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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