ThirdOrder Intermodulation Distortion in Capacitively Driven CCBeam mMechanical Resonators

1
Third-Order Intermodulation Distortion in
Capacitively Driven CC-Beam mMechanical
Resonators

• Reza Navid, John R. Clark, Mustafa Demirci and
  Clark T.-C. Nguyen
• Center for Wireless Integrated MicroSystems
  (WIMS)
• Dept. of Electrical Engineering and Computer
  Science
• University of Michigan
• Ann Arbor, Michigan 48109-2122
• http//www.eecs.umich.edu/rnavid

2
Outline

• Background
• need for linearity in comm. circuits
• metrics for nonlinearity and distortion
• Intermodulation Distortion in CC-Beam mMechanical
  Resonators
• distortion generation mechanisms
• analytical expression for IIP3
• strategies for attaining better IIP3
• Measurement vs. Theory
• Conclusions

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Frequency Division Multiplexed Communications

• Information is transmitted in specific frequency
  channels within specific bands

Transmitted Power
Frequency
4
Ideal Case Perfectly Linear Filter

• An ideal filter removes interferers without
  generating any additional components

Band
Transmitted Power
Frequency
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Ideal Case Perfectly Linear Filter

• An ideal filter removes interferers without
  generating any additional components

GSM Band
Transmitted Power
Ideal Filter

• Need a filter with extremely high frequency
  selectivity

Interferers Removed
Frequency
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Highly-Selective Low-Loss Filters

• For high selectivity, need high Q ? use
  mechanical or acoustic resonance (e.g.,
  present-day SAW filters)
• Micromechanical filters have Q 10,000 (very
  high)
• problem small size and capacitive transduction
• question are they sufficiently linear for comms.?

fo 34.5MHz Qres 5,000
Wong et al., Trans99
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Case Nonlinear Filter

• Due to nonlinearity, interferers outside the
  filter passband can generate in-band distortion
  components

Band
Transmitted Power
Frequency
Nonlinearity Generates 3rd Order Intermodulation
Distortion (IM3)
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Case Nonlinear Filter

• Due to nonlinearity, interferers outside the
  filter passband can generate in-band distortion
  components

Transmitted Power
Actual Filter
Interferers Removed By the Filter
But IM3 Component Cannot Be Filtered Away
Frequency
Nonlinearity Generates 3rd Order Intermodulation
Distortion (IM3)
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Signal Propagation Through a Linear System
Sout
Sin
Linear System
Sout A1Sin
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Signal Propagation Through a Nonlinear System
Sout
Sin
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3rd Order Intermodulation Intercept Point

• Definition

Sout
Fundamental (slope1)
Output Amplitude, So dB
IM3 (slope3)
Input Amplitude, Si dB

• The higher the IIP3, the better the linearity
• IIP3 is independent of the input signal Si,
  making it useful as a metric for nonlinearity
• GSM standard requires a total IIP3 of -18dBm

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Outline

• Background
• need for linearity in comm. Circuits
• metrics for nonlinearity and distortion
• Intermodulation Distortion in CC-Beam mMechanical
  Resonators
• distortion generation mechanisms
• analytical expression for IIP3
• strategies for attaining better IIP3
• Measurement vs. Theory
• Conclusions

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Nonlinearity in Capacitively-Driven CC-Beams

• Typical displacement _at_ 10MHz x 100Å ltlt beam
  length
• neglect mechanical noninearity (e.g., Duffing)
• capacitive nonlinearity most important when gap lt
  0.1mm

CC-Beam Resonator
Anchor
Electrode

• Force

• Taylor series expansion
• gather 3rd order terms

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IM3 Generation in Capacitively-Driven mBeams
Force
Resonator
Displacement, x
Voltage, vi
w
w
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Closed Form IIP3 Expression

• Very strong dependence on electrode-to-resonator
  gap spacing!

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Summary of VIIP3 Dependencies
Overlap Area, Ao
Equivalent Circuit
Gap, do

• To attain better VIIP3 (i.e., better linearity)
• increase electrode-to-resonator gap, do
• increase stiffness, kreff
• decrease DC-bias, VP
• decrease electrode overlap area, Ao

• There is a clear trade-off between VIIP3
  (linearity) and Rx

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Deficiencies of the Simplified Formulation

• Problem The above simplified formulation ignores

(1) Finite Electrode Width
Underestimation in VIIP3
(2) Resonator Bending
Overestimation in VIIP3
kreff and d change across the electrode

• For more accurate results, must
• find the bending curve analytically
• integrate the force to find the effective FIM3
• Above implemented in a mathematica program to
  obtain theoretical numbers to follow

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Outline

• Background
• need for linearity in comm. Circuits
• metrics for nonlinearity and distortion
• Intermodulation Distortion in CC-Beam mMechanical
  Resonators
• distortion generation mechanisms
• analytical expression for IIP3
• strategies for attaining better IIP3
• Measurement vs. Theory
• Conclusions

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Measurement Setup
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Fabricated 10MHz CC-Beam mResonator

• Below CC-beam resonator constructed in boron
  implant-doped polysilicon annealed at 1050oC for
  1 hr.

Design/Performance Lr 40mm, Wr 8mm hr 2mm,
do 1000Å VP 5V, fo 9.65MHz Q 3,668
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Measured Intercept Point VIIP3

• Below measured using interferers 200kHz and
  400kHz below the resonance frequency
• Measured VIIP3 67.8 dBmV 2.45V

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Measurement vs. Theory

• Must know the electrode-to-resonator gap d
  accurately

h
d
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Intercept-Point (IIP3) vs. DC-Bias (VP)
Resonator Series Resistance
Determines insertion loss
Optimum value of VP 7.8V

• There is an optimum value of VP that maximizes
  PIIP3

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Projection to High Frequencies

• Under constant VP and quality factor

• VIIP3 increases monotonically with frequency
• mainly because stiffness kreff increases
• PIIP3 as high as 6dBm for a 70MHz resonator
  optimized for IIP3 ? good enough for many
  receiver applications

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Conclusions

• An analytical formulation capable of predicting
  the out of band IIP3 of a capacitively-driven
  CC-Beam resonator has been derived and found to
  match measurements
• capacitive nonlinearity dominates in distortion
  generation
• discovered a linearity vs. motional resistance Rx
  trade-off
• IIP3 vs. frequency, dc-bias voltage and quality
  factor
• Voltage VIIP3 increases with
• decreasing dc-bias voltage
• increasing frequency
• Power, PIIP3
• increases as quality factor increases
• can increase or decrease with dc-bias voltage
• is optimum for a specific value of dc-bias
  voltage
• It seems that capacitively-driven mmechanical
  resonators should be able to meet requirements
  for many receiver applications, such as GSM
  (requires -18dBm at front-end)