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ENERGY CONVERSION ONE (Course 25741) CHAPTER SEVEN INDUCTION MOTORS … (Induced Torque…)

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…INDUCTION MOTOR INDUCED- TORQUE EQUATION Z TH =R TH +jX TH = jX M (R 1 +jX 1 )/[R 1 +j(X 1 +X M )] Since X M >>X 1 and X M +X 1 >>R 1, Thevenin resistance & reactance can be approximated as: R TH ≈ R 1 ( X M / [X 1 +X M ] ) ^2 X TH ≈ X 1 resulting equivalent circuit:

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…INDUCTION MOTOR INDUCED- TORQUE EQUATION The current I 2 obtained as follows: I 2 = V TH / (Z TH + Z 2 )=V TH /[R TH +R 2 /s + j X TH +j X 2 ] Magnitude of current: I 2 = V TH /√(R TH +R 2 /s)^2+(X TH +X 2 )^2 Air gap power P AG = 3 I 2 ^2 R 2 /s= 3 V TH ^2 R 2 /s / [(R TH +R 2 /s)^2+(X TH +X 2 )^2] and rotor induced torque is: T ind =P AG / ω sync T ind =3V TH ^2 R 2 /s / {ω sync [(R TH +R 2 /s)^2+(X TH +X 2 )^2]} a plot of torque as function of speed (& slip) shown in next figure:

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…INDUCTION MOTOR INDUCED- TORQUE EQUATION A typical induction motor torque-speed characteristic curve

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…INDUCTION MOTOR INDUCED- TORQUE EQUATION A plot of speed above & below normal range shown below:

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INDUCTION MOTOR TORQUE-SPEED CURVE Torque-speed characteristics curve provides several important information 1- induced torque of motor is zero at syn. Speed 2- torque-speed curve is nearly linear between no load and full load. In this range rotor resistance is much larger than its reactance so rotor current, rotor magnetic field & induced torque increase linearly with increasing slip 3- there is a maximum possible torque that cannot be exceeded (pullout torque) is 2 to 3 times rated full-load torque of motor (calculated in next section) 4-starting torque on motor is slightly larger than its full-load torque, so this motor will start carrying any load that it can supply at full power 5- Note: that torque on motor for a given slip varies as square of applied voltage. This is useful in one form of induction motor speed control that will be described

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INDUCTION MOTOR TORQUE-SPEED CURVE 6- if rotor of induction motor driven faster than sync. Speed, direction of T ind reverses & machine become Gen. converting P mech to P elec (discussed later) 7- if motor turning backward relative to direction of magnetic fields, induced torque will stop machine very rapidly & will try to rotate it in other direction since reversing direction of magnetic field rotation is simply a matter of switching any two stator phases, this fact can be used as a way to very rapidly stop an induction motor act of switching two phases in order to stop motor very rapidly is called plugging

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INDUCTION MOTOR TORQUE-SPEED CURVE Power converted to mechanical in an induction motor: P conv =T ind ω m Note: peak power supplied by induction motor occurs at a different speed than maximum torque; and of course no power is converted to mechanical form when rotor is at zero speed

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INDUCTION MOTOR TORQUE-SPEED CURVE Induced Torque & Power Converted versus motor Speed in r/min (4 pole induction motor)

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INDUCTION MOTOR TORQUE-SPEED CURVE Maximum (Pullout) Torque in induction motor T ind =P AG /ω sync maximum possible torque occurs when air gap power is maximum Since air-gap power = power consumed in R 2 /s maximum induced torque will occur when power consumed by this resistor is maximum If angle of load impedance is fixed, maximum power transfer theorem states : maximum power transfer occur when magnitude of that impedance = source impedance magnitude Equivalent source impedance: Z source =R TH +jX TH +jX 2 So maximum power transfer occurs when: R 2 /s=√R TH ^2 + (X TH +X 2 )^2

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