UPSC ESE Electrical Engineering Electrical Engineering Syllabus

Define the coefficient of coupling $k$ between two coupled coils.

$k = \frac{M}{\sqrt{L_{1}L_{2}}}$, where $M$ is mutual inductance. $0 \leq k \leq 1$; $k=1$ means perfect coupling.

For two series-aiding and series-opposing coupled coils, write the total inductance.

Series-aiding: $L = L_{1} + L_{2} + 2M$. Series-opposing: $L = L_{1} + L_{2} - 2M$.

What does the dot convention indicate in coupled circuits?

Currents entering both dotted terminals produce mutual flux that aids the self flux. The sign of the mutually induced voltage $\pm M\frac{di}{dt}$ is determined by the dots.

Define the $Z$-parameters (impedance parameters) of a two-port network.

$V_{1} = Z_{11}I_{1} + Z_{12}I_{2}$, $V_{2} = Z_{21}I_{1} + Z_{22}I_{2}$. Parameters are found with input/output ports open-circuited ($I=0$).

Define the $Y$-parameters (admittance parameters) of a two-port network.

$I_{1} = Y_{11}V_{1} + Y_{12}V_{2}$, $I_{2} = Y_{21}V_{1} + Y_{22}V_{2}$. Parameters are found with ports short-circuited ($V=0$).

Write the ABCD (transmission) parameter equations and the reciprocity condition.

$V_{1} = AV_{2} - BI_{2}$, $I_{1} = CV_{2} - DI_{2}$. Reciprocity requires $AD - BC = 1$.

What is the condition for a two-port network to be reciprocal and symmetrical in terms of Z-parameters?

Reciprocal: $Z_{12} = Z_{21}$. Symmetrical: additionally $Z_{11} = Z_{22}$.

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

Star: $V_{L} = \sqrt{3}\,V_{ph}$ and $I_{L} = I_{ph}$. Line voltage leads phase voltage by $30^{\circ}$.

For a balanced three-phase system, relate line and phase quantities in 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 in terms of line quantities.

$P = \sqrt{3}\,V_{L}I_{L}\cos\phi$ for both star and delta, where $\phi$ is the phase angle of the load impedance.

In the two-wattmeter method, how is total power and power factor angle obtained?

$P = W_{1} + W_{2}$ and $\tan\phi = \sqrt{3}\,\frac{W_{1}-W_{2}}{W_{1}+W_{2}}$.

State Gauss's law in differential and integral form for electrostatics.

Integral: $\oint \vec{D}\cdot d\vec{A} = Q_{enc}$. Differential: $\nabla\cdot\vec{D} = \rho$, where $\vec{D} = \varepsilon\vec{E}$.

Write the electric field and potential due to a point charge $Q$ at distance $r$ in free space.

$\vec{E} = \frac{Q}{4\pi\varepsilon_{0}r^{2}}\hat{r}$ and $V = \frac{Q}{4\pi\varepsilon_{0}r}$.

State Ampere's circuital law (magnetostatics) in integral form.

$\oint \vec{H}\cdot d\vec{l} = I_{enc}$. In differential form $\nabla\times\vec{H} = \vec{J}$ (for steady currents).

Write the energy stored per unit volume in electric and magnetic fields.

Electric: $w_{e} = \frac{1}{2}\varepsilon E^{2}$. Magnetic: $w_{m} = \frac{1}{2}\mu H^{2}$ (or $\frac{B^{2}}{2\mu}$).

State the EMF equation of a transformer.

$E = 4.44\,f\,N\,\Phi_{m}$, where $f$ is frequency, $N$ the number of turns, and $\Phi_{m}$ the maximum core flux.

Write the transformer voltage, current and impedance transformation ratios.

$\frac{V_{1}}{V_{2}} = \frac{N_{1}}{N_{2}} = a$, $\frac{I_{1}}{I_{2}} = \frac{N_{2}}{N_{1}} = \frac{1}{a}$, and referred impedance $Z' = a^{2}Z$.

Name the two main loss types in a transformer and how each varies with load.

Core (iron) losses: hysteresis + eddy current, essentially constant (independent of load). Copper losses: $I^{2}R$, vary with the square of load current.

What is the condition for maximum efficiency in a transformer?

Maximum efficiency occurs when copper losses equal iron losses: $I^{2}R = P_{iron}$. The corresponding load fraction is $\sqrt{\frac{P_{iron}}{P_{cu,fl}}}$.

Define voltage regulation of a transformer.

$\%\text{Reg} = \frac{V_{no\text{-}load} - V_{full\text{-}load}}{V_{full\text{-}load}}\times 100$. Lower regulation indicates a stiffer (better) transformer.

Write the EMF equation of a DC machine.

$E_{b} = \frac{P\Phi Z N}{60 A}$, where $P$=poles, $\Phi$=flux/pole, $Z$=conductors, $N$=speed (rpm), $A$=parallel paths (lap $A=P$, wave $A=2$).

Write the torque equation of a DC motor.

$T = \frac{P\Phi Z I_{a}}{2\pi A}$, i.e. $T \propto \Phi I_{a}$. Electromagnetic torque $T = \frac{E_{b}I_{a}}{2\pi N/60}$.

Give the back-EMF/voltage relation in a DC motor and the speed equation.

$V = E_{b} + I_{a}R_{a}$, so $N \propto \frac{E_{b}}{\Phi} = \frac{V - I_{a}R_{a}}{\Phi}$.

Compare speed-torque behaviour of DC series and shunt motors.

Series: high starting torque ($T \propto I_{a}^{2}$), speed varies widely (runs away on no load). Shunt: nearly constant speed, moderate starting torque ($T \propto I_{a}$).

Define slip $s$ of an induction motor.

$s = \frac{N_{s} - N_{r}}{N_{s}}$, where synchronous speed $N_{s} = \frac{120 f}{P}$ and $N_{r}$ is rotor speed.

Write the relationship between rotor copper loss, air-gap power and slip in an induction motor.

Rotor copper loss $= s \times P_{ag}$ (air-gap power); mechanical power developed $= (1-s)P_{ag}$. Thus $P_{ag} : P_{cu,rotor} : P_{mech} = 1 : s : (1-s)$.

What is the rotor frequency of an induction motor in terms of slip?

$f_{r} = s f$, where $f$ is the stator supply frequency. At standstill ($s=1$) rotor frequency equals supply frequency.

State the condition for maximum torque in an induction motor in terms of rotor parameters.

Maximum torque occurs when rotor resistance equals rotor reactance per phase: $\frac{R_{2}}{s} = X_{2}$, i.e. slip at max torque $s_{m} = \frac{R_{2}}{X_{2}}$. Max torque is independent of $R_{2}$.

Write the synchronous speed formula and EMF equation of a synchronous machine.

$N_{s} = \frac{120 f}{P}$. EMF: $E = 4.44\,f\,\Phi\,T\,k_{w}$, where $k_{w}$ is the winding factor and $T$ turns per phase.

Write the power-angle equation of a cylindrical-rotor synchronous machine.

$P = \frac{E V}{X_{s}}\sin\delta$, where $\delta$ is the load (torque) angle. Maximum (pull-out) power occurs at $\delta = 90^{\circ}$.

How does field excitation affect a synchronous motor's power factor (V-curves)?

Under-excitation draws lagging current; over-excitation draws leading current (acts as a synchronous condenser); normal excitation gives unity PF. V-curves plot armature current vs field current.

Why is a single-phase induction motor not self-starting, and how is it made self-starting?

Its pulsating field gives zero net starting torque (two equal counter-rotating fields). Starting is provided by an auxiliary winding for phase splitting: split-phase, capacitor-start, or shaded-pole methods.

What is a permanent magnet (PM) machine and one key advantage over wound-field machines?

A machine where field excitation is provided by permanent magnets instead of a field winding. Advantages: no field copper loss/excitation supply, higher efficiency and power density. Examples: BLDC and PMSM motors.

Describe the basic operating principle of a stepper motor and its step angle formula.

A stepper rotates in discrete steps, one per input pulse, with position proportional to pulse count. Step angle $= \frac{360^{\circ}}{m \times N_{r}}$ (phases $\times$ rotor teeth), e.g. variable-reluctance, PM, and hybrid types.

State the relationship between mechanical power, torque and speed used in power/machine analysis.

$P = T\omega = T\times\frac{2\pi N}{60}$, where $T$ is in N·m, $N$ in rpm, and $\omega$ in rad/s.