CP1300 Circuits

• Logic gates and circuits
  • In a computer different voltages represent 0's and 1's
  • Logic gates manipulate the voltages (0s and 1s) like Boolean algebra

    

  • Collections of gates (circuits) are small components, collections of gates and small components are ICs

  • Working out what a circuit does
    • Put on the inputs and propagate them through the circuit
    • Use a truth table to show all possible combinations
    • Example

      

• Designing circuits
  (Jason's Notes)
  • For given binary inputs, the output is known
    • Expressable as a truth table

      X Y Z  ?
      0 0 0  0
      0 0 1  1
      0 1 0  0
      0 1 1  1
      1 0 0  1
      1 0 1  1
      1 1 0  1
      1 1 1  1
      

    • OR of the AND of the inputs for rows with output 1

      X Y Z  ?
      0 0 0  0
      0 0 1  1  X'*Y'*Z +
      0 1 0  0
      0 1 1  1  X'*Y*Z  +
      1 0 0  1  X*Y'*Z' +
      1 0 1  1  X*Y'*Z  +
      1 1 0  1  X*Y*Z'  +
      1 1 1  1  X*Y*Z 
      

    • Build the circuit
  • Efficient Boolean functions
    • What is needed?
      • Smallest number of gates
      • Inexpensive combination of gates
      • A problem for Computer Scientists and Computer Systems engineers
    • Using standardized components and ICs
      • Have a standard set of modules to solve problems with
      • Modules have a standard way of being connected
      • Easier for the designer, less likely to get errors
      • Sub-optimal results
    • Simplifying functions
      • Use of Boolean laws (expensive and complicated)
      • Specialized techniques, e.g., Karnaugh Maps, are useful
      • Example:

        X Y Z  ?
        0 0 0  0
        0 0 1  1  X'*Y'*Z +
        0 1 0  0
        0 1 1  1  X'*Y*Z  +
        1 0 0  1  X*Y'*Z' +
        1 0 1  1  X*Y'*Z  +
        1 1 0  1  X*Y*Z'  +
        1 1 1  1  X*Y*Z 
        

  • Example: Design a half adder

• Small Logic Components
  • Adders

    

    • The half adder adds two bits
    • The full adder adds three bits, using two half-adders
    • The parallel adder adds two numbers (64 bits each)
  • Memory
    • A flip-flop is a circuit which remembers its previous state.

      

    • Y = (X'.C) NOR (Y NOR (X.C))
      = (X'.C) NOR (Y+X.C)'
      = ((X'.C) + (Y+X.C)')'
      = (X'.C)' . (Y+X.C)''
      = (X''+C') . (Y+X.C)
      = (X+C') . (Y+X.C)
      = X.Y + X.X.C + C'.Y + C'.X.C
      = X.Y + X.C + C'.Y
      = X.(Y + C) + C'.Y
    • If Control = 1 then
      Y = X.(Y + 1) + 0.Y
      = X (i.e., it stores the input value)
    • If Control = 0 then
      Y = X.(Y + 0) + 1.Y
      = Y (i.e., it remembers itself)
    • A byte is built from eight of these, etc.

• Building logic gates
  • Transistors
    • Logic gates are built using transistors.

      

    • Apply a voltage to the Base
    • The voltage effects the semi-conductors
    • Allows another voltage to flow from the Collector to the Emitter
  • OR gate

    

  • AND gate

    

  • NOT gate

    

  • All functions can be built from AND, OR, and NOT

    

• The Basics of Computer Manufacturing?
  • Organizing the gates on the circuit board
    • Smallest surface area
    • Minimal power requirements
    • Connecting components
    • A problem for Computer Systems engineers

  • Very Large Scale Integration (VLSI)
    • Circuits with ten thousand to one million gates on a single chip
    • Modern CPUs are a good example (ie. Pentiums, ...)
    • How to design solutions for VLSI?
      • Dividing the problem into sub-problems... (like top-down design)
      • Solving each sub-problem on its own
      • Combining the sub-solutions into the overall solution
    • Time-consuming to build: e.g. 6 to 18 months
    • Designers use CAD tools to overcome the problems: heat, power, synchronization, external interference

  • Construct the wafer (tens of thousands of chips imprinted on)

    

  • Select from the wafer the good chips (with no defects)
  • Cut off good chips, and add pins and packaging
  • Test again
  • Construct the circuit boards to hold the chips
  • Combine the chips, etc..., onto the circuit boards
  • Combine the boards and build a system

Self assessments

• Small Components
• Designing Circuits (Q10 to end)

Tutorial Questions

• Logic Gates (Answers)
• Karnaugh Maps (Q6 to end) (Answers)
• More Digital Logic (Answer)

Exam style questions

1. What is an IC?
2. What function do logic gates perform? What are they composed from?
3. Draw simple diagrams which show the structure of the AND, OR, and NOT gates.
4. What are small components?
5. Draw the small component for the expression F = (X * Z') + (Y * Z)
6. What is a full adder? What is a half adder? Draw the structure of each.
7. In terms of adders, how are two 16bit binary numbers added together?
8. What are the biggest problems in circuit design?
9. What are the three approaches to designing the 'best' Boolean functions? Briefly explain an relative advantages/disadvantages between each.
10. What is VLSI?
11. What are the steps of manufacturing a computer?