GATE Electrical Engineering Electrical Machines Syllabus

List the four standard vector group categories by phase shift.

Group 0: $0^{\circ}$ shift (e.g., Yy0, Dd0). Group 6: $180^{\circ}$ (e.g., Yy6, Dd6). Group 1: $-30^{\circ}$, LV lags by $30^{\circ}$ (e.g., Dy1, Yd1). Group 11: $+30^{\circ}$, LV leads by $30^{\circ}$ (e.g., Dy11, Yd11).

State the conditions required for satisfactory parallel operation of single-phase transformers.

Same voltage ratio (same secondary voltages), same per-unit (percentage) impedance, same polarity, and the same X/R ratio of equivalent impedances. For three-phase units additionally: same phase sequence and same vector group (zero relative phase displacement).

How is load shared between two transformers operating in parallel?

Load divides in inverse proportion to their per-unit impedances. For transformers A and B: $$S_A = S\,\frac{Z_B}{Z_A+Z_B}, \quad S_B = S\,\frac{Z_A}{Z_A+Z_B}$$ The one with smaller impedance takes the larger share.

Why must transformers in parallel have the same vector group?

Different vector groups produce different phase displacements between secondaries, creating a large circulating current driven by the resultant voltage difference even at no load, which can damage the windings. Only groups with the same phase shift can be paralleled.

In a separately excited DC machine, how is the field winding supplied?

The field winding is energized from an independent external DC source, completely separate from the armature supply. This allows the field current (and hence flux) to be controlled independently of the armature current.

Write the back-EMF (generated EMF) equation of a DC machine.

$$E_b = \frac{P\,\phi\,Z\,N}{60\,A}$$ where $P$ = poles, $\phi$ = flux per pole, $Z$ = total armature conductors, $N$ = speed in rpm, $A$ = parallel paths. Compactly $E_b = k\phi N$ (or $E_b = k\phi\omega$).

Write the torque equation of a DC machine.

$$T = \frac{P\,\phi\,Z\,I_a}{2\pi A}$$ which simplifies to $T = k\phi I_a$. Electromagnetic (developed) torque is proportional to flux times armature current.

Give the voltage relationship for a DC motor and a DC generator armature circuit.

Motor: $V = E_b + I_a R_a$. Generator: $V = E_g - I_a R_a$. In a motor the back-EMF opposes the supply; in a generator the generated EMF exceeds the terminal voltage by the armature drop.

How is a DC shunt motor connected, and what is its field characteristic?

The field winding is connected in parallel with the armature across the supply, so field current $I_{sh}=V/R_{sh}$ is nearly constant. Flux is essentially constant, giving an almost constant-speed characteristic.

How is a DC series motor connected, and what is the key consequence?

The field winding is in series with the armature, so the same current flows through both ($I_{se}=I_a$). Flux varies with load current, giving high starting torque ($T\propto I_a^{2}$ before saturation) but the motor must never be run unloaded (it would overspeed).

Compare the torque-armature current characteristics of DC shunt and series motors.

Shunt: $\phi$ constant so $T\propto I_a$ (a straight line through the origin). Series: $\phi\propto I_a$ (unsaturated) so $T\propto I_a^{2}$ (a parabola), giving much greater torque at high currents and high starting torque.

Compare the speed-armature current (speed regulation) of DC shunt and series motors.

Shunt: nearly constant speed, slight drop as load increases (good regulation). Series: speed varies inversely with load current ($N\propto 1/I_a$ approximately), runs dangerously fast at no load (poor regulation).

State the speed equation of a DC motor and identify the controllable quantities.

$$N = \frac{V - I_a R_a}{k\phi}$$ Speed can be varied by changing the applied voltage $V$, the armature-circuit resistance (the $I_a R_a$ drop), or the field flux $\phi$.

Explain field (flux) control of DC motor speed and its speed range.

A rheostat in the shunt field reduces $I_f$ and hence $\phi$, increasing speed since $N\propto 1/\phi$. It gives speeds above base/rated speed, is efficient (low field power), but is limited by commutation and reduced torque ($T\propto\phi$) at high speeds.

Explain armature-resistance control of DC motor speed.

A series resistance in the armature circuit increases the $I_a R_a$ drop, lowering $E_b$ and speed below base speed. It is simple but wasteful (large $I_a^{2}R$ loss) and gives poor speed regulation, suited to intermittent loads like cranes.

What is Ward-Leonard speed control and its main advantage?

A motor-generator set supplies a continuously variable DC voltage to the armature of the main motor, allowing smooth, wide-range speed control in both directions below base speed. Advantage: very fine, smooth control with regenerative braking; disadvantage: high cost and low overall efficiency.

Describe the principle of operation of a three-phase induction motor.

Three-phase currents in the stator produce a rotating magnetic field at synchronous speed. This field cuts the rotor conductors, inducing EMF and currents in them; the interaction of rotor currents with the rotating field produces torque (by Lenz's law the rotor chases the field). It is essentially a rotating transformer.

Define synchronous speed and slip of an induction motor.

Synchronous speed $N_s = \dfrac{120 f}{P}$ rpm. Slip $$s = \frac{N_s - N_r}{N_s}$$ the fractional difference between synchronous speed and actual rotor speed $N_r$.

Why can an induction motor never run at synchronous speed?

If the rotor turned at synchronous speed there would be no relative motion between the rotating field and rotor conductors, so no EMF, no rotor current, and no torque. Hence it always runs at slightly less than $N_s$ (it is asynchronous).

What are the two main types of three-phase induction motor rotor?

Squirrel-cage rotor: short-circuited bars in slots, rugged, cheap, low starting torque, no external rotor connections. Slip-ring (wound) rotor: three-phase winding brought out via slip rings, allowing external resistance for higher starting torque and speed control.

How do rotor frequency and rotor-induced EMF vary with slip in an induction motor?

Rotor frequency $f_r = s f$. Rotor EMF at slip $s$ is $E_{2s} = s E_2$, where $E_2$ is the standstill (s=1) rotor EMF. Rotor reactance is $X_{2s} = s X_2$.

Write the torque expression of an induction motor in terms of slip.

$$T \propto \frac{s E_2^{2} R_2}{R_2^{2} + (s X_2)^{2}}$$ At low slip $T\propto s/R_2$ (rising), at high slip $T\propto R_2/(sX_2^{2})$ (falling).

What is the condition for maximum torque in an induction motor, and the slip at which it occurs?

Maximum torque occurs when rotor resistance equals rotor standstill reactance: $R_2 = s_m X_2$, i.e. at slip $$s_m = \frac{R_2}{X_2}$$ The maximum (breakdown) torque value is independent of $R_2$.

How does adding rotor resistance affect the torque-slip characteristic of a wound-rotor motor?

It does not change the maximum torque value but shifts the slip at which maximum torque occurs ($s_m=R_2/X_2$) toward higher slip. This increases starting torque (can make starting torque = maximum torque if $R_2=X_2$) at the cost of efficiency.

What does the No-Load test on an induction motor determine?

Run at rated voltage with no mechanical load; slip is nearly zero. It gives the core (iron) losses, friction and windage losses, and the magnetizing branch parameters ($X_0$, $R_0$), since the rotor copper loss is negligible at very low slip.

What does the Blocked-Rotor (locked-rotor) test on an induction motor determine?

Run with rotor held stationary ($s=1$) at reduced voltage to pass rated current. It gives the equivalent series leakage impedance ($R_{01}$, $X_{01}$) and the short-circuit/full-load copper losses, analogous to a transformer short-circuit test.

Describe the per-phase equivalent circuit of an induction motor.

Stator impedance $R_1+jX_1$ in series with the input; a shunt magnetizing branch $R_0\parallel jX_0$; then the rotor referred branch $R_2'+jX_2'$ in series with the slip-dependent load resistance $R_2'\left(\dfrac{1-s}{s}\right)$, which represents the mechanical power developed.

In the induction motor equivalent circuit, what does the resistance $R_2'\frac{1-s}{s}$ represent?

It is the fictitious dynamic resistance representing the gross mechanical power developed (converted to shaft output). Power dissipated in it equals the mechanical power, while $R_2'$ itself accounts for rotor copper loss.

State the power-flow ratio for an induction motor: air-gap power, rotor copper loss, and mechanical power.

They are in the ratio $$P_g : P_{cu,r} : P_m = 1 : s : (1-s)$$ Thus rotor copper loss $= s P_g$ and gross mechanical power $= (1-s)P_g$. Efficiency of the rotor cannot exceed $(1-s)$.

Why must induction motors and squirrel-cage motors use starters instead of direct switching, and name common methods?

At standstill ($s=1$) the back-EMF is zero, so direct-on-line starting current is very high (5-7 times full load). Starters reduce starting voltage/current: star-delta starter, auto-transformer starter, stator-resistance/reactor starter, soft starter, and rotor-resistance starter (for slip-ring motors).

How does a star-delta starter reduce the starting current of an induction motor, and by what factor?

The motor starts in star (each phase gets $V_L/\sqrt{3}$) and runs in delta. Both starting line current and starting torque are reduced to one-third ($1/3$) of their direct-on-line delta values.