Maurice Tutor

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    Phoniex University
    Oct-2001 - Nov-2016

Category > Management Posted 28 Jan 2018 My Price 8.00

capacitor microphone

3.10.                           A capacitor microphone, shown in Fig. 3-lO(a), may be modeled by the electromechanical system of Fig. 3-1O(b). With v as the input, obtain the linearized dynamical equations of the system. Give the steady-state operating point.

 

 

 

Movable plate

mass = M area = A

+      V

 

le X

 

 

 

 

R

friction coeff . b     Vh'

Diaphrap                                                   Slidina contact, --..._                                                    +

 

V

 

G
 

                                   -

 

(a)                                                                                              (b)

Fig. 3-10

 

For a charge q, the electrical energy stored in a capacitor can be expressed  as

 

 

where  C is the  capacitance.   The electrical  force between  the plates  is then  given  by

 

 

ox
 

- - awe  - - q2            a  ( 1)

Fe -     8x -     2       C

where the minus sign indicates that the force tends to decrease the separation x of the plates.  For the  present case,

 

 

 

 

 

so that

 

 

 

 

The equations of motion are then:

 

C  =  BA

X

 

 

mechanical:         Mi  + bi  +  k(x  - 10)    =                                                                  (])

 

 

 

electrical:

 

Rq  + !!:.. = V

eA

 

 

(2)

 

These  equations are nonlinear  due to the presence  of terms  involving q2 and  qx.

Linearization is accomplished by assuming that a steady-state operating point (X0, Q0, V0) exists

such that

 

                  

Substitution  into (J) and (2), with  neglect  of higher-order  terms,  yields

 

Qo2                                Q

 

 

--
 

Mx 1

+ bx1

 

+ k(X0

 

- l0)  + kx1

 

=  ---

 

oq,

;A;--

 

2 BA      "'

 

Rq  +  Qo-¥0  +  XOql   +  QoX1  =  V  + V

1              BA         BA         BA         O             1

From  these,  it is seen that the  steady-state operating point  is given   by

gg   = k(X   - 1 )

2BA          0            0

The linearized  dynamical  equations  are thus

 

mechanical:      electrical:

 

Answers

Maurice Tutor
(5)
Status NEW Posted 28 Jan 2018 11:01 PM My Price 8.00

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