Electromagnetics
Maxwell equations, plane waves, transmission lines, the Smith chart, waveguides and antennas.
About this subject
Electromagnetics covers Maxwell's equations and what follows from them: plane waves and their power, wave propagation in conductors and dielectrics, transmission lines, waveguides and antennas.
Many questions are direct applications of one formula, so the marks are won or lost on details: which dimension of a waveguide matters, whether a field is a peak or an RMS value, and whether a reflection coefficient is used as a complex number or as a magnitude.
Solved questions
Each question is a practice problem written for this portal at GATE standard. Try it first, then open the full derivation, and read the note on the idea behind it.
- NAT VSWR on a 50 ohm line terminated in 100 ohms Transmission lines · 1 mark
- NAT Dominant-mode cutoff of a rectangular waveguide Waveguides · 2 marks
- NAT Skin depth in copper at 1 MHz Wave propagation · 2 marks
- NAT Average power density of a plane wave in free space Plane waves · 1 mark
Key formulas
The results these questions lean on most. Know what each symbol means, and when each one applies.
| What | Formula |
|---|---|
| Reflection coefficient | $\Gamma_L = \dfrac{Z_L - Z_0}{Z_L + Z_0}$ |
| VSWR | $S = \dfrac{1 + |\Gamma_L|}{1 - |\Gamma_L|}$ |
| Rectangular waveguide cutoff | $f_c = \dfrac{c}{2}\sqrt{(m/a)^2 + (n/b)^2}$; for $\text{TE}_{10}$, $f_c = c/2a$ |
| Skin depth | $\delta = 1/\sqrt{\pi f \mu \sigma}$ |
| Average power density | $S_{avg} = E_0^2 / (2\eta)$ for a peak amplitude $E_0$ |
| Intrinsic impedance of free space | $\eta_0 = \sqrt{\mu_0/\epsilon_0} \approx 377\ \Omega$ |
| Phase velocity in a line | $v_p = 1/\sqrt{LC}$ per unit length |
How to prepare
- Note whether each field amplitude is a peak or an RMS value before computing power.
- Learn the dominant mode of each waveguide shape, and which dimension sets its cutoff.
- Use the magnitude of a reflection coefficient for VSWR, and its phase for where the maxima are.
- Check answers by scaling: skin depth falls as $1/\sqrt{f}$, so four times the frequency halves it.
Common traps
Each of these costs marks every year. The questions above are built to catch them.
- Using both waveguide dimensions for the $\text{TE}_{10}$ cutoff.
- Leaving out the factor of one half in the average power density.
- Using a complex reflection coefficient directly in the VSWR formula.
- Mixing the $\omega$ and $f$ forms of the skin-depth formula.