DC Motors

When a conductor of length $L$ carries current $I$ in a magnetic field of flux density $B$, it experiences a force given by $F = BIL\sin\theta$, where $\theta$ is the angle between the current direction and the field. This is the motor effect. The direction of this force is given by Fleming's Left-Hand Rule: point the First finger in the direction of the magnetic Field, the seCond finger in the direction of conventional Current, and the thuMb points in the direction of the force (Motion). In a motor coil, the two sides parallel to the rotation axis carry current in opposite directions, so they experience forces in opposite directions, producing a net torque.

For a rectangular coil of $N$ turns, width $w$ and length $l$, carrying current $I$ in a uniform field $B$, the torque is $\tau = NBIL w \sin\phi$, where $\phi$ is the angle between the plane of the coil and the magnetic field direction (equivalently, $\tau = NBIA\sin\phi$ where $A = lw$ is the coil area). Torque is maximum when the coil is parallel to the field ($\phi = 90°$) and zero when the coil is perpendicular to the field (the coil plane is perpendicular to B). At the zero-torque position, the coil would stop rotating — this is where the commutator plays its critical role.

The commutator is a split copper ring attached to the axle of the coil. Two carbon brushes press against the commutator segments and connect to the external DC supply. As the coil rotates, each commutator segment slides from one brush to the other exactly every half-turn (180°). This mechanically reverses the direction of current in the coil at the moment it passes through the plane perpendicular to the field (the zero-torque position). By reversing the current at this precise moment, the forces on the coil sides always act in the direction that continues the same sense of rotation. Without the commutator, the torque would reverse every half-turn and the motor would oscillate rather than rotate continuously.

As the motor coil rotates, it cuts through magnetic field lines and acts as a generator, inducing an electromotive force (EMF) that opposes the applied voltage — this is called the back-EMF ($\varepsilon_{back}$). By Kirchhoff's voltage law, the current in the circuit is $I = \frac{V - \varepsilon_{back}}{R}$, where $V$ is the supply voltage and $R$ is the coil resistance. When the motor runs at full speed, the back-EMF is large, reducing the current drawn. When the motor is stalled (not rotating), back-EMF is zero, so current is at its maximum ($I_{stall} = V/R$), which can overheat the motor. The mechanical power output is $P_{mech} = \varepsilon_{back} \cdot I$, and electrical power input is $P_{in} = VI$, giving efficiency $\eta = \varepsilon_{back}/V$ in the ideal case.

Real DC motors use several techniques to improve performance. Multiple coil windings offset at different angles (wound on an armature) reduce torque ripple and ensure smooth rotation at all positions. Electromagnets (field coils) replace permanent magnets in powerful motors, allowing field strength to be varied. Laminated iron cores inside the coil increase the magnetic flux density and therefore the torque. Carbon brushes are used instead of metal contacts because carbon is self-lubricating and has a relatively constant contact resistance, reducing sparking and wear.

• \tau — Torque (N·m): The turning force (moment) acting on the coil about its rotation axis
• N — Number of turns (dimensionless): The total number of loops of wire in the coil; more turns increase torque proportionally
• B — Magnetic flux density (T (tesla)): The strength of the external magnetic field in which the coil rotates
• I — Current (A (ampere)): The electric current flowing through the coil
• A — Coil area (m²): The area enclosed by one turn of the coil ($A = lw$ for a rectangular coil of length $l$ and width $w$)
• \phi — Angle of coil (degrees or radians): The angle between the plane of the coil and the direction of the magnetic field; torque is maximum at $\phi = 90°$

Problem. A DC motor coil has 80 turns, a cross-sectional area of $4.0 \times 10^{-3}$ m², and carries a current of 2.5 A. The coil rotates in a uniform magnetic field of flux density 0.30 T. (a) Calculate the maximum torque on the coil. (b) Calculate the torque when the coil plane makes an angle of 30° with the magnetic field direction.

Answer. Maximum torque = 0.24 N·m; torque at 30° = 0.12 N·m

• The Motor Effect: Force on a Current-Carrying Conductor — The DC motor is a direct application of the motor effect; understanding $F = BIL\sin\theta$ and Fleming's Left-Hand Rule is essential before studying motors.
• Electromagnetic Induction and Faraday's Law — Back-EMF in a DC motor is generated by electromagnetic induction as the coil rotates, directly affecting the motor's current draw and efficiency.
• Magnetic Flux and Flux Density — Torque calculations and back-EMF both depend on the magnetic flux through the coil, making flux a fundamental quantity in motor analysis.
• AC Generators (Alternators) — An AC generator and a DC motor use the same coil-in-magnetic-field geometry; a generator converts mechanical to electrical energy, the reverse of a motor, and uses slip rings instead of a commutator.