Chapter 3
Before you can understand why Tesla's induction motor was a breakthrough, you need to understand what a brushed DC motor is, how it works, and why it was both a triumph of Victorian engineering and a machine with a fundamental mechanical flaw at its core.
The history of electric motors begins not with a machine anyone would call practical, but with a demonstration of a principle. On September 3, 1821, Michael Faraday set up a small apparatus in the basement of the Royal Institution in London and proved that electromagnetic force could produce continuous rotation. His device was startlingly simple: a vessel of mercury with a vertical bar magnet standing at its center. A flexible wire hung from above, its lower tip dipping into the mercury. When Faraday connected a battery, the wire began to rotate around the magnet — continuously, as long as current flowed. He had built the world's first electric motor.
This device is a homopolar motor, also called a unipolar motor. It has no commutator, no brushes, no switching mechanism of any kind. The rotating wire is a conductor carrying current in the presence of the magnet's field. The force on that conductor, F = qv × B, always points in the same rotational direction — because the current direction and field direction are fixed, and their cross product consistently produces a torque around the magnet's axis. Rotation is not oscillation interrupted and redirected by mechanical trickery; it is a genuine continuous rotational force.
William Sturgeon (1823–1832) made the next key advance. In 1823, Sturgeon invented the electromagnet — a soft iron core wound with wire, which became a magnet only when current flowed. This was not a trivial improvement over permanent magnets. Electromagnets could be made far stronger than any permanent magnet of the era, and — crucially — they could be switched. In 1832, Sturgeon demonstrated the first practical commutated DC motor using an electromagnet-based design.
Zenôbe Gramme (1871) produced the ring armature that became the standard configuration for industrial DC machines. The Gramme ring wound continuous wire around a toroidal iron core, with many commutator segments providing relatively smooth torque output. The Gramme machine was exhibited at the Vienna International Exhibition in 1873, where a famous demonstration occurred: a Gramme generator was accidentally connected backward to another Gramme generator, which immediately began running as a motor, demonstrating bidirectional operation and the utility of DC machines as both generators and motors. By the 1870s and 1880s, large DC motors with brush commutators were standard industrial equipment: running printing presses, machine tools, street railways, and elevator hoists.
This was the world Tesla entered when he arrived in the United States in 1884 and went to work for Edison.
A commutator is a cylindrical assembly of copper segments, separated by thin slivers of hard mica (or modern epoxy composites), mounted on and rotating with the motor shaft. Spring-loaded carbon brushes — stationary blocks of graphite-copper composite — press against the rotating commutator surface under constant spring force, maintaining electrical contact as the commutator turns beneath them.
The function of the commutator is to act as a synchronous mechanical switch: it reverses the current direction in each armature coil at precisely the moment the coil is in the position where reversal is required to maintain continuous torque.
Consider a single rectangular coil rotating in a uniform magnetic field between two poles (north and south). When the coil is in one orientation, the left conductor carries current toward you, the right conductor carries current away. The F = BIL forces on the two conductors produce a torque in, say, the clockwise direction. But after the coil rotates 180°, the geometry has flipped: the same coil with the same current now produces forces in the counter-clockwise direction. Net torque over a full revolution: zero. The coil oscillates; it does not rotate continuously.
The commutator solves this by reversing the current in the coil at exactly the 180° point — the moment when the coil plane is perpendicular to the field (passing through its "neutral plane"). After commutation, the current in the coil is reversed, so the forces remain in the clockwise direction. The motor continues to rotate.
In the simplest two-pole, single-coil motor, the sequence is:
| Rotation Angle | Coil Position | Event |
|---|---|---|
| 0° | Coil plane parallel to field | Current flows; torque acts clockwise |
| 90° | Coil plane perpendicular to field | Maximum rate of flux linkage change; peak torque |
| 180° | Coil at neutral plane | Commutation: brush crosses gap; current reverses in coil |
| 270° | Opposite peak position | Torque still clockwise (current reversed, geometry reversed) |
| 360° | Back to start | Another commutation; cycle repeats |
A practical motor has many coils (typically 12–48) and a corresponding number of commutator segments. With N coils, the torque ripple frequency is N commutations per revolution, and the torque variation decreases toward a constant average. A 36-segment commutator produces torque smooth to within a few percent.
The force on a current-carrying conductor in a magnetic field derives directly from the Lorentz force law: F = qv × B. For a straight conductor perpendicular to a uniform field:
F = BILwhere B is in Tesla, I in amperes, L in meters, and F in Newtons. This is the fundamental motor force equation.
Torque is force times moment arm. For a rectangular coil of N turns, width w and length l, with the field B perpendicular to the coil plane, the two active conductors each experience force F = BIlN. These forces act at a moment arm of w/2 from the rotation axis:
T = 2 × F × (w/2) = BIlNw = BIANwhere A = l×w is the coil area. The complete torque equation for a practical DC motor with armature having Z conductors, P poles, and winding constant a:
T = (Z / 2πa) × Φ × Ia = KT × Φ × Iawhere KT = Z/(2πa) is the machine's torque constant, Φ is the total flux per pole (Wb), and Ia is the armature current. Torque is proportional to flux and to armature current. Double the flux (by increasing field current in a wound-field machine) and double the torque. Double the armature current and double the torque. This linearity is one of the great virtues of DC motors — they are straightforward to control.
A spinning motor armature is a collection of conductors moving through a magnetic field. By Faraday's Law, those conductors generate EMF — ε = Blv. The motor is simultaneously a generator, and the voltage it generates appears across the armature terminals in opposition to the applied supply voltage. This is the motor's back-EMF (counter-EMF), and it is Lenz's Law in its most practical form.
εb = KE × Φ × ωwhere KE is the motor's EMF constant (equal to KT in consistent SI units), Φ is flux per pole, and ω is angular velocity in rad/s. For a permanent-magnet DC motor (constant Φ), back-EMF is simply proportional to speed.
The circuit equation for the armature loop:
Vsupply = εb + Ia × Ra + Vbrushwhere Ra is armature resistance and Vbrush ≈ 0.5–2 V for carbon brushes. Rearranging:
Ia = (Vsupply − εb) / RaAt constant supply voltage and constant flux, the speed-torque relationship is linear:
ω = Vsupply / (KEΦ) − [Ra / (KE KT Φ²)] × TA downward-sloping straight line on a speed-torque plot. Key points:
| Operating Point | Condition | Value |
|---|---|---|
| No-load speed ω0 | T → 0 | ω0 ≈ Vsupply / (KEΦ) |
| Stall torque Tstall | ω = 0 | Tstall = KTΦ × (Vsupply / Ra) |
| Max power point | T = Tstall/2 | Pmax = Tstall × ω0 / 4 |
A motor with low armature resistance and high flux has a flat speed-torque curve (good speed regulation — speed barely changes as load varies). A motor with high armature resistance has a steep curve (speed varies significantly with load).
This simple, predictable, linear speed-torque characteristic made DC motors the preferred choice for applications requiring precise speed control: electric railways, rolling mills, and machine tools. The AC induction motor that Tesla would patent in 1888 has a fundamentally different curve — nonlinear, with a torque peak at a specific slip frequency, designed to run near synchronous speed under varying load. For wide-range speed control, the DC motor's linearity was genuinely superior, and it remained the choice for demanding traction applications well into the 1960s.
The commutator and brushes are not incidental components — they are the defining mechanism of the brushed DC motor. They are also its principal engineering liability.
Carbon brushes are consumables. They wear at a rate depending on current density, peripheral velocity of the commutator, and spring pressure. Typical brush wear life in industrial motors is 2,000–5,000 hours of operation, after which the brushes must be replaced. This requires stopping the machine, opening the motor housing, removing and replacing the brushes, and often dressing the commutator surface. On a motor with many hours of service, the commutator diameter has decreased measurably from its original dimension and must be machined in a lathe to restore its cylindrical form.
As a brush transitions from one commutator segment to the next, the armature coil being commutated contains stored inductive energy (LI²/2). The current in that coil must reverse direction in the brief time the brush spans the insulating gap. If reversal is not complete before the brush leaves the segment, the residual inductive voltage appears as a spark — an arc between the brush and the departing segment.
At high rotational speeds, two mechanical constraints limit performance:
A conventional brushed DC motor will not operate from AC supply in its designed form. The commutator is synchronized to rotor position, reversing current at the correct mechanical angle. An AC supply would reverse armature current at the supply frequency (60 Hz = 120 reversals per second) regardless of rotor position, producing oscillating torque, vibration, and heating.
The "universal motor" (series-wound, where field winding and armature are in series from the same supply) bypasses this: because both field and armature reverse simultaneously, the torque T = KΦIa remains positive throughout the AC cycle (negative times negative = positive). Universal motors work on both AC and DC and are found in hand drills, vacuum cleaners, and kitchen appliances. They retain all brush and commutator limitations, and their AC field current wave-form produces significant iron losses — but they are cheap, high-speed, and high-power-density for intermittent use.
For fixed-installation industrial power, where the supply is AC and the motor runs continuously, the universal motor's high losses and brush wear make it unattractive. The demand for a motor that could run directly from an AC supply — with no commutator, no brushes, and no mechanical switching of any kind — was the engineering problem that Tesla solved.
The commutator, brush holders, and end bracket needed to support them add significant axial length to the motor. A brushless motor of equivalent power can be substantially shorter and lighter. In applications where size and weight matter — aircraft, electric vehicles, high-speed spindles — this is a real penalty that motivated the twentieth century's gradual shift away from brushed machines.
What this demonstrates: the Lorentz force F = qv × B producing continuous rotation with no commutator, no switching, no mechanical trickery. The rotating element is a conductor carrying current in a magnetic field, and the force is continuous and unidirectional by the geometry of the cross-product — identical in principle to Faraday's 1821 device.
With a fresh AA battery and an N42 magnet, the wire rotates at roughly 5–15 revolutions per second, depending on wire geometry and contact friction. The rotation is steady as long as the battery supplies current. The wire will become noticeably warm after 30–60 seconds (short-circuit current ≈ 1–3 A through a fresh AA cell is a significant fraction of its short-circuit capacity).
The neodymium magnet's field at its rim is not purely axial — there is a significant radial component in the fringe region. The force on the wire's sections near the magnet, F = I(L × B), has a tangential (rotational) component that is always in the same rotational sense regardless of the wire's angular position. This is the essence of the homopolar geometry: the force direction does not depend on rotational angle. No switching is needed. This is identically the motor Faraday demonstrated in 1821.
What this demonstrates: the commutator as a mechanism. The coil-and-magnet system that would oscillate without switching is converted to continuous rotation by breaking the circuit for the half-revolution where torque would be reversed. This is the conceptual core of every commutated DC motor ever built.
Use a cylindrical former approximately 2–3 cm in diameter (a AA battery, a film canister, a small pill bottle). Wind approximately 25–35 turns of magnet wire, keeping the turns tight and uniform. Leave approximately 5–7 cm of wire as leads at each end. Slip the coil off the former and wrap a few turns of the wire tightly around the coil body to hold it together. Gently shape it into a compact flat oval.
The two lead wires act as the shaft. Straighten them carefully so they are collinear (on the same axis) and centered on the coil. The coil must be balanced — the center of gravity must lie on the shaft axis, or the coil will wobble and stall. Test by resting the shaft wires across two fingers: the coil should rest in a stable horizontal position without tipping.
This is the key step. It is the most common point of failure if done incorrectly.
Result: as the coil rotates, the fully-stripped lead always makes contact. The half-stripped lead makes contact for half the revolution (stripped side facing down) and breaks contact for the other half (insulated side facing down). This is a single-segment, half-cycle commutator.
Give the coil a gentle push to start it. Once rotating, it should sustain itself indefinitely. Operating sequence:
A well-built Beakman motor on 1.5 V (single AA) runs at roughly 500–2,000 RPM, visible as a blur of rotation. On 3 V (two AA cells), it runs noticeably faster. The coil should start reliably with a gentle push from any starting position.
Without the half-insulated lead, this is a simple coil in a magnetic field: it would oscillate around the equilibrium position (aligned with the magnet). The mechanical switch converts oscillation into rotation by eliminating the braking torque on the back-swing. The switch does not generate force — it selectively removes the opposing force. This is the conceptual core of every commutated DC motor ever built. The commutator is a switch synchronized to rotor position. The rest is geometry.
Beakman's Motor was developed as an educational demonstration for the PBS science program Beakman's World in the early 1990s. Quantitative analysis in: Williams, M. (2005). "The simplest electric motor." The Physics Teacher, 43, 565–567.