The ideal transformer
- - VP/VS = EP/ES = a is the voltage ratio and NP/NS = a is the winding turns ratio, the value of these ratios being respectively higher and lower than unity for step-down and step-up transformers,[a][b].
- - VP designates source impressed voltage,
- - VS designates output voltage, and,
- - EP & ES designate respective emf induced voltages.[c]
The real transformer
Real transformer deviations from ideal
- Core losses collectively called magnetizing current losses consisting of:
- Hysteresis losses due to nonlinear application of the voltage applied in the transformer core
- Eddy current losses due to joule heating in core proportional to the square of the transformer's applied voltage.
- Whereas the ideal windings have no impedance, the windings in a real transformer have finite non-zero impedances in the form of:
- Joule losses due to resistance in the primary and secondary windings
- Leakage flux that escapes from the core and passes through one winding only resulting in primary and secondary reactive impedance.
- Primary winding: RP, XP
- Secondary winding: RS, XS.
- Core or iron losses: RC
- Magnetizing reactance: XM.
Basic transformer parameters and construction
Effect of frequency
- Winding joule losses
- Current flowing through winding conductors causes joule heating. As frequency increases, skin effect and proximity effect causes winding resistance and, hence, losses to increase.
- Core losses
- Hysteresis losses
- Each time the magnetic field is reversed, a small amount of energy is lost due to hysteresis within the core. According to Steinmetz's formula, the heat energy due to hysteresis is given by
- , and,
- hysteresis loss is thus given by
- where, f is the frequency, η is the hysteresis coefficient and βmax is the maximum flux density, the empirical exponent of which varies from about 1.4 to 1 .8 but is often given as 1.6 for iron.
- Eddy current losses
- Ferromagnetic materials are also good conductors and a core made from such a material also constitutes a single short-circuited turn throughout its entire length. Eddy currents therefore circulate within the core in a plane normal to the flux, and are responsible forresistive heating of the core material. The eddy current loss is a complex function of the square of supply frequency and inverse square of the material thickness. Eddy current losses can be reduced by making the core of a stack of plates electrically insulated from each other, rather than a solid block; all transformers operating at low frequencies use laminated or similar cores.
- Magnetostriction related transformer hum
- Magnetic flux in a ferromagnetic material, such as the core, causes it to physically expand and contract slightly with each cycle of the magnetic field, an effect known as magnetostriction, the frictional energy of which produces an audible noise known as mains hum ortransformer hum. This transformer hum is especially objectionable in transformers supplied at power frequencies[i] and in high-frequency flyback transformers associated with PAL system CRTs.
- Stray losses
- Leakage inductance is by itself largely lossless, since energy supplied to its magnetic fields is returned to the supply with the next half-cycle. However, any leakage flux that intercepts nearby conductive materials such as the transformer's support structure will give rise to eddy currents and be converted to heat. There are also radiative losses due to the oscillating magnetic field but these are usually small.
- Mechanical vibration and audible noise transmission
- In addition to magnetostriction, the alternating magnetic field causes fluctuating forces between the primary and secondary windings. This energy incites vibration transmission in interconnected metalwork, thus amplifying audible transformer hum.