Chapter 5
No brushes, no contact, no problem — and the most important motor ever built.
Chapter 4 established the rotating magnetic field: two or three stationary stator coils, fed with phase-shifted alternating currents, produce a magnetic field of constant magnitude that sweeps continuously around the stator interior at the supply frequency. Something must be placed inside that rotating field to extract mechanical work. That something is the squirrel cage rotor.
The squirrel cage rotor is deceptively simple. It consists of a cylindrical stack of laminated silicon steel punchings — thin iron sheets, stamped to shape and stacked axially, the lamination necessary to suppress eddy current losses in the iron. Around the perimeter of this iron cylinder, evenly spaced slots are cut. Into those slots are inserted copper or aluminum bars running the full axial length of the rotor. At each end of the rotor, all the bars are connected together by a solid copper or aluminum ring — the end ring.
That is the complete electrical circuit of the rotor: bars connected at both ends by rings. No external connections. No slip rings. No brushes. No wires entering or leaving the rotor. The rotor is electrically isolated from the stator except through electromagnetic induction.
If you removed the iron laminations and held just the copper or aluminum conductors up to the light, you would see a cylinder of parallel bars connected at each end by rings — precisely the geometry of the exercise wheel sold as a squirrel cage pet toy. The name is in common use among electrical engineers since the 1890s and is technically exact: it describes the geometry without ambiguity.
The sequence of events from field rotation to shaft torque involves four distinct physical steps, each governed by a well-established law of electromagnetics. Understanding all four is essential to understanding why the motor is self-regulating.
The rotating stator field sweeps past the stationary rotor bars. From the reference frame of a rotor bar, the magnetic field appears to be moving. This is the condition for motional EMF. The voltage induced in a conductor moving relative to a magnetic field is:
ε = Blv
where B is flux density (tesla), l is the active bar length (meters), and v is the relative velocity between field and bar (m/s). At startup, the rotor is stationary and the field rotates at full synchronous speed, so v is maximum and the induced EMF is maximum.
The induced EMF drives current through the conducting path: down the bar, through one end ring, back through adjacent bars, through the other end ring, and back. The current magnitude is:
I = ε / Z_bar
where Z_bar is the rotor circuit impedance. The direction is determined by Lenz's Law: the induced current will create a field that opposes the relative motion between the stator field and the rotor. At startup, the rotor frequency equals the supply frequency (the field moves at full synchronous speed relative to the stationary rotor), so rotor reactance is at maximum.
Each current-carrying rotor bar now sits in the stator's rotating magnetic field. The Lorentz force on the bar is:
F = BIl
The force is perpendicular to both current and field — tangential to the rotor periphery. This is torque. By Lenz's Law, the force direction is in the direction of the stator field's motion: the rotor is pushed to follow the rotating field.
The rotor accelerates. As it does, the relative velocity v between the rotating field and the rotor bars decreases. Reduced v reduces the induced EMF (Step 1). Reduced EMF reduces current (Step 2). Reduced current reduces force and torque (Step 3). The torque output automatically decreases as the rotor speeds up.
The rotor reaches equilibrium where developed torque equals load torque. If load increases, the rotor slows slightly, relative velocity increases, EMF increases, current increases, torque increases — the motor responds automatically. This is a passive, inherent feedback mechanism requiring no control electronics. The motor regulates its own torque output through the physics of induction.
Slip is the fractional difference between synchronous speed and rotor speed — the fundamental operating variable of the induction motor:
s = (Ns − Nr) / Ns
where Ns is synchronous speed and Nr is rotor speed (in the same units — RPM or rad/s). Slip is dimensionless; expressed as a decimal or percentage.
| Operating Condition | Typical Slip | Physical Meaning |
|---|---|---|
| No load | 0.5–3% | Nearly synchronous; tiny induced current, just enough to overcome friction |
| Full rated load | 3–6% | Normal operating point; nameplate RPM is at this condition |
| Breakdown torque | 10–25% | Maximum torque point; beyond this torque the motor stalls |
| Stall (locked rotor) | 100% (s = 1.0) | Rotor stationary; maximum induced EMF; starting condition |
This is why motor nameplates give rated speed as slightly below the round-number synchronous speed — 1750 RPM or 1725 RPM, not 1800 RPM. That difference encodes the full-load slip.
The relationship between torque and slip is not linear. The torque equation derived from the per-phase equivalent circuit (Chapman, Ch. 6–7; Fitzgerald et al., Ch. 6) is:
T = (3 / ωs) × (s × R₂ × |E₂|²) / (R₂² + (sX₂)²)
where ωs is synchronous angular velocity (rad/s), s is slip, R₂ is rotor resistance referred to the stator (Ω), X₂ is rotor leakage reactance at standstill (Ω), and E₂ is rotor open-circuit voltage referred to the stator (V, RMS).
When s is small, (sX₂)² ≪ R₂² and the denominator ≈ R₂². The equation simplifies to:
T ≈ (3 / ωs) × (s × |E₂|²) / R₂ [for small s]
Torque is approximately proportional to slip. This is the stable operating region: the characteristic behaves like a very stiff torsional spring, with large torque changes for small speed deviations. All properly loaded industrial motors operate here.
Differentiating with respect to s and setting equal to zero gives the slip at maximum torque:
s_max_T = R₂ / X₂
At this slip, the maximum (breakdown) torque is:
T_max = (3 / 2ωs) × |E₂|² / X₂
Critically: T_max is independent of R₂. Rotor resistance determines where maximum torque occurs on the speed axis, not how large that maximum torque is. This fact underlies the wound-rotor motor design (Section 9).
For standard NEMA Design B motors, breakdown torque occurs at roughly 15–25% slip and is approximately 200–300% of rated (full-load) torque.
Beyond the breakdown torque point, (sX₂)² dominates the denominator and torque falls inversely with increasing slip. This is an unstable equilibrium: if load exceeds breakdown torque, the motor accelerates into this region, torque drops further, and the motor stalls. Engineers size motors with adequate torque margin below breakdown; a well-applied motor never enters Region 2 during normal operation.
| Torque Type | Typical Value (% of full-load torque) | Notes |
|---|---|---|
| Locked rotor (starting) torque | 150–200% | NEMA Design B; adequate for most loads |
| Breakdown (maximum) torque | 200–300% | Short-term peak; beyond this = stall |
| Full-load torque | 100% (by definition) | Nameplate operating point |
| No-load torque | ~5–10% | Overcomes friction and windage only |
At stall (s = 1), the stator applies full line voltage while the rotor presents a low-impedance load — similar electrically to a short-circuited transformer. Typical starting inrush current is 6 to 8 times full-load current.
The consequences are real and must be engineered against:
Voltage sag. The large inrush current flowing through the finite impedance of the supply circuit causes a momentary voltage dip. Neighboring equipment on the same distribution bus experiences reduced voltage — motors slow, lights dim, sensitive instruments may trip. Large motors starting on weak grids can produce observable disturbances across an entire facility.
Thermal stress. I²R heating is proportional to the square of current. Starting at 7× full-load current produces 49× the thermal load of normal operation, concentrated in the rotor bars and stator windings during the acceleration period. Motor ratings include a maximum number of starts per hour precisely because of cumulative thermal damage.
Mechanical shock. Starting torque applied instantaneously to flexible couplings, gearboxes, and belt drives causes fatigue damage on repeated starts.
Standard mitigation methods:
Star-delta (wye-delta) starting. Windings connected in star (Y) at startup, reducing phase voltage by 1/√3 and therefore current and torque to 1/3 of across-the-line values. A contactor switches to delta at approximately 75–80% of synchronous speed. Limitation: starting torque is also reduced to 1/3; may be inadequate for loaded starts.
Autotransformer starting. A tapped autotransformer reduces applied voltage to 65–80% of nominal. Line current reduced by the square of the voltage ratio. More flexible than star-delta; used for larger motors.
Soft starters. SCR-based solid-state devices that ramp the applied voltage from low to full over a programmable time, controlling both current rise and torque rise. No mechanical contactors; standard in modern industrial installations.
Variable frequency drives (VFDs). The most complete solution — see Section 6.
Tesla's induction motor had one significant limitation in 1887: its speed was determined by the supply frequency, which was fixed. Within a given pole configuration, the motor ran at essentially one speed. The fundamental equation:
Ns = 120f / P
Vary f and you vary Ns. The obstacle was generating variable-frequency AC power efficiently. Rotating frequency changers existed but were large and expensive. The solution waited for semiconductor power electronics.
A variable frequency drive (VFD) consists of three stages:
1. Rectifier. The AC supply (typically 480 V three-phase at 60 Hz) is rectified to DC by a diode or SCR bridge, producing a DC bus at approximately 650–700 V.
2. DC bus. A bank of electrolytic capacitors smooths the DC bus voltage and buffers momentary transients.
3. Inverter. An array of IGBTs (insulated gate bipolar transistors) synthesizes three-phase AC of any desired frequency and voltage from the DC bus using pulse-width modulation (PWM). Output voltage and frequency are independently programmable.
A modern VFD synthesizes AC from 0 Hz to 200+ Hz, with voltage proportional to frequency (constant V/Hz ratio) to maintain constant flux in the motor. The result: full variable-speed control of a standard squirrel cage motor, from zero to above base speed, with smooth torque delivery at any speed. The motor requires no modification.
VFDs are ubiquitous in modern industrial and commercial facilities: HVAC fans and pumps, compressors, conveyors, machine tools, elevators, industrial processes. This is Tesla's motor, completed by semiconductor technology developed three decades after his death. He designed the engine; the 20th century built the throttle.
The AC induction motor is among the most efficient energy conversion devices ever built at practical scale. NEMA Premium efficiency (NEMA MG-1 Table 12-12) and IEC IE3 class at full rated load for three-phase, 4-pole, 60 Hz motors:
| Motor Rating | NEMA Premium Efficiency |
|---|---|
| 1 HP (0.75 kW) | 85.5% |
| 2 HP (1.5 kW) | 86.5% |
| 5 HP (3.7 kW) | 89.5% |
| 10 HP (7.5 kW) | 91.7% |
| 25 HP (18.7 kW) | 93.6% |
| 50 HP (37 kW) | 94.5% |
| 100 HP (75 kW) | 95.4% |
| 200 HP (149 kW) | 96.2% |
| 500 HP (373 kW) | 96.2% |
Source: NEMA MG-1 (current edition), Table 12-12; and de Almeida et al., IEEE Transactions on Industry Applications, 2014.
These figures are for a machine with no brushes, no commutator, no oil to change, and no scheduled maintenance beyond bearing lubrication every two to five years. Capable of continuous operation for 20+ years. The losses — stator copper loss (I²R), rotor copper/aluminum loss, core loss, friction and windage, stray load losses — are minimized through winding geometry, core material selection, and mechanical tolerances. They cannot be eliminated, only reduced.
Compare to a brushed DC motor at equivalent size: 80–88% efficiency, brush replacement every few thousand hours, commutator turning required periodically, brush contact generating sparks that create fire hazard in many industrial environments. The induction motor has none of these failure modes and exceeds the DC motor's efficiency at virtually every power level where both have been built.
Most household electrical supply is single-phase. A single-phase stator produces a pulsating magnetic field — a field oscillating on a single spatial axis, not a rotating field. A pulsating field produces no net starting torque. The forward and backward rotating-field components (into which any pulsating field can be decomposed) cancel exactly at standstill. A pure single-phase induction motor will not self-start.
If you manually spin the rotor in either direction, it will accelerate — because now one rotating component aligns with the motion and dominates the opposing one. But at rest it goes nowhere. The solution in all household single-phase motors is to provide a starting circuit that synthesizes an approximation of a second phase:
Capacitor-start: A capacitor in series with an auxiliary winding (at 90° spatial offset from the main winding) shifts the auxiliary current by approximately 90°, producing a rough two-phase rotating field at startup. A centrifugal switch disconnects the auxiliary circuit at ~75% of synchronous speed. Starting torque: 200–350% of full-load torque. Found in compressors, pumps, tools.
Capacitor-start, capacitor-run: Two capacitors — large for starting, small for continuous running. Improves both starting torque and running power factor. Used where both are important.
Permanent-split capacitor (PSC): Single capacitor always in circuit. No starting switch; lower starting torque; quieter; longer-lived. Standard in HVAC fan motors and small circulation pumps.
Shaded-pole: A copper ring embedded in part of each stator pole creates a phase lag in that flux portion, producing a weak asymmetric rotating field. Low starting torque, very low efficiency (20–30%). Used only in small low-duty-cycle applications: small fans, cheap appliances.
The squirrel cage has one engineering constraint: rotor resistance R₂ is fixed by the bar geometry and material. From the torque equation (Section 4), the slip at maximum torque is s_max_T = R₂ / X₂. A motor with higher R₂ reaches its maximum torque at higher slip — at lower rotor speed. For applications requiring high torque at low speed (cranes, hoists, ball mills), this shift is advantageous.
The wound rotor induction motor solves this by replacing the squirrel cage bars with actual windings on the rotor, brought out to three slip rings at one end of the shaft. Carbon brushes riding on the slip rings allow an external resistor bank to be connected to the rotor circuit. Adding resistance increases the effective R₂, shifting the breakdown torque to higher slip values — to lower speed. At startup, maximum resistance is inserted; the motor develops high torque at low speed while drawing moderate current (the resistance limits the current). As the motor accelerates, resistance is progressively cut out.
The penalties: slip rings and brushes require periodic maintenance — exactly the maintenance-intensive arrangement Tesla designed the squirrel cage to eliminate. The external resistors dissipate rotor energy as heat rather than converting it to mechanical work. Modern installations almost universally replace wound-rotor motors with squirrel cage motors plus VFDs, achieving the same torque-speed flexibility with higher efficiency and no brushes.
By 1895, Westinghouse had deployed AC induction motors in hundreds of industrial installations across the United States. The advantages were unambiguous.
No brush maintenance. A DC motor brush set requires inspection and replacement every few thousand hours of continuous operation. An induction motor's only serviceable components are the bearings.
No commutator. The commutator on a DC motor is a precision-machined copper assembly requiring the contact surface to be kept round, smooth, and properly undercut. In dirty or wet environments, commutator failure was a common cause of motor replacement. The squirrel cage rotor is a cast or welded structure with no precision surfaces requiring maintenance.
No spark. Brush-commutator contact produces arcing. In mines, grain elevators, flour mills, textile mills — wherever flammable dust or vapors are present — DC motors were a chronic fire and explosion hazard. Induction motors run with complete electrical isolation of the rotor and generate no arc under normal operation. They are explosion-safe by design.
Near-constant speed under varying load. The stiff torque-speed characteristic of the induction motor in the normal operating region (Region 1 of the torque-slip curve) means that large changes in load produce very small changes in speed. For conveyors, pumps, compressors, machine tool spindles — virtually all industrial drives — near-constant speed was exactly what was needed.
Higher efficiency than DC at equivalent power levels. Brush-commutator friction losses and copper losses in the commutator segments added 5–15% to DC motor losses at industrial scale. Induction motors, with no such mechanical losses, consistently outperformed them in efficiency measurements.
AC everywhere. Once the decision was made to distribute power as AC, the induction motor was the natural prime mover. No rectification required. The same transformer that stepped down transmission voltage fed the motor directly.
Objective: Measure slip as a function of load on a real induction motor and verify the torque-slip relationship quantitatively. Calculate shaft efficiency. Observe passive self-regulation.
Three-phase supply at 208–480 V is lethal. Make all connections with power locked out. If you are not experienced with three-phase electrical work, use a motor test stand from an educational supplier (Hampden, Lab-Volt/Festo Didactic) which packages motor, load, and instrumentation in a safe enclosure.
s_NL = (Ns − Nr_NL) / Ns × 100%
s_load = (Ns − Nr) / Ns × 100%
P_out = τ × ω_rotor = τ × (2π × Nr / 60) (watts)η = P_out / P_in × 100%
As load increases, the motor slows slightly. That slight slowing — the increase in slip — is the mechanism by which the motor "requests" more torque from the physics of induction. You can watch it happen in real time on the tachometer. The motor has no control system; slip is both the signal and the actuator. Pure self-regulating feedback through electromagnetic induction.
Given the torque-slip equation from Section 4, predict the slip at your measured full-load torque using nameplate impedance data (if available) or estimate R₂ from nameplate efficiency and rated speed. Compare to measured slip. Agreement within 20–30% is good for a nameplate-only estimate. Closer agreement requires the motor's test-report data sheet, which manufacturers provide on request for industrial motors.