Relay feedback systems are strongly nonlinear due to their switching properties. Some nonlinear properties of relay feedback systems have been verified to be preferable to modern control engineering, whereas others might drive the system to be more complex or even unpredictable. An alternative criterion is proposed to investigate the pitchfork bifurcations of the limit cycle of relay feedback systems in this paper. The proposed critical criterion is explicitly formulated by the coefficients of the characteristic polynomial equation instead of the eigenvalues of the Jacobian matrix. It is more convenient and efficient for detecting the existence of this type of bifurcation than the classical critical criterion. Numerical simulations show the pitchfork bifurcation behaviors in relay feedback systems and demonstrate that the proposed criterion is a general and exact analytic method for determining pitchfork bifurcations in maps.