GATE E&C Engineering Electromagnetics Syllabus

Write the general propagation constant $\gamma$ for a lossy medium, naming its parts.

$\gamma = j\omega\sqrt{\mu\varepsilon}\sqrt{1 + \frac{\sigma}{j\omega\varepsilon}} = \alpha + j\beta$, where $\alpha$ is the attenuation constant (Np/m) and $\beta$ is the phase constant (rad/m).

Give the attenuation and phase constants for a good conductor.

For a good conductor ($\sigma \gg \omega\varepsilon$): $\alpha = \beta \approx \sqrt{\frac{\omega \mu \sigma}{2}} = \sqrt{\pi f \mu \sigma}$.

Define skin depth and give its formula for a good conductor.

Skin depth $\delta$ is the distance over which a wave's amplitude decays to $1/e$ ($\approx 37\%$). $\delta = \frac{1}{\alpha} = \sqrt{\frac{2}{\omega\mu\sigma}} = \frac{1}{\sqrt{\pi f \mu \sigma}}$.

How does skin depth vary with frequency and conductivity?

$\delta \propto \frac{1}{\sqrt{f}}$ and $\delta \propto \frac{1}{\sqrt{\sigma}}$. Higher frequency or higher conductivity gives a thinner skin depth (more confinement to the surface).

What is the intrinsic impedance of a good conductor (magnitude and phase)?

$\eta_c = \sqrt{\frac{\omega\mu}{\sigma}}\,\angle 45^\circ = (1+j)\sqrt{\frac{\pi f \mu}{\sigma}}$. It has a $45^\circ$ phase angle, so $\vec{H}$ lags $\vec{E}$ by $45^\circ$.

Distinguish phase velocity and group velocity.

Phase velocity $u_p = \frac{\omega}{\beta}$ is the speed of a constant-phase point. Group velocity $u_g = \frac{d\omega}{d\beta}$ is the speed of energy/information (envelope). In a dispersive medium $u_p \neq u_g$.

What is a dispersive medium, and what is the relation $u_p u_g$ for a waveguide?

A dispersive medium is one where phase velocity depends on frequency, so different frequencies travel at different speeds. In a hollow waveguide, $u_p u_g = c^2$ (with $u_p > c > u_g$).

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).

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.

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 $\frac{\sin\theta_i}{\sin\theta_t} = \frac{n_2}{n_1} = \sqrt{\frac{\mu_2\varepsilon_2}{\mu_1\varepsilon_1}}$.

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 $1 + \Gamma = \tau$.

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: $\tan\theta_B = \frac{n_2}{n_1} = \sqrt{\frac{\varepsilon_2}{\varepsilon_1}}$.

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.

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.

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.

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

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 $\beta = \omega\sqrt{LC}$.

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

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

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.

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 $Z_0$.

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.

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 $\Gamma(z) = \Gamma_L e^{-2j\beta z}$.

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.

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 ($a_k=0$).

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.

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.

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 $z = \frac{Z}{Z_0} = r + jx$.

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.

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.

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 $\text{NA} \approx n_1\sqrt{2\Delta}$.

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.