__Transformer Losses __

The ratio of the active power *P*_{2} at the output of a transformer to the active power *P*_{1} at its input (see Fig. 4) is called its efficiency

ƞ = *P*_{2}/ *P*_{1}

or

ƞ% = (*P*_{2}/*P*_{1}) X 100 (8.23)

When a transformer is operating at its rated primary voltage, *V*_{1} = *V*_{1},_{rtd }, its rated primary current, *I*_{1} = I_{1},_{rtd }, and a load power factor ( cos φ_{2}) of over 0.8. the efficiency is very high, being in excess of 99 % for power transformers. For this reason it is never found by directly measuring *P*_{1} and P_{2} . The point is that one would have then to measure *P*_{1} and *P*_{2} to an extremely high level of accuracy not attainable at the existing state of the art. Fortunately, the efficiency of a transformer can be found indirectly from the transformer losses which can be measured directly. Since the transformer losses are

Δ*P* = *P*_{1} – *P*_{2}

it follows then that the efficiency is

The transformer losses are the sum of the core losses, *P*_{c} , and the winding losses, *P*_{w} . Given *V*_{1} = *V*_{1},_{rtd} and *I*_{1} = *I*_{1},_{rtd }, the core losses are equal to the active power of the transformer in the open circuit test (see Sec. 8.6), and the winding losses (traditionally called the copper losses) are equal to the active power of the transformer in the short-circuit test (see. Sec. 8.7).

Consider how the efficiency of a transformer varies in operation at the rated primary voltage *V*_{1} = *V*_{1},_{rtd} into a load whose impedance *Z*_{2} is varying and whose power factor cos φ_{2} remains constant. A change in *Z*_{2} brings about a change-In the load power, in the winding currents and. as a consequence, in the winding losses and the transformer efficiency. The winding losses are also referred to as the transient losses of a transformer. and the core losses are called its *steady-state losses *.

The winding losses (see Fig.17 a) are given by

*r*_{sc} (*I*'_{2})^{2} = *k*^{2}_{load}r_{sc} (*I*'_{2,rtd})2 = *k*^{2}_{load}*P*_{sc,rtd }

where k_{10ad} is the load current ratio, Eq. (8.22), and *P*_{sc,rtd }is the winding losses at rated currents.

As the secondary current is varied from zero to its rated value, it may be assumed that

*V*_{2} ≈ const ≈ *V*_{2},rtd = *V*_{1},_{rtd} *n*_{21}

The output active power of a transformer is given by

*P*_{2} = *V*_{2}*I*_{2} cos φ_{2} = k_{10ad} *V*_{2}*I*_{2,rtd }cos φ_{2} ≈ *k*_{load}*S*_{rtd} cos φ_{2}

and the efficiency, in accord with Eq. (8.24), is

As is seen, the efficiency of a transformer depends on the load power factor cos φ_{2} and the load current ratio *k*_{load} . If we hold the load power factor constant and equate to zero the derivative of ƞ with respect to *k*_{load }, we will find that the efficiency of a transformer is a maximum at

*k*_{load }= *√P*_{c}/*P*_{sc,rtd}_{ }

It follows then that in the case of a maximum load current ratio (*k*_{load} = 1) a maximum efficiency can be obtained when the core losses are equal to the winding (or copper) losses.

Actually, the load current ratio of a real transformer is never unity. For this reason, transformers are designed so that their efficiency (Fig. 19) would be a maximum at some average load. For example, for *P*_{c}/*P*_{sc,rtd }= 0.25-0.5 the efficiency will be a maximum at

*K*_{load} = *√P*_{c}/*P*_{sc.rtd }= 0.5-0.7