $V(x, y)$.
(b) Plot the magnitude of the electric field $|E|$.

For two cases:

• $n = 10$ (number of series terms)
• $n = 50$ (number of series terms)

📓 Description of the notebook electrostatic_potential.ipynb

This notebook develops the analytical solution for the electrostatic potential in a rectangular domain using the method of separation of variables applied to Laplace's equation. It includes:

• Symbolic derivation of the potential function $V(x, y)$.
• Computation and visualization of $V(x, y)$.
• Calculation and plotting of the electric field magnitude $|E(x, y)|$ derived from $\vec{E} = -\nabla V$.

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⚙️ Wave Reflection on Dielectric Structures

📐 Reflection and Transmission Model

This figure illustrates the boundary conditions of an electromagnetic wave encountering a dielectric interface. The incident, reflected, and transmitted fields are represented along with their respective directions and media properties.

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📌 Applications

This analysis is fundamental in designing anti-reflection coatings, radar absorbing materials, stealth technology, and wireless communication systems, where signal reflection needs to be controlled.

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Problem 01 — Single Dielectric Slab (Air–Dielectric–Air)

A uniform plane wave at 10 GHz is incident normally on a dielectric plate with $\epsilon_r = 3.7$, bounded by air on both sides.

(a) Determine the thickness $d$ such that the input reflection coefficient is zero at 10 GHz. Repeat for 20 GHz.
(b) Plot the reflection coefficient magnitude for frequencies from 5 GHz to 30 GHz for $d = 0.9375$ cm, considering both design frequencies.

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Problem 02 — Dielectric Slab with Semi-Infinite Medium

A plane wave is incident normally on a dielectric slab of thickness $\lambda_{2_{0}}/4$ (at 10 GHz), bounded by air on one side and a semi-infinite medium with $\epsilon_{r_3} = 5.7$ on the other.

(a) Compute the intrinsic impedance $\eta_2$ and dielectric constant $\epsilon_2$ to achieve zero reflection at both 10 GHz and 20 GHz.
(b) Plot the reflection coefficient magnitude for 5 GHz ≤ f ≤ 30 GHz based on the design in (a).
(c) Using the ray-tracing model, compute the first and next two higher-order contributions to the reflection coefficient at 10 GHz and 20 GHz.

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Problem 03 — Binomial Matching with Two Dielectric Slabs

Two lossless dielectric plates, each with thickness $\lambda_0/4$ at $f_0 = 10$ GHz, are placed between air and a semi-infinite medium with $\epsilon_L = 6.7$. A binomial impedance-matching design is applied with a fractional bandwidth of 0.375.

(a) Calculate the intrinsic impedances, dielectric constants, and physical thicknesses needed for zero reflection at 10 GHz and 20 GHz.
(b) Determine the maximum reflection coefficient and the standing wave ratio (SWR) within the bandwidth.
(c) Plot the reflection coefficient from 5 GHz to 30 GHz, comparing the binomial design (two sections) to the single-section design.

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📓 Description of the notebook dielectric_reflection.ipynb

This notebook performs an analytical study of wave reflection and transmission in single and multilayer dielectric structures. It includes:

• Design of matching layers for zero reflection at specific frequencies.
• Reflection coefficient analysis over frequency ranges.
• Comparison between single-section and binomial multi-section matching techniques.
• Clear visualizations of the frequency-dependent behavior of the reflection coefficient.

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⚙️ Waveguide Propagation and Simulation

📌 Applications

Waveguides are the backbone of high-frequency signal transmission in radars, satellite communication, microwave circuits, and optical fiber technologies. Understanding dispersion and attenuation is critical in designing low-loss, high-performance transmission systems.

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Problem 9.10 — WR-159 Rectangular Waveguide

Computer Experiment:
Reproduce the plots in Figure 9.8.2, showing:

• Power Transmission vs Frequency
• Attenuation Constant vs Frequency
  for the WR-159 rectangular waveguide operating in the dominant TE(_{10}) mode.

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Problem 9.11 — Dielectric Slab Waveguide

Computer Experiment:
Reproduce all results and plots from Examples 9.11.1 and 9.11.2, including:

• Dispersion curves (frequency vs propagation constant $\beta$).
• Normalized parameter plots $v(u)$.
• Electric field profiles $E_y(x)$ for supported TE and TM modes.

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📓 Description of the notebook waveguide_simulation.ipynb

This notebook analyzes:

• Power transmission and conductor loss attenuation in rectangular waveguides.
• Computation of TE and TM mode propagation constants in dielectric slab waveguides.
• Plotting dispersion relations, normalized frequency curves, and transverse electric field distributions.
• Validation against the textbook Electromagnetic Waves and Antennas by Sophocles J. Orfanidis.

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⚙️ Transmission Line Analysis

📐 Microstrip Transmission Line Model

This figure depicts the electric (E) and magnetic (H) field distributions in a microstrip transmission line, which is a common planar transmission line used in high-frequency circuits. The fringing effect of the fields in the air above the substrate is clearly visible.

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📌 Applications

Transmission lines are ubiquitous in RF circuits, antennas, microwave circuits, and printed circuit boards (PCBs). Understanding impedance matching, reflection, and signal integrity is essential for reliable and efficient communication and sensing systems.

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📓 Description of the notebook transmission_line_analysis.ipynb

This notebook develops a complete analysis of transmission lines, covering theoretical modeling and computational solutions.

🔍 Theoretical Background

Transmission lines are described by the Telegrapher's Equations:

$$ \frac{\partial V(z)}{\partial z} = - (R + j\omega L) I(z) $$

$$ \frac{\partial I(z)}{\partial z} = - (G + j\omega C) V(z) $$

Where:

• $R$ = resistance ($\Omega/m$)
• $L$ = inductance ($H/m$)
• $G$ = conductance ($S/m$)
• $C$ = capacitance ($F/m$)

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🔗 Fundamental Parameters

• Characteristic Impedance:

$$ Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}} $$

• Propagation Constant:

$$ \gamma = \alpha + j\beta = \sqrt{(R + j\omega L)(G + j\omega C)} $$

where:

• $\alpha$ = attenuation constant ($Np/m$)
• $\beta$ = phase constant ($rad/m$)

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🔸 Input Impedance:

For a line of length $l$ terminated with a load $Z_L$:

$$ Z_{in} = Z_0 \frac{Z_L + j Z_0 \tan(\beta l)}{Z_0 + j Z_L \tan(\beta l)} $$

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🔸 Reflection Coefficient and SWR:

• Reflection Coefficient:

$$ \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} $$

• Voltage Standing Wave Ratio (SWR):

$$ SWR = \frac{1 + |\Gamma|}{1 - |\Gamma|} $$

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📊 Notebook Features

• Frequency-domain analysis of lossless and lossy transmission lines.
• Computation of input impedance, reflection coefficient, and SWR.
• Visualization of voltage and current distributions.
• Smith chart generation for impedance matching and transformations.
• Application examples with coaxial and microstrip lines.

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📚 References

• C. A. Balanis, Advanced Engineering Electromagnetics, 2nd Edition, John Wiley & Sons, 2012.
• Matthew N. O. Sadiku, Computational Electromagnetics with MATLAB, 4th Edition, CRC Press, 2019.
• Sophocles J. Orfanidis, Electromagnetic Waves and Antennas, Rutgers University, 2016.
  🔍 Focus on Chapter 9 — Waveguides, especially Sections 9.8 and 9.11, Examples 9.11.1, 9.11.2, and Problems 9.10 and 9.11.