Electrical air conditioning system for a cabin of an aircraft comprising a source of fresh air, a circulation duct for dynamic air taken from outside the aircraft, a network of pipes (30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47) and control valves (20, 21, 22, 23, 24, 25, 26, 27, 28) configured to be able to regulate the flow of air circulating through said pipes according to the flight conditions of the aircraft, a heat exchanger, called the primary cooling exchanger (PHx), housed in said dynamic air circulation duct, a heat exchanger, called the main cooling exchanger (MHx), housed in said dynamic air circulation duct, the system further comprises: - an air injector opening into said dynamic air duct upstream of said cooling exchangers, - a port for discharging stale air from said cabin, - an electrically driven motorized compressor comprising an air inlet fluidically connected to said source of fresh air by said network of pipes, and an air outlet connected to said primary cooling exchanger (PHx), housed in said dynamic air circulation duct, - an air cycle turbomachine comprising at least a first compressor and a first turbine mechanically coupled to one another by a mechanical shaft, - said first compressor comprising an air inlet adapted to be able to be fluidically connected by said network of pipes, by means of said control valves, either to said primary cooling exchanger (PHx), or to said source of fresh air, and an air outlet fluidically connected to said main cooling exchanger (MHx), housed in said circulation duct for dynamic air taken from outside the aircraft, - said first turbine comprising an air inlet adapted to be able to be fluidically connected by said network of pipes, by means of said control valves, either to said discharge port for discharging stale air from said cabin, or to said main cooling exchanger (MHx), and an air outlet adapted to be able to be fluidically connected, by said network of pipes, by means of said control valves, either to said cabin in order to be able to supply it with air at a controlled pressure and temperature, or to said air injector opening into said air circulation duct, upstream of said cooling exchangers (MHx, PHx). - an air injector opening into said dynamic air duct upstream of said cooling exchangers, - a port for discharging stale air from said cabin, - an electrically driven motorized compressor comprising an air inlet fluidically connected to said source of fresh air by said network of pipes, and an air outlet connected to said primary cooling exchanger (PHx), housed in said dynamic air circulation duct, - an air cycle turbomachine comprising at least a first compressor and a first turbine mechanically coupled to one another by a mechanical shaft, - said first compressor comprising an air inlet adapted to be able to be fluidically connected by said network of pipes, by means of said control valves, either to said primary cooling exchanger (PHx), or to said source of fresh air, and an air outlet fluidically connected to said main cooling exchanger (MHx), housed in said circulation duct for dynamic air taken from outside the aircraft, - said first turbine comprising an air inlet adapted to be able to be fluidically connected by said network of pipes, by means of said control valves, either to said discharge port for discharging stale air from said cabin, or to said main cooling exchanger (MHx), and an air outlet adapted to be able to be fluidically connected, by said network of pipes, by means of said control valves, either to said cabin in order to be able to supply it with air at a controlled pressure and temperature, or to said air injector opening into said air circulation duct, upstream of said cooling exchangers (MHx, PHx). - an air cycle turbomachine comprising at least a first compressor and a first turbine mechanically coupled to one another by a mechanical shaft, - said first compressor comprising an air inlet adapted to be able to be fluidically connected by said network of pipes, by means of said control valves, either to said primary cooling exchanger (PHx), or to said source of fresh air, and an air outlet fluidically connected to said main cooling exchanger (MHx), housed in said circulation duct for dynamic air taken from outside the aircraft, - said first turbine comprising an air inlet adapted to be able to be fluidically connected by said network of pipes, by means of said control valves, either to said discharge port for discharging stale air from said cabin, or to said main cooling exchanger (MHx), and an air outlet adapted to be able to be fluidically connected, by said network of pipes, by means of said control valves, either to said cabin in order to be able to supply it with air at a controlled pressure and temperature, or to said air injector opening into said air circulation duct, upstream of said cooling exchangers (MHx, PHx).