Chemistry 454

Lecture Notes
February 2, 2001

These notes may often differ from your class notes. This occurs frequently when I rely on the blackboard for illustrations. I do not check or follow my notes carefully, so material may change slightly. This is often the case with the order that topics appear. Also, class may end before I cover everything I intended so the notes may have additional material. This material usually shows up at the start of the next class period.

When I generate Web pages several days later, I may rely more on my notes than on my memory of what sequence we followed in class.

2/2/01--

• I'd like to finish Chapter 4 (Digital Electronics)
• Then cover main points of Chapter 5 (Signal/Noise)
  • we actually have already covered, less formally, most of the ideas in this chapter

• I want to start Chapter 6
  • (Optics, Spectroscopy and Instruments based on light)
• We really need to get to looking a real instruments and real analytical measurements
• The earlier chapters are essential ideas and tools, but not really Instruments or Analysis

Binary Numbers

• a few additional remarks
• typically on ways computers store numbers

• Integers are easy
  • we often group 4 bits into a byte
    • a byte can represent numbers from 1 to 15
    • computer memory is usually a multiple of 4 bits
    • A "word" is 8 bits or two bytes
    • a word can have 256 different values
      • (enough to code each letter a-z, A-Z; 0-9 and most punctuation in one byte)
    • easier to read, write if binary is broken into bytes
      • (analogous to writing 1,231,384 with the commas every three digits)
    • Simple number: (msb) 1101 1010 1011 (lsb=least significant bit)
    • We can convert to decimal (see text, earlier notes)

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  Hexadecimal Numbers

  

  • Obviously 1 byte (4 bits) can count to 15 (decimal)
  • For the first 8 values the MSB is 0
  • The next two values, 1000= 8 and 1001= 9 are easy too
  • At 1010 there's a difficulty
    • we don't want to say "10"
    • we need a single digit to express "ten"
    • we use A as the next hexadecimal digit
    • 1011=B 1100=C 1101=D 1110=E 1111=F

  • Computer data and addresses are often given in Hexadecimal Form
    • (You don't need to be skilled in hexadecimal)
    • (You shouldn't be surprised by it)

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  Binary Representation of Numbers that are not integers

  • decimal: 731.546 is really 731 + 546 /1000 stored as two integers
    • second integer tells the value of a fraction
  • binary: 1100 0110 . 0111
    • (try it in hexadecimal)
    • really A6 + 7/ (16) (0r 7/16 as fraction)
  • computer usually use several words to store longer numbers
  • floating point numbers have a 3rd integer for the exponent to cover very large to very small values
    • 6.023 E23 decimal
    • analogous binary form

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  A few useful lessons to remember
  • integers take much less computer memory than floating point
  • integer arithmetic is much faster
  • transferring integers is much faster
  • most experimental measurements are made as binary integers
    • may be useful to do as much as possible in the integer form
    • faster, less memory demand
  • However, few experiments collect more than a few thousand data points
    • perhaps 32K or even 64K points
    • memory comes in M (millions, not thousands of bytes)
    • we seldom are worried about the amount of memory that must be used for stored
      • this used to be a serious concern, but the cost of memory has dropped greatly in the past decade
    • we are often concerned with the rate at which data must be transferred.

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  Digital electronics

  typically has one wire per digital bit
  • simple signals (on/off) need one wire
  • complex signals will need several wires
  • circuit is said to be parallel (all data bit are simultaneous)
  • the collection of lines may be called a BUS
    • The alternative is a serial signal
    • all data bit appear on the same wire
    • each bit must occur at a specified time

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  Simple digital gates or devices

  process digital signals

  

  • Inverter (changes 1 to 0, or 0 to 1)
  • AND (output is 1 only if all inputs are 1's)
  • OR (output is 1 if any input is a "1")
  An AND can be used to turn on/off another line
  • If lower terminal is zero, output is zero
    • no matter what happens at input #1
    • If lower terminal is a "1", out put matches input #1
      • If input is a 1, then we get a "1"
      • If input is a "0", then output is a zero
    • If we send a square wave to input #1
      • ii passes if terminal 2 is "1"
      • it stops if terminal 2 is a "0"

  ---

  We can also build a device that responds only to a specific numerical input
  • example responds to 5(dec) or 0101(binary)
  • the example uses two inverters (Will convert 0101 to 1111)
  • then the AND gate (four inputs) gives a "1" output
  • any other input combination yields a "0" output

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  Another type of device responds not to the value of the input but to changes in the input
  • These are edge-triggered devices
    • we'll consider a + edge device
    • it recognizes a signal changing from 0 to 1
      • it ignores steady values of 1 or 0
      • it ignores the change from 0 to 1
  Most useful form is a Flip Flop
  • (really a monostable multivibrator)
  • output changes state whenever a 0->1 transition occurs
  • if we send in a square wave
    • signal changes state once per cycle
    • needs two cycles to change back
    • basically converts 2 pulses into one pulse
    • it divides the signal by two
  A binary counter could be made from, say, four FF's
  • output of one connected to input of the next
  • the first output changes rapidly (is LSB)
  • the last output changes only after 8 pulses
  • four wires have a binary value= # pulses in
  • a binary counter capable of counting 0-15 (0-F hex)

  Aside: if you add additional wiring between stages 3 and 4

  • you can get it to count to 10
  • these are called BCD (Binary Coded Decimal ) counters
  • useful when mere mortals want to read the count

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  Analog to Digital Converters

  • ways to measure and digitalize analog voltages
  • (my examples will focus on a 0-1.000 voltage range)
  • (most real devices cover -5 to +5 or -10 to +10 volt ranges)

  Simplest device uses a voltage controlled oscillator (VCO)

  • this produces a square wave
  • the frequency is proportional to the input voltage
    • say we count for 1 second
    • we can use our binary counter
    • we can use a AND gate to limit the count to 1 second
  • if we set it up right
    • a count of 337 means 0.337 volts input

    This isn't a very good way to measure voltages

    • but all computers can count pulses on their serial data port
    • some have game ports that do this
    • so we can use this with any computer

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  It's easier to Begin with the DAC
  • Digital to Analog Converter
    • easy to build with an Op Amp
      • Assume we have a -1.000 volt signal
    • If I connect R(0) to this signal, I get 0.0625 V out
      • gain is -1/16 when R(f) and R(0) are involved
    • If I connect R(1) to this signal, I get 0.125 V out
      • gain is -1/8 when R(f) and R(1) are involved
    • If I connect the signal to R(2)
      • gain is -1/4
      • so I get 0.250 volts out
    • If both R(1) and R(2) are connected
      • then I get 0.125 + 0.250 = 0.375 v
    • If I use R(3)
      • the gain is -1/2
      • signal changes by 0.500V
    • If I use R(4)
      • the gain is -1
      • signal changes by 1.000 volt
    • By a combination of these four resistors
      • I can create voltages from 0 - 1.9375 volts
      • With a resolution of 0.0625 V
  • A typical DAC has 10, 12 or 16 input lines
    • so we get a much wider choice of output voltages
  
  We let a computer or other digital electronic circuit control our "switches"

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  Now for an Analog to Digital Converter

  
  • Start with a 10 bit counter
    • one wire lets us set the counter to zero (reset)
  • Connect the counter output lines to the DAC input
  • assume the LSB drives a 1 mV output
  • DAC voltage = "count" in mV
• Feed a square wave in through one of our AND gates
• As long as the other line on the AND is a 1
• the count increases and the voltage advances in 1 mV steps
• this can continue to a total count of 1023
• or an output voltage of 1.023 volts

two drawbacks worth considering

• 1. it's slow-- we need 1023 steps to make a measurement
  • if we count each microsecond (typical)
  • it takes over 1 msec to read a voltage
    • slow for many purposes
• 2. it's subject to errors due to noise
  • we only require V(DAC)= signal
    • at any instance
  • negative noise can trigger the comparator early
  • (in fact, will read low more often than high due to noise)

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The Successive Approximation Converter

• Uses a DAC and a Comparator as before
• Needs a more elaborate controlling device
• Start with a 1 on DAC's MSB
  • in our example, that generates 0.500V
• Compare signal to DAC output
  • IF V(DAC) < V(signal)
    • then keep a "1" in this digit
  • If V(DAC) > V(signal)
    • then reset this digit to a "0"
• Now shift to the next digit and repeat the process
  • we find the voltage to the nearest 1.000/n volts each time
• After 10 steps, we know the voltage to 10 bit accuracy
  • That's 100 times faster than our earlier example
• Good ADC based on successive approximation converters
• available
• compact, automatic
• fast (5-20 microseconds for 12 bit accuracy)
• modest cost ($20-50 typical)
• available at 8, 10, 12, 16 bit accuracy
• some can measure in 0.05 microseconds

Most ADC devices are susceptible to noise they measure the voltage at some very brief instant they do not perform any signal averaging there's no filtering of high frequency noise most voltmeters, pH meters and recorders filter out high frequency components

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There's another ADC that is less susceptible to noise

• it is much slower
• this is the way most digital voltmeters work
• this is mainly an analog circuit
• start with a precision voltage source
  • we'll stick with a -1.000 reference
• we will send our signal into an integrator
  • we get a voltage that increases with time
  • the slope is proportional to the input voltage
  • we also start a time (counter)
• we integrate for 1.0 seconds
  • disconnect the signal
  • connect the reference
  • restart a timer
• continue to integrate until the voltage is back to 0.000V
• evaluate the voltage as 1.000 V * (time_to_down / 1.0 sec)