Positive Feedback, Negative Feedback
A closer look at phase margin
Bernhard E. Boser
Berkeley Sensor & Actuator CenterDept. of Electrical Engineering and Computer Sciences
University of California, Berkeley
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MEMS Gyroscope
Electrostatic Drive
Electrostatic Sense Pickup
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Vibratory Gyroscope
• Vibrate along drive axis withoscillator @ fdrive
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• Detect vibration @ fdrive about sense axis with accelerometer
Angstrom4000
1≅x
Operation at Resonance
Proof Mass Am
plitu
de
Gain Q
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Frequency
• Signal amplification at resonance• Qgyro > 1000
fres
Gyro Sensitivity
• Limited bandwidth
• Sensitivity is a function
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• Sensitivity is a function of Q, temperature
Feedback
Open-loop
Feedback
5 15 25
Frequency (kHz)
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Stability?
Virtually no phase margin
Sensor Frequency Response
• Smaller parasitic modes all over
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• Main mode near 15kHz
• Big parasitic modes near 95kHz and 300kHz
Example Frequency Response
Mag
nitu
de (
dB)
Pha
se (
deg)
Mag
nitu
de (
dB)
Pha
se (
deg)
Frequency (kHz)
Pha
se (
deg)
Frequency (kHz)
Pha
se (
deg)
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• Main resonance• Single parasitic mode
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System Model
sensedynamics
front-end
over-sampleddigital output
compen-sator
coriolisforce
fs ≈ 480 kHz
sensedynamics
front-end
over-sampleddigital output
compen-sator
coriolisforce
fs ≈ 480 kHz
Filter
Gyro
• Two-level (Σ∆) feedback (linearize)
actuatorfeedback
force
actuatorfeedback
force
• Sampler• MEMS Gyroscope
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Sampled Frequency Response
Mag
nitu
de (
dB) aliased
resonance
Frequency (kHz)
Pha
se (
deg)
excess lag Nyquist frequency
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Negative Feedback
Mag
nitu
de (
dB)
Mag
nitu
de (
dB)
Frequency (kHz)
Pha
se (
deg)
Frequency (kHz)
Pha
se (
deg)
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Negative Feedback w/ Lead Comp.
First & secondcrossover okay
Mag
nitu
de (
dB)
First & secondcrossover okay
Mag
nitu
de (
dB)
Large negative marginat third crossover
Frequency (kHz)
Pha
se (
deg)
Large negative marginat third crossover
Frequency (kHz)
Pha
se (
deg)
unstable
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Mag
nitu
de (
dB)
Positive Feedback
DC gain < 0dB
Huge phasemargins
Frequency (kHz)
Pha
se (
deg) No phase marginat first crossover
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Stable from-180o to +180ostable
Positive Feedback w/ Lag Comp.
Mag
nitu
de (
dB)
Mag
nitu
de (
dB)
Frequency (kHz)
Pha
se (
deg)
Frequency (kHz)
Pha
se (
deg) Lag
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small lag
Results
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Comparison to previous work
ReferencePower(mW)
Noise(°/sec/√Hz)
BW(Hz)
Tuning Time(sec)
[1] 30 0.05 20 -
[2] 13 1 40 -
[3] 31 0.05 36 -
[4] 6 - 0.2 140[4] 6 - 0.2 140
This work 1 0.004 50 0.3
[1] Geen, JSSC 2002[2] Petkov, ISSCC 2004[3] Saukoski, ESSCIRC 2006[4] Sharma, ISSCC 2007
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Conclusions• Stable phase range is -180o to +180o
• Negative feedback phase starts at 0o
– Accommodates only 180o phase lag
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– Accommodates only 180 phase lag
• Positive feedback phase starts at +180o
– Accommodates up to 360o phase lag– Unstable for DC gain ≥ 1
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Acknowledgements
• Chinwuba EzekweChristoph LangVladimir PetkovVladimir Petkov
• Robert Bosch CorporationGyroscope and financial support
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Parasitic Resonances
NormalizedMagnitude
(dB)
Phase(°)
NormalizedMagnitude
(dB)
Phase(°)
NormalizedMagnitude
(dB)
Phase(°)
NormalizedMagnitude
(dB)
Phase(°)
Frequency(Hz)
Frequency(Hz)
Collocated Control(same electrode)
Frequency(Hz)
Frequency(Hz)
Non-collocated Control(separate electrodes)
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