We develop a theoretical framework for local stability analysis of active microwave circuits around an equilibrium. We first characterize linearized (partial) transfer functions of ideal circuits around such an equilibrium, and show they can be unstable even though there are no unstable poles. Next, we consider linearized transfer functions of realistic circuits, comprising components which are passive at sufficiently high frequency. We establish that such realistic transfer functions are stable if and only if they have no poles in the closed right half-plane, and that there are at most finitely many such poles. This suggests that anti-analytic projection-based methods and meromorphic approximation techniques in Hardy spaces should be helpful to check for stability.