Electrical Machines: Transformers, DC, Induction and Synchronous — ESE Electrical

Weightage: Electrical machines are the single largest block of the Electrical papers. The topics repeat predictably: transformer efficiency and regulation, DC motor speed control, induction motor power stages and torque-slip, and synchronous machine power angle.

1. The transformer

A transformer changes voltage and current by electromagnetic induction. The EMF equation is:

$E=4.44fN{\Phi }_m$

and the turns ratio gives $V_1/V_2=N_1/N_2=I_2/I_1$. The core flux depends on $V/f$, so a transformer must not be run at lower frequency with the same voltage, since the flux would rise and the core saturate.

Tests. The open-circuit test (rated voltage on the LV side, HV open) gives the iron loss and the magnetising branch. The short-circuit test (reduced voltage, rated current) gives the copper loss at full load and the equivalent impedance. Core loss is constant with load, while copper loss varies as the square of the load.

Regulation is the fall in secondary voltage from no load to full load, as a fraction of the no-load voltage. Approximately:

$\text{Reg}\approx \frac{I(R\cos \varphi \pm X\sin \varphi )}{V}$

with plus for a lagging power factor and minus for leading. Regulation can be zero or negative with a leading load.

Maximum efficiency occurs when the variable (copper) loss equals the constant (iron) loss. The load fraction at which this occurs is $x=\sqrt{P_i/P_{cu,fl}}$.

Worked example. A 100 kVA transformer has iron loss 1 kW and full-load copper loss 2 kW. Then $x=\sqrt{1/2}=0.707$, which is 70.7 kVA. At 0.8 power factor the output is 56.6 kW and total loss is $1+1=2$ kW, so $\eta =56.6/58.6=96.6\%$.

An autotransformer shares one winding, so for a ratio $a$ close to 1 it saves copper and is more efficient, but offers no isolation. Three-phase connections (star-delta, delta-star) are named by vector group, such as Dyn11. Parallel operation needs the same voltage ratio, polarity, phase sequence, phase angle shift and a matching per-unit impedance.

2. DC machines

The induced EMF in a DC machine is $E=\frac{P\Phi ZN}{60A}$, where $P$ is the number of poles, $Z$ the conductors, $A$ the parallel paths ($A=P$ for lap and $A=2$ for wave windings). The torque is $T=\frac{P\Phi Z{I}_a}{2\pi A}$.

For a motor, $E_b=V-I_a{R}_a$ and the speed is:

$N\propto \frac{E_b}{\Phi }$

Speed control: (a) armature voltage control below base speed (constant torque), (b) field flux control above base speed (constant power) and (c) the Ward-Leonard scheme for smooth wide-range control. A starter limits the large starting current, since $E_b=0$ at start.

If the field of a running shunt motor opens, the flux falls to its residual value and the speed rises dangerously, so protection is needed. A Swinburne test finds no-load losses and so efficiency, without loading the machine, but cannot be used for series motors.

3. The induction motor

The stator field rotates at the synchronous speed $N_s=120f/P$. The rotor runs slower, and the slip is $s=(N_s-N)/N_s$. Rotor current frequency is $sf$.

Worked example. A 4-pole, 50 Hz motor has $N_s=1500$ rpm. At 1440 rpm, $s=60/1500=0.04$, and the rotor frequency is $0.04\times 50=2$ Hz.

Power flow from the air gap is the rotor input $P_g$, divided as:

$P_g:P_{cu,rotor}:P_{mech}=1:s:(1-s)$

So with $P_g=10$ kW at $s=0.04$, rotor copper loss is 0.4 kW and mechanical power developed is 9.6 kW. This also shows that an induction motor at high slip is inefficient.

Torque-slip curve. The torque is proportional to the square of the supply voltage. The maximum torque occurs at $s_m=R_2/X_2$, and its value is independent of rotor resistance, so adding rotor resistance moves the peak toward higher slip and raises starting torque. At starting ($s=1$) torque is low in a squirrel-cage motor.

Starting:

Speed control: vary the supply frequency (V/f control keeping flux constant), the number of poles, the rotor resistance (slip-ring) or the voltage. A single-phase induction motor has no self-starting torque and needs a split-phase, capacitor or shaded-pole arrangement. An induction machine run above synchronous speed acts as a generator.

4. The synchronous machine

An alternator generates EMF at frequency $f=P{N}_s/120$. The EMF per phase is $E=4.44K_{w}f\Phi T_{ph}$, with $K_w$ the winding factor.

The synchronous impedance method gives the regulation from the open-circuit and short-circuit tests. For a lagging load the regulation is positive and large, and for a leading load it can be negative (the Ferranti-like rise in voltage).

The power developed in a cylindrical-rotor machine is:

$P=\frac{EV}{X_s}\sin \delta$

where $\delta$ is the load angle. Maximum power is at $\delta =90^{\circ }$, and the stability limit is reached there. For a salient-pole machine an extra reluctance term appears in $\sin 2\delta$.

Worked example. With $E=1.2$ pu, $V=1.0$ pu, $X_s=1.0$ pu and $\delta =30^{\circ }$, $P=1.2\times 0.5=0.6$ pu.

A synchronous motor is not self-starting and needs damper windings or a pony motor. Its speed is exactly synchronous. V-curves plot armature current against field current. An over-excited motor takes a leading current and so can correct the system power factor, acting as a synchronous condenser. Hunting is a periodic oscillation of the rotor about its mean position, reduced by damper windings.

Parallel operation of alternators needs the same voltage, frequency, phase sequence and phase. After synchronising, changing the prime mover input changes the real-power share, and changing the excitation changes the reactive share.

Common traps

• Thinking iron loss changes with load. It is constant; copper loss varies with the square.
• Assuming rotor resistance changes the maximum torque. It moves the slip at which it occurs.
• Forgetting that a series motor must not run unloaded.
• Treating torque as proportional to $V$ in an induction motor. It goes as $V^2$.
• Ignoring leading power factor in regulation. The sign changes.

Memory aids

• "1, s, 1 minus s": air-gap power split in an induction motor.
• "Equal losses for best efficiency": transformer.
• "EV over X sine delta": power angle.

Summary

The transformer is analysed through its EMF equation, equivalent circuit, tests and efficiency condition. DC machines are governed by $E\propto \Phi N$ and torque proportional to flux and armature current, with speed controlled by voltage and flux.

The induction motor splits air-gap power by slip, has maximum torque independent of rotor resistance, and starts by reduced voltage. The synchronous machine delivers power by $EV\sin \delta /X_s$, can correct power factor and needs a starting aid as a motor.

Exam protocol

• Check which losses vary with load before using efficiency formulas.
• Draw the equivalent circuit before applying a test result.
• Use the $1:s:(1-s)$ split for all induction power questions.
• State whether a synchronous machine is over- or under-excited.