Single-phase induction motors

Why it matters

Fans, pumps, compressors, washing machines, small blowers and laboratory equipment run on single-phase induction motors because homes, offices and small workshops have only a single-phase supply. Knowing why they are not self-starting, and how each starting arrangement fixes that, is what you need to choose, troubleshoot (the classic failed fan capacitor) or replace one.

Key ideas

A single winding gives a pulsating field. One stator winding fed with single-phase AC produces a field that alternates along a fixed axis; it does not rotate. At standstill it produces no net torque, so the motor hums but does not start. Once spun in either direction, however, it keeps running in that direction.

Double-revolving-field theory. A pulsating field of peak Φm is equivalent to two fields of Φm/2 each rotating in opposite directions at synchronous speed Ns.

• If the rotor runs at speed N in the forward direction, its slip with respect to the forward field is s = (Ns − N)/Ns, and with respect to the backward field it is (Ns + N)/Ns = 2 − s.
• Each field produces a torque–slip curve like a three-phase motor. At standstill (s = 1 for both) the two torques are equal and opposite, so net starting torque is zero.
• When the rotor turns forward, the forward torque dominates and the motor accelerates. The backward field induces rotor currents at frequency (2 − s)·f (close to 2f), which cause extra rotor loss and a double-frequency torque pulsation. Hence single-phase motors are noisier, less efficient and have a lower power factor than three-phase motors of the same size, and maximum torque is reached at a speed lower than in the equivalent three-phase motor.
• The equivalent circuit splits the rotor into a forward half (0.5·R2'/s, 0.5·X2', 0.5·Xm) and a backward half (0.5·R2'/(2 − s), 0.5·X2', 0.5·Xm) in series. Net air-gap power = Pf − Pb; torque = (Pf − Pb)/ωs.
• (The cross-field theory gives the same results by a different argument.)

Making it self-starting: a second winding displaced 90° in space carrying a current displaced in time — together the two windings produce a (roughly elliptical) rotating field.

• Split-phase (resistance-start): auxiliary winding of fewer turns of thinner wire, so higher R/X ratio; its current leads the main-winding current by about 25–30°. Moderate starting torque; a centrifugal switch disconnects the auxiliary winding at about 70–80 % of synchronous speed. Used in fans, small pumps, washing machines.
• Capacitor-start: a capacitor (electrolytic, short-duty) in series with the auxiliary winding brings its current nearly 90° ahead of the main current — high starting torque. Centrifugal switch removes it. Used for compressors, refrigerators, pumps.
• Capacitor-start capacitor-run: a large start capacitor plus a smaller run capacitor that stays in circuit: good starting torque, better running power factor and efficiency, quieter.
• Permanent-split capacitor (PSC): one run capacitor permanently in circuit, no switch; modest starting torque; ceiling fans and blowers. A failed fan capacitor gives exactly the symptom described above: it hums but turns only if pushed.
• Shaded-pole: part of each salient pole is surrounded by a copper shading ring; induced ring current delays the flux in the shaded part, giving a field that sweeps from the unshaded to the shaded part. Very cheap and robust; very low starting torque and efficiency; direction fixed by construction. Small fans, hair dryers.
• Reversing a split-phase or capacitor motor: reverse the connections of either the main or the auxiliary winding (not both).

Formulas

• Ns = 120·f / P — synchronous speed (rpm).
• s_f = s = (Ns − N)/Ns ; s_b = 2 − s — slips with respect to forward and backward fields.
• f_rotor,f = s·f ; f_rotor,b = (2 − s)·f — rotor current frequencies (Hz).
• Zf = 0.5·[jXm ∥ (R2'/s + jX2')] ; Zb = 0.5·[jXm ∥ (R2'/(2 − s) + jX2')] — forward and backward impedances (Ω).
• Pf = I²·Re(Zf) ; Pb = I²·Re(Zb) — forward and backward air-gap powers (W).
• T = (Pf − Pb) / ωs ; Pm = (1 − s)·(Pf − Pb) — net torque (N·m) and gross mechanical power (W).

Worked examples

Example 1 (standard): slips and rotor frequencies. Given: 4-pole, 50 Hz single-phase induction motor running at 1440 rpm.

1. Ns = 120 × 50/4 = 1500 rpm.
2. Forward slip s = (1500 − 1440)/1500 = 0.04; backward slip = 2 − 0.04 = 1.96.
3. Rotor current frequency due to forward field = 0.04 × 50 = 2 Hz; due to backward field = 1.96 × 50 = 98 Hz.

Example 2 (GATE level): equivalent-circuit performance. Given: 230 V, 50 Hz, 4-pole single-phase motor, main winding only (running). R1 = 2 Ω, X1 = 3 Ω, R2' = 4 Ω, X2' = 2 Ω, Xm = 60 Ω; slip 0.05. Neglect core loss.

1. R2'/s = 80 Ω; R2'/(2 − s) = 4/1.95 = 2.05 Ω.
2. Zf = 0.5 × [j60 ∥ (80 + j2)] = 14.06 + j19.11 Ω; Zb = 0.5 × [j60 ∥ (2.05 + j2)] = 0.96 + j1.00 Ω.
3. Total Z = (2 + j3) + Zf + Zb = 17.02 + j23.11 Ω, |Z| = 28.70 Ω.
4. Current I = 230/28.70 = 8.02 A at power factor 17.02/28.70 = 0.593 lagging.
5. Pf = 8.02² × 14.06 = 903 W; Pb = 8.02² × 0.96 = 61.6 W.
6. Torque T = (903 − 61.6)/157.08 = 5.36 N·m; gross mechanical power = 0.95 × 841.4 = 799 W.

Common mistakes

• Saying the single-phase field "rotates slowly" — it only pulsates; rotation comes from the second winding.
• Using s instead of 2 − s for the backward field.
• Forgetting the factor 0.5 on each half of the equivalent circuit.
• Thinking a capacitor-start motor stops if its capacitor fails while running; it keeps running on the main winding but will not restart.
• Reversing both windings to change direction (that changes nothing).
• Expecting a shaded-pole motor to be reversible without mechanical changes.

For GATE IN

Expect conceptual MCQs (why not self-starting, which motor has the highest starting torque, what the centrifugal switch does, phase angle in split-phase versus capacitor motors) and numericals on forward/backward slip, rotor frequencies and torque from the double-revolving-field equivalent circuit.

Quick check

1. A 2-pole, 50 Hz single-phase motor runs at 2850 rpm. What is the backward slip?
2. Which single-phase motor has the lowest starting torque?
3. Why does a capacitor give more starting torque than a split-phase resistance?
4. At what frequency do rotor currents due to the backward field flow at slip s?

Answers: 1. s = 0.05, backward slip = 1.95. 2. Shaded-pole. 3. It makes the auxiliary current lead the main current by nearly 90°, giving a near-circular rotating field. 4. (2 − s)·f.