VM312

Overview

Reference:	Any basic Electrostatics textbook
Analysis Type(s):	Static Analysis (ANTYPE = 0) Linear Perturbation Static Analysis (ANTYPE = 0) Linear Perturbation Modal Analysis (ANTYPE = 2) Linear Perturbation Harmonic Analysis (ANTYPE = 3)
Element Type(s):	3D 20-Node Coupled-Field Solid (SOLID226) Spring-Damper (COMBIN14) Structural Mass (MASS21)
Input Listing:	vm312.dat VM312 requires a supplemental .cdb input file which is too long to include full input listings. This file must be downloaded and placed in your working directory for the test case to run properly. Additionally, the geometry and mesh should be regenerated. Download link: MAPDL Test Case Files for 2024 R2 vm312.cdb

Test Case

A capacitor made of two concentric conducting spheres of radii a and b (a < b) has its inner surface fixed and the outer surface attached to an evenly distributed set of mass-springs as shown in the problem sketch below. The space between the spherical electrodes is filled with air. A voltage differential is applied first between the conductors to determine the outer sphere displacement until the pull-in voltage (V_PI) is attained. Then the model is restarted from a selected bias voltage (V_DC), and a series of linear perturbation analyses are performed to determine the static, modal, and harmonic response of the DC-biased capacitor.

Figure 567: Spherical Capacitor Problem Sketch

Figure 568: Finite Element Model

Material Properties

Geometric Properties

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Free space permittivity:

ϵ₀ = 8.854x10^-6 pF/μm

Spring constant:

k = 60 kg/s²

Plate mass:

m = 1x10^-4 kg

Inner radius:

a = 1 μm

Outer radius:

b = 2 μm

Static Analyses:

Pull-in voltage:

V_PI ≈ 397 V

DC-bias voltage:

V_DC = 402.48 V

Linear Perturbation Analyses:

Voltage = 20 V

Displacement = 0.02 μm

Force = 1.3 μN

Frequency = 50 Hz

Analysis Assumptions and Modeling Notes

The space between the spherical electrodes is modeled using the electrostatic-structural analysis option (KEYOPT(1) = 1001) of SOLID226, to allow for the deformation of the air region induced by the electrostatic force acting between the electrodes. A negligible elastic modulus is assigned to this region. KEYOPT(4) is set to 1 to avoid applying the electrostatic forces to the interior nodes of the air region. Each node of SOLID226 located on radius b is attached to a spring modeled by spring-damper (COMBIN14). The stiffness of each spring is k/n where n is the total number of springs. Likewise, to account for the mass of the electrode, a point mass is defined by structural mass (MASS21) at each node of radius b. The non-radial components of displacement, UY and UZ, are all suppressed. The radial component UX is constrained to remain uniform on the outer sphere while set to zero on the inner sphere.

The analytical value for the pull-in voltage is 399.73 V. The nonlinear analysis will run until reaching 397 V. The test case is voltage-controlled, but the time steps are bisected based on the increments of displacement using CUTCONTROL,ULIMIT so that the convergence is retained as the pull-in voltage is approached. The displacement-voltage curve is shown in the figure below.

For a given value of displacement, the corresponding value of voltage is obtained from the analytical solution, and the ratio of numerical and analytical voltages averaged along the displacement-voltage curve is obtained by

where is the number of substeps.

Figure 569: Displacement vs. Applied Voltage

Results Comparison

Nonlinear Static Analysis till Pull-In Voltage:

	Target		Mechanical APDL	Ratio
Disp-volt curve		—	—	1.001746
Capacitance at V_DC		0.23571E-03	0.23572E-03	1.000077
Electrostatic force at V_DC		6.33582	6.35691	1.003329