In a 3-phase induction motor, the resultant flux is of a constant nature and is __________.

1️⃣ Equal to ${\varphi }_m$, where ${\varphi }_m$ is the maximum flux due to any phase
2️⃣ 1.5 times maximum value of flux due to any phase
3️⃣ ${\displaystyle \frac{\sqrt{3}}{2}}$ times maximum value of flux due to any phase
4️⃣ $3{\varphi }_m$

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✅ Answer (Detailed Solution Below)

✔️ Option 2: 1.5 times maximum value of flux due to any phase

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📖 Detailed Solution

🔹 In a 3-phase induction motor, the stator has three windings displaced by 120° electrically in space.
🔹 When a balanced 3-phase AC supply is applied:

✔️ Each phase produces an alternating magnetic flux.
✔️ These fluxes are time-displaced by 120°.
✔️ They are also space-displaced by 120°.

🔹 Due to vector addition of these three fluxes:

👉 A rotating magnetic field (RMF) is produced.
👉 The RMF has constant magnitude.
👉 It rotates at synchronous speed.

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📐 Magnitude of Resultant Flux

Let:

$$$$ϕm=Maximum flux due to one phase

Then, resultant flux:

$$$$ϕr=1.5 ϕm

🔹 Mathematical vector addition of three equal sinusoidal fluxes displaced by 120° gives:

$$$$ϕr=32ϕm

👉 Therefore, the magnitude of resultant flux is 1.5 times that of any single phase.

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⚙️ Why Resultant Flux is Constant

🔹 At every instant:

• Individual phase fluxes change.
• Their vector sum remains constant.

🔹 Direction changes continuously.
🔹 Magnitude remains constant.

👉 This produces smooth torque in the motor.

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❌ Why Other Options are Incorrect

(1) Equal to ${\varphi }_m$ ❌

🔹 Resultant is higher than single-phase flux.
🔹 It is not equal to ${\varphi }_m$.

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(3) ${\displaystyle \frac{\sqrt{3}}{2}}{\varphi }_m$ ❌

🔹 This value does not represent 3-phase flux summation.
🔹 Used in other vector relationships, not RMF magnitude.

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(4) $3{\varphi }_m$ ❌

🔹 This is scalar sum, not vector sum.
🔹 Fluxes are 120° apart, so they cannot add directly.

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📘 Comparison with Two-Phase System (For Understanding)

🔹 In 2-phase motors (90° apart):

✔️ Resultant flux magnitude = ${\varphi }_m$
✔️ Direction rotates
✔️ Magnitude remains constant

🔹 In 3-phase motors:

✔️ Resultant flux magnitude = $1.5{\varphi }_m$
✔️ Better torque
✔️ Higher efficiency

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📈 Synchronous Speed

The rotating magnetic field moves at synchronous speed:

$$$$Ns=120fP

Where:
Ns = Synchronous speed (rpm)
f = Supply frequency (Hz)
P = Number of poles

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⭐ Important Exam Points

✔️ 3-phase supply → Rotating magnetic field
✔️ RMF → Constant magnitude
✔️ RMF → Rotates at synchronous speed
✔️ Resultant flux = 1.5 φm

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📌 Key One-Line Exam Statement ⭐

In a 3-phase induction motor, the resultant rotating flux is constant and equal to 1.5 times the maximum flux of any phase.

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📝 Final Conclusion

📍 In a balanced 3-phase induction motor, three phase fluxes combine vectorially to produce a rotating magnetic field of constant magnitude equal to 1.5 times the maximum flux of any phase.

👉 Hence, Option 2 is Correct.