Piezoelectricity

Direct and converse effects

Direct effect: Mechanical stress on a piezoelectric crystal → electric polarization (voltage) develops.

Converse effect: Electric field applied to a piezoelectric crystal → mechanical strain (deformation).

Both effects arise from the same underlying physics — coupling between mechanical and electrical degrees of freedom in non-centrosymmetric crystals.

Common piezoelectric materials

Material	Type	Key Use
Quartz (SiO₂)	Crystal, naturally piezo	Watches, oscillators, sensors
PZT (Pb-Zr-Ti-O ceramic)	Synthetic ferroelectric	Sonar, ultrasound, actuators
Barium titanate (BaTiO₃)	Ferroelectric	Capacitors, ceramics
PVDF (polyvinylidene fluoride)	Polymer	Flexible sensors, sonar arrays
Aluminum nitride (AlN)	Thin-film	RF filters in cell phones
Lithium niobate	Single crystal	SAW devices, optical modulators
Bone (collagen)	Biological	Possible role in bone remodeling

Piezoelectric coefficients

Piezoelectric coupling characterized by constants like d (charge per unit force, C/N) and g (voltage per unit stress, V·m/N):

Material	d₃₃ (10⁻¹² C/N)
Quartz	2.3
PZT (typical)	375
BaTiO₃	190
PVDF	~30
AlN thin film	5

Higher d → more responsive material.

Applications

Application	How it uses piezo
Quartz watches	Quartz crystal oscillator at 32,768 Hz
Microphones (some)	Sound pressure → voltage signal
Speakers (tweeters)	Voltage → diaphragm vibration → sound
Ultrasound (medical)	Piezo transducer emits/receives MHz waves
Ultrasonic cleaners	Piezo creates cavitation in liquid
Inkjet printers	Voltage pulse contracts piezo, ejects ink drop
Igniters (lighters, BBQ)	Mechanical impact → high voltage spark
Force sensors	Pressure → voltage; many automotive applications
Atomic force microscopes	Piezo positioners for sub-Å precision
SAW filters (cell phones)	Surface acoustic waves on piezo substrate filter signals

JavaScript — piezoelectric calculations

// Direct effect: voltage from applied force
function piezoVoltage(force_N, d33, area_m2, thickness_m, epsilon_r = 1700) {
  // Surface charge density σ = d_33 · stress = d_33 · F/A
  // V = σ · t / ε₀ε_r (for charge on capacitor)
  const epsilon_0 = 8.854e-12;
  const stress = force_N / area_m2;
  const charge_density = d33 * stress;  // C/m²
  return charge_density * thickness_m / (epsilon_0 * epsilon_r);
}

// PZT, 100 N applied to 1 cm² × 1 mm thick
const d33_PZT = 375e-12;
console.log(`PZT voltage: ${piezoVoltage(100, d33_PZT, 1e-4, 0.001).toFixed(0)} V`);

// Converse effect: deformation from voltage
function piezoStrain(voltage, d33, thickness) {
  // Strain = d_33 · E = d_33 · V/t
  return d33 * voltage / thickness;
}

// PZT actuator: 100 V on 1 mm thick
console.log(`Strain: ${piezoStrain(100, d33_PZT, 0.001).toExponential(2)}`); // 3.75e-5
console.log(`Displacement: ${(piezoStrain(100, d33_PZT, 0.001) * 0.001 * 1e9).toFixed(1)} nm`); // 37.5 nm

// Quartz crystal oscillator frequency
function quartzFrequency(thickness_mm, mode_constant = 1660) {
  // f = const / thickness for thickness mode
  return mode_constant / thickness_mm;  // kHz
}

// 50 µm thick AT-cut quartz
console.log(`50 µm quartz: ${quartzFrequency(0.050).toFixed(0)} kHz`); // 33,200 Hz

// Watch crystal: 32.768 kHz tuning fork
const watch_f = 32768;
const ticks_per_second = watch_f / 32768;
console.log(`Watch crystal: ${watch_f} Hz, divides by ${watch_f/1} for 1 Hz`);

// Energy stored in piezo capacitor
function piezoEnergy(C_pF, V) {
  return 0.5 * C_pF * 1e-12 * V * V;
}

console.log(`100 pF at 100V: ${piezoEnergy(100, 100).toExponential(2)} J`);  // 5e-7 J

// Piezoelectric energy harvesting (rough)
function harvestPower(force_amplitude_N, frequency_Hz, d33, area_m2, thickness_m, R_load_ohm) {
  // Very rough: P ~ ω · (d33 · F)² / (2 · C_load)
  const C = 8.854e-12 * 1700 * area_m2 / thickness_m;  // capacitance
  const omega = 2 * Math.PI * frequency_Hz;
  // Charge per cycle: Q = d33 · F
  const Q = d33 * force_amplitude_N;
  const V = Q / C;
  return V * V / R_load_ohm;
}

// 1 N at 1 Hz on PZT
console.log(`Harvest: ${harvestPower(1, 1, d33_PZT, 1e-4, 0.001, 1e6).toExponential(2)} W`);

Where piezo matters

• Sensors and actuators. Force, pressure, vibration sensors; precision positioning.
• Audio. Microphones, tweeters, ultrasonic transducers.
• Medical. Ultrasound imaging, lithotripsy (kidney stones), diagnostic sensors.
• Communications. Quartz oscillators (every digital device), SAW filters in phones.
• Industrial. Ultrasonic cleaning, welding, drilling, inkjet printing.
• Automotive. Knock sensors, fuel injectors, anti-skid systems.
• Energy harvesting. Vibration-powered IoT sensors, structural health monitoring, wearables.

Common mistakes

• Confusing piezoelectric with pyroelectric. Piezo — strain → voltage. Pyro — temperature change → voltage. Different physics, both useful.
• Treating piezo as battery. They generate brief pulses with stress changes; not steady DC. Don't last long with constant load.
• Ignoring temperature limits. Most piezos have Curie temperature above which piezoelectric properties vanish. Quartz ~573°C; PZT typically 250-350°C.
• Mismatched impedance. Piezos have very high source impedance. Need high-impedance amplifiers for sensing.
• Forgetting non-linear behavior. At high stress or voltage, piezo response saturates and can have hysteresis. Linear approximations break down.
• Applying DC voltage long-term. Can degrade many piezos via depolarization. AC or pulsed operation safer.