Circle diagram

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]

where

R_s, X_s: Stator resistance and leakage reactance
R_r′, X_r′, ..._s: Rotor resistance and leakage reactance referred to the stator and rotor slip
R_c, X_m, : Core and mechanical losses, magnetization reactance
V_s, Impressed stator voltage
I₀ = OO′, I_BL = OA, I₁ =OV: No load current, blocked rotor current, operating current
Φ₀, Φ_BL : No load angle, blocked rotor angle
P_max, s_Pmax, PF_max, T_max, s_Tmax: 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.

References

^ Behrend, Bernhard Arthur Behrend (1921). The Induction Motor and Other Alternating Current Motors, their Theory and Principles of Design. McGraw-Hill. p. ix. OL 7033483M.
^ Heyland, Alexander Heinrich [in German] (1894). "A Graphical Method for the Prediction of Power Transformers and Polyphase Motors". ETZ. 15: 561–564. Retrieved 2013-01-04.
^ ^a ^b Terman, Frederick Emmons; Freedman, Cecil Louis; Lenzen, Theodore Louis; Rogers, Kenneth Alfred (January 1930). "The General Circle Diagram of Electrical Machinery". Transactions of the American Institute of Electrical Engineers. 49. American Institute of Electrical Engineers.
^ ^a ^b ^c Bhattacharya, S. K. (2008-08-27). Electrical Machines (2008 ed.). Tata McGraw-Hill Education. p. 359. ISBN 978-0-07-066921-5.
^ Heyland, Alexander Heinrich [in German] (1906). A Graphical Treatment of the Induction Motor. G. H. Rowe, R. E. Hellmund (translators). McGraw Publishing Company. Retrieved 2013-01-10.
^ "The Asynchronous Motor Model" (PDF). Phase to Phase BV. 2006. pp. 5–6. Archived from the original (PDF) on 2014-08-10. Retrieved 2013-01-10.
^ ^a ^b ^c ^d Alger, Philip L.; et al. (1949). "Induction machines [in § 7, Alternating-current generators and motors]". In Knowlton, A. E. (ed.). Standard Handbook for Electrical Engineers (8 ed.). McGraw-Hill. pp. 710–711.
^ ^a ^b Fernandez, Francis M. "Construction of Circle Diagram" (PDF). College of Engineering Trivandrum. Archived from the original (PDF) on 2014-08-08. Retrieved 2013-01-10.

[Behrend_1921-1] Behrend, Bernhard Arthur Behrend (1921). The Induction Motor and Other Alternating Current Motors, their Theory and Principles of Design. McGraw-Hill. p. ix. OL 7033483M.

[Heyland_1894-2] Heyland, Alexander Heinrich [in German] (1894). "A Graphical Method for the Prediction of Power Transformers and Polyphase Motors". ETZ. 15: 561–564. Retrieved 2013-01-04.

[Terman-Freedman-Lenzen-Rogers_1930-3] Terman, Frederick Emmons; Freedman, Cecil Louis; Lenzen, Theodore Louis; Rogers, Kenneth Alfred (January 1930). "The General Circle Diagram of Electrical Machinery". Transactions of the American Institute of Electrical Engineers. 49. American Institute of Electrical Engineers.

[Bhattacharya_2008-4] Bhattacharya, S. K. (2008-08-27). Electrical Machines (2008 ed.). Tata McGraw-Hill Education. p. 359. ISBN 978-0-07-066921-5.

[Heyland_1906-5] Heyland, Alexander Heinrich [in German] (1906). A Graphical Treatment of the Induction Motor. G. H. Rowe, R. E. Hellmund (translators). McGraw Publishing Company. Retrieved 2013-01-10.

[Ph2Ph_2006-6] "The Asynchronous Motor Model" (PDF). Phase to Phase BV. 2006. pp. 5–6. Archived from the original (PDF) on 2014-08-10. Retrieved 2013-01-10.

[Knowlton_1949-7] Alger, Philip L.; et al. (1949). "Induction machines [in § 7, Alternating-current generators and motors]". In Knowlton, A. E. (ed.). Standard Handbook for Electrical Engineers (8 ed.). McGraw-Hill. pp. 710–711.

[Fernandez-8] Fernandez, Francis M. "Construction of Circle Diagram" (PDF). College of Engineering Trivandrum. Archived from the original (PDF) on 2014-08-08. Retrieved 2013-01-10.

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Circle diagram

See also

References

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