NANOHUB TOOL MANUAL
Micro-Electro-Mechanical Simulation Lab
(MEMSLab)
User’s Guide
Oluwatosin Adeosun
Sambit Palit
Ankit Jain
Muhammad A. Alam
Sponsors:
Network for Computational Nanotechnology (NCN)
Purdue University Center for Prediction of Reliability,
Integrity and Survivability of Microsystems (PRISM)
Summer Undergraduate Research Fellowship (SURF)
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Introduction
Electro-mechanical (EM) actuators have diverse applications both as an analog (tunable) and a digital
(switch) element. Analog applications involve continuous position control of a movable electrode, e.g., external
cavity tunable lasers, reflective diffraction grating, deformable mirrors for adaptive optics, RF-MEMS varactors, etc.
Digital operation requires only a binary position control of the movable electrode e.g., RF-MEMS capacitive/ohmic
switches, interferrometric Mirasol displays, etc.
An electro-mechanical actuator can be modeled as a spring-mass system, with a point mass of value 𝑚,
attached to a linear spring of spring constant 𝑘. The figure below illustrates the system.
The dielectric on the bottom electrode is present to avoid electrode-electrode contact. The dynamic
behavior of the spring mass system is as follows –
𝑚
𝑑2 𝑦
𝑑𝑦
= 𝐹𝑒𝑙 − 𝐹𝑠 − 𝑏 .
2
𝑑𝑡
𝑑𝑡
Here 𝐹𝑒𝑙 is the downward acting electrostatic force, 𝐹𝑠 is the restoring spring froce, 𝑦 is the displacement
of the mass from the equilibrium position and 𝑏 is the damping coefficient. In this case, 𝐹𝑠 = 𝑘(𝑦0 − 𝑦) and 𝐹𝑒𝑙 =
𝜖0 𝐴𝑉 2
2
𝑦
2( 𝑑 +𝑦)
, where 𝐴 is the elecrode area, 𝑉 is the applied voltage, 𝑦𝑑 is the dielectric thickness, 𝜖𝑟 is the dielectric
𝜖𝑟
constant and 𝑦0 is the air-gap.
In general, the top electrode of electromechanical actuators have a specific support configuration, and
the electrode does not move freely unlike the spring mass system. In order to determine the shape of the beam for
a given voltage, the Kirchhoff-Love plate theory is used. The dynamic equation for a differential area element on
the electrode is given by –
𝜌𝐻
𝑑2𝑦
𝜖0 𝑉 2
𝐸𝐻3
𝑑4𝑦
𝑑𝑦
=
−
−𝑏 .
2
𝑦𝑑
𝑑𝑡 2
12(1 − 𝜈 2 ) 𝑑𝑥 4
𝑑𝑡
( + 𝑦)
𝜖𝑟
This equation is solved using the Finite Difference method for boundary conditions imposed based on the
electrode geometry and support configurations. Here 𝜌 is the density of the top electrode material, 𝐻 is the
thickness of the electrode, 𝐸 is the Young’s modulus, and 𝜈 is the Poisson’s ratio.
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The actuation behavior of a typical electromechanical actuator is hysteretic in nature, primarily defined by
four performance metrics – the pull-in voltage (𝑉𝑃𝐼 ), the pull-out voltage (𝑉𝑃𝑂 ), the below pull-in capacitive
response (𝐶𝑏𝑝 − 𝑉) and the post pull-in capacitive response (𝐶𝑝𝑝 − 𝑉). These are indicated in the figure below.
In the compact model developed by Palit et. al., these four performance metrics can be represented by
simple analytical scaling expressions, and can be estimated without the need of computationally expensive finite
element simulations and/or experimental fabrication and subsequent characterization.
The scaling relations for the pull-in voltage (𝑉𝑃𝐼 ), pull-out voltage (𝑉𝑃𝑂 ), below pull-in 𝐶𝑏𝑝 − 𝑉 response
and post pull-in 𝐶𝑝𝑝 − 𝑉 response are given by –
𝐾0 𝑧02
𝑉𝑃𝐼 = 𝛼𝑃𝐼 √
(𝑛 − 1)𝐶𝑂𝐹𝐹
𝑉𝑃𝑂 = 𝛼𝑃𝑂 √
= 𝛼𝑃𝐼 𝑉0 ,
𝐾0 𝑧02
𝑧0 𝛾
( ) = 𝛼𝑃0 𝑉1 ,
(𝑛 − 1)𝐶𝑂𝑁 𝑇𝑑
𝐶𝑏𝑝
𝑉
= 𝑓̌ (
),
𝐶𝑂𝐹𝐹
𝑉𝑃𝐼
𝐶𝑂𝑁 − 𝐶𝑃𝑂
𝑉 𝜅
= (
) .
𝐶𝑂𝑁 − 𝐶𝑃𝑃
𝑉𝑃𝑂
Here, 𝛼𝑃𝐼 , 𝛼𝑃𝑂 , 𝛾, 𝑓̃ and 𝜅 are scaling variables/functions which depend purely on the top electrode
geometry and support configuration, but are independent of the specific physical dimensions and material
properties. Here 𝐾0 =
𝐸𝐻 3
6(1−𝜈2 )𝐿3
, 𝑧0 = 𝑦0 +
𝑦𝑑
𝜖𝑟
, 𝐶𝑂𝐹𝐹 is the off-state capacitance, 𝐶𝑂𝑁 is the theoretically possible
maximum on-state capacitance (when the entire top electrode is in contact with the dielectric), 𝐶𝑃𝑂 is the
capacitance at the point of pull-out instability. For exact values of these scaling parameters for cantilever, fixedfixed and circular configurations, please refer to Palit et. al.
MEMS actuators have multiple design applications. Understanding their behavior as well as the ability to
predict their actuation characteristics and voltage response is important when designing these actuators. In order
to determine how these devices will behave, designers have to perform computationally expensive finite element
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simulations or design experiments that can be very time consuming. This tool is created to allow users to enter
basic information about a MEMS actuator and obtain a reasonably accurate estimation of the actuation response
of the actuator. A compact model based on a scaling theory provides an almost instantaneous estimation of the
actuation voltages (pull-in voltage and pull-out voltage) and the hysteretic CV characteristics (for both below pullin and post pull-in states). A more comprehensive numerical simulation option, which employs the Kirchoff-Love
plate equation, is also available. Quasi-static and dynamic response of the cantilever, fixed-fixed beam and circular
shapes can be obtained using the numerical simulation. A method-of-moments based 3D Poisson solver can also
be invoked if necessary. For users who want to understand the basics of MEMS actuator operation, quasi-static
and dynamic simulations can be performed for a simple spring-mass model.
In this manual, we will explain how to use the tool, describing the input pages and providing illustrative
examples of outputs.
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Inputs and Settings
Input page 1: Simulation settings
The following input options are available in this input panel:
a)
Electrode geometry: This lets the user choose the geometrical shape (and support configuration) of the
actuating top electrode. There are four available choices:
a. Cantilever: Rectangular beam, fixed support at one end, free on the other end.
b. Fixed-Fixed: Rectangular beam, fixed support at both ends.
c. Circular: Circular beam, fixed support along the circumference.
d. Spring-mass: A point mass attached to a linear spring.
These options for the available electrode geometries have been explained pictorially on the tool.
b) Simulation type: There are two available options – “Compact model” and “Numerical model”. Choosing
the compact model will utilize the scaling theory developed by Palit et. al. for obtaining actuation voltages
and CV characteristics for the cantilever, fixed-fixed beam and circular geometries. The numerical model
on the other hand will solve the Kirchhoff-Love plate equation instead. Note that choosing this option will
increase the computational time significantly. If the electrode geometry is chosen to be “Spring-mass”,
this option is disabled because a spring-mass system is always solved numerically in this tool.
c)
3D Poisson: This switch enables the use of a method-of-moments based Poisson solver for calculating
the electrostatic force on the actuating membrane. This option is available only for numerical simulation.
Note that choosing this option will increase the computational time significantly. When disabled, the
electrostatic force is calculated by the parallel-plate assumption i.e., 𝐹𝑒𝑙 =
𝜖0 𝑉 2
2
𝑦𝑑
2( +𝑦)
𝜖𝑟
.
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Input page 2: Voltage settings
This page allows the user to select the type of analysis to be performed on the geometry configuration based on a
simulation mode selected by the user on input page 1. Even though all the analysis modes are initially turned off by
default, at least one analysis mode must be enabled. Out of the following, only the “DC analysis” mode is enabled
if compact model simulation is invoked on input page 1.
The following input options are available in this input panel:
a)
DC analysis: Perform quasi-static forward and reverse voltage sweeps on the actuator. There are three
user input parameters for DC analysis.
a. Voltage range: User can chose between a “Default” sweep, or specify a “Custom” sweep. The
default sweep performs a forward and reverse sweep between 0 and an estimated value for
1.5𝑉𝑃𝐼 , using approximately 200 voltage points for calculation.
b. Increment by: Voltage step (Δ𝑉) for custom sweep. Units: V. Default: 0.1V.
c. Stop at: Maximum voltage (𝑉𝑚 ) for custom sweep. Units: V. Default: 10V.
Both forward and reverse sweeps between 0 and 𝑉𝑚 are performed if custom sweep is selected. The
latter two options are disabled if default sweep has been selected.
b) Transient analysis: Performs the time-dependent dynamic simulation on the actuator. There are a total
of five parameters available to the user for specifying the time-dependence of input voltage with time.
a. Voltage type: User can choose between a “Step voltage” or “Ramp voltage”.
b. Run Time: Duration for which the step voltage is applied (𝑡𝑠𝑡𝑒𝑝 ). Units: s. Default: 0.01s.
c. Voltage: Step voltage value (𝑉𝑠𝑡𝑒𝑝 ). Units: V. Default: 3V.
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d.
e.
Ramp rate: Rate of change of applied voltage (𝑅𝑟𝑎𝑚𝑝 ). Units: V/s. Default: 100V/s.
Final voltage: Maximum ramped voltage value (𝑉𝑟𝑎𝑚𝑝 ). Units: V. Default: 3V.
Inputs (b) and (c) are enabled if step voltage has been selected as the “Voltage type”. On the other hand,
inputs (d) and (e) are enabled if ramp voltage has been selected. For step voltage analysis, the time
dependence of applied voltage is given by:
𝑉(𝑡) = {𝑉
0
𝑠𝑡𝑒𝑝
𝑡<0
𝑡𝑠𝑡𝑒𝑝 ≥ 𝑡 ≥ 0
For ramp voltage analysis, the time dependence of applied voltage is given by:
0
𝑉(𝑡) = {𝑅
𝑟𝑎𝑚𝑝 𝑡
𝑡<0
𝑉𝑟𝑎𝑚𝑝
≥𝑡≥0
𝑅𝑟𝑎𝑚𝑝
Note that a nominal damping coefficient (100 N.s/m3 – value is per unit area) is used for transient analysis
so that any resulting oscillations do not continue forever. The simulation is stopped as soon as a pull-in
event is detected and no bouncing of the electrode at the dielectric surface has been accounted.
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c)
AC analysis: Performs time-dependent small-signal dynamic simulation on the actuator. There are a total
of four input parameters available to the user for specifying the variance of input voltage with time.
a. DC bias: DC biasing point for small-signal analysis (𝑉𝑑𝑐 ). Units: V. Default: 2V.
b. AC amplitude: Small signal sinusoid amplitude (𝑉𝑎𝑐 ). Units: V. Default: 0.1V.
c. Frequency: Small signal sinusoid frequency (𝑓). Units: Hz. Default: 1000Hz.
d. Time: Duration for which small signal voltage is applied (𝑡𝑎𝑐 ). Units: s. Default: 0.01s.
For this analysis mode, the time dependence of the applied voltage is given by:
0
𝑉(𝑡) = {
𝑉𝑑𝑐 + 𝑉𝑎𝑐 sin(2𝜋𝑓𝑡)
𝑡<0
𝑡𝑎𝑐 ≥ 𝑡 ≥ 0
Note that a nominal damping coefficient (100 N.s/m3 – value is per unit area) is used for AC analysis. AC analysis is
performed only in the below pull-in state. The simulation is stopped as soon as a pull-in event is detected and no
bouncing of the electrode at the dielectric surface has been accounted.
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Input page 3: Dimensions and Material Properties
The following input parameters are available in the two tabs of this input page:
a) Length of Electrode: Length scale of the top actuating electrode (𝐿). Units: m. Default: 400 𝜇m.
b) Width of Electrode: Width of the top actuating electrode (𝑊). This option is disabled if the circular
geometry is selected on input page 1. Units: m. Default: 100 𝜇m.
c) Thickness of Electrode: Electrode thickness (𝐻). Units: m. Default: 2 𝜇m.
d) Air-gap: Initial air-gap between actuating top electrode and top of the dielectric (𝑦0 ). Units: m. Default: 3
𝜇m.
e) Dielectric thickness: Thickness of the dielectric layer deposited on the bottom electrode (𝑦𝑑 ). Units: nm.
Default: 250 nm.
f) Number of grids: Number of grids placed along 𝐿 (𝑁𝑥 ). The same grid spacing is used along the width
direction. This input is enabled only if Numerical model is selected in input page 1. Default: 40.
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g)
Choose electrode material: Select between a set of pre-defined materials, or define a “Custom” material
for the top actuating electrode. Default: Nickel.
h) Young’s modulus: Young’s modulus (𝐸) of the top actuating electrode. Units: GPa. Default: 200GPa.
i) Poisson’s ratio: Poisson’s ratio (𝜈) of the top actuating electrode. Units: none. Default: 0.31.
j) Density: Density (𝜌) of the top actuating electrode. Units: kg/m3. Default 8900 kg/m3.
k) Choose dielectric material: Select between a set of pre-defined materials, or define a “Custom” material
for the dielectric layer on the bottom electrode. Default: Silicon nitride.
l) Dielectric constant: Dielectric constant (𝜖𝑟 ) of the dielectric layer. Units: none. Default: 7.9.
Inputs (h), (i) and (j) are enabled only if “Custom” option is selected while specifying the top electrode material.
Similarly, input (l) is enabled only if “Custom” option is selected while selecting the dielectric material.
If the “Spring-Mass” geometry is selected from input page 1, the effective spring constant (𝑘) and the mass (𝑚) is
estimated using the inputs on this page as follows:
𝑘=
𝐸𝐻3 𝑊
(1 − 𝜈 2 )𝐿3
𝑚 = 𝜌𝐻𝐿𝑊
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Illustrative examples
DC analysis: Spring mass model
This section summarizes the results obtained for DC analysis performed on the spring-mass model.
The figure below shows the 𝐶 − 𝑉 characteristics of the system. Default parameters are used. The actuation
voltages 𝑉𝑃𝐼 and 𝑉𝑃𝑂 and indicated on the figure. The blue curve is the forward sweep, the red curve is the reverse
sweep.
VPO
VPI
The figure below plots the steady state position of the actuating electrode as a function of voltage for the forward
sweep. Observe that at 𝑉𝑃𝐼 , the electrode snaps down and comes into contact with the dielectric layer. The initial
beam position is given by 𝑦0 .
VPI
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DC analysis: Cantilever with compact model
This section summarizes the results obtained for DC analysis performed on the cantilever geometry, using the
compact model based on scaling equations derived by Palit et. al. Similar qualitative results may be obtained for
fixed-fixed beam and circular geometries as well.
The figure below shows the 𝐶 − 𝑉 characteristics of the system. Default parameters are used. The actuation
voltages 𝑉𝑃𝐼 and 𝑉𝑃𝑂 are indicated on the figure. The blue curve is the forward sweep; the red curve is the reverse
sweep. Observe that the 𝐶 − 𝑉 behavior in the pulled-in state is markedly different from that obtained using the
spring-mass model because the electrode area in contact with the dielectric varies with applied voltage.
VPO
VPI
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DC analysis: Cantilever with numerical model
This section summarizes the results obtained for DC analysis performed on the cantilever geometry using
numerical simulation based on the Kirchhoff-Love plate theory. Similar qualitative results may be obtained for
fixed-fixed beam and circular geometries as well.
The figure below shows the 𝐶 − 𝑉 characteristics of the system. Default parameters are used. The blue curve is the
forward sweep; the red curve is the reverse sweep.
VPO
VPI
The figure below plots the steady state position of the maximum deflection point on the actuating electrode as a
function of voltage for the forward sweep and reverse sweep.
VPO
VPI
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If we increase the number of grid points, the steps appearing on the 𝐶 − 𝑉 characteristics disappear, and we get
smoother curves. The figure below compares the results obtained from using the default parameters (𝑁𝑥 = 40)
and those obtained by using 𝑁𝑥 = 100. Also plotted are 𝐶 − 𝑉 characteristics as predicted by the compact model.
The steps, therefore are a result of coarse gridding. However, increasing the number of grid-points will increase
the simulation time drastically. However, one observes that even by using a coarse grid, the actuation voltages are
predicted reasonably well.
Nx=100
Nx=40
Compact
model
In the figure below, we evaluate the effect of using the 3D Poisson solver over the simple parallel-plate assumption
for computing electrostatic force. Again, we observe that even though using the 3D solver gives us a slightly
different value for the actuation voltage; the post pull-in 𝐶 − 𝑉 characteristics are identical to the results from the
parallel plate assumption. One must therefore evaluate the tradeoff between getting a more accurate result using
the 3-D Poisson solver and the massively high computational cost required for the simulation.
Without 3D
With 3D
Compact
model
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In the DC analysis mode, it is possible to observe the actual deflections on the beam for all voltage points on the
𝐶 − 𝑉 sweep. The figures below show the beam shapes at voltage just below 𝑉𝑃𝐼 (voltage point = 66) during the
forward sweep, and at voltage just above 𝑉𝑃𝑂 (voltage point = 170) during the reverse sweep. The beam shapes
can be viewed in different modes as shown. For these simulations, default parameters were used.
Voltage point 66:
Voltage point 170:
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Transient analysis: Spring-mass with step voltage
This section summarizes the results obtained for transient step voltage simulation using the spring-mass model.
The figure below shows the 𝐶 − 𝑡 characteristics for the simulation. In this case, 𝑉𝑠𝑡𝑒𝑝 = 1.5𝑉, and other
parameters were default. Observe that the electrode eventually stabilizes in the air-gap.
The figure below plots the position of the actuating electrode as a function of time.
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The figure below shows the 𝐶 − 𝑡 characteristics for the simulation. Default parameters were used in this case.
Observe that the electrode eventually snaps down to the dielectric. Simulation stops as soon as the pull-in event is
detected.
The figure below plots the position of the actuating electrode as a function of time.
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Transient analysis: Spring-mass with ramp voltage
This section summarizes the results obtained for transient ramp voltage simulation using the spring-mass model.
The figure below shows the 𝐶 − 𝑡 characteristics for the simulation. Default parameters were used in this case.
Observe that the electrode eventually snaps down to the dielectric. Simulation stops as soon as the pull-in event is
detected.
The figure below plots the position of the actuating electrode as a function of time.
The figure below plots the variation of applied voltage as a function of time.
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For this simulation, one may also plot the variation of capacitance and beam position with respect to voltage.
These results are not shown here.
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Transient analysis: Cantilever with numerical model and step voltage
This section summarizes the results obtained for transient step voltage simulation on the cantilever geometry.
Similar qualitative results may be obtained for fixed-fixed beam and circular geometries as well.
The figure below shows the 𝐶 − 𝑡 characteristics for the simulation. Default parameters were used. Observe that
the electrode eventually stabilizes in the air-gap.
The figure below plots the position of the point on actuating electrode with maximum displacement as a function
of time.
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The figure below shows the 𝐶 − 𝑡 characteristics for the simulation. In this case, 𝑉𝑠𝑡𝑒𝑝 = 8𝑉, and other parameters
were default. Observe that the electrode eventually snaps down to the dielectric. Simulation stops as soon as the
pull-in event is detected.
The figure below plots the position of the point on actuating electrode with maximum displacement as a function
of time.
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Transient analysis: Cantilever with numerical model and ramp voltage
This section summarizes the results obtained for transient ramp voltage simulation on the cantilever geometry.
Similar qualitative results may be obtained for fixed-fixed and circular geometries as well.
The figure below shows the 𝐶 − 𝑡 characteristics for the simulation. Default parameters were used in this case.
Observe that the electrode eventually snaps down to the dielectric. Simulation stops as soon as the pull-in event is
detected.
The figure below plots the position of the point on actuating electrode with maximum displacement as a function
of time.
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The figure below plots the variation of applied voltage as a function of time.
For this simulation, one may also plot the variation of capacitance and beam position with respect to voltage.
These results are not shown here.
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AC analysis: Spring-mass
This section summarizes the results obtained for AC voltage simulation using the spring-mass model.
The figure below shows the 𝐶 − 𝑡 characteristics for the simulation. In this case, 𝑉𝑑𝑐 = 1𝑉, and other parameters
were default.
The figure below plots the position of actuating electrode as a function of time.
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The figure below plots the variation of applied voltage as a function of time.
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AC analysis: Cantilever with numerical model
This section summarizes the results obtained for AC voltage simulation on the cantilever geometry. Similar
qualitative results may be obtained for fixed-fixed and circular geometries as well.
The figure below shows the 𝐶 − 𝑡 characteristics for the simulation. Default parameters were used.
The figure below plots the position of the point on actuating electrode with maximum displacement as a function
of time.
The figure below plots the variation of applied voltage as a function of time.
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Exercises
Switch design
Design a cantilever switch having a pull-in voltage of approximately 40𝑉, pull-out voltage of
approximately 3𝑉 and off-capacitance of 100𝑓𝐹. Change only the beam thickness (𝐻), dielectric
thickness (𝑇𝑑 ) and width (𝑊), keeping other parameters as default. Use the “compact model” simulation
mode to iterate between guesses. Verify by running the “numerical simulation” that your designed
switch works as expected.
Hint1: Changing 𝑇𝑑 will affect 𝑉𝑃𝑂 , but has minimal effect on 𝑉𝑃𝐼 .
Hint2: Changing 𝑊 affects capacitance, but does not change the actuation voltages.
Answer: 𝑊 = 85𝜇𝑚, 𝑇𝑑 = 50𝑛𝑚, 𝐻 = 3.75𝜇𝑚.
Resonance
Estimate the natural resonance frequency of a spring-mass system (using the default parameters) at
𝑉 = 0.5𝑉 and 𝑉 = 2.0𝑉. For this, apply a step of the given voltage. The initial oscillation frequency
(before it gets damped out) is the natural resonance frequency at the given voltage.
Verify the correctness of the estimated resonance frequencies by observing the AC capacitance change
(Δ𝐶) in the cyclo-stationary state for AC simulation with frequencies of 0.9𝑓𝑅𝐸𝑆 , 𝑓𝑅𝐸𝑆 and 1.1𝑓𝑅𝐸𝑆 , and
using 𝑉𝐴𝐶 = 0.01𝑉. For this, measure the peak-to-peak difference in capacitance when the oscillation
frequency stabilizes to the forcing input frequency (also called the cyclo-stationary state). Δ𝐶 at
resonance should be maximum because the forcing input frequency matches the natural oscillation
frequency, resulting in positive feedback.
Answer:
At 𝑉 = 0.5𝑉, 𝑓𝑅𝐸𝑆 = 2852𝐻𝑧.
Δ𝐶(0.9𝑓𝑅𝐸𝑆 ) = 0.199𝑓𝐹, Δ𝐶(𝑓𝑅𝐸𝑆 ) = 0.212𝑓𝐹 and Δ𝐶(1.1𝑓𝑅𝐸𝑆 ) = 0.163𝑓𝐹. Note that Δ𝐶 maximizes
near 𝑓𝑅𝐸𝑆 .
At 𝑉 = 2.0𝑉, 𝑓𝑅𝐸𝑆 = 2210𝐻𝑧.
Δ𝐶(0.9𝑓𝑅𝐸𝑆 ) = 0.189𝑓𝐹, Δ𝐶(𝑓𝑅𝐸𝑆 ) = 0.206𝑓𝐹 and Δ𝐶(1.1𝑓𝑅𝐸𝑆 ) = 0.180𝑓𝐹.
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