The document outlines fundamental torque equations and dynamics of electrical drives, focusing on the relationship between developed motor torque and load torque. It describes the multi-quadrant operation of drives, including modes of operation like motoring and braking, and explains the sign conventions for speed and torque. Additionally, examples involving hoist operations are provided to illustrate the practical applications of multi-quadrant operations.

1. DYNAMICS OF ELECTRICAL
DRIVES
Prof. S. Y.
Gadgune

2. FUNDAMENTAL TORQUE EQUATION
J- Polar moment of inertia of motor load
system, kg-𝑚2
𝑤 𝑚 - Angular velocity of motor, rad/sec
T- Developed Motor electromagnetic torque,
N-m
𝑇𝑙 - Load torque, N-m

3. FUNDAMENTAL TORQUE EQUATION
Fundamental torque equation of above motor load system is given as-
𝑇 − 𝑇𝑙 =
𝑑
𝑑𝑡
𝐽𝑤 𝑚 = 𝐽
𝑑𝑤 𝑚
𝑑𝑡
+ 𝑤 𝑚
𝑑𝐽
𝑑𝑡
….(1)
For most of the drives inertia is constant.
i.e.
𝑑𝐽
𝑑𝑡
= 0
∴Equation (1) becomes,
𝑇 − 𝑇𝑙 = 𝐽
𝑑𝑤 𝑚
𝑑𝑡
∴ 𝑇 = 𝑇𝑙 + 𝐽
𝑑𝑤 𝑚
𝑑𝑡
……(2)
i.e. Motor developed torque has to supply load torque and dynamic
torque present during transient operations (starting, acceleration,
deceleration).

4. MULTI-QUADRANT OPERATION OF
DRIVES
Sign Convention
Speed : Motor Speed is considered to be positive, if it is rotating in
forward direction. If we consider a particular direction of rotation as
forward (or +ve), then rotation in opposite direction will be Reverse
direction (-ve). In applications involving up and down motion, the
direction of rotation of motor causing upward motion will be the
forward.
Torque : In case of forward motion, if the motor torque causes it to
accelerate, then it will be positive torque. The load torque operates in
opposite direction that of the motor torque.

5. MULTI-QUADRANT OPERATION OF
DRIVES
Modes of Operation
Motoring- Electrical energy is converted into mechanical energy. i.e.
power flows from supply to motor.
Braking- Motor works as generator converting mechanical energy
into electrical energy. i.e. power flows from motor to supply.

6. MULTI-QUADRANT OPERATION OF
DRIVES
Power developed by motor is given by product torque and speed.
i.e. 𝑃 = 𝑇 × 𝑤 𝑚
Quadrant I (Forward Motoring)- Speed is +ve and Power is +ve
Quadrant II (Forward Braking)- Speed is +ve and Power is –ve
Quadrant III (Reverse Motoring)- Speed is –ve and Power is +ve
Quadrant IV (Reverse Braking)- Speed is –ve and Power is -ve
https://circuitglobe.com/four-quadrant-operation-
of-dc-motor.html

7. MULTI-QUADRANT OPERATION OF
DRIVES
Example of Hoist
𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑒𝑚𝑝𝑡𝑦 𝑐𝑎𝑔𝑒 < 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑜𝑢𝑛𝑡𝑒𝑟 𝑤𝑒𝑖𝑔ℎ𝑡
𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑙𝑜𝑎𝑑𝑒𝑑 𝑐𝑎𝑔𝑒 > 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑜𝑢𝑛𝑡𝑒𝑟 𝑤𝑒𝑖𝑔ℎ𝑡
Forward direction will be the one causing upward
motion of hoist.
Quadrant I (Forward Motoring)- To move the loaded
cage upward, motor has to rotate in anticlockwise
direction. So it is the forward direction. The motor
torque should act in the same direction to lift the cage
against gravity.
Quadrant II (Forward Braking)- The empty cage is
moved upward due to counter weight itself. For safe
operation, motor torque should act in opposite
direction of rotation.
https://sciamble.com/Resources/pe-drives-lab/basic-
drives/dc-four-quadrant

8. MULTI-QUADRANT OPERATION OF
DRIVES
Example of Hoist
Quadrant III (Reverse Motoring)- To move the empty
cage in downward direction, motor has to rotate in
clockwise direction. The motor torque also has to act
in clockwise direction against the gravity due to
counter weight.
Quadrant IV (Reverse Braking)- To move fully loaded
cage in downward direction, motor will rotate in
clockwise direction due to gravity of its weight. For
safe operation, motor torque will act in anticlockwise
direction.

9. REFERENCES
 G. K. Dubey, “Fundamentals of Electrical Drives”, Second Edition,
Narosa Publishing House