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368    IDT MICROSENSORS

   with different  amplitudes  B  and angular frequencies.  The  frequencies  and phases of these
   two signals  depend  on the delay  times.  The two signals  can be separated  in the  frequency
   domain.  Because  (wo is  usually much  greater  than  , the  phase  shift  is  more  sensitive  to
   the  variation  of  the  delay  time  than  the frequency.
     The  difference  of  the  two  phases can be  written  as

                     <P = (<Pi-  <P2) = N) -  M('i + '2)/2][/ 2  -  ti]  (13.16)

   where  the  extra  delay  time  of  the  second  echo  compared  with  the  first  is  equal  to  the
   round  trip  time  of  the  acoustic  wave  traveling  from  the  first  reflector  to  the  second  and
   is  given by
                                T  = (t 2-ti)  = —                     (13.17)
                                              v
  where  d  is the  distance  between  the  two  reflectors.
     The  phase  difference is  sensitive  to  the  change  in the  delay  time.  The  variation  of  the
  phase  difference because  of  the  change  in  the  delay  time  is expressed  as

                                                and   Ar  =  2AJ/U     (13.18)

  Because  COQ is  usually much  larger  than the  other  term,  the  equation  reduces  to

                                                                       (13.19)

  The  wave traveling  time  T is proportional  to  the  distance  between  the  two  reflectors  and
  is  inversely  proportional  to  the  velocity.  If  the  possible  velocity  variation  of  the SAW
  under  stress  is  neglected  and  we  take  into account only the  direct  effect  on  the  distance,
  we  have
                                      2sd
                                                                      (13.20)

  where  e  is  the  mechanical  strain  (Ad/W)  and  r 0  is  the  traveling  time  when  the  strain
  is  zero.
     Thus,  the sensitivity of this wireless  sensor  system depends  on the operating  frequency
  and  the  round traveling  time  of  the  SAW between  the  two  reflectors.


  13.4.1.1  Sensor dimensions and  reading system

  A  strain  sensor  was  developed  by  Varadan et al.  (1997),  wherein YZ-cut  lithium  niobate
  (LiNbO 3)  crystal  of  dimensions  3.7 mm x  6.7  mm x  0.5  mm  was  used.  An  IDT  was
  designed  with  a  central  frequency  of  912.5  MHz.  The  IDT  is  connected  directly  to  a
  small  antenna of dimensions  25 mm x  75 mm x  0.3  mm. The  distance  between  the  two
  reflectors  was  0.15  mm and the round trip traveling time of the surface waves was 0.1  us
  at zero  stress. The wireless  reading system had a patch antenna to send out FM microwave
  signals  and  to  receive  the  echoes  coming  from  the  reflectors  through  the  antenna-IDT.
  The  transmitted  signal  was  linearly  modulated  from  905  to  920  MHz  in  a  pulse  lasting
  1760 th  of  a  second.  The  resolution  of  the  phase  detector  of  the  system  is  1°. According
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