This paper presents a theoretical formulation on the stability criteria for non-linear oscillator circuits. We extend Kurokawa's discriminant to a general free-running oscillator employing an n-port active device. It is shown that the product of device admittance and passive circuit impedance matrices dominates the oscillation behavior. The frequency- and amplitude-partial derivatives of the matrix determinant play a key role in the stability analysis. The obtained results are especially effective for designing broadband micro- and millimeter-wave signal generators and frequency synthesizers.