Relays, magnetic force, and stepper motor principle

Relay: isolation through an electromagnet. A coil wound on a soft-iron core is fed a small control current. The resulting field pulls a hinged, spring-restrained iron armature across an air gap toward the core; the armature's motion mechanically opens or closes a separate set of contacts, which can carry a far larger current than the coil itself ever sees. Cut the coil current and a return spring pulls the armature back to rest. The design payoff is that a weak control signal — from a microcontroller pin, say — can switch a heavy load (a motor, a heater, mains power) without the two circuits ever touching electrically.

Where the pull comes from. The air gap between armature and core stores magnetic field energy. As the gap narrows, reluctance drops and, for a fixed mmf, the arrangement "prefers" the lower-energy, lower-reluctance state — exactly as a compressed spring prefers to relax. That preference shows up as a mechanical pull on the armature. At this course's level the pull is stated qualitatively as

where: $B$ is the flux density in the air gap and $A$ is the pole face area presented to the gap. Two design facts follow directly. First, force is far more sensitive to flux density than to area — doubling $B$ roughly quadruples the pull, so relay cores are usually driven close to saturation to squeeze out maximum force for a given size. Second, at fixed $B$, a larger pole face gives proportionally more pull. The full expression carries a factor of $\mu_0$ from field-energy theory, but deriving it is outside this course; you only need the proportionality.

Stepper motor: the same alignment, repeated. A stepper motor has several separately wound stator phases arranged around a rotor. Energising one phase sets up a field in a fixed direction, and the rotor swings to align with it — minimising reluctance, just like the relay armature aligning to the core. Switching excitation to the next phase shifts the field direction by a fixed increment; the rotor follows, advancing by one step angle. Cycling through the phases in sequence therefore produces continuous rotation built from discrete, countable steps, with

where: $\theta_{total}$ is the total angle turned, $n$ is the number of drive pulses applied, and $\theta_{step}$ is the fixed angle turned per pulse. Because each pulse produces one known step, a controller can track shaft position purely by counting pulses — no separate position sensor is required, which is why stepper motors dominate open-loop positioning in printers, disk drives, and small robotic stages.

A simple stepper motor has 4 stator phases, and each single-phase energisation step rotates the rotor by a fixed step angle of $\theta_{step} = 1.8^\circ$. The motor must complete exactly 5 full revolutions. (a) How many step pulses are required? (b) If the driving pulse rate is a constant 200 pulses per second, how long does the motion take?

Step 1. Convert the required rotation into a total angle in degrees, since the step angle is given in degrees and one revolution is $360^\circ$: $\theta_{total} = 5 \times 360^\circ = 1800^\circ$.

Step 2. Divide the total angle by the step angle to get the pulse count, using $\theta_{total} = n\theta_{step}$ rearranged for $n$: $n = \dfrac{1800^\circ}{1.8^\circ} = 1000\ \text{pulses}$.

Step 3. Divide the pulse count by the pulse rate to get the time taken: $t = \dfrac{n}{\text{pulse rate}} = \dfrac{1000}{200} = 5\ \text{s}$.

The motor needs 1000 pulses and completes the 5 revolutions in 5 seconds. This pulse-counting relationship — a fixed, known angle per pulse — is exactly why stepper motors are chosen for open-loop position control: the driving controller only has to count pulses sent, with no separate shaft encoder needed, provided the motor never "misses" a step under load.

• Assuming relay pull scales linearly with $B$ — it scales as $B^2$, so a modest rise in flux density produces a much larger rise in force than intuition suggests.
• Confusing the small coil (control) current that creates the field with the much larger contact (load) current the relay switches — the entire point of a relay is that these two circuits are electrically isolated.
• Believing a stepper motor needs continuous position feedback like a servo motor — its defining feature is open-loop operation, valid only as long as no steps are lost (lost synchronism under excessive load or speed).
• Trying to derive the full field-energy formula $F = B^2A/2\mu_0$ or detailed stepper drive-sequencing and torque behaviour — both are explicitly out of scope; this course requires only the qualitative relation and the step-counting idea.

Ask the tutor: Curious how the field-energy method actually derives $F = B^2A/2\mu_0$, or want a worked comparison of a permanent-magnet versus a variable-reluctance stepper rotor?