GATE E&C Engineering Electromagnetics Flashcards

24 sample cards from the Electromagnetics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Define linear, circular, and elliptical polarization of a plane wave.

   Polarization describes the tip of $\vec{E}$ over time. Linear: two orthogonal components in phase (or $180^\circ$). Circular: equal amplitudes, $90^\circ$ phase difference. Elliptical: unequal amplitudes and/or arbitrary phase difference (general case).

2. What conditions on the two orthogonal $E$ components give right-hand vs left-hand circular polarization?

   Equal amplitudes with a $\pm 90^\circ$ phase difference. Using $e^{j\omega t}$ convention and propagation in $+z$: a $-90^\circ$ ($E_y$ lagging) typically gives right-hand circular, $+90^\circ$ gives left-hand circular polarization.

3. State the law of reflection and Snell's law of refraction for an oblique incidence.

   Reflection: $\theta_i = \theta_r$. Snell's law: $n_1 \sin\theta_i = n_2 \sin\theta_t$, equivalently $\frac{\sin\theta_i}{\sin\theta_t} = \frac{n_2}{n_1} = \sqrt{\frac{\mu_2\varepsilon_2}{\mu_1\varepsilon_1}}$.

4. Define the reflection and transmission coefficients for normal incidence between two media.

   $\Gamma = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1}$ and $\tau = \frac{2\eta_2}{\eta_2 + \eta_1}$, with the relation $1 + \Gamma = \tau$.

5. What is the Brewster angle and its condition for parallel (p) polarization?

   The Brewster angle is the incidence angle at which the parallel-polarized wave is fully transmitted (zero reflection). For nonmagnetic media: $\tan\theta_B = \frac{n_2}{n_1} = \sqrt{\frac{\varepsilon_2}{\varepsilon_1}}$.

6. Define the critical angle and the condition for total internal reflection.

   Critical angle $\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)$ (with $n_1 > n_2$). For $\theta_i > \theta_c$, total internal reflection occurs and an evanescent wave exists in medium 2.

7. What is the standing wave ratio (VSWR) in terms of the reflection coefficient?

   $$\text{VSWR} = S = \frac{1 + |\Gamma|}{1 - |\Gamma|}$$ Range $1 \leq S \leq \infty$; $S=1$ means a perfectly matched (no reflection) condition.

8. Write the transmission line telegrapher's equations (phasor form).

   $\frac{dV}{dz} = -(R + j\omega L)I$ and $\frac{dI}{dz} = -(G + j\omega C)V$, where $R, L, G, C$ are per-unit-length parameters.

9. Give the propagation constant and characteristic impedance of a general transmission line.

   $\gamma = \sqrt{(R + j\omega L)(G + j\omega C)} = \alpha + j\beta$ and $Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}}$.

10. What is the characteristic impedance of a lossless transmission line?

    $Z_0 = \sqrt{\frac{L}{C}}$ (purely real). The line is also dispersionless, with $u_p = \frac{1}{\sqrt{LC}}$ and $\beta = \omega\sqrt{LC}$.

11. State the input impedance formula of a lossless transmission line of length $l$.

    $$Z_{in} = Z_0 \frac{Z_L + jZ_0\tan(\beta l)}{Z_0 + jZ_L\tan(\beta l)}$$

12. What is the input impedance of a quarter-wave ($l = \lambda/4$) lossless line, and its use?

    $Z_{in} = \frac{Z_0^2}{Z_L}$ (an impedance inverter). Used for impedance matching: a quarter-wave transformer of $Z_0 = \sqrt{Z_{in}\,Z_L}$ matches a real load to a line.

13. What is the input impedance of a half-wave ($l = \lambda/2$) lossless line?

    $Z_{in} = Z_L$. A half-wavelength line (or its multiples) repeats the load impedance regardless of $Z_0$.

14. Give the input impedance of short-circuited and open-circuited lossless line stubs.

    Short-circuit ($Z_L=0$): $Z_{in} = jZ_0\tan(\beta l)$. Open-circuit ($Z_L=\infty$): $Z_{in} = -jZ_0\cot(\beta l)$. Both are purely reactive, used as tuning stubs.

15. Define the reflection coefficient at the load of a transmission line.

    $$\Gamma_L = \frac{Z_L - Z_0}{Z_L + Z_0}$$ Its magnitude is constant along a lossless line; phase rotates as $\Gamma(z) = \Gamma_L e^{-2j\beta z}$.

16. What is single-stub matching and what does it match?

    A short/open-circuited stub of suitable length placed at a specific distance from the load. It cancels the reactive part and matches the conductance/susceptance so that $Z_{in} = Z_0$ (eliminates reflections), typically designed on the Smith chart.

17. Define the scattering (S) parameter $S_{ij}$ of a two-port network.

    $S_{ij} = \frac{b_i}{a_j}\Big|_{a_k = 0,\, k\neq j}$, the ratio of the outgoing wave at port $i$ to the incident wave at port $j$ with all other ports terminated in matched loads ($a_k=0$).

18. What do $S_{11}$ and $S_{21}$ represent physically in a two-port?

    $S_{11}$ = input reflection coefficient (port 2 matched). $S_{21}$ = forward transmission coefficient (gain/insertion) from port 1 to port 2 with port 2 matched.

19. State the reciprocity and losslessness conditions on the S-matrix.

    Reciprocal network: $[S] = [S]^{T}$ ($S_{ij} = S_{ji}$). Lossless network: $[S]$ is unitary, $[S]^{*T}[S] = [I]$, i.e. columns are orthonormal.

20. What is the Smith chart and what two families of circles does it contain?

    A graphical tool plotting the complex reflection coefficient on the unit circle ($|\Gamma|\leq 1$). It superimposes constant-resistance circles and constant-reactance arcs of normalized impedance $z = \frac{Z}{Z_0} = r + jx$.

21. On the Smith chart, what does one full rotation correspond to, and which direction moves toward the generator?

    One full $360^\circ$ rotation corresponds to $\lambda/2$ along the line. Clockwise rotation moves toward the generator (input); counterclockwise moves toward the load.

22. For an optical fiber, define the numerical aperture and acceptance angle.

    $\text{NA} = \sin\theta_a = \sqrt{n_1^2 - n_2^2}$, where $n_1$ = core index, $n_2$ = cladding index, and $\theta_a$ is the maximum acceptance (half-)angle for guided rays.

23. What principle guides light in an optical fiber, and what is the fractional refractive index difference?

    Light is guided by total internal reflection at the core-cladding boundary ($n_1 > n_2$). The fractional index difference is $\Delta = \frac{n_1^2 - n_2^2}{2n_1^2} \approx \frac{n_1 - n_2}{n_1}$, so $\text{NA} \approx n_1\sqrt{2\Delta}$.

24. Compare single-mode and multimode optical fibers.

    Single-mode: small core ($\sim$8–10 µm), carries one mode, very low dispersion, long-haul high bandwidth. Multimode: larger core ($\sim$50–62.5 µm), many modes, suffers intermodal dispersion, used for shorter distances.

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