Good Afternoon,
The GFCI breakers I have contain a current transformer (coil) around both the phase and neutral. This in effect forms the differential transformer. This configuration is much more cost effective and smaller then a separate differential transformer.
The operation is simple; since the power is AC a conductor will cause current to flow in the secondary of a transformer (coil) because of the field surrounding it. The strength of this field is in direct relationship to the current the conductors are carrying. The Phase and Return (neutral) are 180 degrees out of phase canceling common mode current fields.
The transfer into the secondary of the coil (current transformer) is zero as long as both conductors are carrying the same current. If there is any unbalance in either wire current will be introduced into the current transformer which with some electronics causes the device to open. You can add turns to the primary, this will multiply the field strength and increase the sensitivity.
Note these are not load sensitive; they are just sensitive to the current difference between the wires. For example if you touch one of the wires (sometimes even the white) current will flow from the wire to ground if you are grounded. That current is not going back through the GFCI which results in it tripping, that is what they were designed for.
If you short the white wire and earth ground with a tester it will probably trip even though these two wires are connected together in the main panel, this depends on the IR drop in your system. If you connect a voltage tester between the safety ground and the hot (smaller blade) it should trip. Doing the same thing between the neutral and phase will have no effect. If this works connect a large load and watch it work again, if not your voltage tester is pulling minimal current. I use a Square D tester with a solenoid indicator.
Be Safe
Gil Shultz