CSE 1520 Computer Use: Fundamentals

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• Week 6 Gates and Circuits PART I
• READING Chapter 4

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Gates and Circuits
EECS 1520 -- Computer Use Fundamentals

• What is a gate?
• A gate is a device that performs a basic
  operation on electrical signals

• What is circuit?
• Gates are combined to form different circuits
  to perform more complicated tasks

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Gates and Circuits
EECS 1520 -- Computer Use Fundamentals

• Three notational methods to describe the behavior
  of gates
• Boolean expressions A form of algebra in which
  variables and functions take on only one of two
  possible values (0 and 1)
• Logic diagrams graphical representation of a
  circuit
• Truth tables defines the function of a gate by
  listing all possible input combination and the
  corresponding output.

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Gates and Circuits
EECS 1520 -- Computer Use Fundamentals

• A gate or logic gate performs only one logical
  function. Each gate accepts one or more input
  values and produces a single output value.

• Six types of logic gates
• NOT
• AND
• OR
• XOR
• NAND
• NOR

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Gates and Circuits NOT Gate
EECS 1520 -- Computer Use Fundamentals

• Also referred to as an inverter
• If the input value is 1, the output is 0 if the
  input value is 0, the output is 1

Logic diagram Symbol
Truth Table
Boolean Expression

• Sometimes the mark is replaced by
  horizontal bar placed over the value

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Gates and Circuits AND Gate
EECS 1520 -- Computer Use Fundamentals

• If the two input values are both 1, the output is
  1 otherwise, the output is 0

Logic diagram Symbol
Truth Table
Boolean Expression

• Sometimes the . mark is replaced by the
  asterisk symbol

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Gates and Circuits OR Gate
EECS 1520 -- Computer Use Fundamentals

• If both input values are both 0, the output is 0
  otherwise, the output is 1

Logic diagram Symbol
Truth Table
Boolean Expression
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Gates and Circuits XOR or exclusive OR Gate
EECS 1520 -- Computer Use Fundamentals

• If the two inputs are the same, the output is 0
  otherwise, the output is 1

Logic diagram Symbol
Truth Table
Boolean Expression

• Not the difference between the XOR gate and the
  OR gate they only differ in one input situation
• When both input signals are 1, OR gate produces a
  1 and the XOR gate produces a 0

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Gates and Circuits NOR Gate
EECS 1520 -- Computer Use Fundamentals

• The NOR gate is essentially the opposite of the
  OR gate. That is, the output of a NOR gate is
  the same as if you took the output of an OR gate
  and put it through a NOT gate

Logic diagram Symbol
Truth Table
Boolean Expression
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Gates and Circuits NAND Gate
EECS 1520 -- Computer Use Fundamentals

• The NAND gate is the opposite of the AND gate.

Logic diagram Symbol
Truth Table
Boolean Expression
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Transistors
EECS 1520 -- Computer Use Fundamentals

• How do we implement the gates?
• A gate uses one or more transistors to establish
  how the input values map to the output value
• A transistor acts like a switch.
• It either turns on to conduct electricity or
  turns off to block the flow of electricity

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Transistors
EECS 1520 -- Computer Use Fundamentals

• A transistor has three terminals source, base
  and emitter

source
output
base
emitter

• When an electrical signal is grounded, it has 0
  volts!
• If the source signal is pulled to ground, the
  output signal is low output is 0
• If the source signal remains high, the output
  signal is high output is 1

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Transistors NOT Gate
EECS 1520 -- Computer Use Fundamentals

• The output is determined by the base electrical
  signal.

source
Vout
Vin Vout
1 0
0 1
base
Vin
emitter

• If Vin is high, the source is pulled to ground
  and Vout is low (i.e. 0)
• If Vin is low, the source is not grounded and
  Vout is high (i.e. 1)

NOT Gate needs 1 transistor
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Transistors NAND Gate
EECS 1520 -- Computer Use Fundamentals
source
Vout
Vin1 Vin2 Vout
0 0 1
0 1 1
1 0 1
1 1 0
Vin1
Vin2
emitter

• If Vin1 and Vin2 are high, the source is pulled
  to ground and Vout is low (i.e. 0)
• If Vin1 and Vin2 are low, the source is not
  grounded and Vout is high (i.e. 1)
• If either Vin1 or Vin2 is low, the source is not
  grounded and Vout is high (i.e. 1)

NAND Gate needs 2 transistors
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Transistors NOR Gate
EECS 1520 -- Computer Use Fundamentals
source
Vin1 Vin2 Vout
0 0 1
0 1 0
1 0 0
1 1 0
Vout
Vin1
Vin2
emitter
emitter

• If Vin1 and Vin2 are high, the source is pulled
  to ground and Vout is low (i.e. 0)
• If Vin1 and Vin2 are low, the source is not
  grounded and Vout is high (i.e. 1)
• If either Vin1 or Vin2 is low, the source is
  grounded and Vout is low (i.e. 0)

NOR Gate needs 2 transistors
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Transistors OR Gate
EECS 1520 -- Computer Use Fundamentals

• Since OR gate is the opposite of NOR gate, how
  many transistors would you think will be required
  to implement the OR gate?

OR Gate needs 3 transistors
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Combinational Circuits
EECS 1520 -- Computer Use Fundamentals

• Gates are combined into circuits by using the
  output of one gate as the input for another gate.
• For example

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Combinational Circuits
EECS 1520 -- Computer Use Fundamentals

• For example

Logic diagram Symbol
Truth Table

• Since there are 3 inputs, there are 8 possible
  outcomes

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Combinational Circuits
EECS 1520 -- Computer Use Fundamentals

• For example

Logic diagram Symbol
Boolean expression

• D A B
• E AC
• X AB AC

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Combinational Circuits
EECS 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
EECS 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
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Combinational Circuits
EECS 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
EECS 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)
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Properties of Boolean Algebra
EECS 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