A regulator circuit maintaining an approximate constant current output characteristic. In one aspect, a disclosed regulator controls a switch that has a current limit threshold. A supply terminal and feedback terminal of the regulator are connected together as a control terminal such that a shunt regulator current is the control terminal current in excess of the internal supply current consumed by the regulator. The current limit threshold of the switch is changed as a function of the shunt regulator current. In another aspect, a control input of the regulator circuit receives a current that is the sum of the internal supply current consumed by the regulator circuit and a feedback current. The feedback current is a fraction of the consumption current of the regulator circuit and the current limit threshold of the switch is changed as a function of the feedback current.