18. Alternating Currents

The root-mean-square value of an alternating current is equivalent to the steady direct current that would dissipate heat at the same rate as the alternating current in a given resistor.
For a sinusoidal source,
(a) the root mean square value of the current is given by $I_{r m s} = \frac{I_{o}}{\sqrt{2}}$ .
(b) the mean or average power < P > absorbed by a resistive load is half the maximum power.
$< P >= \frac{1}{2} P_{o} = \frac{1}{2} I_{o} V_{o} = \frac{1}{2} {I_{o}}^{2} R = \frac{V_{o}^{2}}{2 R}$ .
An a.c. transformer is a device for increasing or decreasing an a.c. voltage. It consists of a primary coil of N_p turns and voltage V_p and secondary coil of N_s turns and voltage V_s wrapped around an iron core.
For an ideal transformer (assuming no energy is lost), the following equation is obeyed
$\frac{N_{s}}{N_{p}} = \frac{V_{s}}{V_{p}} = \frac{I_{p}}{I_{s}}$ .
Power loss in the transmission lines is minimized if the power is transmitted at high voltages (i.e. low currents) since $P_{l o s s} = I^{2} R$ where I is the current through the cables and R is the resistance of the cables.
The equation $P = \frac{V^{2}}{R}$ is often mistakenly used to suggest that power lost is high when voltage of transmission is high. In fact, V refers to the potential difference across the cables, which often have but a fraction of the overall resistance through which the current passes.

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