Normal approximation of power for fixed effects is obtained from the following algorithm, due to Winer et al. (1991, pp 136-138):
is distributed as the standardized normal distribution, with β obtained from the cumulative probability at z. Power = 1 - β.
Conversely, knowing z = -0.842 at β = 0.2, it is possible to iterate the value of λ, and hence θ/σ, at 80% power to detect an effect for any given p and q, and hence F[α],p,q, and n.
Winer, B. J., Brown, D. R. and Michels, K. M. (1991) Statistical Principles in Experimental Design 3rd edn. New York: McGraw-Hill.
Doncaster, C. P. & Davey, A. J. H. (2007) Analysis of Variance and Covariance: How to Choose and Construct Models for the Life Sciences. Cambridge: Cambridge University Press.