You are correct, on the Tesla Supercharger Website https://www.tesla.com/supercharger under "How it Works" and "Engineered for performance"; it states, "Tesla is the only EV manufacturer capable of charging vehicles at up to 120 kW, (375 Volts x 320 Amps = 120 kW) which equates to about 170 miles of range in as little as 30 minutes."
The battery voltage for both the "S" and "X" is 375 Volts and that's what the Superchargers are built for. Since the Charge Point can go up to 400 kW (1000 Volts x 400 Amps = 400 kW) its only because they have the capability of going up to 1000 Volts. This may be for future vehicles like trucks or hybrid RV's.
To confuse you even further, the 240V AC (single phase) is referring to the charger input voltage, the charger is actually in your car, which has an output of 375V DC to charge the battery pack. The device in your garage is not really a charger but rather a means of applying the AC to the car's charger.
The Supercharger has an input voltage of 480V AC (three phase). It then produces an output of 375V DC that connects to the car and goes direct to the battery pack bypassing the smaller charger in the car.
The Chademo and CCS Combo kW outputs are dependent on their max output voltage and current. The battery voltage of other EV's range between 200 & 400 volts and would also be in the ball park of 120 kW.
If there was sufficient interest in this area I would be happy to give a presentation (dry-white board with simple block diagrams) to expand on all this.
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