Derivation capacitance is passed feeling the transfer function of accelerometer

Derivation capacitance is passed feeling the transfer function of accelerometer

[introduction] in the part in this series, we discussed quality – bedspring – damper (or quality – damper – bedspring) the structure can be used at measuring acceleration. To make pledge gauge block displacement and the acceleration that bring to bear on become direct ratio, should choose qualitative gauge block appropriately – bedspring – the different parameter of damper system. The article will use classical and mechanical notional derivation quality – bedspring – the transfer function of damp system.

In the part of this series, we discussed quality – bedspring – damper (or quality – damper – bedspring) the structure can be used at measuring acceleration. To make pledge gauge block displacement and the acceleration that bring to bear on become direct ratio, should choose qualitative gauge block appropriately – bedspring – the different parameter of damper system. The article will use classical and mechanical notional derivation quality – bedspring – the transfer function of damp system.

Transfer function makes we can be described how qualitative gauge block answers exterior acceleration and move. The different parameter that explaining accelerometer (for example limits of sensor linear work and bandwidth norms) when, derived transfer function will be used in the follow-up article in this series.

However, before trying derivation sensor transfer function, make us brief introduce system of personal computer cable (MEMS) technology, this technology makes current small-sized low cost inertial sensor becomes a possibility.

MEMS accelerometer: Structure of damper of service mass bedspring measures acceleration

The quality that is used at feeling to measure acceleration – bedspring – damper structure is shown 1 times like the graph.

The equipment of signal adjustment electron that MEMS technology makes we can realize this very little version of mechanical system and place to need on same silicon chip, have thereby pass thoroughly feeling solution.

Derivation capacitance is passed feeling the transfer function of accelerometer

Graph 1.  Quality – bedspring – damper structure.

MEMS technology drew lessons from the is based on photoetching small processing technique of microelectronics industry, combine they and other and special production technology look, can found portable component on silicon chip thereby.

The ascensive help of small production technology realized today’s small-sized, low cost small mechanical accelerometer, graph 2 showed a give a demonstration.

Derivation capacitance is passed feeling the transfer function of accelerometer
Graph 2.  The scanning electron metallograph of CMOS MEMS accelerometer (SEM) . Picture by K. Zhang is offerred

In an article, we are brief there is crucial effect in mentioning damper to be operated in accelerometer. Trying derivation quality – bedspring – before the transfer function of damper system, now is the inning of this one substantial of farther familiar system.

The damp mechanism in MEMS accelerometer

Force of damper imitate dissipation, this dissipation power can reduce quality – bedspring – the mechanical energy of damper system is decelerated examine the motion of quality.

One of main damp mechanisms in MEMS accelerometer are motion quality and all round the internal clash that produces between air element. In fact, the accelerometer that is based on MEMS can be enclosed below extremely low pressure, in order to reduce the effect of air damp. However, generally speaking, air damp is losing of the energy in MEMS accelerometer main.

Other and common damp cause is structural damp and hot damp.

Structural damp considered the energy loss that causes by the package structure that uses in MEMS parts of an apparatus.

Thermal resistance Buddhist nun is corresponding the stress at MEMS structure – the deviation that relation of meet an emergency changes along with temperature. Damper brings to bear on to build a model to be the speed proportionate force with quality of test and verify normally in the total decelerative force on quality of test and verify.

The effect way of this force and mass movement are contrary, give out by next type:

[F_{damper} = Bv]

Among them B expresses damp coefficient, v shows the rate of qualitative gauge block.

Ask an attention, when the object special hour, the speed of wind resistance and object becomes direct ratio, small treatment structure is this kind of circumstance.

Generally speaking, the speed of wind resistance and object is having complex concern. For example, a big object, the parachute that moves in sky for example person, can be become the resistance of direct ratio by the square with object speed.

Damp effect: Need to still be fed up with?

Because damp comes from dissipation force, accordingly it may be a kind of trouble that should avoid it seems that. In fact, the design is a lot of MEMS accelerometer to have a few damp only (in order to reduce systematic noise) .

But those who need an attention is, the system of good quality bedspring that does not have damp is an oscillator actually, cannot use as accelerometer.

If us will ” ideal ” the quality of bedspring quality system removeds to release it next from balance position, although did not bring to bear on to the system exterior acceleration, quality also can move back and forth forever. This is why to accelerometer, we need to introduce a few damp to bedspring quality system at least.

Law of use Newton sports proves quality displacement

If pursue,assume 3 are shown, master piece is used at sensor frame outside.

Derivation capacitance is passed feeling the transfer function of accelerometer

Graph 3.  The quality that makes response to outside force – bedspring – frame of damper structure sensor.

The accelerometer that brings to bear on for the basis calculates displacement of qualitative gauge block, we use Newton the 2nd motion is mensurable. You may know no less than in that way, this law points out, the size of the object acceleration that join forces produces and join forces becomes direct ratio, become inverse ratio with the quality of the object.

This expresses by the equality that is familiar with below:

[F= horse]

Formulary 1

Among them F is the join forces that brings to bear on to go up in the body, m is the quality of the body, a indicates acceleration.

Want to use this equality correctly our system, here should notice to be nodded delicately. Law of the 2nd sports applies to Newton only inertial coordinate is, the coordinate that does not quicken namely is.

The plan differs twice of the 3 accelerometer that depicted us coordinate is. When orange coordinate fastens correspondence to solve physical problem at be in, assume for inertial earth reference is fastened.

However, magenta coordinate department states the reference that secures sensor frame is fastened.

This coordinate is dispute is inertial, because should bring to bear on to sensor,it can be quickened when outside force. Accordingly, want to find the athletic equation of qualitative gauge block, we should use inertial reference is (orange coordinate is fastened) .

What master piece is used at proving quality?

If pursue,assume 3 are shown, the position of the dormant position that X 0 and X M represent qualitative gauge block respectively and aleatoric hour qualitative gauge block. Be in below the outside force action of X direction, sensor frame is quickened right. First, qualitative gauge block because its are inertial and apt ” regressive ” . This meeting change detects quality is opposite the relative position at sensor frame, compress bedspring X 0 – X M. Compression bedspring confrontations to gauge block applies thrust augmentation and push its right.

The force that bedspring brings to bear on gives out by next type:

[F_s = K (X_0 – X_m) ]

Formulary 2

When deviate of qualitative gauge block is balanced, damper brings to bear on an as opposite as qualitative gauge block Yu Jing stops the force of the position that becomes direct ratio relative to speed, we get:

[F_d = B (dot X_0 – Dot X_m) ]

Formulary 3

In above equality, drop sign is used at showing variable is opposite the first-order derivative at time. Ask an attention, positional derivative is speed.

Apply equality 1, we get:

[F_s + F_d = Ma_{proves quality} ]
Era enters equation 2 with 3, we get
[K (X_0 – X_m) + B (dot X_0 – Dot X_m) = M Ddot X_m]

Equational 4 

In this equality, double drop sign states X M is opposite at time 2 rank derivative. Ask an attention, ? M is the acceleration of quality of test and verify.

Beg the athletic equation in be not inertial reference to fasten

The displacement that need basis pledges gauge block balances the position from its will rescript equational 4. Because we are in,this is practice is medium those who pass feeling method to measure is deviate of qualitative gauge block the displacement that its balance position.

For example, if the graph is shown 4 times, method of capacitance type induction is measured detect qualitative gauge block stops relative to Yu Jing positional displacement.

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Graph 4. Measure the setting of displacement of qualitative gauge block through capacitance induction. About more information, examine an article of this series please.

Convey equation to come according to quality displacement 4, we need to use a figure 3 in the mobile reference that carmine coordinate department shows is. We use lowercase X and Y to state this coordinate is fastened.

If what you see, displacement of quality of test and verify by X M – X 0 = X gives out.

Below this kind of circumstance, formulary 4 brief melt into:

[- Kx – B Dot X = M (Ddot X_0 + Ddot X) ]

As a result of 0 one be sensor coordinate strung does not move a point, its 2 rank the acceleration that derivative is equal to sensor coordinate to fasten A. This is the parameter that we should measure actually.

Accordingly, afore-mentioned equality bring about:

[M Ddot X + B Dot X + Kx = -ma]

Search transfer function

Apply Laplace commutation, we can find the transfer function of accelerometer to be:

[H(s) = Frac{x(s)}{a(s)} = Frac{-1}{s^2 + Frac{b}{m}s + Frac{k}{m}} ]

This is 2 rank system. According to the value of systematic parameter, namely M, K and B, the system is answered will different.

For example, if sensor frame acceleration is sudden from 0 turn into finite value (rank jump an input) , criterion systematic output will be close to his to be worth eventually, character is answered to be decided by systematic parameter between meantime.

Graph the 5 ringing that showed how change system parameter changes output and stabilization period. In our discussion, output is proof quality displacement.

Derivation capacitance is passed feeling the transfer function of accelerometer

Graph 5. 2 rank of the system rank jump answer the meeting value according to systematic parameter to happen showing change. Picture by David L of Masschusetts Institute of Technology. Trumper is offerred

In an article, we will use derived transfer function to discuss a few important system parameter, for example limits of sensor linear work, answer bandwidth of sum of errors.

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