Electric Motor Principles

Predict first: a coil carrying current sits between a north and a south pole. At which angles does the field give it no turning push at all, and where is the push largest?

Torque on the coil. For an ideal rectangular coil in a uniform field, the torque about the shaft is τ = N·B·I·A·sin θ. N is the number of turns, B the field (T), I the current (A) and A the coil's area (m²), so τ is in N·m. θ is the angle from the field B to the coil's normal n, the line perpendicular to the coil's face. The coil's magnetic moment lies along n or against it, depending on which way the current circulates. That sets the direction of the torque, not its size.

Why the angle matters. Whenever current flows, each long side of the coil feels a force F = BIL, the same size at every angle. What changes is its lever arm about the shaft, r·sin θ, where r is half the coil's width. As the coil approaches θ = 0° or 180°, with its face across the field in the neutral plane, the forces point almost straight through the shaft. The lever arm shrinks to zero, and so does the torque. With slip rings the current still flows at those angles and the forces point straight through the shaft. With an ideal split ring the brushes sit on the gaps at that instant, so there is momentarily no current and no force at all. Either way the torque there is zero. At θ = 90° and 270° the lever arm, and the torque, are largest.

Why a commutator. If the coil were fed through slip rings, its current would never reverse. With the polarity drawn here, past 180° the torque would reverse too, pushing the coil back toward that neutral-plane position, and over a whole turn it would average zero. Reverse the supply or the magnets and the coil's resting position moves half a turn. A split-ring commutator turns with the coil. Its brushes change segments exactly as the coil crosses the neutral plane, which reverses the coil current. The torque becomes N·B·I·A·|sin θ|: it never points backward, but it still falls to zero twice a turn.

Two coils. Add a second coil 90° ahead of the first, each with its own split ring. When one coil is in the neutral plane, the other is at its peak. The total, |sin θ| + |cos θ| in units of one coil's peak, runs from 1 to √2. Its smallest value is 1/√2 ≈ 0.71 of its largest, so the ripple shrinks and the zero-torque dead spot disappears.

Worked example, with illustrative numbers rather than a real motor rating: N = 50 turns, B = 0.20 T, I = 2.0 A, and a coil 40 mm × 30 mm, so A = 1.2 × 10⁻³ m². The peak torque is N·B·I·A = 50 × 0.20 × 2.0 × 0.0012 = 0.024 N·m. At θ = 30°, τ = 0.024 × sin 30° = 0.012 N·m.

Direction. Reversing the current reverses every force (F = I L × B), so the motor turns the other way. Swapping the magnets reverses B and does the same. Reversing both restores the original direction.

Model limits. This is an ideal teaching model: one or two rectangular coils, a uniform field, and current switched exactly at the neutral plane. Torque is shown as a fraction of one coil's peak, N·B·I·A. It does not model back EMF, speed, magnetic saturation or brush sparking.