Four quadrant operation in DC motor control

Four-quadrant operation means the dc motor can work with either direction of speed and either direction of torque. The speed–torque plane has four quadrants, and a four-quadrant drive can run the motor forward or reverse and can both drive (motor) and brake in each direction.

            +speed
   II: Fwd braking  |  I: Fwd motoring
   T<0, w>0         |  T>0, w>0
   ------------------+------------------ torque
   III: Rev motoring |  IV: Rev braking
   T<0, w<0         |  T>0, w<0
            -speed

How it gives motoring and braking

For a separately excited motor, speed sets the sign of back emf $E$ and torque sets the sign of armature current $I_a$ ($T = K I_a$).

• Motoring: $E$ and $I_a$ have the same sign, so $E I_a > 0$ and power flows from supply to load.
• Braking (regenerative): the converter makes the terminal voltage less than $E$ (or reverses current), so $I_a$ reverses while speed is the same direction. Now $E I_a < 0$; the motor acts as a generator and kinetic or potential energy is returned to the supply (or burnt in a resistor in dynamic braking).

A four-quadrant converter (dual converter, or four-quadrant H-bridge chopper) can reverse both the voltage and the current, so a hoist, for example, raises the load in quadrant I and lowers it under control in quadrant IV, and a reversing mill can brake quickly and reverse.