Polarization of electrostatic actuator
Introduction
DC polarization
AC polarization
Introduction
Applying a pure sinusoidal voltage
What happen if we add an offset voltage to the sinusoidal
voltage?
And what about a square, triangular, or patatoidal
voltage?
Conclusion
A ±30V voltage amplifier
Introduction
Polarizing an electrostatic actuator seems quite simple but may be hindered
by quite a few mistakes that only experience can learn... (Put in a simpler
way, I've done all of these mistakes but I probably overlooked many things,
thus if my explanations suit you, take them number ,
but otherwise tell
me what you think is the truth :-)
First of all, electrosatic actuators dislike stray charge that may
come from airborne particule, or by improper grounding. To lessen these
effects you must ground everything than can be grounded and leave a ground
plane just below the actuator. If you forget this you may expect unreliable
behaviour and, for example, actuator working for a while and then stopping
(build-up of charge in insulating layer that stick the actuator), or actuator
operation varying with humidity level (a film of water helps to reduce
the static charge by increasing the conductivity) or even actuator not
working at all... Actually even with well grounded structure you may already
be affected by these problems...
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DC polarization
This is the simplest one: you apply voltage between the two electrodes
of the actuator and you increase it slowly while watching the structure
through your microscope. If it moves you're quite a happy man! An interesting
point with the electrostatic actuator is that the force generateed is proportional
to the square of the voltage, ie:
F ~ V²
Hence when the voltage is multiplied by 3 the force increase almost by
an order of magnitude.
For a comb-drive actuator, where the force does not change with the
displacement, and with a linearly increasing restoring force (ie, a spring)
it means that the displacement also increases with the square of the voltage.
Thus take care not to increase to rapidly your voltage:, when you go from
1 to 10 V, you increase the force by a factor of 100 and at 31.5V it is
a factor of 1000!
Moreover, for a gap-closing actuator, where the force increase quadratically
with the displacement, associated with a linearly increasing restoring
force (ie, a spring), after the actuator has moved 1/3 of the initial gap
width, it will snap through and 'close' the gap. This may cause a short-circuit
if no insulator or landing point has been designed. However this problem
does not always appear and I've noted that for doped polysilicon actuator
short-circuit do not appear for moderate voltage, presumably because of
the native oxide layer and the large rouhghness of the side that yields
a very small contact area.
And finally, if the structure does not move? Hum, hum. Actually this
is the most common case :-( Then either your structure is stuck (check
your release process, and why not try SAM coating?)
or more probably the voltage is not high enough (Actually, and to be honest,
it means that your structure is too stiff or the electrodes surface too
small :-). Then, you need a DC power supply which can deliver higher voltage
or you may try to excite your structure by applying AC signal, which will
give you more information on your structure.
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AC polarization
Introduction
The main interest of applying AC voltage to an electrostatic actuator is
the measurement of its resonnant frequency. But it may also be simply used
to check if a stiff structure is moving, because at the resonnant frequency
the displacement is increased by the Q factor of the structure, often more
than ten, and more than enough to see something with the optical microscope
without investing in laser vibrometer...
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Applying a pure sinusoidal voltage
As we have already pointed out the force delivered by an electrostatic
actuator vary with the square of the voltage. Thus what happen if we apply
a sinusoidal voltage V = V0 sin(wt + Ø
) to it? We have:
F ~ V² = (V0 sin(wt + Ø
))² = V0²(½ - ½cos(2wt
+ 2Ø)) = ½V0²(1 + sin(-2wt
- 2Ø + ¶/2))
Thus we see that the force has a constant component of magnitude proportional
to ½V0² and a varying component
with frequency twice the frequency of the applied voltage
and a complex phase factor. Thus under the microscope we will see the actuator
oscillates at a frequency twice that of the exciting voltage around a deflected
position different from the rest position. It should be noted that the
existence of a deflected position will certainly result in a different
mode shape than the one corresponding to the tested mode, and will add
to the innacuracy of this method to check the resonant frequency (actually
even if the structure is driven by the actuator around its rest position
the deformation will not correspond exactly to the shape of the corresponding
mode :-).
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What happen if we add an offset voltage to the sinusoidal
voltage?
We return back to the expression of the force, which gives in this case:
F ~ V² = (V0 sin(wt + Ø
) + V1)² = ½V0²
+ V1² + 2V0V1sin(wt
+ Ø ) - ½V0²cos(2wt
+ 2Ø) =
F ~ ½V0² + V1²
+ 2V0V1sin(wt
+ Ø ) + ½V0²sin(-2wt
- 2Ø + ¶/2)
As you see there is a larger DC component and now there is a signal driving
the structure at a pulsation equal to the pulsation of the
voltage. The interest of such arrangement is that it provides a mean to
control the offset displacement (ie, the DC component) and that it will
provide a displacement synchronized with the driving signal. Actually,
if we choose V0 « V1
then the expression of the force may be simplified as:
F ~ V1² + 2V0V1sin(wt
+ Ø )
then the driving force is at the same pulsation than the applied voltage
which may be very helpful if we want to use synchronous detection
of the movement. Once more the start from a deflected position will result
in a different mode shape than the one corresponding to the tested mode.
Moreover, we should not forget the existence of the second order harmonic
(2wt) which may create spurious excitation if its amplitude (½V0²)
is too large (or simply if the Q-factor is large).
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And what about a square, triangular, or patatoidal
voltage?
Actually the whatever periodic signal you use may be decomposed in Fourier
series and, for example, you will find that a square signal is the sum
of sinusoidal signals whose pulsations ar an odd multiple of the original
pulsation (ie, the odd harmonics) and whose amplitude decrease as the inverse
of the rank of the harmonic. Generally, signals will have a more complex
decomposition, including odd and even harmonics with amplitude varying
as a complex function of the harmonic rank (it is similar to the signal
with offset case, where we had only the two first harmonics).
Then the force is once more proportional to the square of the signal
and thus it will have frequency components which will be a multiple of
the original signal frequency, and including this frequency. The amplitude
of the different harmonics will depend on the signal and may be computed
using Fourier analysis. However it is not a simple calculus, as not only
square of the signal will appear but also cross-product between harmonics
which result in the generation of signal having a frequency corresponding
to the difference of the two signal mixed.
Practically you will see your structure being excited in resonance
with many different frequencies that you may mistakenly take for higher
order mode. However, if you find that they are an integer multiple of the
lowest resonnance frequency (which will be equal to the signal
frequency and not twice its value as with a pure sinusoidal
signal)... then it is most probably just the harmonics of the AC signal
that excite one after the other the resonance of your structure.
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Conclusion
Generally speaking, except for special case, you should avoid signal other
than sinusoidal when you want to measure resonnant frequency of your structures.
If you want to use synchronous detection of displacement you need to add
an offset voltage, preferably with an amplitude much larger than the amplitude
of the sinusoidal component, and the structure will be driven at the same
frequency than the AC signal. If you use a pure sinusoidal signal (without
offset) the structure will be driven at a frequency twice that of the AC
signal.
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A ±30V voltage amplifier
Have you ever wanted to have just that extra boost that this damned function
generator made for electronics refuse to give? Just wanted to obtain more
than the ±5V but don't want to play with hundred volts either? Hey
here is what you were looking for, a simple voltage amplifier that can
deliver up to ±30V (or 0-60V) using a standard lab power supply
and a function generator. No need to buy a new ±30V function generator!
The trick is just to use a power audio amplifier in a low-gain configuration
and a pair of 30V DC supply connected in series, a very common piece of
equipment in most lab. To keep the circuit simple and robust we used the
operational amplifier LM675 from National Semiconductor which can be powered
up to ±30V and is stable for gain above 10. It is supposed to be
mandatory to attach a heat sink to the amplifier but as the electrostatic
actuator does not need much current (the capacitance is very small) it
was found to be not necessary. Don't forget to ground everything with the
0V output of the power supply (ie, the common point connecting the two
30V DC power supply in series). One more thing Vin is supposed to remain
lower than VCC and VEE
otherwise you may damage irremediably the circuit (actually we found that
the circuit was quite tolerant to this point... but don't try it expressly
:-)
For example, we have used the following values for the components, soldering
everything 'on the fly' owing to the simplicity of the circuit:
R1 = 18kO; R2 = 1.2kO; R3 = 12k; C = 0.1µF, two BNC plugs for
VIN and VOUT
and three 4mm socket for VCC, VEE
and the ground.
We obtained a gain of 11, an output voltage of 58Vpp (VCC
= VEE = 30V), and a bandpass of about 1MHz
for small signal but limited by the slew rate (typical 8V/µs, but
larger for our component :-) for large output swing (±29V) to about
80kHz without too much distorsion. For most structure 80kHz is more than
enough, but don't try to power your 1MHz resonator with this device...
you won't get more voltage than with the function generator alone :-)
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