Circle Diagram of The Induction Motor - Electrical Engineering Gate

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Sunday, December 7, 2014

Circle Diagram of The Induction Motor

In a particular circuit, if one of the circuit elements is variable, then depending upon its value, the circuit characteristics varies. As the value of the variable element is changed, the circuit parameters like current, power factor, power losses etc. also change. The locus of the extremity of the current phasor, obtained for various values of a variable element is called a locus diagram.
       From the equivalent circuit of an induction motor, the motor can be treated as series R-L circuit where the element resistance of the circuit is variable which varies as slip s. Thus for variable load conditions, the resistance changes and hence the current drawn by the motor also changes. The locus diagram of such a current phasor is circular in nature and hence called circle diagram of a three phase induction motor. Using this diagram, all the performance characteristics of an induction motor like power factor, efficiency, stator losses, rotor losses, maximum output, maximum torque etc. can be predicted. Thus, a circle diagram is a graphical approach of predetermining the operation characteristics of an induction motor.
       Let us prove that the locus diagram obtained for a current phasor is a circle, for a series R-L circuit with an element R as variable.

By using the data obtained from the no load test and the blocked rotor test, the circle diagram can be drawn using the following steps :

Step 1 : Take reference phasor V as vertical (Y-axis).
Step 2 : Select suitable current scale such that diameter of circle is about 20 to 30 cm.
Step3 : From no load test, Io and are Φo obtained. Draw vector Io, lagging V by angle Φo. This is the line OO' as shown in the Fig. 1.
Step 4 : Draw horizontal line through extremity of Io i.e. O', parallel to horizontal axis.
Step 5 : Draw the current ISN calculated from Isc with the same scale, lagging V by angle Φsc, from the origin O. This is phasor OA as shown in the Fig. 1.
Step 6 : Join O'A is called output line.
Step 7 : Draw a perpendicular bisector of O'A. Extend it to meet line O'B at point C. This is the centre of the circle.
Step 8 : Draw the circle, with C as a center and radius equal to O'C. This meets the horizontal line drawn from O' at B as shown in the Fig. 1.
Step 9 : Draw the perpendicular from point A on the horizontal axis, to meet O'B line at F and meet horizontal axis at D.
Step 10 : Torque line.
      The torque line separates stator and rotor copper losses.
      Note that as voltage axis is vertical, all the vertical distances are proportional to active components of currents or power inputs, if measured at appropriate scale. 
       Thus the vertical distance AD represents power input at short circuit i.e. WSN, now which consists of core loss and stator, rotor copper losses.
      Now          FD = O'G
                             = Fixed loss
Where O'G is drawn perpendicular from O' on horizontal axis. This represents power input on no load i.e. fixed loss.
Hence            AF α Sum of stator and rotor copper losses
       Then point E can be located as,
      AE/EF = Rotor copper loss / Stator copper loss
      The line O'E under this condition is called torque line.
 Power scale : As AD represents WSN i.e. power input on short circuit at normal voltage, the power scale can be obtained as,
      Power scale = WSN/l(AD)   W/cm
      where l(AD) = Distance AD in cm
Location of Point E : In a slip ring induction motor, the stator resistance per phase R1 and rotor resistance per phase R2 can be easily measured. Similarly by introducing ammeters in stator and rotor circuit, the currents I1 and I2 also can be measured.
...                         K = I1/I2 = Transformation ratio
Now   AF/EF = Rotor copper loss / Stator copper loss = (I22R2)/(I12R1) = (R2/R2)(I22/I12) = (R2/R2).(1/K2)
But      R2'= R2/K2 = Rotor resistance referred to stator
...        AE/EF = R2'/R1
       Thus point E can be obtained by dividing line AF in the ratio R2to R1.
In a squirrel cage motor, the stator resistance can be measured by conducting resistance tset.
...        Stator copper loss = 3ISN2 R1  where ISN is phase value.
       Neglecting core loss, WSN = Stator Cu loss + Rotor Cu loss
...    Rotor copper loss = WSN - 3ISN2 R1
...        AE/EF = (WSN - 3ISN2 R1)/(3ISN2 R1)
       Dividing line AF in this ratio, the point E can be obtained and hence O'E represents torque line.
1.1 Predicting Performance From Circle Diagram
       Let motor is running by taking a current OP as shown in the Fig. 1. The various performance parameters can be obtained from the circle diagram at that load condition.
       Draw perpendicular from point P to meet output line at Q, torque line at R, the base line at S and horizontal axis at T.
       We know the power scale as obtained earlier.
       Using the power scale and various distances, the values of the performance parameters can be obtained as,
       Total motor input = PT x Power scale
       Fixed loss = ST x power scale
      Stator copper loss = SR x power scale
       Rotor copper loss = QR x power scale
      Total loss = QT x power scale
      Rotor output = PQ x power scale
      Rotor input = PQ + QR = PR x power scale
      Slip s = Rotor Cu loss = QR/PR
      Power factor cos = PT/OP
      Motor efficiency = Output / Input = PQ/PT
      Rotor efficiency = Rotor output / Rotor input = PQ/PR
      Rotor output / Rotor input = 1 - s = N/Ns = PQ/PR
      The torque is the rotor input in synchronous watts.
1.2 Maximum Quantities
       The maximum values of various parameters can also be obtained by using circle diagram.
1. Maximum Output : Draw a line parallel to O'A and is also tangent to the circle at point M. The point M can also be obtained by extending the perpendicular drawn from C on O'A to meet the circle at M. Then the maximum output is given by l(MN) at the power scale. This is shown in the Fig. 1.
2. Maximum Input : It occurs at the highest point on the circle i.e. at point L. At this point, tangent to the circle is horizontal. The maximum input given l(LL') at the power scale.
3. Maximum Torque : Draw a line parallel to the torque line and is also tangent to the circle at point J. The point J can also be obtained by drawing perpendicular from C on torque line and extending it to meet circle at point J. The l(JK) represents maximum torque in synchronous watts at the power scale. This torque is also called stalling torque or pull out torque.
4. Maximum Power Factor : Draw a line tangent to the circle from the origin O, meeting circle at point H. Draw a perpendicular from H on horizontal axis till it meets it at point I. Then angle OHI gives angle corresponding to maximum power factor angle.
...      Maximum p.f. = cos {OHI}
                               = HI/OH
5. Starting Torque : The torque is proportional to the rotor input. At s = 1, rotor input is equal to rotor copper loss i.e. l(AE).
...         Tstart = l(AE) x Power scale   synchronous watts
1.3 Full load Condition 
       The full load motor output is given on the name plates in watts or h.p. Calculates the distance corresponding to the full load output using the power scale.
       Then extend AD upwards from A onwards, equal to the distance corresponding to full load output, say A'. Draw parallel to the output line O'A from A' to meet the circle at point P'. This is the point corresponding to the full load condition, as shown in the Fig. 2.

       Once point P' is known, the other performance parameters can be obtained easily as discussed above.

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