The previous section illustrated the
fact that reducing a winding conductor’s length enables a
corresponding reduction in the conductor’s cross-sectional area to
maintain the same total I 2R losses.

Therefore, to maintain constant losses,
the required volume of copper is proportional to the square of the
conductor length L.

Vcu directly proportional to L^2

The required number of turns in a
winding N is inversely proportional to the volts per turn generated
by the core. The volts per turn are proportional to the total
magnetic flux, and the flux is proportional to the cross-sectional
area of the core AFe for a given allowable peak flux density,
expressed as volts per turn.

N = inverselt proportional to Afe

From simple geometry, we know that the
conductor’s length is equal to the number of turns times the
circumference of the coil. If the cross section of the core is nearly
circular and the winding is placed directly over the core, the
circumference of the coil is roughly proportional to the square root
of the core’s cross-sectional area.

Assuming that the core’s volume is
roughly proportional to the core’s cross sectional area, The
relationships given indicate that the volume of copper required to
limit I 2R losses is inversely proportional to the volume of the core
for a given KVA rating, winding configuration, and applied voltage.

In other words, adding 25% more core
steel should permit a 25% reduction in the quantity of copper used in
a transformer. This results in a 1:1 trade-off in copper volume vs.
core volume.

However, that the total core losses are
proportional to the core volume for a given flux density. For
example, if we decide to reduce the volume of copper by 25% by
increasing the volume of core steel by 25%, the core losses will
increase by 25% even though the conductor losses remain constant.

In order to maintain the same core
losses, the flux density must be reduced by increasing the
cross-sectional area of the core, meaning that additional iron must
be added. Therefore, the 1:1 trade off in copper volume vs. core
volume is only a very rough approximation.

There are also other practical physical
limitations in selecting the dimensions of the core and windings;
however, this exercise does illustrate the kinds of trade-offs that a
transformer design engineer can use to optimize economy.

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