The　breakdown　conditions　of　short　air-insulated　gaps　under　alternating　non-uniform　electric　fields　are　investigated　for　providing　an　accurate　prediction　of　the　breakdown　voltages　of　commonly　used　electrode　systems.　The　breakdown　voltages　with　the　tip　radius　of　the　highly　stressed　electrode　varying　between　0.1　cm　and　2　cm　and　the　gap　distance　ranging　from　1　cm　to　5　cm　were　experimentally　investigated,　and　the　breakdown　electric　field　distributions　in　inter-electrode　gaps　were　analyzed　by　a　finite　element　method.　An　abnormal　phenomenon　was　observed　that　the　breakdown　voltage　increased　in　a　highly　non-uniform　electric　field　with　the　same　gap　distance.　For　different　electrode　arrangements,　the　breakdown　conditions　could　not　be　characterized　by　the　relationship　between　the　maximum　breakdown　electric　field　at　the　electrode　tip　and　the　field　non-uniform　coefficient.　Therefore,　a　combined　breakdown　criterion　was　proposed　by　introducing　a　geometrical　parameter　of　the　electrode　system.　The　breakdown　conditions　can　be,　respectively,　evaluated　by　the　streamer　inception　criterion　and　a　fitting　formula　when　the　value　of　the　geometric　parameter　is　on　opposite　sides　of　a　critical　value.　Published　by　AIP　Publishing.