Q2Electrical Machine Design

Procedure to draw the circle diagram of an induction motor: the circle diagram is a graphical method (an alternative to the equivalent circuit calculation method) for predicting an induction motor's complete performance characteristics from two simple tests, exploiting the fact that under the standard approximate circuit model, the locus of the stator current phasor (as slip varies from 0 to 1, at constant applied voltage) traces out a circle.

Step 1 — No-load test: the motor is run at no load (rated voltage, no mechanical load), and the no-load current I0 and no-load power factor cos φ0 are measured. This establishes the point O' on the diagram (representing the near-zero-slip operating condition) — plotted at a distance OO'=I0 from the origin O, at an angle φ0 below the voltage reference axis.

Step 2 — Blocked-rotor (short-circuit) test: the rotor is locked (prevented from rotating), and a reduced voltage is applied to circulate approximately rated current; the blocked-rotor current Isc and power factor cos φsc are measured (and then scaled up proportionally to correspond to what the current would be if rated voltage, rather than the reduced test voltage, were applied, since the blocked-rotor test is normally performed at reduced voltage to avoid excessive current/heating during the test). This establishes point A on the diagram, plotted at the (voltage-scaled) distance OA=Isc(rated voltage equivalent) from the origin, at angle φsc.

Step 3 — Construct the circle: draw a line O'A (called the output line reference, after further constructions), and construct the circle passing through points O' and A with its center on a horizontal line through O' (parallel to the voltage reference axis), determined using the perpendicular bisector of O'A intersected with this horizontal line — this circle represents the complete locus of the stator current phasor tip as slip varies continuously from 0 (at O') through 1 (at, or near, point A) and beyond.

Step 4 — Draw auxiliary lines: from point A, draw a line back to O' (the output line) and further construct the torque line (separating the diagram into the fixed loss/rotor-copper-loss and stator-copper-loss/output regions, based on the no-load and blocked-rotor test power measurements, using established geometric constructions involving perpendiculars dropped from various points on the circle to the O'-A output/torque reference lines).

Information obtainable from the circle diagram: once fully constructed, the circle diagram allows direct graphical (ruler-measurement based) determination, for any operating point on the circle (corresponding to a particular slip/load condition), of: the stator input current and power factor (from the length and angle of the line from O to that point); the output power (from the vertical distance from that point down to the output line); the torque developed (from the vertical distance down to the torque line, converted to torque units via the diagram's power-scale factor and synchronous speed); the rotor copper loss and stator copper loss separately (from further vertical distance segments between the various constructed reference lines); the efficiency (as the ratio of appropriate vertical distances representing output power to input power); and the maximum torque and maximum output power points (found geometrically as the points on the circle where a line parallel to the torque or output line respectively is tangent to the circle), making the circle diagram a comprehensive, single graphical construction from which essentially all standard induction motor performance characteristics can be read off directly, without needing to repeat the underlying equivalent-circuit calculation for every different operating condition of interest.

Theoretical basis for the circular locus: the reason the stator current phasor traces a circle is found in the approximate equivalent circuit of the induction motor: with the magnetizing branch moved to the input terminals (the standard approximation), the load branch consists of a fixed leakage impedance (R1+R2'/s viewed as varying only through the s-dependent resistance, in series with the fixed total leakage reactance X1+X2'). As slip s varies, the current through a series circuit of fixed reactance and variable resistance is known (from circle-diagram theory of such circuits) to trace a semicircular locus whose diameter equals V/(X1+X2') — the applied voltage divided by the total fixed leakage reactance. This also immediately explains a useful design insight: the circle's diameter, and hence the motor's maximum power and maximum torque capability, is inversely proportional to the total leakage reactance, directly linking the winding/slot design choices that determine leakage reactance to the motor's peak capability as displayed in the diagram.

Precautions and limitations in practice: the circle diagram's accuracy depends on the validity of the approximate equivalent circuit and the quality of the two underlying tests — errors arise because the magnetizing branch is not truly at the terminals (introducing small errors at high current), because rotor resistance R2' varies with slip frequency due to skin effect in deep-bar/double-cage rotors (making the blocked-rotor test at line frequency somewhat unrepresentative of the low-frequency rotor conditions at running slip, which is why blocked-rotor tests at reduced frequency are sometimes specified for such machines), and because saturation of the leakage flux paths at the high blocked-rotor current can alter the measured reactance relative to normal running conditions. For these reasons, the circle diagram is treated in modern practice as an excellent instructional and estimation tool — providing quick, visual, reasonably accurate full-range performance prediction from just two simple tests — while precise performance guarantees for commercial machines are established using more detailed equivalent-circuit calculations with slip-dependent parameters or direct load testing, positioning the circle diagram as the standard classroom and preliminary-design method rather than the final certification method for machine performance. Its enduring instructional value lies precisely in making the complete performance envelope of the machine — from no-load through full-load to standstill — visible in a single geometric picture, so that the consequences of any equivalent-circuit parameter change can be understood visually before any detailed recalculation is undertaken, which is precisely why this elegant graphical construction still remains a standard fixture of machine design curricula and early-stage preliminary design work even today, despite the ready availability of fast direct numerical computation tools.