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Negative cyclic feedback has been considered to be a core circuit to produce sustained oscillations in gene regulatory networks. In this paper, we investigate quantitative properties of the periodic oscillations observed in cyclic gene regulatory networks with negative feedback, and we provide analytic estimates of frequency, phase and amplitude. We employ the harmonic balance method, which is one of the frequency domain techniques to examine nonlinear oscillatory behaviors by approximating with bias and first order harmonic components. We then solve the harmonic balance equations by utilizing the structure of gene expression dynamics. The presented estimates are analytically written only in terms of essential biochemical parameters proposed in authors' previous work, and hence they can be easily applied to large-scale cyclic gene regulatory networks involving any number of genes. Our results are demonstrated with illustrative numerical examples, and some novel biological insights are presented.