Role of Sample Size in Power Analysis
For any given effect size and alpha, increasing the sample size will increase the power (ignoring for the moment the case of power for a single proportion by the binomial method). As is true of effect size and alpha, sample size cannot be viewed in isolation but rather as one element in a complex balancing act. In some studies it might be important to detect even a small effect while maintaining high power. In this case it might be appropriate to enroll many thousands of patients (as was done in the "Physicians" study that found a relationship between aspirin use and cardiovascular events).
Typically, though, the number of available cases is limited. The researcher might need to find the largest N that can be enrolled, and work backwards from there to find an appropriate balance between alpha and beta. She may need to forgo the possibility of finding a small effect, and acknowledge that power will be adequate for a large effect only.
Note: For studies that involve two groups power is generally maximized when the subjects are divided evenly between the two groups. When the number of cases in the two groups is uneven the "effective N" for computing power falls much closer to the smaller sample size than the larger one.