GATE Electrical Engineering (EE) Syllabus

Write the complex power $S$ and its components.

$$S = VI^{*} = P + jQ$$ where $P = VI\cos\theta$ is real (average) power in watts, $Q = VI\sin\theta$ is reactive power in VAR, and $|S| = VI$ is apparent power in VA. Also $|S| = \sqrt{P^{2}+Q^{2}}$.

Define the quality factor $Q$ of a series RLC resonant circuit.

$$Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 R C} = \frac{1}{R}\sqrt{\frac{L}{C}}$$ A higher $Q$ means sharper resonance and a narrower bandwidth.

Relate bandwidth, resonant frequency, and quality factor for a resonant circuit.

$$\text{BW} = \omega_2 - \omega_1 = \frac{\omega_0}{Q} = \frac{R}{L}\ \text{(series RLC)}$$ The bandwidth is the range between half-power (−3 dB) frequencies.

At resonance in a series RLC circuit, what happens to the impedance and current?

Inductive and capacitive reactances cancel ($X_L = X_C$), so impedance is purely resistive and minimum ($Z = R$). The current is maximum and in phase with the source voltage (pf = 1).

Compare series resonance and parallel (anti-) resonance in terms of impedance.

Series resonance: impedance is minimum ($=R$), current maximum. Parallel resonance: impedance is maximum (theoretically infinite for ideal elements), current minimum. Both occur near $\omega_0 = \frac{1}{\sqrt{LC}}$.

In a balanced three-phase system, relate line and phase quantities for a star (Y) connection.

Star: $V_L = \sqrt{3}\,V_{ph}$ (line voltage leads phase by $30^{\circ}$), and $I_L = I_{ph}$.

In a balanced three-phase system, relate line and phase quantities for a delta ($\Delta$) connection.

Delta: $V_L = V_{ph}$, and $I_L = \sqrt{3}\,I_{ph}$.

Write the total power in a balanced three-phase load.

$$P = \sqrt{3}\,V_L I_L \cos\theta$$ where $V_L$ and $I_L$ are line voltage and current and $\theta$ is the phase angle of the per-phase load. Reactive power $Q = \sqrt{3}\,V_L I_L \sin\theta$.

Define the ABCD (transmission) parameters of a two-port network.

They relate input to output: $V_1 = A V_2 - B I_2$ and $I_1 = C V_2 - D I_2$. $A$ is dimensionless (voltage ratio), $B$ has units of ohms (impedance), $C$ has units of siemens (admittance), $D$ is dimensionless.

Write the defining equations of the Z (open-circuit impedance) parameters of a two-port.

$$V_1 = Z_{11}I_1 + Z_{12}I_2, \qquad V_2 = Z_{21}I_1 + Z_{22}I_2$$ Each parameter is found with one port open-circuited, e.g. $Z_{11} = \left.\frac{V_1}{I_1}\right|_{I_2=0}$.

What condition makes a two-port network reciprocal in terms of its parameters?

Reciprocity requires $Z_{12} = Z_{21}$ (Z-params), $Y_{12} = Y_{21}$ (Y-params), or $AD - BC = 1$ (transmission params). Networks made only of passive bilateral elements (R, L, C, transformers) are reciprocal.

What condition makes a two-port network symmetrical?

Symmetry requires $Z_{11} = Z_{22}$, $Y_{11} = Y_{22}$, or $A = D$ in transmission parameters. The network looks identical from either port.

State Coulomb's law for the force between two point charges.

$$F = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^{2}}$$ directed along the line joining them, with $\frac{1}{4\pi\varepsilon_0} \approx 9\times10^{9}\ \text{N·m}^2/\text{C}^2$. The force is repulsive for like charges, attractive for unlike.

Write the electric field of a point charge and the relation between field and potential.

$$\vec{E} = \frac{1}{4\pi\varepsilon_0}\frac{q}{r^{2}}\hat{r}, \qquad \vec{E} = -\nabla V$$ The field points away from a positive charge.

State Gauss's law in integral and differential form.

Integral: $$\oint_S \vec{D}\cdot d\vec{A} = Q_{enc}$$ (or $\oint \vec{E}\cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0}$). Differential: $\nabla\cdot\vec{D} = \rho_v$. The net flux out of a closed surface equals the enclosed charge.

Using Gauss's law, write the electric field magnitude for an infinite line charge and an infinite sheet of charge.

Infinite line charge (density $\lambda$): $E = \frac{\lambda}{2\pi\varepsilon_0 r}$. Infinite sheet (density $\sigma$): $E = \frac{\sigma}{2\varepsilon_0}$ (independent of distance).

Define capacitance and write the capacitance of a parallel-plate capacitor.

Capacitance $C = \frac{Q}{V}$ (farads). Parallel-plate: $$C = \frac{\varepsilon_0 \varepsilon_r A}{d}$$ where $A$ is plate area, $d$ the separation, and $\varepsilon_r$ the relative permittivity of the dielectric.

How do capacitances combine in series and in parallel?

Series: $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ (smaller than the smallest). Parallel: $C_{eq} = \sum C_i$. This is opposite to how resistors combine.

What effect does inserting a dielectric of relative permittivity $\varepsilon_r$ have on a capacitor?

It increases capacitance by the factor $\varepsilon_r$ ($C' = \varepsilon_r C$). For fixed charge, voltage and field drop by $\varepsilon_r$; for fixed voltage, stored charge and energy increase. The dielectric reduces the internal field via polarization.

Write the energy stored in a capacitor and in an inductor.

Capacitor: $W = \frac{1}{2}CV^{2} = \frac{Q^{2}}{2C}$. Inductor: $W = \frac{1}{2}LI^{2}$. Electrostatic energy density is $w = \frac{1}{2}\varepsilon E^{2}$; magnetic energy density is $w = \frac{1}{2}\frac{B^{2}}{\mu}$.

State the Biot–Savart law.

$$d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{l}\times\hat{r}}{r^{2}}$$ It gives the magnetic field contribution from a current element $I\,d\vec{l}$ at a point distance $r$ away.

State Ampère's circuital law (magnetostatic form).

$$\oint \vec{H}\cdot d\vec{l} = I_{enc}$$ (differential form $\nabla\times\vec{H} = \vec{J}$). The line integral of $\vec{H}$ around a closed loop equals the net current enclosed.

Using Ampère's law, write the magnetic field of an infinite straight wire and inside a long solenoid.

Infinite wire: $B = \frac{\mu_0 I}{2\pi r}$. Long solenoid (n turns/length): $B = \mu_0 n I$ inside, nearly zero outside.

Define self-inductance and write the inductance of a long solenoid.

Self-inductance $L = \frac{N\Phi}{I}$ (henrys), the flux linkage per unit current. Long solenoid: $$L = \frac{\mu_0 N^{2} A}{l}$$ with $N$ turns, cross-section $A$, length $l$.

Define mutual inductance and the coupling coefficient.

Mutual inductance $M = \frac{N_2 \Phi_{21}}{I_1}$, relating flux in coil 2 to current in coil 1. Coupling coefficient $k = \frac{M}{\sqrt{L_1 L_2}}$ with $0 \le k \le 1$; $k=1$ means perfect coupling.

State Faraday's law of electromagnetic induction.

$$\text{emf} = -\frac{d\Phi}{dt} = -N\frac{d\Phi}{dt}$$ The induced emf equals the negative rate of change of magnetic flux linkage. The minus sign (Lenz's law) means the induced current opposes the change in flux.

Write Maxwell's four equations in differential form.

Gauss (E): $\nabla\cdot\vec{D} = \rho_v$. Gauss (B): $\nabla\cdot\vec{B} = 0$. Faraday: $\nabla\times\vec{E} = -\frac{\partial \vec{B}}{\partial t}$. Ampère–Maxwell: $\nabla\times\vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}$.

What is displacement current and why did Maxwell introduce it?

Displacement current density is $\vec{J}_d = \frac{\partial \vec{D}}{\partial t}$. Maxwell added it to Ampère's law so the law would hold for time-varying fields (e.g. between capacitor plates where no conduction current flows), ensuring continuity of current and predicting electromagnetic waves.

Write the EMF equation of a transformer.

$$E = 4.44\, f\, N\, \Phi_m = 4.44\, f\, N\, B_m A$$ where $f$ is frequency, $N$ the number of turns, $\Phi_m$ the peak flux, $B_m$ peak flux density, and $A$ core area.

Write the turns ratio relations for an ideal transformer.

$$\frac{V_1}{V_2} = \frac{N_1}{N_2} = \frac{I_2}{I_1} = a$$ Voltage transforms in proportion to turns, current inversely. Impedance reflects as $Z_1 = a^{2} Z_2$, and an ideal transformer is $100\%$ efficient (input VA = output VA).

Name the two main types of losses in a transformer and how each varies.

Core (iron) losses: hysteresis loss $\propto f B_m^{n}$ and eddy-current loss $\propto f^{2} B_m^{2} t^{2}$ — roughly constant with load. Copper ($I^{2}R$) losses: vary with the square of load current. Maximum efficiency occurs when copper loss equals core loss.