6 hours · 6 marks
Three stator windings displaced 120° in space, fed by currents displaced 120° in time, produce a magnetic field of constant amplitude that rotates at synchronous speed:
$$N_s = \frac{120 f}{P}\quad(\mathrm{rpm})$$where $f$ = supply frequency (Hz), $P$ = number of poles. Direction of rotation reverses if any two supply phases are interchanged.
Slip:
$$s = \frac{N_s - N_r}{N_s},\qquad N_r = N_s(1-s)$$Rotor frequency: $f_r = s f$. Rotor induced EMF at slip $s$: $E_r = s E_2$ (where $E_2$ = standstill rotor EMF).
General torque expression (per phase model):
$$T = \frac{3}{2\pi N_s}\cdot\frac{s E_2^2 R_2}{R_2^2 + (s X_2)^2}\quad(N_s\text{ in rps})$$or in proportional form:
$$T \propto \frac{s E_2^2 R_2}{R_2^2 + (s X_2)^2}$$At start, $f_r=f$, rotor reactance $X_2$ is high → starting torque of ordinary cage motors is moderate; slip-ring motors add $R_{\mathrm{ext}}$ to raise $T_{\mathrm{st}}$.
For small slip (normal running), $s X_2 \ll R_2$:
$$T \propto \frac{s E_2^2}{R_2}\quad\Rightarrow\quad T\propto s$$(near-linear torque–slip region used for load sharing / stability).
Differentiate $T$ w.r.t. $s$ (or use $R_2 = s X_2$):
$$s_m = \frac{R_2}{X_2},\qquad T_{\max} \propto \frac{E_2^2}{2 X_2}$$Important: $T_{\max}$ is independent of $R_2$; only the slip at which $T_{\max}$ occurs changes with $R_2$.
| Increase $R_2$ | Effect |
|---|---|
| $s_m = R_2/X_2$ | Shifts toward $s=1$ (max torque nearer to start) |
| $T_{\mathrm{st}}$ | Increases (up to $T_{\max}$ when $R_2=X_2$) |
| $T_{\max}$ | Unchanged (ideally) |
| Running efficiency / slip | Higher full-load slip → more Cu loss, lower $\eta$ |
Hence: high $R_2$ for starting (or external $R$ in wound rotor), low $R_2$ for efficient running (deep-bar / double-cage designs compromise both).
Stator has a single-phase main winding (plus usually an auxiliary winding for starting). Rotor is typically squirrel-cage.
Used in fans, blowers, washing machines, refrigerators, small pumps — fractional-kW domestic/commercial drives.
A pulsating MMF along a fixed axis with amplitude $F_m\cos\omega t$ is equivalent to two rotating MMFs of amplitude $F_m/2$:
At standstill, slips w.r.t. both fields are 1 → equal opposite torques → $T_{\mathrm{net}}=0$.
If rotor is given a push in the forward direction, slip to forward field $s_f = (N_s-N_r)/N_s$ becomes < 1, while slip to backward field $s_b = (N_s+N_r)/N_s$ becomes > 1. Forward torque exceeds backward torque → motor continues in that direction.
Conclusion: single-phase IM is not self-starting; needs a second (auxiliary) winding or shaded poles to create a rotating (or elliptical) field at start.
Main + auxiliary windings displaced in space, currents displaced in phase → produce a revolving (or elliptical) field → nonzero starting torque. After start, auxiliary may be disconnected (centrifugal switch) or left in (capacitor-run).
Capacitor in series with auxiliary winding → nearly 90° phase shift → high $T_{\mathrm{st}}$ (2–3× FL). Centrifugal switch cuts out aux + capacitor above ~75% speed. Good for compressors, pumps needing high start torque.
Run capacitor remains in circuit for better pf, quieter running, and improved efficiency; start capacitor (if used) is short-time rated and switched out.
Salient pole with a copper shading ring on part of the pole face. Induced current in the ring delays flux in the shaded portion → sweeping field from unshaded to shaded → weak starting torque. Simple, cheap, low efficiency — small fans, timers.
Split-phase (R-start): resistance aux winding, moderate $T_{\mathrm{st}}$. Repulsion-start, etc., rarely examined beyond name.
| Motor | Idea / use |
|---|---|
| PM BLDC | Permanent-magnet rotor; electronic commutation (Hall/encoder); high efficiency, long life — fans, drones, EVs, HDDs. |
| Hysteresis motor | Rotor of hysteresis material; synchronous speed; smooth quiet torque — clocks, recorders, gyros. |
| Stepper motor | Moves in discrete steps with pulsed excitation; open-loop position control — printers, CNC, robotics. |
| Servo motor | Closed-loop position/speed control with feedback; DC or AC — robotics, CNC, automation. |
| Universal motor | Series-wound; runs on AC or DC; high speed, high starting torque — drills, mixers, vacuum cleaners. |
Q. A 4-pole, 50 Hz induction motor runs at 5% slip. Find (i) synchronous speed $N_s$, (ii) rotor speed $N_r$. Also find rotor frequency when the rotor runs at 600 rpm.
(i) Synchronous speed:
$$N_s = \frac{120 f}{P} = \frac{120\times 50}{4} = \mathbf{1500\,\mathrm{rpm}}$$(ii) Rotor speed at 5% slip:
$$N_r = N_s(1-s) = 1500\times(1-0.05) = 1500\times 0.95 = \mathbf{1425\,\mathrm{rpm}}$$(iii) Rotor frequency at $N_r=600\,\mathrm{rpm}$:
$$s = \frac{N_s - N_r}{N_s} = \frac{1500-600}{1500} = 0.6$$ $$f_r = s f = 0.6\times 50 = \mathbf{30\,\mathrm{Hz}}$$Torque requires induced rotor current, which needs relative motion between stator RMF and rotor conductors. If $N_r=N_s$, slip $s=0$, relative speed = 0 → $E_r=0$, $I_r=0$ → $T=0$. The motor cannot sustain load (or even overcome friction) at synchronous speed. Hence it runs asynchronously with $N_r < N_s$ so that slip produces the required torque. (Ideal unloaded speed approaches but never quite reaches $N_s$.)
At the instant of starting, $N=0$ so back EMF $E_b=0$. Armature current $I_a=(V-E_b)/R_a=V/R_a$. Since $R_a$ is intentionally very small (to reduce running Cu loss), $I_{\mathrm{st}}$ is many times full-load current and can damage the armature / commutator. Hence a starter (external resistance) is used to limit $I_{\mathrm{st}}$, then cut out as $E_b$ builds up with speed.
With a single winding the air-gap field is purely pulsating. By the double-field revolving theory this equals two equal opposite rotating fields → net starting torque is zero. A second (auxiliary) winding, spatially displaced and carrying a phase-shifted current (via capacitor or resistance), produces a rotating/elliptical field and thus starting torque. After start, the aux winding may be disconnected (capacitor-start) or retained (PSC / capacitor-run).
A 6-pole, 50 Hz IM has full-load speed 960 rpm. Find slip and rotor frequency at full load.
$$N_s=\frac{120\times 50}{6}=1000\,\mathrm{rpm},\quad s=\frac{1000-960}{1000}=0.04=4\%,\quad f_r=sf=2\,\mathrm{Hz}$$