Regulating Power with Frequency

For the current to lag the voltage by 45 degrees, the resistance ( $R$ ) and inductive reactance ( $X$ ) must have the same magnitude at frequency $f_0$ . Find the power dissipated initially.

$I = \frac{V}{\sqrt{R^2 + X^2}} \\ P = I^2 R = \frac{V^2 R}{R^2 + X^2} \\ P_R = \frac{V^2 R_0}{R_0^2 + R_0^2}$

When the frequency is increased, the inductive reactance increases by the same ratio. The resistance retains its value (if we neglect the skin effect). Compare the new power to the old one.

$\frac{R_0}{R_0^2 + R_0^2} = 2 \frac{R_0}{R_0^2 + X_N^2} = 2 \frac{R_0}{R_0^2 + \alpha^2 R_0^2} \\ R_0^2 + \alpha^2 R_0^2 = 2 R_0^2 + 2 R_0^2 \\ 1 + \alpha^2 = 4 \\ \alpha = \sqrt{3}$

The new frequency is therefore $\sqrt{3}$ times the initial frequency.