Solenoids, relays and piezo actuators

Why it matters

Solenoids open valves, lock doors and push parts off conveyors; relays let a 5 V microcontroller switch a 230 V motor or heater; piezo actuators position lenses, inject diesel fuel and drive inkjet nozzles with nanometre precision. Sizing these actuators, and driving them safely from electronics, is one of the most common small design tasks in mechatronics.

Key ideas

Solenoid actuators. A solenoid is a coil with a soft-iron frame and a movable iron plunger. Current in the coil sets up flux across an air gap, and the plunger is pulled in the direction that shortens the gap (reduces reluctance). The force is always attractive, whatever the current direction, so a return spring brings the plunger back. Key properties:

• Force rises steeply as the gap closes (roughly as 1/g² while the iron is unsaturated). The force–stroke curve is therefore far from flat: weakest at the start of the stroke, strongest at the end. Strokes are short, typically a few millimetres to about 25 mm.
• Pull-in and hold. A large current is needed to start the stroke; far less is needed to hold the closed plunger. "Peak-and-hold" drivers or PWM reduce the current after pull-in to cut heating.
• Duty cycle rating. Coils are rated for continuous or intermittent duty; running an intermittent coil continuously overheats it.
• Types: pull, push (a rod through the plunger), latching (a permanent magnet holds the position with no current), rotary, and proportional solenoids (shaped pole pieces give a flatter force–stroke curve, used in hydraulic proportional valves).

Electromagnetic relays. A relay is a solenoid that moves an armature carrying switch contacts. Contact forms: SPST-NO (normally open), SPST-NC, SPDT (changeover), DPDT. Coil data: rated voltage, coil resistance, pick-up (must-operate) voltage, drop-out (must-release) voltage. Contact data: rated current and voltage, different for AC and DC loads because a DC arc does not self-extinguish at a current zero. Relays give galvanic isolation between the control and load circuits.

• Driving a relay coil. A microcontroller pin cannot supply the coil current, so a transistor (or driver IC such as a Darlington array) switches it. A flyback (freewheeling) diode across the coil is essential: when the transistor turns off, the coil inductance tries to keep the current flowing (v = L·di/dt) and would produce a voltage spike large enough to destroy the transistor.
• Solid-state relays (SSRs) use a triac/thyristor pair (AC) or MOSFETs (DC) with an optocoupler for isolation. No moving parts, silent, fast and long-lived; zero-cross types switch on at a voltage zero to reduce interference. Drawbacks: on-state voltage drop and heat (heat sink needed), off-state leakage current, and failure usually in the shorted (on) state.
• Reed relays have contacts sealed in a glass tube, operated by the coil field: fast and low-power, for small signals.

Piezoelectric actuators. Piezoelectric ceramics (usually PZT, lead zirconate titanate) strain when an electric field is applied: the converse piezoelectric effect. One layer gives a strain of only about 0.1 %, so practical actuators are either stacks (many thin layers in series mechanically and in parallel electrically, giving tens of micrometres of stroke with kilonewtons of force) or benders (bimorphs, giving larger motion but little force).

• Force–stroke trade-off. The free stroke Δx₀ is obtained with no load; the blocked force F_b is obtained when the actuator is clamped rigidly. Between them the relation is linear. Against a spring load the stroke is reduced by the stiffness ratio.
• Electrically a capacitor. A static position draws almost no current, but fast motion needs current i = C·dV/dt, so dynamic drives need power amplifiers.
• Non-idealities. Hysteresis of about 10–15 % of stroke and slow creep in open loop; precision stages use strain-gauge or capacitive sensors in closed loop. Stacks are weak in tension and must be preloaded.
• Typical d₃₃ values for PZT are a few hundred pm/V; take the exact value from the manufacturer's data sheet.

Formulas

B = μ₀ · N · I / g

• B: flux density in the air gap (T); μ₀ = 4π × 10⁻⁷ H/m (permeability of free space — the gap is air); N: turns; I: current (A); g: air-gap length (m). Assumes the iron reluctance is negligible and the iron is unsaturated.

F = B² · A / (2 · μ₀) = μ₀ · (N · I)² · A / (2 · g²)

• F: pull force across one gap (N); A: pole-face area (m²). Use only while B stays below saturation (about 1.5 T for soft iron).

F = ½ · I² · dL/dx

• General form: force from the rate of change of coil inductance L (H) with plunger position x (m).

I_coil = V / R_coil, I_B ≥ I_coil / β_min

• Relay coil current (A) and the transistor base current needed to saturate it (β_min: minimum current gain); use a 2–5× overdrive.

Δx₀ = n · d₃₃ · V

• Free stroke of a piezo stack (m); n: number of layers; d₃₃: piezoelectric coefficient (m/V, numerically equal to C/N); V: voltage across each layer (V).

F_b = k_A · Δx₀ and Δx = Δx₀ · k_A / (k_A + k_L)

• F_b: blocked force (N); k_A: actuator stiffness (N/m); k_L: stiffness of an external spring load (N/m); Δx: stroke against that spring.

i = C · dV/dt

• Current to charge a piezo actuator of capacitance C (F) at voltage slew rate dV/dt (V/s).

Worked examples

Example 1 (standard): driving a relay from a microcontroller. Given: a 12 V relay with coil resistance 400 Ω, driven by an NPN transistor (β_min = 100, V_BE = 0.7 V) from a 5 V logic pin. Use 2× overdrive.

1. Coil current: I_coil = V/R_coil = 12/400 = 30 mA.
2. Minimum base current: 30 mA / 100 = 0.3 mA; with 2× overdrive I_B = 0.6 mA.
3. Base resistor: R_B = (5 − 0.7)/I_B = 4.3/0.0006 = 7.17 kΩ, so choose the next lower standard value, 6.8 kΩ.
4. Fit a flyback diode (e.g. a 1N4148 or 1N4007) across the coil, cathode to +12 V.

Example 2 (GATE level): solenoid force versus gap. Given: N = 500 turns, I = 2 A, pole-face area A = 1 cm² = 1 × 10⁻⁴ m², single air gap. Neglect iron reluctance. Find the force at g = 2 mm and at g = 1 mm, and comment.

1. Formula: F = μ₀·(N·I)²·A/(2·g²), with N·I = 1000 A.
2. At g = 2 mm: F = 4π × 10⁻⁷ × 1000² × 1 × 10⁻⁴ / (2 × (2 × 10⁻³)²) = 1.2566 × 10⁻⁴ / 8 × 10⁻⁶ = 15.7 N.
3. At g = 1 mm: halving the gap multiplies the force by 4, giving 62.8 N.
4. Check the flux density: at 1 mm, B = μ₀·N·I/g = 1.2566 × 10⁻⁶ × 1000 / 0.001 = 1.26 T — close to saturation of soft iron, so at smaller gaps the real force rises less steeply than 1/g² predicts.

Example 3: piezo stack against a spring. Given: a 200-layer stack, d₃₃ = 500 pm/V, 100 V per layer, actuator stiffness k_A = 100 N/µm, capacitance 2 µF. Find the free stroke, blocked force, stroke against a spring of k_L = 50 N/µm, and drive current for a 100 V ramp in 1 ms.

1. Δx₀ = n·d₃₃·V = 200 × 500 × 10⁻¹² × 100 = 1 × 10⁻⁵ m = 10 µm.
2. F_b = k_A·Δx₀ = 100 × 10⁶ N/m × 10 × 10⁻⁶ m = 1000 N.
3. Δx = Δx₀·k_A/(k_A + k_L) = 10 × 100/150 = 6.67 µm.
4. i = C·dV/dt = 2 × 10⁻⁶ × 100/0.001 = 0.2 A during the ramp.

Common mistakes

• Using the core permeability in the gap-force formula. The energy is stored in the air gap, so use μ₀.
• Applying F ∝ 1/g² down to zero gap. Saturation, and the residual non-magnetic gap, cap the force.
• Omitting the flyback diode on a relay or solenoid coil driven by a transistor.
• Choosing a relay by its AC contact rating for a DC load; DC ratings are much lower at the same voltage.
• Assuming an SSR is a perfect switch. It drops about 1–1.5 V, needs a heat sink, and leaks current when off.
• Treating a piezo actuator as a resistive load, or expecting both full stroke and full force at the same time.

For GATE ME

This topic produces short conceptual questions (relay contact forms, SSR versus electromechanical relay, piezoelectric effect, why a flyback diode is needed) and numericals on electromagnetic force across an air gap, relay-coil and transistor-drive currents, and piezo stroke and force. Practise the B²A/(2μ₀) force, the 1/g² scaling, and the force–stroke line of a piezo stack.

Quick check

1. Why is a solenoid's pull force smallest at the start of its stroke?
2. A 24 V relay coil has a resistance of 960 Ω. What is the coil current?
3. What does the flyback diode across a relay coil protect?
4. A piezo stack has a free stroke of 20 µm and stiffness 50 N/µm. What is its blocked force?
5. Name one drawback of a solid-state relay.

Answers: 1. The air gap is largest there, and force varies roughly as 1/g². 2. 25 mA. 3. The driving transistor, from the L·di/dt voltage spike at turn-off. 4. 1000 N. 5. Any of: on-state voltage drop and heating, off-state leakage, fails short.