How come I can vary the voltage output on an unloaded alternator by varying the field?
Just as it does on a DC generator.
The regulator keeps the Voltage at 14.5 volts loaded or unloaded.
Max.

Silly argument really. The field voltage and or current controls the output current and voltage. You can't have current without voltage! But it is also true, what our learned friend said, there is a current transfer function with alternators.
Sorry if I've missed, further down the thread, someone else unruffling feathers!

 

Actually field current controls output current, not output voltage.

In a typical automotive alternator, the field resistance (measured at DC) is about 5 to 8Ω, so the max field current when connected to 12V is ~2A.

At a given RPM, the alternator acts as a current-controlled current-source; the output current is proportional to field current, so if you connect 12Vdc to the field, the field current will be about 2A, the output current will be about 50A, so the current gain is 50/2 = 25. If the output side of the alternator is open circuited with full-on field excitation, the output voltage will shoot up to hundreds of V (hence current source, not voltage source).

The typical automotive voltage regulator senses battery voltage, and controls the field current using PWM, which determines the output current such that it just exactly matches total load current, keeping the battery at a nominal 14.2V. A simulation of this behavior is shown here.

Your claim that a car alternator is a current controlled current source is not a good model of how an alternator work.
A better model would be a current controlled voltage source with some resistance in series with the output.
The output voltage does not rise without limit when the load changes or the battery is disconnected. The voltage induced in the stator winding is strictly controlled by rate of change of magnetic flux the stator coil is exposed to. The flux coming from the poles of the rotor is proportional to the rotor current. The rate of change of flux experienced by the stator coils depends on how fast the rotor pole passes by the stator pole.
For any given current in the rotor the unloaded stator coil voltage output will be linearly proportional to the RPM.
For any given RPM, the unloaded output voltage will be linearly related to the rotor current.
In a real alternator, the winding resistance (which is significant) makes the loaded voltage less than the unloaded voltage.
If I recall correctly, the winding resistance is about 0.05 ohm. If the alternator is putting out 55 amps into a 14 volt load, the induced voltage on the stator will need to be nearly 17 volts to account for the I*R drop in the wires, and the forward voltage drop of the rectifier.