CSE 1520 Computer Use: Fundamentals

1
CSE 1520 -- Computer Use Fundamentals

• Week 5
• Glade Manual Chapter 4
• Gates and Circuits (Dale Lewis Ch. 4)

2
Combinational Circuits
CSE 1520 -- Computer Use Fundamentals

• Recall the following logic circuit (called
  circuit 1)

Logic diagram Symbol
Boolean expression

• D A B
• E AC
• X AB AC

3
Combinational Circuits
CSE 1520 -- Computer Use Fundamentals

• Now, we want to investigate the following Boolean
  expression

X A(BC)

• How do we want to create the logic diagram
  (called circuit 2) of the above Boolean
  expression?

- We have an inner function which consists of an
OR gate between B and C - We then have an
outer function which is an AND gate between
A and (BC)
Logic diagram Symbol (circuit 2)
A(BC)
BC
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Combinational Circuits
CSE 1520 -- Computer Use Fundamentals

• We have the following

X A(BC)
Boolean expression
Logic diagram Symbol
A(BC)
BC
A B C BC A(BC)
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
A B C BC A(BC)
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
A B C BC A(BC)
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 1 0
1 0 0 0 0
1 0 1 1 1
1 1 0 1 1
1 1 1 1 1
Truth table
5
Combinational Circuits
CSE 1520 -- Computer Use Fundamentals

• Circuit 1

• Circuit 2

A(BC)
BC
A B C BC A(BC)
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 1 0
1 0 0 0 0
1 0 1 1 1
1 1 0 1 1
1 1 1 1 1
A B C D E X
0 0 0 0 0 0
0 0 1 0 0 0
0 1 0 0 0 0
0 1 1 0 0 0
1 0 0 0 0 0
1 0 1 0 1 1
1 1 0 1 0 1
1 1 1 1 1 1

• Their results are identical!

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Combinational Circuits
CSE 1520 -- Computer Use Fundamentals

• We have therefore demonstrated circuit equivalence

• That is, both circuits produce the same results
  for each input combination

• Boolean algebra allows us to apply provable
  mathematical principles to help us design logical
  circuits

• From the previous example

X AB AC A(BC)
7
Properties of Boolean Algebra
CSE 1520 -- Computer Use Fundamentals

• DeMorgans law, in particular, is very useful in
  Boolean algebra.

• For instance, it means that

___ ___ ___
1 NAND gate is equivalent to 2 NOT gates with an
OR gate
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Properties of Boolean Algebra
CSE 1520 -- Computer Use Fundamentals

• So, what does it imply?

• Suppose we have the following logic diagram

3T
A
2T
Requires 8 transistors in total to implement
B
3T
C
D

• Recall that a NAND gate needs 2 transistors

Vout
Vin1
Vin2
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Properties of Boolean Algebra
CSE 1520 -- Computer Use Fundamentals

• If we apply DeMorgans law

___ ___ ___

• So, we will obtain the following logic diagram

2T
A
3T
B
Requires 7 transistors in total to implement
2T
C
D
10
Addition
CSE 1520 -- Computer Use Fundamentals

• One of the most basic operations a computer can
  perform is to add two numbers together

• Addition operations in binary are carried out by
  special circuits called adders

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Adder
CSE 1520 -- Computer Use Fundamentals

• A circuit that computes the sum of two single
  bits and produces the correct carry bit is called
  a half adder

• How do we implement the circuit?

• Recall adding two binary digits

Sum
Carry
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Half Adder
CSE 1520 -- Computer Use Fundamentals

• Based on the previous results, we obtain 2 output
  results Sum, Carry

• The next step is to create a truth table that
  consists A, B, Sum and Carry

Sum
Carry
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Half Adder
CSE 1520 -- Computer Use Fundamentals

• Based on the previous results, we obtain the
  following truth table with 2 output results Sum,
  Carry

Corresponds to AND gate
A B Sum Carry
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
Corresponds to XOR gate
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Half Adder
CSE 1520 -- Computer Use Fundamentals

• Based on the previous results, the circuit for a
  half adder is

A B Sum Carry
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1

• Because the circuit produces two distinct output
  values, we represent the half adder with 2
  Boolean expressions

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Half Adder
CSE 1520 -- Computer Use Fundamentals

• A half adder does not take into account a
  possible carry value into the calculation
  (carry-in)

• For example if we want to perform another
  addition based on the following result, the Carry
  bit is ignored

Sum
Carry

• Half adder is only good for adding 2 single bits,
  but cannot be used to compute the sum of 2 binary
  values with multiple digits each

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Full Adder
CSE 1520 -- Computer Use Fundamentals

• A circuit called full adder takes the carry-in
  value into account

• Based on the logic diagram, we should then create
  the truth table for the full adder