Steinmetz Capacitor: Three-Phase Motor on Single Phase

Sequence circuit and capacitor sizing

This tool derives its terminal solution from Kirchhoff's laws and symmetrical components. Popescu et al., attached source s2, pp. 1–4, establishes the Steinmetz delta/wye sequence framework; the nodal equations and bounded optimization below are derived here.

All motor impedances are per physical stator winding, rotor values referred to that winding. Reactances scale with supply/test frequency. The winding-voltage ceiling uses a separate motor nameplate frequency, not the frequency at which reactances were identified. The linear T circuit is

Use slips $s$ and $2-s$ for the forward and backward circuits. Let $a=e^{j120^\circ}$ and $k=3$ for delta, $k=1$ for isolated wye. Terminal sequence admittances are $Y_+=k/Z(s)$ and $Y_-=k/Z(2-s)$.

The supply fixes $V_A=U$, $V_B=0$. The capacitor connects A to the otherwise floating terminal C. With $b=2\pi fC$, $D=(Y_++Y_-)/3$ and $H=(aY_++a^2Y_-)/3$, its KCL equation gives

Multiply these terminal phase voltages by $1-a^2$ and $1-a$ respectively for delta winding sequence voltages; wye needs no multiplier. The parallel branch voltage gives each air-gap EMF. Their magnitude ratio is the backward/forward flux ratio because both EMFs are at the supply frequency.

For a fixed slip, this ratio differs from $|V_-/V_+|$ by a constant. The squared voltage ratio is a ratio of quadratics in $b$. Evaluate its stationary points and both endpoints within $0\le C\le C_{\max}$ to find the global bounded minimum. Full balance requires a positive-sequence impedance angle of 60° for the drawn connection, and is generally unavailable at an arbitrary load.

Power, current and load

Reconstruct the three physical winding currents from their sequences. Capacitor RMS voltage is $|U-V_C|$ and capacitor current is $2\pi f C|U-V_C|$. Electrical angular frequency is reported in rad/s; dividing by one radian treats radians as dimensionless when forming scalar capacitor susceptance. If $I'_{r+}$ and $I'_{r-}$ are the rotor branch currents,

Power mode brackets the first rising crossing of the target on 81 samples over slip 0.0005–0.25, then solves the load equation. The power ceiling is the largest positive sample before the sampled power peak satisfying the rated winding current and a voltage screen of $1.10\,U_{w,\mathrm{rated}}f_{\mathrm{line}}/f_{\mathrm{rated}}$. In automatic mode the curve retunes C at each slip; it is not the torque-speed curve of one fixed capacitor. The ceiling is capped by neither thermal measurements nor a cooling model.

Practical comparisons and starting

TPU Lab 5, source s1, pp. 5–6, gives the approximate 50 Hz current rules $C_{\Delta}=4800I/U$ and $C_Y=2800I/U$ with C in µF, I in A and U in V. Here I is the rated winding current: delta nameplate line current divided by √3, or line current directly for wye. This follows the manual's current definition using three phases and phase voltage; inverse-frequency scaling is an extension based on capacitor susceptance. The separate 70 µF/kW comparison is a 230 V, 50 Hz delta rule and is not extrapolated as a physical motor model.

The starting choice follows TPU p. 6: total bank = run + extra, with 2–3 × run in total as a loaded-start trial. Standstill torque and voltage use the same sequence model at s = 1. These estimates do not determine a cutout time or prove acceleration.

• Sinusoidal stiff single-phase mains, three identical windings, isolated motor neutral and an ideal lossless capacitor. The diagram is conceptual; protective, switching and discharge equipment is omitted.
• A single-cage linear T circuit with constant resistances, no core-loss resistance, no saturation, rotor skin effect, harmonics, temperature iteration or mechanical transient. The additional loss allowance is subtracted from converted power.
• Default R and X values are the SA-ERWCA Machine 4 values in Ćalasan et al., source s3, Table 18, p. 17. The publication reports a 0.75 kW, 380 V, 50 Hz, one-pole-pair motor. Treating those per-phase values as physical windings of a reconnectable 220/380 V motor for the 230 V delta example is an explicit illustrative reconnection assumption, not a manufacturer-approved conversion. The default 1.86 A winding-current limit is an illustrative assumption based on the published s = 0.06 test point, not a reported nameplate rating; its delta line-current equivalent is approximately 3.22 A. The 20 W loss allowance is assumed, not reported by that paper.
• Replace every equivalent-circuit and nameplate value for your motor. kW, efficiency and power factor alone do not identify a unique motor circuit.
• Flux vectors use the air-gap EMF divided by jω; the animation normalizes the maximum field envelope to unity. Winding current phasors share a common RMS scale. Electrical angle rotates the picture only and does not rerun sizing.
• Published checks cover the balanced T circuit at two slips, the TPU starting-bank arithmetic and the TPU delta formula's winding-current convention. They are not measured validation of the complete Steinmetz predictions. Nodal, energy and optimization residual properties provide additional internal verification.

Use for preliminary sizing and teaching with small, reconnectable squirrel-cage motors and an operating slip between 0.0005 and 0.25. Results depend strongly on identified circuit parameters and actual load. Near resonance or saturation, calculated capacitor voltages and currents can be misleading. A minimum backward field is not an optimum-efficiency or minimum-heating guarantee.

The sampled power ceiling can miss a narrow feasible region and underestimates the continuous optimum between samples. Automatic mode permits a different capacitor at each sampled load. It does not certify continuous duty: negative-sequence rotor heating, cooling, hot resistance, starting duty and winding temperature require motor-specific checks. Never increase winding current to match a desired output.

The TPU manual's p. 7 voltage guidance contains inconsistent margins (1.5 and 1.15 times supply) and mixes discussion of AC and DC markings. Neither rule is used as a component rating here. Use the actual continuous AC RMS rating, capacitor current rating and manufacturer's duty/temperature limits. The 15% modeled run-voltage margin is only a screening reminder; include supply tolerance, resonance during acceleration and switching transients separately. A start capacitor must be intended for AC motor starting, with a suitable automatic cutout and discharge provision. Do not leave it connected during normal running.

Installation work belongs to a qualified person. Isolate and verify dead, discharge capacitors safely, provide protective earthing, correct motor overload protection and guarded terminals. Live commissioning and verification of every winding current require qualified personnel and suitable equipment.

No jurisdiction-specific installation or grid-connection rule is implemented. The 1.5 kW reminder is an advisory threshold from the task brief, not a legal limit or connection approval. Check your grid operator's current connection agreement and starting-current limits. Scope statement dated 27 October 2026; check the rules in force.