2012 Homework 1
2012 Homework 2

1.
Consider  the  following  circuit:   a
0
a
1 a
2
2R 2R 2R a
3
2R
V out
2R R R R
a) Calculate  V out   as  a  function  of  voltages  a
3
,  a
2
,  a
1
,  and  a
0
.   b) Assuming  a n
only  take  binary  values  0V  or  1V,  show  a  table  of  V out
for  all  possible   combinations  of  the  bits  a n
(i.e.  [0000,  0001,  0010,  0011…]  where  a
0
is  the  least   significant  bit).   c) What  is  the  resolution  of  the  output  V out
?   d) What  is  the  worst-­‐case  accuracy  of  the  output  V out
when  the  leftmost  resistor  2R  has  a   relative  error  of  1%?  What  are  the  binary  values  of  a
3
,  a
2
,  a
1
,  and  a
0
?   e) What  is  the  worst-­‐case  accuracy  of  the  output  V out
when  the  rightmost  resistor  2R  has  a   relative  error  of  1%?  What  are  the  binary  values  of  a
3
,  a
2
,  a
1
,  and  a
0
?   f) In  one  sentence,  what  is  the  function  of  this  circuit?

2.
A  photodiode  is  a  semiconductor  device  that  can  be  used   as  a  light  transducer.  It  behaves  ideally  in  the  following
fashion:
!
= − !
!
!
!"!
+ !
!
!
!
!
!
− 1
Where:   !
!
!
!"!
is  the  positive  photocurrent,  linearly  proportional  to  the  irradiance;   !
!
is
the  thermal  voltage,  25mV  at  room  temperature;     !
!
is  the  reverse  saturation  current,
100nA.   a) Find  the  open  circuit  voltage  V  (when  the  photodiode  is  not  loaded)  as  a   function  of   !
!
!
!"!
.    Plot  V  for   !
!
!
!"!
ranging  from  0  to  1uA.     b) At  what  photocurrent  and  output  voltage  is  the  sensitivity  maximum?  What   is  the  sensitivity  there?   c) Find  the  range  of   !
!
!
!"!
for  which  the  system  is  linear  within  5%.

3.
A  patient  is  very,  very  cold.  You  design  a  thermal  blanket  with  equivalent  circuit:
R
1
V s
R
2
R
B
Where:   !
!
is  the  source  voltage;   blanket  heating  resistor.
!
!
is  series  resistor;   !
!
is  shunt  resistor;   !
!
is  the   a.
Find  an  expression  for  the  heating  power  delivered  to  the  patient  via   !
!
b.
For     !
!
= 120 !"#$ ,   !
!
= 100Ω  ,   !
!
= 1 !
Ω  and   !
!
.
= 1 !
Ω ,    find  the  power   delivered  to  the  patient.   c.
Is  this  a  power  efficient  design?    Quantify  the  power  efficiency,  defined  as  the   ratio  of  delivered  power  over  sourced  power.   d.
How  would  you  modify  the  values  for  resistances   !
!
delivery?    What  power  efficiency  do  you  obtain?
and   !
!
to  maximize  power

4.
In  the  following  circuit  :
R
1
V in
C
1
R
2
V out
C
2
a.
Find  the  transfer  function  V out
(jω)  /  V in
(jω)  of  this  circuit.     b.
Show  a  Bode  plot  of  the  transfer  function,  with  log  amplitude  and  phase  as  a   function  of  log  frequency.   c.
Find  the  time  domain  output  of  this  circuit  for  an  input  of  a  linear  ramp:
!
!"
= !
for   !
> 0 ;   0   for   !
≤ 0
Hint:  you  may  use  Laplace  transforms.    Knowing  the  Laplace  transform  of  a  delta
function  and  a  step  function,  what  is  the  Laplace  transform  of  the  linear  ramp?

5.
Design  Problem:   a.
Design  a  second-­‐order  RLC  notch  filter,  with  input   !
!"
and  output   !
!"#
,  to   remove  60Hz  line  noise  in  the  input.  The  filter  must  have  minimum  input   impedance  of   !
!"
= 1 !"   .
b.
What  happens  if  you  operate  this  circuit  in  other  countries  in  Asia,  Europe  and   other  parts  in  the  world,  and  why?    How  would  you  change  your  circuit  to   operate  properly  in  these  countries?

1.
Using the following circuit:
V ref
V in
R
BENG 186B
Homework 2
SOLUTIONS
R
R
+
-
3.3V
+
-
3.3V
+
-
3.3V
R
B
1
B
0 a.
Make a table showing each combination of B and
0
B
1
values for regions of V in from 0 to V ref
. b.
What is the function of the circuit? c.
What is the resolution of the circuit? d.
What is the worst case accuracy of the ranges of V in
(from part a.) when the resistors are rated to 10%.

2.
Using the following circuit of a LVDT, where the shaded area represents a ferromagnetic rod that can be displaced in the +x or –x directions. Assume all three coils are of equal inductance. d
+x d d
-x d
+
V in
-
R
1
R
2
R
2
-
+
R
1
V o d a.
Find an expression for V as a function of o
V in for
− ≤ ≤ d . Hint: Assume ideal
LVDT and op-amp and use KCL. Also assume that the values for resistances R
1 and R are very large.
2 b.
Sketch the ratio
V o
V in input. as a function of x ranging from -2d to +2d for an AC sine

3.
Using the following circuit (assume ideal op-amp):
C
1
R
1
R
2
V in
+
-
V o a.
Find the transfer function (
ω
) as a function of radial frequency
ω
in terms of
R R C C
1
,
2
,
1
,
2
. b.
Find the input impedance and output impedance as a function of radial frequency
ω
. c.
Assuming values of R
1
= Ω
, R
2
= Ω
, C
1
=
10 nF , C
2
=
1 nF , sketch Bode plots of the transfer function (magnitude and phase), input impedance, and output impedance as a function of radial frequency
ω
.
C
2

4.
You are testing the bending properties of a bone using two identical strain gauges in the following setup. The nominal resistance at zero strain is R
G o
.
R
G
R
G
V s
R
-
V o
+
R
(top)
R
G
(bottom)
R
G
F a.
Derive the sensitivity
V o
ε
of the strain measurement in terms of V R R s
Assume gauge factor, G, for both strain gauges. Hint: You can make a
G o
, G . simplification with the small strain case. b.
Assuming that R
=
R
G o
, what does the sensitivity expression simplify to? What is the effect of a temperature increase on the sensitivity? c.
You now want to find a proper strain gauge to use for this test. You expect your strain at these locations to be ~5%. What should your gauge factor be in order to achieve a 1V output with a 10V source, using the expression you derived in part
(b)?

5.
Design a baby alarm circuit that monitors breathing and body temperature, sounding an alarm if breathing ceases for more than 5 seconds or temperature rises above 39°C. The alarm circuit should be mounted on an elastic waistband with integrated thermistor and strain gauge. The device must be powered by a single 3.3V battery. You may also use any other circuit elements that we have studied, but you need to specify all parameters such as values of resistance and part numbers of active circuit (e.g., OP27, LM311,
TLC555, etc.). Choose your components to minimize the power consumption.
The alarm speaker can be simplified to an 8
Ω
resistor that sounds when a non-zero voltage is applied to V : in
V in
Hint: You will need to research certain circuit component specifications to ensure appropriate temperature/strain response.