We consider circulant graphs G(r, N) where the vertices are the integers modulo N and the neighbours of 0 are . The energy of G(r, N) is a trigonometric sum of terms. For low values of r we compute this sum explicitly. We also study the asymptotics of the energy of G(r, N) for . There is a known integral formula for the linear growth coefficient, we find a new expression of the form of a finite trigonometric sum with r terms. As an application we show that in the family G(r, N) for there is a finite number of hyperenergetic graphs. On the other hand, for each there is at most a finite number of non-hyperenergetic graphs of the form G(r, N). Finally, we show that the graph minimizes the energy among all the regular graphs of degree 2r.