Four quadrant operation of a hoist

A hoist (lift/mine winder) has a cage, a counterweight and a rope over a drum driven by a motor. The load torque from gravity is an active, constant torque: its direction does not change when the motor reverses. Taking upward motion of the cage as positive speed and motor torque in the hoisting direction as positive, the motor works in all four quadrants of the speed-torque plane.

                 +N (cage up)
                  |
   II: Forward    |   I: Forward
   braking        |   motoring
   (empty cage up)|   (loaded cage up)
   T<0, N>0       |   T>0, N>0
 -T --------------+--------------- +T
   III: Reverse   |   IV: Reverse
   motoring       |   braking
   (empty cage    |   (loaded cage
    down)         |    down)
   T<0, N<0       |   T>0, N<0
                  |
                 -N (cage down)

 Load torque lines (constant):
   loaded cage : Tl = +Tl1 (vertical line, right)
   empty cage  : Tl = -Tl2 (vertical line, left;
                 counterweight heavier than empty cage)

Assumption: counterweight is heavier than the empty cage but lighter than the loaded cage, so the load torque is positive for a loaded cage and negative for an empty cage.

Quadrant I: forward motoring

• Loaded cage is raised. Motor torque and speed are both positive (upward).
• Power flows from supply to motor to load: motor supplies the energy to raise the load ($P = T\omega > 0$).

Quadrant II: forward braking

• Empty cage is moving up, but the heavier counterweight would pull it up faster. The motor torque is reversed (negative) to hold the speed.
• Speed positive, torque negative: $P < 0$, the motor works as a generator; energy can be returned to the supply (regenerative braking) or dissipated (dynamic braking).

Quadrant III: reverse motoring

• Empty cage is lowered. Since the counterweight is heavier, the motor must drive the cage downward against it.
• Torque and speed both negative: $P > 0$, motoring in reverse direction.

Quadrant IV: reverse braking

• Loaded cage is lowered. Gravity tends to accelerate it downward, so the motor produces positive (upward) torque to limit the speed.
• Speed negative, torque positive: $P < 0$, generating (regenerative braking, or dynamic/plugging).

With a constant load torque, steady operation in each quadrant occurs where the motor characteristic crosses the vertical load-torque line. Drives such as DC motors with dual converters or induction motors with regenerative VFDs provide all four quadrants and save energy by regeneration in II and IV.