HW  6    

  Behroozpour    

  Summer  2015         Posted:  Wed  7/22/2015       Due:  Fri  7/31/2015  -­‐  11pm    

1. Consider  the  following  parallel  RLC  circuit:  

 a) Write  the  equation  for  the  transfer  function  from  input  current  to  output  voltage.  b) Evaluate  the  transfer  function  at  𝜔! =

!!"  when  𝑅 → ∞.  

c) Now  find  the  equivalent  resistance  seen  from  node  X  in  the  following  circuit.  

   

d) Using  the  result  from  previous  parts  write  the  frequency  response  from  input  current  to  the  

output  voltage  for  the  following  circuit  and  evaluate  it  at  𝜔! =!!".  

 e) According  to  the  results  from  previous  parts  what  would  be  your  prediction  as  the  behavior  

of   this   circuit   knowing   that   in   practice   there   will   always   be   a   little   bit   of   noise   at   all  frequencies  that  can  be  modeled  as  a  current  being  fed  into  node  X?    

f) What   do   you   expect   to   see   as   the   amplitude   of   the   voltage   at   the   output   node   of   the  opamp?  How  would  you  explain  the  apparent  drop  in  the  gain?  

2. Generate  Bode  magnitude  and  phase  plots  for  the  following  transfer  functions.    

a) 𝐻 𝜔 = !"×!"!(!"!!!!)(!"!!!""!)!!!" !(!!!!!)

 

 

b) 𝐻 𝜔 = !"#!!!"!!!!

!!(!"!!")(!""!!!!!!"!)!  

 

c) 𝐻 𝜔 = !"!!(!!!")(!"!!")(!"!!")(!"#!!")

 

 

d) 𝐻 𝜔 = !!(!"#!!"/!)(!"""!!")(!"""!!!!!!"!)!

 

 3. The  following  Bode  magnitude  plot  is  given  for  𝐻(𝜔).  How  many  possibilities  are  there  for  𝐻(𝜔)  

to  result  in  this  magnitude  plot?  Write  down  one  of  those  possibilities  and  determine  its  phase  at  𝜔 =1M  rad/s.  

   

4. Design   an   active   band   reject   filter   using   RC   elements   and   an   opamp.   The   center   frequency  should  be  at  100  kHz  and  the  rejection  band  should  be  100  kHz  wide.    

5. Find  the  value  𝑣!"#(𝑡)  in  the  following  circuit  for  the  following  input  source  values:  -­‐ 𝑣! 𝑡 = 10!! ∙ cos  (20×10!𝑡 + 78°)  -­‐ 𝑖! 𝑡 = 10!! ∙ cos  (32×10!𝑡 + 78°)