Define Initial Per Unit Impedance

The formula for per unit impedance is $Z_{pu} = \frac{Z_{ohms} \times S_b}{(V_b)^2}$. Therefore, $X = \frac{Z_{ohms} \times S_{b1}}{(V_{b1})^2}$.

$X = \frac{Z_{ohms} \times S_{b1}}{(V_{b1})^2}$

Define New Per Unit Impedance

Given $S_{b2} = 2S_{b1}$ and $V_{b2} = 2V_{b1}$, substitute these into the per unit formula for the new state.

$Z_{pu, new} = \frac{Z_{ohms} \times (2S_{b1})}{(2V_{b1})^2}$

Calculate and Simplify

Expanding the denominator gives $(2V_{b1})^2 = 4(V_{b1})^2$. Substituting this back leads to $Z_{pu, new} = \frac{2 \times Z_{ohms} \times S_{b1}}{4 \times (V_{b1})^2}$.

$Z_{pu, new} = \frac{2}{4} \times \left( \frac{Z_{ohms} \times S_{b1}}{(V_{b1})^2} \right) = 0.5X$