A new feedback neural network model is proposed. The network has the sinusoidal basis functions as its weights. Neuronal activation function is a linear function. The network connection form is feedback structure. An energy function is defined for the feedback neural network. And then, the network stability issue in operation is analyzed. In the Liapunov sense, the proposed feedback network stability is proved. During the operation of the network, the network states are changed ceaselessly but network weights vary according to time-dependent sinusoidal law. As the network state changes continuously, its energy will be reduced. Finally, when network comes to a stable state, its energy arrivs at a minimum value. The network is particularly suited for the adaptive approximation and the detection for periodic signals because of its sinusoidal basis function weights. It is, in practice, a new and effective way for periodic signal detection and processing. The very good detection results are obtained in the detection of power system voltage sag characteristics. Simulation examples show that the dynamic response speed of the network is very high.