$$\\Power = \Phi \Bigg\{ \frac{\Delta}{\sqrt{p_1q_1/n_1 + p_2q_2/n_2}} - z_{1-\alpha/2}*\frac{\sqrt{\bar{p}\bar{q}(1/n_1+1/n_2)}}{\sqrt{p_1q_1/n_1 + p_2q_2/n_2}} \Bigg\}\\q_1 = 1-p_1\\q_2 = 1-p_2\\\bar{p} = \frac{p_1+kp_2}{1+K}\\\bar{q} = 1-\bar{p}\\Power = \Phi \Bigg\{ \frac{0.15}{\sqrt{0.35*0.65/150 + 0.2*0.8/148}} - 1.96*\frac{\sqrt{0.276*0.724(1/150+1/148)}}{\sqrt{0.35*0.65/150 + 0.2*0.8/148}} \Bigg\}\\Power = \Phi(0.953) = 0.83 = 83\% \; power$$
p_{1}, p_{2} = proportion (incidence) of groups #1 and #2
Δ = |p_{2}-p_{1}| = absolute difference between two proportions
n_{1} = sample size for group #1
n_{2} = sample size for group #2
α = probability of type I error (usually 0.05)
z = critical Z value for a given α or β
K = ratio of sample size for group #2 to group #1
Φ() = function converting a critical Z value to power