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

2. …INDUCTION MOTOR INDUCED-TORQUE EQUATION
ZTH=RTH+jXTH = jXM(R1+jX1)/[R1+j(X1+XM)]
Since XM>>X1 and XM+X1>>R1 , Thevenin
resistance & reactance can be approximated as:
RTH ≈ R1 ( XM/ [X1+XM] ) ^2
XTH ≈ X1
resulting equivalent circuit:

3. …INDUCTION MOTOR INDUCED-TORQUE EQUATION
The current I2 obtained as follows:
I2= VTH / (ZTH + Z2)=VTH/[RTH +R2/s + j XTH +j X2]
Magnitude of current:
I2= VTH /√(RTH+R2/s)^2+(XTH+X2)^2
Air gap power
PAG = 3 I2^2 R2/s=
3 VTH^2 R2/s / [(RTH+R2/s)^2+(XTH+X2)^2]
and rotor induced torque is:
Tind=PAG/ ωsync
Tind=3VTH^2 R2/s / {ωsync [(RTH+R2/s)^2+(XTH+X2)^2]}
a plot of torque as function of speed (& slip) shown in
next figure:

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

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

6. 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

7. INDUCTION MOTOR TORQUE-SPEED CURVE
6- if rotor of induction motor driven faster than sync. Speed, direction of Tind reverses & machine become Gen. converting Pmech to Pelec
(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

8. INDUCTION MOTOR TORQUE-SPEED CURVE
Power converted to mechanical in an induction motor:
Pconv=Tindωm
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

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

10. INDUCTION MOTOR TORQUE-SPEED CURVE
Maximum (Pullout) Torque in induction motor
Tind=PAG/ωsync maximum possible torque occurs when air gap power is maximum
Since air-gap power = power consumed in R2/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:
Zsource=RTH+jXTH+jX2
So maximum power transfer occurs when:
R2/s=√RTH^2 + (XTH+X2)^2