A prime mover drives a 6-pole, 3-phase induction frequency converter. The converter is connected to 60 Hz supply. If the prime mover speed is 3000 rpm, the frequencies of possible outputs are:

A) 120 Hz, 60 Hz
B) 90 Hz, 210 Hz
C) 186 Hz, 86 Hz
D) 180 Hz, 210 Hz

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✅ Answer:

➡️ B) 90 Hz, 210 Hz

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🧠 Detailed Solution

◆ Step 1: Synchronous Speed of Stator Field

$N_s=\frac{120f}{P}$

$${\mathrm{<span\; style="font-size:\; medium;">N</span>}}_{\mathrm{<span\; style="font-size:\; medium;">s</span>}}<span\; style="font-size:\; medium;">=</span>\frac{\mathrm{<span\; style="font-size:\; medium;">120</span>}<span\; style="font-size:\; medium;">\times </span>\mathrm{<span\; style="font-size:\; medium;">60</span>}}{\mathrm{<span\; style="font-size:\; medium;">6</span>}}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;">1200</span>}\text{<span style="font-size: medium;">\hspace{0.17em}</span>}\text{<span style="font-size: medium;">rpm</span>}$$

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◆ Step 2: Mechanical Frequency of Rotor

$${\mathrm{<span\; style="font-size:\; medium;">f</span>}}_{\mathrm{<span\; style="font-size:\; medium;">m</span>}}<span\; style="font-size:\; medium;">=</span>\frac{\mathrm{<span\; style="font-size:\; medium;">P</span>}<span\; style="font-size:\; medium;">\times </span>\mathrm{<span\; style="font-size:\; medium;">N</span>}}{\mathrm{<span\; style="font-size:\; medium;">120</span>}}<span\; style="font-size:\; medium;">=</span>\frac{\mathrm{<span\; style="font-size:\; medium;">6</span>}<span\; style="font-size:\; medium;">\times </span>\mathrm{<span\; style="font-size:\; medium;">3000</span>}}{\mathrm{<span\; style="font-size:\; medium;">120</span>}}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;">150</span>}\text{<span style="font-size: medium;">\hspace{0.17em}</span>}\text{<span style="font-size: medium;">Hz</span>}$$

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◆ Step 3: Key Concept (Very Important 🚨)

➡️ Output frequency depends on relative speed between RMF and rotor

Two cases:

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◆ Case 1: Opposite Direction

→ Relative speed increases

$${\mathrm{<span\; style="font-size:\; medium;">f</span>}}_{\mathrm{<span\; style="font-size:\; medium;">2</span>}}<span\; style="font-size:\; medium;">=</span>{\mathrm{<span\; style="font-size:\; medium;">f</span>}}_{\mathrm{<span\; style="font-size:\; medium;">m</span>}}<span\; style="font-size:\; medium;">+</span>{\mathrm{<span\; style="font-size:\; medium;">f</span>}}_{\mathrm{<span\; style="font-size:\; medium;">1</span>}}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;">150</span>}<span\; style="font-size:\; medium;">+</span>\mathrm{<span\; style="font-size:\; medium;">60</span>}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;">210</span>}\text{<span style="font-size: medium;">\hspace{0.17em}</span>}\text{<span style="font-size: medium;">Hz</span>}$$

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◆ Case 2: Same Direction

→ Relative speed decreases

$${\mathrm{<span\; style="font-size:\; medium;">f</span>}}_{\mathrm{<span\; style="font-size:\; medium;">2</span>}}<span\; style="font-size:\; medium;">=</span>{\mathrm{<span\; style="font-size:\; medium;">f</span>}}_{\mathrm{<span\; style="font-size:\; medium;">m</span>}}<span\; style="font-size:\; medium;">-</span>{\mathrm{<span\; style="font-size:\; medium;">f</span>}}_{\mathrm{<span\; style="font-size:\; medium;">1</span>}}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;">150</span>}<span\; style="font-size:\; medium;">-</span>\mathrm{<span\; style="font-size:\; medium;">60</span>}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;">90</span>}\text{<span style="font-size: medium;">\hspace{0.17em}</span>}\text{<span style="font-size: medium;">Hz</span>}$$

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❌ Why Other Options Are Incorrect

◆ A) 120, 60 → Not derived from relative speed
◆ C) 186, 86 → No valid calculation
◆ D) 180, 210 → 180 Hz is incorrect assumption

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🔑 Key Points to Remember

◆ Induction frequency converter works on relative motion principle
◆ Output frequency = sum or difference of frequencies
◆ Mechanical frequency: $f_m=\frac{PN}{120}$
◆ Two outputs always possible → (f_m ± f_1)

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🎯 Final Conclusion

➡️ Possible output frequencies are:
90 Hz and 210 Hz ✔️