MEMS

Resonant microsensors

TF Consulting provides specialized consulting and management services for high-tech industries, with a core focus on MEMS (Micro-Electro-Mechanical Systems) and microsystems technology.We support our clients across the entire product lifecycle with tailored Product Lifecycle Management (PLM) services for sensors, actuators, and complex microsystems. Our expertise includes:

Development & Innovation – End-to-end support for implementing deep-tech and micromechanics solutions, from concept to series production.
Technology Fields – Deep know-how in numerical simulation (FEM, CFD, multiphysics), software engineering, sensor technology, semiconductor processes, and data science (AI).

Si-Kraftsensoren mit piezoelektrischem Antrieb

How to create a good sensor design

Developing silicon microsensors from scratch is extremely time-consuming and costly. Achieving the required accuracy while keeping error rates low is particularly challenging due to the complex multi-physical interactions at the microscale.That is why TF Consulting supports you with a smart combination of analytical methods and high-performance numerical simulation techniques – especially FEM (Finite Element Method) and CFD (Computational Fluid Dynamics).

Our Approach

For simple structures such as a micro-resonator, we can quickly estimate the desired operating range and the sensor’s characteristic curve using analytical models.
For complex, coupled systems, we apply advanced numerical methods from mathematical approximation theory to deliver precise, reliable, and optimization-ready results.

This hybrid methodology significantly reduces development time and risk, while enabling you to reach market-ready designs faster and with higher confidence.

Analytical description

For a geometrically simplified resonance device – such as a beam or membrane – the fundamental resonance frequency can be estimated analytically using the following function:

where:

x = length of the resonator (m)
y = thickness (or characteristic thickness) of the structure (m)
$Ered \approx1.696\times1011 Pa ($ reduced modulus for silicon, plane stress approximation)
$ρ(Si) \approx 2330 kg/m3$ (density of silicon)

General form

More general form for beams and plates (Euler-Bernoulli / Kirchhoff plate theory):

where:

λ(mn)

is a dimensionless coefficient that depends on the boundary conditions (clamped, free, simply supported, etc.) and the mode indices (m,n).

Geometric influence

This formulation clearly shows the strong dependence of resonance frequency on geometry: linearly proportional to thickness and inversely proportional to length squared – a key insight for rapid design iteration of MEMS resonators.

Analytical Scaling

The fundamental resonance frequency of a thin beam or plate follows a strong geometric scaling law. Fundamental mode (1,1):

This scaling results directly from Euler-Bernoulli beam theory (for beams) or Kirchhoff plate theory (for plates).

3D plot

This 3D plot shows how the resonance frequency scales with length (L) and thickness (h) for different modes. Higher modes sit significantly above the fundamental mode (1,1).

Modal Analysis

Beams (1D):

Analytical solutions exist for many boundary conditions.

Plates (2D):

More complex; often require numerical methods (like FEM) or approximations.
Two indices (m,n) for vibration modes.
Clamped boundaries increase frequencies significantly compared to simply-supported.

Mode Shapes

This visualization shows the first four normalized cantilever beam mode shapes (fixed-free) according to Euler-Bernoulli beam theory – ideal for MEMS cantilevers and resonators.

Features

Mode 1: Fundamental bending mode (no nodes along the beam except at the fixed end).
Mode 2–4: Higher-order modes with increasing number of nodes.
Vertical offset applied for clarity.

These mode shapes are essential when designing resonators, as each mode has a significantly higher natural frequency (scaling roughly with the square of the eigenvalue βₙ).

3D vibration mode shapes

3D Vibration Mode Shapes of a clamped square plate MEMS resonator. This visualization uses 3D surfaces to show the actual physical deflection of the plate for the first four modes:

Mode (1,1) – Fundamental
Mode (2,1)
Mode (1,2)
Mode (2,2)

Color scale: Red = upward displacement, Blue = downward displacement.

Micromechanical sensors

To optimize the performance of resonant silicon and quartz microsensors, a deep understanding of their dynamic behavior is essential. Accurate modeling and simulation are key to achieving this efficiently.

Analytical Modeling

For simple resonator geometries with well-defined (ideal) boundary conditions, analytical models deliver fast and precise predictions. These models assume homogeneous and isotropic material properties and are particularly useful for rapid design iterations and first-order optimization of key parameters such as resonance frequency and sensitivity.

Numerical Simulation

For complex resonator structures, arbitrary boundary conditions, anisotropic materials, temperature-dependent properties, or coupled multiphysics effects (e.g., electro-mechanical, thermo-mechanical, or fluid-structure interaction), we employ advanced numerical methods:

Finite Element Method (FEM) – for structural mechanics, eigenfrequency analysis, stress/strain distribution, and mode shapes.
Computational Fluid Dynamics (CFD) – for damping behavior, squeeze-film effects, or fluidic interactions.

This hybrid approach (analytical + numerical) enables TF Consulting to significantly reduce development time while delivering highly accurate performance predictions for MEMS resonant sensors.

Analytical modeling vs. FEM simulation

When to use Which approach:

Aspect	Analytical Modeling	FEM Simulation
Best suited for	Simple geometries (beams, membranes, basic plates)	Complex 3D geometries, irregular shapes
Boundary conditions	Ideal / simple (fixed, free, simply-supported)	Arbitrary, mixed, or real-world clamping
Material behavior	Homogeneous, isotropic	Anisotropic, temperature-dependent, layered materials
Speed	Very fast (seconds)	Slower (minutes to hours)
Accuracy	High for ideal cases	High for complex real-world conditions
Multiphysics coupling	Limited	Excellent (thermo-mechanical, piezo-electric, fluid-structure, etc.)
Typical use cases	Quick frequency estimation, scaling laws, initial design optimization	Detailed mode shapes, stress analysis, damping, sensitivity, frequency drift
Computational resources	Negligible	Moderate to high
Insight into internal behavior	Good global parameters (e.g. f₀)	Full field results (stress/strain, temperature, deformation maps)
Development stage	Early concept & rapid iteration	Detailed design validation & optimization

Piezoelectric layers

The above beam/plate modal analysis and f₀(x, y) ∝ thickness / length² scaling directly apply here. Piezoelectric layers on silicon cantilevers or plates convert the mechanical resonance into electrical signals. Optimizing geometry (thickness, length) and material stack, Si + piezoelectric thin film, i.e. AlN or ZnO, is critical for resonance frequency and sensitivity.

Recommendation

As TF Consulting we use a hybrid methodology:

Start with analytical models for fast exploration and understanding of fundamental scaling laws.
Switch to FEM + CFD for complex structures, multiphysics effects, and final performance validation.

This combination delivers both speed and accuracy – minimizing development time while ensuring reliable sensor performance.

Focus areas of our Services

Development & Implementation

We support you from the initial idea through the entire development process — including conceptual design, modeling, simulation, and successful implementation of microsystems technology within your organization. Our goal is to turn innovative concepts into reliable, manufacturable products.

Multiphysics Simulation

Using state-of-the-art computer-aided methods such as Finite Element Method (FEM) and Computational Fluid Dynamics (CFD), we accurately predict the behavior of micro sensors and actuators before physical prototyping. This enables early optimization of performance, reliability, and robustness under real operating conditions.

Cross-Industry Applications

We optimize MEMS components and microsystems for demanding sectors, including:

Defence technology
Automotive industry
Internet of Things (IoT)
Semiconductor and sensor technology

Our expertise helps clients in high-tech industries achieve competitive advantages through superior sensor and actuator solutions.

Analytical + Numerical Excellence

We follow a smart hybrid methodology:

Analytical models for quick insights and fundamental understanding.
High-fidelity FEM & CFD simulations for complex, real-world conditions and final validation.

This combination enables faster development cycles, lower prototyping costs, and superior product performance.

Our Approach: Analytical + Numerical Excellence: We combine the best of both worlds through our hybrid methodology.

GitHub Repositories

GitHub Software repositories

Patent

MEMS Patent

Testimonial

Dr.rer.nat. Franz Lärmer | Robert Bosch GmbH, Gerlingen

Retrospective

Development of Microsensors

FE modeling of resonant sensors

Resonant Microsensors

Reliable Product Development