6.2 Transformers & Rotating Machines
Key Takeaways
- Ideal transformer turns ratio a = N1/N2 sets V1/V2 = a and I1/I2 = 1/a, so power is conserved and impedance reflects by a^2.
- Synchronous speed of an AC machine is Ns = 120·f / P revolutions per minute, where f is frequency in hertz and P is the number of poles.
- Induction-motor slip s = (Ns - N) / Ns; the rotor turns slightly slower than synchronous speed, and slip rises with mechanical load.
- A synchronous machine runs exactly at Ns and is used for generators and constant-speed motors; over-excitation lets it supply reactive power.
- The same DC machine acts as a motor (electrical input to mechanical output) or generator (mechanical input to electrical output) depending on energy-flow direction.
Ideal transformer ratios
A transformer couples two windings magnetically to change AC voltage levels. Define the turns ratio a = N1/N2, where N1 is the primary turns and N2 is the secondary turns. For an ideal (lossless) transformer:
- Voltage ratio: V1/V2 = N1/N2 = a
- Current ratio: I1/I2 = N2/N1 = 1/a
- Power conservation: S1 = S2, so V1·I1 = V2·I2
Because voltage scales by a and current by 1/a, an impedance on the secondary side reflects to the primary as Z_ref = a^2·Z_2. This impedance-scaling-by-a-squared result is a frequent FE answer for impedance matching and reflected-load problems.
Synchronous speed and AC machines
Every AC machine has a rotating magnetic field whose speed is fixed by line frequency and pole count. The synchronous speed is:
Ns = 120 · f / P (rpm)
where f is the supply frequency in hertz and P is the number of magnetic poles (always even). For 60 Hz: a 2-pole machine runs at 3600 rpm, a 4-pole at 1800 rpm, and a 6-pole at 1200 rpm. The factor 120 converts (2f/P) revolutions per second to rpm. A synchronous machine rotor locks to this field and turns exactly at Ns under all loads.
Induction motors and slip
The induction motor is the workhorse of industry. Its rotor is not externally excited; the stator field induces rotor currents, which requires the rotor to turn slightly slower than synchronous speed. The fractional speed difference is slip:
s = (Ns - N) / Ns
where N is the actual rotor speed in rpm. At no load, slip is near zero; as mechanical load increases, slip grows (typically a few percent at full load). Rotor electrical frequency equals s·f. A higher slip means more rotor current and torque, up to the breakdown torque. If a question states the rotor runs at synchronous speed exactly, the machine is synchronous, not induction.
Motors, generators, and DC machines
All rotating machines are reversible energy converters. A motor converts electrical input to mechanical output; a generator converts mechanical input to electrical output. The same physical machine can do either depending on the direction of power flow.
| Machine | Speed behavior | Typical use |
|---|---|---|
| Synchronous | Exactly Ns; over-excitation supplies VARs | Generators, constant-speed drives |
| Induction | Below Ns; slip increases with load | General industrial motors |
| DC machine | Speed set by armature voltage and field | Variable-speed drives, traction |
For a DC machine, generated EMF is E = k·phi·omega (proportional to flux and speed), and torque is T = k·phi·I_a (proportional to flux and armature current). Reversing power flow turns a DC motor into a DC generator using the same windings.
An ideal transformer has a 240 V primary and a 24 V secondary. If the secondary delivers 5 A to a load, what is the primary current?
A four-pole AC machine is supplied at 60 Hz. What is its synchronous speed?
A 60 Hz, 6-pole induction motor runs at 1140 rpm at full load. What is its slip?