The extensive usage of open communication networks in power system control causes inevitable time delays. This paper studies impacts of such delays on the stability of one-area and two-area load frequency control (LFC) systems and proposes an analytical method to determine delay margins, the upper bound on the delay for stability. The proposed method first eliminates transcendental terms in characteristic equation of LFC systems without making any approximation and transforms the transcendental characteristic equation into a regular polynomial. The key result of the elimination process is that real roots of the new polynomial correspond to imaginary roots of the transcendental characteristic equation. With the help of new polynomial, it is also possible to determine the delay-dependency of system stability and root tendency with respect to the time delay. An analytical formula is then developed to compute delay margins in terms of system parameters. For a large set of controller gains, delay margins of LFC systems are computed to investigate the qualitative effect of controller gains on the delay margin. Finally, simulations studies are carried out to verify the effectiveness of the proposed method.