This document discusses testing capacitive sensors using electrical measurements. It describes an electrical model of a capacitive sensor and how its capacitance varies with applied bias voltage. Methods are presented for testing for leakage currents, measuring the capacitance-voltage relationship, and determining the sensor's dynamic behavior by analyzing its resonant frequency and quality factor. Requirements for the test setup include using a capacitance meter, waveform generator, and digitizer configured for Fourier analysis to isolate the sensor's motional signal from background terms in its output current spectrum. Calibration procedures are also outlined to compensate for stray capacitances in the test circuit.

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

18. CALIBRATION HARDWARE
+
-
CF
C
Cstray
Vb
vs,,fT
vo
CDUT
RF
CcoupleRcouple
WFG DIZ
clock
CCAL
X+
m-q
CDUT
value

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