Varying the field:
Some DC generators use an electromagnet to make the magnetic field, rather than a permanent magnet, so changing the current in the "field" coil changes the field.
The power dissipated in this field coil should be quite a bit less than the output current of the generator.
(A permanent magnet participates in a PM motor without losing energy to the output.)
Another application is the "Ward Leonard" control system, used to regulate the DC output of a motor-generator.
https://en.wikipedia.org/wiki/Ward_Leonard_controlThat article includes more complicated equations including inertial effects in the mechanics.
The calculation of the constants
kV and
kI requires engineering calculations of the coil geometry.
The basis is that the back emf is the time derivative of the flux linked by the rotor windings as it rotates through the field, and that the force on the rotor windings (that gives you the torque) comes from the product of the current through the coil wires and the field at the rotor coil.
For details, one should consult textbooks on electric machinery.
In general, you have two electrical variables (
V and
I), two mechanical variables (
T and
S), and the magnetic field
B.
The two constants
kV and
kI are determined by the construction geometry and dimensions.
You can't control all five variables at once: however, you can specify, e.g.,
V (applied voltage) and
T (load torque), for a given
B, and see what speed
S and current
I values result.