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

B. Boser 1

MEMS Gyroscope

Electrostatic Drive

Electrostatic Sense Pickup

B. Boser 2

Vibratory Gyroscope

• Vibrate along drive axis withoscillator @ fdrive

B. Boser 3

• Detect vibration @ fdrive about sense axis with accelerometer

Angstrom4000

1≅x

Operation at Resonance

Proof Mass Am

plitu

de

Gain Q

4 of 17

Frequency

• Signal amplification at resonance• Qgyro > 1000

fres

Gyro Sensitivity

• Limited bandwidth

• Sensitivity is a function

B. Boser 5

• Sensitivity is a function of Q, temperature

Feedback

Open-loop

Feedback

5 15 25

Frequency (kHz)

B. Boser 6

Stability?

Virtually no phase margin

Sensor Frequency Response

• Smaller parasitic modes all over

B. Boser 7

• 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)

B. Boser 8

• Main resonance• Single parasitic mode

9 of 17

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

B. Boser 9

Sampled Frequency Response

Mag

nitu

de (

dB) aliased

resonance

Frequency (kHz)

Pha

se (

deg)

excess lag Nyquist frequency

B. Boser 10

Negative Feedback

Mag

nitu

de (

dB)

Mag

nitu

de (

dB)

Frequency (kHz)

Pha

se (

deg)

Frequency (kHz)

Pha

se (

deg)

B. Boser 11

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

B. Boser 12

Mag

nitu

de (

dB)

Positive Feedback

DC gain < 0dB

Huge phasemargins

Frequency (kHz)

Pha

se (

deg) No phase marginat first crossover

B. Boser 13

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

B. Boser 14

small lag

Results

B. Boser 15

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

B. Boser 16

Conclusions• Stable phase range is -180o to +180o

• Negative feedback phase starts at 0o

– Accommodates only 180o phase lag

17 of 17

– Accommodates only 180 phase lag

• Positive feedback phase starts at +180o

– Accommodates up to 360o phase lag– Unstable for DC gain ≥ 1

B. Boser 17

Acknowledgements

• Chinwuba EzekweChristoph LangVladimir PetkovVladimir Petkov

• Robert Bosch CorporationGyroscope and financial support

B. Boser 18

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)

B. Boser 20