A three-phase 440 V, 6-pole, 50 Hz squirrel cage induction motor is running at a slip of 5%. The speed of stator magnetic field with respect to rotor magnetic field and speed of rotor with respect to stator magnetic field are

1️⃣ zero, –50 rpm

2️⃣ zero, 950 rpm
3️⃣ 1,000 rpm, –50 rpm
4️⃣ 1,000 rpm, 950 rpm

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✅ Answer (Detailed Solution Below)

✔️ Option 1: zero, –50 rpm

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📖 Detailed Solution

🔷 Step 1: Calculate Synchronous Speed

Synchronous speed formula:

$${\mathrm{<span\; style="font-size:\; medium;">N</span>}}_{\mathrm{<span\; style="font-size:\; medium;">s</span>}}<span\; style="font-size:\; medium;">=</span>\frac{\mathrm{<span\; style="font-size:\; medium;">120</span>}\mathrm{<span\; style="font-size:\; medium;">f</span>}}{\mathrm{<span\; style="font-size:\; medium;">P</span>}}$$

Given:

✔ Frequency (f) = 50 Hz
✔ Number of poles (P) = 6

$${\mathrm{<span\; style="font-size:\; medium;">N</span>}}_{\mathrm{<span\; style="font-size:\; medium;">s</span>}}<span\; style="font-size:\; medium;">=</span>\frac{\mathrm{<span\; style="font-size:\; medium;">120</span>}<span\; style="font-size:\; medium;">\times </span>\mathrm{<span\; style="font-size:\; medium;">50</span>}}{\mathrm{<span\; style="font-size:\; medium;">6</span>}}$$$${\mathrm{<span\; style="font-size:\; medium;">N</span>}}_{\mathrm{<span\; style="font-size:\; medium;">s</span>}}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;">1000</span>}\text{<span style="font-size: medium;"> rpm</span>}$$

✔ Synchronous speed = 1000 rpm

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🔷 Step 2: Calculate Rotor Speed

Slip formula:

$${\mathrm{<span\; style="font-size:\; medium;">N</span>}}_{\mathrm{<span\; style="font-size:\; medium;">r</span>}}<span\; style="font-size:\; medium;">=</span>{\mathrm{<span\; style="font-size:\; medium;">N</span>}}_{\mathrm{<span\; style="font-size:\; medium;">s</span>}}<span\; style="font-size:\; medium;">(</span>\mathrm{<span\; style="font-size:\; medium;">1</span>}<span\; style="font-size:\; medium;">-</span>\mathrm{<span\; style="font-size:\; medium;">s</span>}<span\; style="font-size:\; medium;">)</span>$$

Given slip:

$$\mathrm{<span\; style="font-size:\; medium;">s</span>}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;">5</span>}\mathrm{<span\; style="font-size:\; medium;">\%</span>}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;">0.05</span>}$$$${\mathrm{<span\; style="font-size:\; medium;">N</span>}}_{\mathrm{<span\; style="font-size:\; medium;">r</span>}}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;">1000</span>}<span\; style="font-size:\; medium;">(</span>\mathrm{<span\; style="font-size:\; medium;">1</span>}<span\; style="font-size:\; medium;">-</span>\mathrm{<span\; style="font-size:\; medium;">0.05</span>}<span\; style="font-size:\; medium;">)</span>$$$${\mathrm{<span\; style="font-size:\; medium;">N</span>}}_{\mathrm{<span\; style="font-size:\; medium;">r</span>}}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;">1000</span>}<span\; style="font-size:\; medium;">\times </span>\mathrm{<span\; style="font-size:\; medium;">0.95</span>}$$$${\mathrm{<span\; style="font-size:\; medium;">N</span>}}_{\mathrm{<span\; style="font-size:\; medium;">r</span>}}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;">950</span>}\text{<span style="font-size: medium;"> rpm</span>}$$

✔ Rotor speed = 950 rpm

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🔷 Step 3: Speed of Stator Magnetic Field w.r.t Rotor Magnetic Field

✔ Stator magnetic field rotates at 1000 rpm
✔ Rotor magnetic field also rotates at 1000 rpm (relative to stator)

Therefore,

$$\mathrm{<span\; style="font-size:\; medium;">1000</span>}<span\; style="font-size:\; medium;">-</span>\mathrm{<span\; style="font-size:\; medium;">1000</span>}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;">0</span>}$$

✔ Relative speed = 0 rpm

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🔷 Step 4: Speed of Rotor w.r.t Stator Magnetic Field

Rotor speed = 950 rpm
Stator field speed = 1000 rpm

$$\mathrm{<span\; style="font-size:\; medium;">950</span>}<span\; style="font-size:\; medium;">-</span>\mathrm{<span\; style="font-size:\; medium;">1000</span>}<span\; style="font-size:\; medium;">=</span><span\; style="font-size:\; medium;">-</span>\mathrm{<span\; style="font-size:\; medium;">50</span>}\text{<span style="font-size: medium;"> rpm</span>}$$

✔ Negative sign means rotor lags behind stator field.

✔ Speed = –50 rpm

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🎯 Final Answers

✔ Stator field w.r.t rotor field = 0 rpm
✔ Rotor w.r.t stator field = –50 rpm

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❌ Why Other Options Are Incorrect

❌ 2️⃣ zero, 950 rpm

Rotor speed is 950 rpm w.r.t stator, not w.r.t stator field.

❌ 3️⃣ 1,000 rpm, –50 rpm

Stator field does not move at 1000 rpm relative to rotor field.

❌ 4️⃣ 1,000 rpm, 950 rpm

Incorrect relative relationships.

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⭐ Important Exam Points

✔ Rotor mechanical speed = (1 − s)Ns
✔ Rotor magnetic field speed = Ns
✔ Relative speed between stator & rotor fields = 0
✔ Rotor always lags behind stator field by slip speed

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📌 Key One-Line Exam Statement ⭐

In a 3-phase induction motor, stator and rotor magnetic fields rotate at synchronous speed; hence their relative speed is zero, while rotor lags behind stator field by slip speed.

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📝 Final Conclusion

For a 6-pole, 50 Hz motor at 5% slip:

$$\boxed{{\displaystyle \text{<span style="font-size: medium;">Relative speeds = 0 rpm, –50 rpm</span>}}}$$

👉 Correct Answer: Option 1 (zero, –50 rpm)