Circle diagram

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

• 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.

See also

• Steinmetz equivalent circuit