The problem requires finding a combination of poles for a synchronous motor and a synchronous generator to convert a grid frequency of 50 Hz to an output frequency of 16.67 Hz. The motor is connected to the grid, and the generator is driven by the motor.

Synchronous Speed Calculation

The synchronous speed ($N_s$) of a synchronous machine is determined by the frequency ($f$) of the power supply and the number of poles ($P$) according to the formula:

$ N_s = \frac{120 \times f}{P} \quad \text{(in RPM)} $

Motor and Generator Speed Relationship

Since the synchronous generator is driven by the synchronous motor, their speeds must be equal:

$ N_{s\_motor} = N_{s\_generator} $

Substituting the speed formula for both the motor and the generator:

$ \frac{120 \times f_{grid}}{P_{motor}} = \frac{120 \times f_{out}}{P_{generator}} $

Where:

• $f_{grid}$ = Grid frequency = 50 Hz
• $f_{out}$ = Required output frequency = 16.67 Hz
• $P_{motor}$ = Number of poles in the motor
• $P_{generator}$ = Number of poles in the generator

Pole Ratio Determination

Simplifying the speed equality equation by cancelling out 120:

$ \frac{f_{grid}}{P_{motor}} = \frac{f_{out}}{P_{generator}} $

Rearranging to find the ratio of poles:

$ \frac{P_{generator}}{P_{motor}} = \frac{f_{out}}{f_{grid}} $

Plugging in the given frequencies:

$ \frac{P_{generator}}{P_{motor}} = \frac{16.67 \text{ Hz}}{50 \text{ Hz}} \approx \frac{1}{3} $

This implies that the number of poles in the motor must be approximately three times the number of poles in the generator:

$ P_{motor} \approx 3 \times P_{generator} $

Evaluating the Options

We need to find the option where the number of motor poles is roughly three times the number of generator poles.

• Option 1 (2, 6): $P_{motor}=2$, $P_{generator}=6$. Ratio = $6/2 = 3$. This implies $P_{generator} = 3 \times P_{motor}$, which is incorrect.
• Option 2 (6, 2): $P_{motor}=6$, $P_{generator}=2$. Ratio = $2/6 = 1/3$. This satisfies $P_{motor} = 3 \times P_{generator}$.
• Option 3 (4, 6): $P_{motor}=4$, $P_{generator}=6$. Ratio = $6/4 = 1.5$. Incorrect.
• Option 4 (6, 4): $P_{motor}=6$, $P_{generator}=4$. Ratio = $4/6 = 2/3$. Incorrect.

Conclusion

The combination where the motor has 6 poles and the generator has 2 poles correctly matches the required ratio derived from the frequencies.