![fR AND Q MEASURE
Rotor current spectrum is the preferred method
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![fR AND Q MEASURE
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![INTERMODULATION
Intermodulation phenomenon is due to the intrinsic non-linearity
of the voltage-variable capacitance
Consider a bitone consisting of 2 tones of N points, with
frequency bins M1 and M2:
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CAPACITIVE SENSORS ELECTRICAL WAFER SORT
- 1. CAPACITIVE SENSORS ELECTRICAL WAFER SORT M. Garavaglia – Test Systems Development, SensorsSensors
- 2. CAPACITIVE SENSOR EXAMPLE FIXED PARTS MOBILE PART
- 3. ELECTRICAL MODEL V+ Silicon Substrate V- C1 C2 C1,C2 ~ 2pF ∆C ~ 10fF
- 4. C(V) PLOT EXAMPLE C (stator to rotor) measured values theory Pull-in V0V C0 (2pF) (2pF) (300fF) ~2V
- 5. CAPACITIVE SENSORS APPLICATIONS Accelerometers > Rotational or linear 1 axis > 2 axis C(V) C(V) S1 S2 R C(V) C(V) S1_1 S2_1 R_1 C(V) C(V) S1_2 S2_2 R_2 µ-actuators µ –mirror arrays R_1R_2 If 1 mass only is used C(V) C(V) S1 S2 R C(V) S1S2 R C(V)C(V)C(V) S3SN
- 6. WHAT TO TEST Absence of resistive paths between fingers > Leakage current Capacitance dependence on Bias VoltageCapacitance dependence on Bias Voltage > C vs V relationship check > C(V) smoothness > C(0) mismatch between Stators. Dynamical behaviour > Resonance frequency, Q
- 7. HOW TO TEST Leakage > Tie all pads to GND but the one under test > Measure the current flowing> Measure the current flowing from it. C(V) S1S2 R C(V)C(V)C(V) S3SN A
- 8. HOW TO TEST C vs V relationship check > Given two or more bias voltages V0, V1, Vn check the couples (C0=C(0),V0=0V), (C1, V1), (Cn, Vn) against the theoretical values. C against the theoretical values. V V1V0=0V C1 C0
- 9. HOW TO TEST C(V) smoothness > Repeat the previous test back and forth and check for histeresis (C0’ – C0), (C1’ – C1) C C1 2C C V V1 1 2 V10V V 0V C1 C0 C1’ C0’
- 10. HOW TO TEST C(0) mismatch between two stators > The initial offset of the packaged device will depend on this parameter. > Repeat the measurements on S1 and S2 and compare the two C(0). C C V S1 S2 0V V 0V C(0) C(0) MISMATCH
- 11. C(V) MEASURE SETUP REQUIREMENTS Testers do not have built-in capacitance meters. An external jig is required. It must be: > Able to resolve the femtoFarad> Able to resolve the femtoFarad > Stray-capacitance free > Scalable for multi-site > Able to be calibrated with a secondary capacitance sample > As easy as possible (less components = less failures)
- 12. V(C) METER JIG (IDEAL) vo + - CF CDUT DUT Vb vs,,fT CDUT
- 13. NON IDEALITIES Integrator must have a DC feedback or it will saturate. > Add an RF Guard/shield/bootstrap is needed to switch-off stray capacitance.capacitance. > Surround the amplifier input track with a shield/guard track connected to the + terminal. > Connect all the CDUTs but the one under test to the + terminal.
- 14. NON IDEALITIES Amplifier output offset voltage is not controlled (risk to saturate the digitizer) > AC couple the measurement system. Amplifier Input Bias current masks DUT currentAmplifier Input Bias current masks DUT current > Choose an OPAMP with low bias current (e.g. Ibias<100pA) > Choose the test tone frequency and the amplitude such that the DUT current is much higher than Ibias (e.g. 10nA). > Use an OPAMP with enough bandwidth .
- 15. REAL CIRCUIT + - CF DUT Cstray (switched-off) AD544 GUARD/ SHIELD Vb vs,,fT vo CDUT RF CcoupleRcouple WFG DIZ clock CDUT value Cstray PAD PROB E SHIELD PROBES
- 16. TRANSFER FUNCTION F strayDUT DUT F s o C CC Z Z v v + == Provided that f 1>> FF T CR f 1>> For example use RF=10MΩ, CF=10pF, fT=300kHz (1/RFCF=10kHz). Thus strayF s o DUT CC v v C −= (1)
- 17. NEED FOR CALIBRATION Variable stray capacitances (probes bending, humidity) sum-up with the CDUT CF is not known with enough precision (1% capacitors are used, tracks and solder stray caps). > Add a compensation block after the DIGITIZER. > Add an on-board calibration capacitor > Before to start probing: Measure and null the stray capacitances Insert and measure the calibration capacitor
- 19. CALIBRATION EQUATIONS stray s F Cq v C m = = : : Putting: equation (1) becomes: qvmC oDUT −⋅= strayCq =:
- 20. CALIBRATION PROCEDURE 1) Measure #1 (no DUT, no calibration caps): qvmC o −⋅= 11 2) Measure #2 (no DUT, calibration cap inserted): qvmC o −⋅= 22 3) Solve the system in m and q: = = CALCC C 2 1 0
- 21. CALIBRATION PROCEDURE The obtained calibration factors are: − = CAL vv C m ⋅ − = − 1 12 12 o oo CAL oo v vv C q vv
- 22. CALIBRATION TO A SECONDARY CAPACITANCE SAMPLE CCAL value itself cannot be known a-priori (1% capacitor, relay, solder and tracks stray capacitances). Thus it must be measured IN-CIRCUIT using an external Secondary Sample CREF2. Secondary Sample: a capacitor whose value is determined bySecondary Sample: a capacitor whose value is determined by measurement using an instrument that is calibrated against a primary standard (e.g. NIST traceable).
- 23. CALIBRATION TO A SECONDARY CAPACITANCE SAMPLE 1) CAP-METER CALIBRATION ONCE A YEAR CAP-METER UNDER CALIBRATION CREF1 PRIMARY CAPACITANCE SAMPLE
- 24. CALIBRATION TO A SECONDARY CAPACITANCE SAMPLE 2) SECONDARY SAMPLE MEASURE SECONDARY CREF2 actual value CAP-METER SECONDARY CAPACITANCE SAMPLE (10pF nominal) CREF2 9.97654 pF value
- 25. CALIBRATION TO A SECONDARY CAPACITANCE SAMPLE + - CF Mechanical I/F Cstray Secondary sample (9.97654 pF) CREF2 3) CCAL indirect measure Vb vs,,fT vo RF Ccouple Rcouple WFG DIZ clock CCAL X+ m-q CDUT value Cstray CREF2
- 26. CCAL INDIRECT MEASUREMENT Insert mech I/F for cap box START Calibrate m, q Insert CREF2 box CREF2 measure New CCAL guess CCAL initial guess box measure Extract CREF2 boxCREF2meas = CREF2actual ± ε N END guess Y CCAL value
- 27. SIGNAL ANALYSIS vo value cannot be estimated from time domain signal because of noise picked-up by transimpedance amplifier. 6496128 LSBs STDWW11 Discrete Fourier Transform must be performed 0 0.13e3 0.26e3 0.38e3 0.51e3 0.64e3 0.77e3 0.9e3 1.02e3 Time -128-96-64-32032 LSBs
- 28. DFT REQUIREMENTS DFT can fully represent a periodic signal provided that the acquired time-slice lasts exactly one or more periods of the signal itself. In that case:In that case: > The periodic component of the acquired signal is coincident with one of the orthogonal base-vectors of the DFT. > Windowing is not necessary
- 29. DFT ORIENTED TEST SYSTEM To obtain this the stimulus signal is digitally build as an array of points: > Let N be the number of these points> Let N be the number of these points > Let M be the number of periods of the stimulus signal over all the N points > Let fs be the frequency at which the array is scanned
- 30. DFT ORIENTED TEST SYSTEM Basic relations: s N N f f = fN: frequency resolution. Distance between two spectral lines. f : frequency of the generated N Nt f UTP Mff N 1= ⋅= ft: frequency of the generated signal (tone). UTP: Unit Test Period. Amount of time needed to acquire N points (regardless of ft).
- 31. DFT base vectors M=1 M=2 M=3 1 N M=N/2-1 M=N/7
- 32. WAVEFORM GENERATOR 1 2 3 MUX D/A A vs, fT RAM mod N COUNTER N CLOCK
- 33. WAVEFORM DIGITIZER 1 2 3 DE MUX A/D vo, fT RAM FILT mod N COUNTER N MUX CLOCK
- 34. UTP EXAMPLE vs, fT From WFG RAM vo, fT To WFD RAM N=2048N=2048 points, M=7M=7 periods STOPSTART
- 35. OBTAINED DFT
- 36. TIP & TRICKS No ESD protections > Insure that all the probes and the wafer chuck are tied to GND both BEFORE touch-down and take-off High impedance tracks from “R” pads and TransimpedanceHigh impedance tracks from “R” pads and Transimpedance amplifier. > PCB and Probe card layout MUST be carefully designed. Guarding and shielding is mandatory to avoid stray capacitance to inject noise.
- 37. DYNAMIC TEST C-V plot depends only on the geometry and on the potential difference between plates. DOES NOT depend on: > the mass (momentum of inertia) of the shuttle. > the damping of the system (air vs vacuum)> the damping of the system (air vs vacuum) Resonance frequency fR and Q depend on both these features.
- 38. fR AND Q MEASURE Rotor current spectrum is the preferred method ( )[ ])(),()( tvCtxC dt d ti DStrayDUTd += ( )[ ])(),()( tvCtxC dt ti DStrayDUTd += id CDUT + _ vD ~ Cstray
- 39. fR AND Q MEASURE )cos()()( tvVtvVtv dPDdPDD ω⋅+=+= Assuming the following drive voltage: And assuming that x and CDUT are linearly dependent (valid only for comb finger capacitors): )(),(),( 00 tx x C CtxCCtxC DUT mDUT ∂ ∂ +=+=
- 40. fR AND Q MEASURE We obtain: [ ] [ ] )()( ),()(),()( 0 t x x C tvCtx x C C dt dv CtxC dt d tvCtxC dt dv ti DUT DStray DUTD StrayDUTDStrayDUT D d = ∂ ∂ ⋅ ∂ ∂ ⋅+ +⋅ ∂ ∂ +⋅= =+⋅++⋅= FEEDTHROUGH TERM INTERMODULATION TERM ( ) ( ) ( ))()( )()( 0 0 tvtx dt d x C t x x C V dt dv CC t x x C tv dt dv x C tx dt dv CC txxdt d DUT DUT PD d Stray DUT D DDUTD Stray ⋅⋅ ∂ ∂ + + ∂ ∂ ⋅ ∂ ∂ ⋅+⋅+= = ∂ ∂ ⋅ ∂ ∂ ⋅+⋅ ∂ ∂ ⋅+⋅+= ∂∂ ∂
- 41. fR AND Q MEASURE Note that: 1) Every constant capacitance (e.g. Cstray) can only give rise to a current signal at the same frequency as the excitation voltage vd(t). 2) The non-linear term (2) is caused by the fact that CDUT is not constant. It represents a double-frequency line in the id(t) spectrum whose amplitude is proportional to the shuttle displacement. 3) “Motional” component is usually negligible wrt to “Static” one. Frequency doubling allows its detection.
- 42. SPECTRA EXAMPLE Output rotor current spectrum Term #1 “STATIC” dominant Term #2 “MOTIONAL” smaller Input tone spectrum id ω 2ω vd ω
- 43. PROBLEMS Test tone MUST BE NON-DISTORTED (very difficult if large signal voltages have to be used) > its 2nd harmonics would mask the frequency-doubled term Multitones cannot be used to speed-up the test > intermoduation products would appear:
- 44. INTERMODULATION Intermodulation phenomenon is due to the intrinsic non-linearity of the voltage-variable capacitance Consider a bitone consisting of 2 tones of N points, with frequency bins M1 and M2: tMfj d tMfj dd NN evevtv 21 2 2 2 1:)( ππ += Then, the intermodulation term becomes: ( ) ( ) ( )[ ]tMfj d tMfj d tMfj d tMfj d DUT d DUT NNNN evevexex dt d x C tvtx dt d x C 2121 2 2 2 1 2 2 2 1)()( ππππ +⋅+⋅ ∂ ∂ =⋅⋅ ∂ ∂
- 45. INTERMODULATION Considering a multitone with K frequency bin spanning from M to M+K-1 then the intermodulation term would contains: > All the double-frequency components > All the sum components > All the difference components> All the difference components The intermodulation terms would span from K-1 to 2(M+K-1) and would overlap each others
- 46. INTERMODULATION Double-frequency terms and intermodulation terms sum-up scrambling the spectrum: Output rotor current spectrum Input K- M+(M+1), (M+1)+M 2(M+1), M+(M+2), (M+2)+M (M+1)+(M+2), (M+2)+(M+1), id spectrum vd tone spectrum M M+K-1 M+K-1 2M 2(M+K-1) (M+1)+(M+2), (M+2)+(M+1), M+(M+3), (M+3)+M M+(M+4), (M+1)+(M+3), 2(M+2), (M+3)+(M+1), (M+4)+M MK-1
- 47. ELECTRO-MECHANICAL AMPLITUDE MODULATION We talk about EAM when you use a drive signal composed of one “carrier” tone (at ωc) and one or more modulant tones (at ωd) such that ωc >> ωd, ω tones are choosen around the resonance frequency.ωd tones are choosen around the resonance frequency. Due to intermodulation the ωd tones will fold as sidebands around ωc
- 48. ELECTRO-MECHANICAL AMPLITUDE MODULATION ADVANTAGES > Sidebands are independent on stray caps (they are CDUT non-linearity effect). > Test tone distortion has no more negative effect.> Test tone distortion has no more negative effect. > MULTITONE testing is possible because spectrum overlapping is avoided by construction (provided that carrier frequency is much higher than test tones and that carrier tone is UNIQUE). > Sidebands amplitude is proprtional to ωc times ωd. Thus using a large ωc it is possible to amplify id
- 49. EAM SPECTRA EXAMPLE Input multitone spectrum Output rotor current spectrum Term #1 “STATIC” EAM sidebands EAM sidebands BELL-shaped around resonance Term #2 “MOTIONAL” vd ωd1 2ωd1-1 ωc id ωc ωc-ωd ωc+ωd “MOTIONAL” + intermodulation
- 50. ELECTRO-MECHANICAL AMPLITUDE MODULATION DISADVANTAGES > Higher frequencies means worse resolution (higher fN) for the same UTP, or same resolution with a longer UTPor same resolution with a longer UTP
- 51. CONCLUSIONS A selection of test methods for MEMS capacitive sensors and actuators have been presented A suitable hardware have been presented The ability to measure capacitances in the order of theThe ability to measure capacitances in the order of the picoFarads with a repeatability of few femtoFarads have been shown The calibration process has been explained Dynamic testing methods such as double-frequency and EAM were also introduced
- 52. REFERENCES Double-frequency and EAM methods: > Electromechanical Characterisation of Microresonators for Circuit Applications. Tu-Cuong (Clark) Nguyen, Research Project, University of Berkeley, 1991