Question

Two 3-\(\phi\) induction motors with \( p_1 \) and \( p_2 \) poles are cascaded together. Then the possible number of speeds are

• A. \( \text{two} \)
• B. \( \text{one} \)
• C. \( \text{four} \)
• D. \( \text{three} \)

Hint:

The four possible speeds in a cascade connection correspond to the four possible pole configurations: $$p_1, \quad p_2, \quad (p_1 + p_2), \quad \text{and} \quad |p_1 - p_2|$$ This results in exactly four operational speeds.

Answer 1

The Correct Option is C

Concept: The cascading method is a speed control technique used for wound-rotor (slip-ring) induction motors. Two induction motors are mechanically coupled to the same shaft. The stator winding of the main motor is connected to the primary 3-phase supply lines. The slip frequency rotor output of this main motor is then fed into the stator winding of the auxiliary motor. Depending on how the phase sequences are connected between the two machines, their respective rotating magnetic fields can either assist or oppose each other, creating different synchronous speeds for the combined set.

Step 1: Analyze the standard cumulative cascade connection.
When the phase sequences are connected such that the torque produced by both motors acts in the same direction, it is called a cumulative cascade connection. The effective number of poles of the system is the sum of their individual poles ($p_1 + p_2$). The resulting synchronous speed is: $$N_{s1} = \frac{120 \cdot f}{p_1 + p_2}$$

Step 2: Analyze the differential cascade connection.
When the phase connections are reversed such that the rotating fields oppose each other, it is called a differential cascade connection. The effective number of poles is the difference between their individual poles ($|p_1 - p_2|$). The resulting synchronous speed is: $$N_{s2} = \frac{120 \cdot f}{|p_1 - p_2|}$$

Step 3: Analyze independent motor operation.
In addition to the combined configurations, the system can be operated by running either motor independently while leaving the other un-energized:
• Operating the main motor alone yields a synchronous speed of: $$N_{s3} = \frac{120 \cdot f}{p_1}$$
• Operating the auxiliary motor alone (if configured to connect directly to the mains) yields a synchronous speed of: $$N_{s4} = \frac{120 \cdot f}{p_2}$$

Step 4: Count the total number of discrete speeds.
Summing up all the unique speed configurations: 1. Cumulative speed ($p_1 + p_2$) 2. Differential speed ($|p_1 - p_2|$) 3. Main motor speed ($p_1$) 4. Auxiliary motor speed ($p_2$) This provides a total of four distinct discrete synchronous speeds, matching option (3).