Electronic devices may have batteries that power internal circuitry. A power converter may connect to an input-output port in an electronic device to deliver power to the electronic device. Battery charging circuitry in the electronic device may be used to charge the battery in an electronic device while power is delivered from the power converter. The power converter may have load detection circuitry. When an output load is present, the power converter operates in an active mode and delivers power to the electronic device. When the output load is not present, the power converter enters a low-power standby mode. The electronic device has switching circuitry that periodically either electrically couples or electrically isolates the input-output port from internal circuitry. When the input-output port is isolated, the power converter senses that no output load is present and enters the standby state to conserve power.