= Electric power transmission
<Electric power transmission> transfers bulk electrical <power> between generating areas, major network junctions and receiving systems. <Transmission lines> carry <electric current>; <electrical transformers> change the operating <electric potential difference> in <alternating current> networks. Raising that potential can reduce the <electric current> required for a specified transferred <power>, and therefore reduce resistive heating. It also changes insulation and equipment requirements.
For an original fixed-output <direct current> comparison, deliver $P_L=10\,\mathrm{MW}$ through a line with total conductor <electrical resistance> $R=2\,\Omega$. At receiving-end <electric potential difference> $V_L$, use
$$
I=\frac{P_L}{V_L},\qquad P_{\rm loss}=\frac{P_L^2R}{V_L^2},\qquad V_s=V_L+IR.
$$
At $V_L=10\,\mathrm{kV}$, the <electric current> is $1000\,\mathrm A$, conductor heating is $2\,\mathrm{MW}$, and the source must supply $12\,\mathrm{MW}$ at $12\,\mathrm{kV}$. At $100\,\mathrm{kV}$, the corresponding values are $100\,\mathrm A$, $20\,\mathrm{kW}$ and $10.02\,\mathrm{MW}$ at $100.2\,\mathrm{kV}$. This compares the same receiving load and line <electrical resistance>; it does not assume equal source <power> in the two cases.
The resulting loss reduction by a factor of one hundred follows from squaring the tenfold <electric current> reduction. It does not imply lossless transfer: <electrical transformers>, converters and conductors have their own losses. The DOE treatment explicitly separates winding copper loss from useful transformer output.<DOE Fundamentals Handbook Electrical Science>[Volume 4, Module 13, Transformer Theory, printed pp.11–13, copper loss and efficiency]{ref}
Long <alternating current> lines also have distributed <inductance> and <capacitance>, so a resistive balance alone cannot determine their phase relationships or operating limits. An <electric power transmission> calculation must specify whether its <electric potential differences> are sending-end or receiving-end values, and whether stated <power> is <active power> or <apparent power>. Those conventions keep conductor-current calculations consistent with <power factor>.
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