Calculate d33/d31 coefficients, charge sensitivity, thickness-mode resonance frequency, and electromechanical coupling factor k33. Material presets for PZT-4, PZT-5H, PVDF, and BaTiO₃.
Material & Geometry
Material Preset
d33 coefficient
pC/N
d31 coefficient
pC/N
Rel. permittivity ε33/ε₀
Compliance s11 [pm²/N]
pm²/N
Density ρ [kg/m³]
kg/m³
Element thickness t [mm]
mm
Electrode area A [mm²]
mm²
Applied voltage V [V]
V
Results
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Displacement δ₃₃ [nm]
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Charge sensitivity [pC/N]
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Resonance fr [kHz]
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Coupling factor k33
Freq
▲ Thickness vs Resonance Frequency (d33 mode)
▲ Applied Voltage vs Actuator Displacement (d33 vs d31)
CAE Note
Piezoelectric elements are modeled in ANSYS Coupled-Field analysis (SOLID226) and ABAQUS using piezoelectric element formulations. The d-matrix is entered as a material constant for coupled electromechanical simulation. Used in MEMS design, AE sensor calibration, and ultrasonic transducer development.
What exactly is a piezoelectric material? I know it can turn electricity into motion, but how does that work on the inside?
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Basically, it's a special crystal or ceramic where the atoms are arranged in a non-symmetrical lattice. When you apply a voltage, the positive and negative ions shift slightly, causing the whole crystal to physically stretch or shrink. In this simulator, the `d33 coefficient` you see is the key number that tells you how much it moves per volt. Try setting it to 500 pm/V for a typical PZT ceramic and see the displacement change.
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Wait, really? So if it works one way, does it work in reverse? Can motion create electricity?
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Exactly! That's the sensor effect. Squeeze the material, and you push those ions out of place, generating a voltage across it. The same `d33` coefficient governs this effect too. For instance, in a knock sensor in your car engine, vibrations create a voltage signal. The simulator's "Charge Sensitivity" output, $S_q$, calculates the total charge generated when you press on the electrode area `A`.
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That makes sense. But I see a `Resonance Frequency` output. Why is that so important for these devices?
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Great question. Like a tuning fork, a piezoelectric element has a natural frequency where it vibrates most efficiently. This is critical for applications like ultrasonic cleaners or medical imaging. The frequency depends on the material's stiffness (`Compliance s11`) and `Density`, and the `Element Thickness`. Slide the thickness `t` control and watch the resonance frequency change dramatically—it's inversely proportional!
Physical Model & Key Equations
The fundamental actuator equation describes how much a piezoelectric stack expands or contracts in the direction of the applied electric field (the "33" mode). The displacement is directly proportional to the voltage applied.
$$\delta_{33}= d_{33}\cdot V$$
$\delta_{33}$: Thickness displacement [m] $d_{33}$: Piezoelectric charge coefficient [m/V] (a key material property) $V$: Applied voltage across the element [V]
When used as a sensor, the total electrical charge generated on the electrodes for a given stress is calculated. The resonance frequency determines the upper limit for efficient actuation or sensing and is defined by the element's geometry and material properties.
$S_q$: Charge sensitivity [C/N] $A_e$: Electrode area [m²] $f_r$: Fundamental resonance frequency [Hz] $t$: Element thickness [m] $\rho$: Material density [kg/m³] $s_{33}^E$: Elastic compliance at constant electric field [m²/N]
Frequently Asked Questions
PZT-4 is suitable for high-voltage, high-power actuators with high mechanical Q, while PZT-5H is intended for high-sensitivity sensors with a large d33 constant. PVDF is flexible and suitable for thin-film sensors and broadband applications. Each preset automatically sets representative physical property values (density, elastic compliance, piezoelectric constants).
d33 is used when calculating displacement and charge sensitivity in the thickness direction. d31 is required when utilizing in-plane expansion and contraction (e.g., in bimorph actuators). In sensor design, d33 directly contributes to charge sensitivity, while in actuator design, d33 is used for thickness displacement and d31 for bending displacement calculations.
The calculation formula assumes a simple thickness vibration mode, but actual elements are affected by factors such as electrode mass, adhesive layers, support conditions, and temperature changes. Additionally, material density and elastic compliance values can vary depending on manufacturing lots and polarization treatment conditions, so please note that the preset values are representative values.
Theoretically, k² represents the conversion efficiency between electrical and mechanical energy, so it falls within the range of 0 to 1. If a value exceeding 1 is displayed, there may be an inconsistency in the combination of input piezoelectric constants, elastic constants, or dielectric constants. Please especially check whether the unit system (SI units) is consistent.
Real-World Applications
Ultrasonic Cleaners & Medical Imaging: A high-frequency AC voltage drives the piezoelectric element at its resonance frequency, creating intense ultrasonic waves in a fluid. This is used to clean jewelry or, in medical probes, to send and receive sound waves for imaging internal organs.
Precision Positioning & Microscopy: The precise, voltage-controlled motion of piezoelectric actuators is perfect for aligning optical fibers or positioning samples in atomic force microscopes (AFM). Movements can be controlled down to the nanometer scale.
Knock Sensors & Vibration Monitoring: In automotive engines, a piezoelectric sensor mounted on the engine block converts mechanical vibrations from "knocking" into an electrical signal for the engine computer. This principle is also used in structural health monitoring of bridges and aircraft.
Energy Harvesting: Piezoelectric patches can be placed in high-vibration environments, like inside a shoe or on machinery, to convert otherwise wasted mechanical energy into small amounts of usable electrical power for sensors or wireless transmitters.
Common Misconceptions and Points to Note
When you start using this simulator, there are a few points you need to be careful about, or your design might deviate from reality. First and foremost, the idea that "a larger d33 value always means a better actuator" is a misconception. While a large d33 does allow for large displacements with small voltages, the material also tends to be "softer." For example, PZT-5H has a large d33 but low stiffness, making it unsuitable for applications requiring high output force (blocking force). Conversely, PZT-4 has a slightly smaller d33 but is harder and capable of more powerful actuation. It's crucial to choose your material based on whether your application prioritizes "displacement" or "force."
Next, don't forget that resonant frequency calculations assume "free vibration." The formula $$f_r = \frac{1}{2t}\sqrt{\frac{1}{\rho \cdot s_{33}^E}}$$ provided by the simulator is for the ideal state where the element is not fixed to anything. In reality, you'll bond it with adhesive or clamp it in a holder, so the resonant frequency will be lower than the calculated value. For instance, a calculation might yield 100 kHz, but an actual measurement of 80 kHz is not uncommon. Aim for a design with a good safety margin.
Finally, remember that charge sensitivity Sq is an indicator you must consider alongside "noise." Higher sensitivity allows detection of minute signals, but it also makes the system more susceptible to external electrical noise. Especially for sensors using high-impedance materials like PVDF, shielding of wiring and selection of amplifier input impedance are extremely important. Before you rejoice that "the sensitivity is sufficient!" in the simulator, you need a design that considers the noise floor of the entire signal processing chain.