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 $$I_{rms}=\frac{I_o}{\sqrt{2}}$$.
(b) the mean or average power < P > absorbed by a resistive load is half the maximum power.
$$<P>=\frac{1}{2}P_o=\frac{1}{2}{I_o}{V_o}=\frac{1}{2}{I_o}^2R =\frac{V_o^2}{2R}$$.
• An a.c. transformer is a device for increasing or decreasing an a.c. voltage. It consists of a primary coil of Np turns and voltage Vp and secondary coil of Ns turns and voltage Vs 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 $$P_{loss}=I^2R$$ 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.