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An analytical approach to unconditional and conditional instabilities of a three-port network is presented herein. This approach begins with inequality expressions of the input reflection coefficients Γin and Γout of a three-port network defined at ports 1 and 2 having port 3 terminated with load Γ3. The requirements of the conditional instabilities are derived to show that there is a largest conditional instability region in the Γ3 -plane as ports 1 and 2 are terminated with the matched loads. Then the explicit expressions are derived for the boundary of the conditional instability region and the maximum instability curves in the Γ3-plane to lead one to properly select the Γ3 value or design the terminating network at port 3. For the selected Γ3 value, the conditional instability regions in the Γ1- and Γ2-planes are also given for the design consideration of the loads at ports 1 and 2. In order to fully characterize the instability of a three-port network, the similar procedure can be conducted by treating port 1 or port 2 as the terminating network. The resulting instability expressions not only provide useful design equations but also are implemented in Agilent Advanced Design System software to enhance its computer-aided capability of the instability analysis of a three-port network, especially the microwave oscillator.