Engineering:Circle diagram

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For the chart with marked sectors of a circle, see Pie chart.
For other circle diagrams, see Smith chart and Nyquist plot.

The circle diagram (also known as a Heyland, Ossanna, or Sumec diagram or ... circle) is the graphical representation of the performance of an electrical machine[1][2][3] in terms of the locus of the machine's input voltage and current.[4] It was first conceived by Alexander Heyland (de) in 1894 and Bernhard Arthur Behrend in 1895, and subsequently improved by Johann Ossanna (de) in 1899 and Josef Sumec (d) in 1910.

In particular, Sumec's contribution was to incorporate the rotor resistance.

The Heyland diagram is an approximate representation of a circle diagram applied to induction motors, which assumes that stator input voltage, rotor resistance and rotor reactance are constant and stator resistance and core loss are zero.[3][5][6]

The theory of the Heyland diagram begins with Steinmetz's analysis of an induction motor as a real transformer attached to a varying resistance:

Steinmetz equivalent circuit for an induction motor

As the motor speed varies, so does the resistance, as does the current through the motor.[4] The circle diagram obtains its name because the real and imaginary parts of the current phasor from a circle in the complex plane.[7][4]

Further information can be obtained through additional geometric constructions on the same plot.[7][8] The appropriate scale identifies current with power, multiplying the current by the phase voltage and the number of phases.[7]

A complete diagram, with all possible information marked, is:[7][8]

Constant air-gap induction motor circle diagram

where

  • Rs, Xs: Stator resistance and leakage reactance
  • Rr′, Xr′, ...s: Rotor resistance and leakage reactance referred to the stator and rotor slip
  • Rc, Xm, : Core and mechanical losses, magnetization reactance
  • Vs, Impressed stator voltage
  • I0 = OO′, IBL = OA, I1 =OV: No load current, blocked rotor current, operating current
  • Φ0, ΦBL : No load angle, blocked rotor angle
  • Pmax, sPmax, PFmax, Tmax, sTmax: Maximum output power & related slip, maximum power factor, maximum torque & related slip
  • η1, s1, PF1, Φ1,: Efficiency, slip, power factor, PF angle at operating current
  • AB: Represents rotor power input, which divided by synchronous speed equals starting torque.

In practice, the circle diagram is drawn from the data obtained from no load and either short-circuit or, in case of machines, blocked rotor tests by fitting a half-circle in points O' and A.

Beyond the error inherent in the constant air-gap assumption, the circle diagram introduces errors due to rotor reactance and rotor resistance variations caused by magnetic saturation and rotor frequency over the range from no-load to operating speed.

See also

  • Steinmetz equivalent circuit

References

  1. ↑ The Induction Motor and Other Alternating Current Motors, their Theory and Principles of Design. McGraw-Hill. 1921. p. ix. https://archive.org/details/inductionmotorot00behrrich. 
  2. ↑ "A Graphical Method for the Prediction of Power Transformers and Polyphase Motors". ETZ 15: 561–564. 1894. http://translate.googleusercontent.com/translate_c?depth=1&hl=en&rurl=translate.google.ca&sl=de&tl=en&u=http://www.regione.taa.it/GIUNTA/enel/Heilbronnertr_td.pdf&usg=ALkJrhjSZbhRQLilTGDiDl0ISV9eO9k-tQ. Retrieved 2013-01-04. 
  3. ↑ 3.0 3.1 "The General Circle Diagram of Electrical Machinery". Transactions of the American Institute of Electrical Engineers (American Institute of Electrical Engineers) 49. January 1930. 
  4. ↑ 4.0 4.1 4.2 Electrical Machines (2008 ed.). Tata McGraw-Hill Education. 2008-08-27. p. 359. ISBN 978-0-07-066921-5. 
  5. ↑ A Graphical Treatment of the Induction Motor. G. H. Rowe, R. E. Hellmund (translators). McGraw Publishing Company. 1906. https://archive.org/stream/graphicaltreatme00heylrich#page/n3/mode/2up. Retrieved 2013-01-10. 
  6. ↑ "The Asynchronous Motor Model". Phase to Phase BV. 2006. pp. 5–6. http://www.phasetophase.nl/pdf/Asynchronous_machine_model.pdf. 
  7. ↑ 7.0 7.1 7.2 7.3 Knowlton, A. E., ed (1949). "Induction machines [in § 7, Alternating-current generators and motors]". Standard Handbook for Electrical Engineers (8 ed.). McGraw-Hill. pp. 710–711. 
  8. ↑ 8.0 8.1 "Construction of Circle Diagram". College of Engineering Trivandrum. http://www.ee.cet.ac.in/Downloads/Circle%20Diagram.pdf. 



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