Transformers and AC machines
A four-pole induction motor never reaches 1500 rpm: at rated load it turns at 1448. That 3.5% difference, the slip, is what produces the torque.
01The real transformer
The ideal transformer, seen in coupled circuits, relates voltages by the turns ratio and conserves power. The real one has losses and flux that does not link the two windings, and it is modeled with an equivalent circuit: the resistances of the windings and their leakage reactances in series, and a parallel branch that represents the core, with a resistance for the iron losses and a reactance for the magnetizing current.
| Test | How it is done | What it gives |
|---|---|---|
| No-load | Rated voltage on one winding, the other open | The iron losses and the magnetizing branch |
| Short-circuit | Secondary short-circuited, reduced voltage until rated current | The copper losses and the series impedance; the short-circuit voltage in % |
The iron losses are almost constant and the copper losses grow with the square of the current: the efficiency is maximum when the two are equal, and in a power transformer it exceeds 98%. Installation and basic calculation are covered in transformers, in the technician program.
02The rotating field
Three windings displaced by 120° in the stator, fed with three-phase currents shifted by 120°, produce a magnetic field of constant amplitude that rotates. Its speed, the synchronous speed, depends only on the frequency and the number of pole pairs \( p \):
03The asynchronous machine
In the induction motor, the rotating field induces currents in the rotor —short-circuited bars, the “squirrel cage”— and those currents produce the torque. If the rotor turned at the synchronous speed there would be no relative motion, no induction and no torque: the rotor always turns a little slower. That relative difference is the slip:
The torque curve as a function of speed has a starting torque, rises to a maximum torque and falls to zero at the synchronous speed. The useful region is the one to the right of the maximum, almost straight: if the load increases, the motor slows a little and produces more torque. To the left of the maximum the operation is unstable. In direct starting the current reaches 6 or 8 times the rated current, which is why wye-delta starting, soft starters and drives are used.
A 5.5 kW, four-pole motor, wye-connected at 380 V (220 V per phase), with R₁ = 0.8 Ω, R₂' = 0.75 Ω, X₁ = 1.6 Ω, X₂' = 1.8 Ω and Xm = 40 Ω at 50 Hz. A drive changes the frequency while keeping V/f constant. The dot marks where the motor curve crosses the load curve.
04The synchronous machine
In the synchronous machine the rotor has its own field, from a winding fed with DC or from permanent magnets, and turns locked to the rotating field, exactly at the synchronous speed. Almost all the electrical energy in the world is generated by synchronous alternators in power plants. As a motor, by changing the excitation current the power factor is controlled: overexcited, it delivers reactive power, like a capacitor. Permanent-magnet synchronous motors, controlled by electronics, are those of electric vehicles and of many modern appliances.
05Variable-frequency drive control
Since the synchronous speed is proportional to the frequency, a drive that generates three-phase power of variable frequency controls the speed of the induction motor. To keep the flux in the core, the voltage has to drop in the same proportion: this is constant V/f control. At low frequencies the drop across the stator resistance weighs more and the maximum torque falls; drives compensate with a voltage boost. The drive circuits are covered in variable-speed drives, in the technician program.
06In the laboratory
Perform the no-load and short-circuit tests of a single-phase transformer and calculate its equivalent circuit and its full-load efficiency.
With a tachometer, measure the speed of an induction motor at no load and with increasing load on a brake, and calculate the slip in each case.
Run the same motor with a drive between 10 and 50 Hz and measure the current and the speed with a constant load.
07Common mistakes
- Expecting an induction motor to turn at the synchronous speed. Without slip there is no torque.
- Starting a large motor directly on a weak grid. The starting current drags the voltage down.
- Running a motor at low speed for a long time with its own fan. It is poorly cooled.
- Confusing shaft power with electrical power. A motor’s nameplate gives the mechanical output power.
08Self-assessment
What synchronous speed does a 6-pole motor have at 50 Hz?
\( 60 \cdot 50 / 3 = 1000 \) rpm.
A 4-pole motor turns at 1440 rpm at 50 Hz. What is the slip?
\( (1500 - 1440)/1500 = 4 \)%.
What voltage must a drive give at 25 Hz for a 380 V, 50 Hz motor, with constant V/f?
190 V.
When is a transformer’s efficiency at its maximum?
When the copper losses equal the iron losses.
09Further reading
- Stephen J. Chapman. Electric Machinery Fundamentals (Spanish edition: “Máquinas eléctricas”). 5th ed., McGraw-Hill, 2012. The course text: transformers and induction, synchronous and DC machines.
- Jesús Fraile Mora. Máquinas eléctricas (in Spanish). 8th ed., Garceta, 2016. A complete treatise in Spanish, with many solved problems.
- A. E. Fitzgerald, Charles Kingsley and Stephen D. Umans. Electric Machinery. 7th ed., McGraw-Hill, 2014. The classic, with the field approach and the models for control.