Logic, Circuits, and Truth Tables

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Logic, Circuits,and Truth Tables

• CS 147Dr. Sin-Min Lee
• Presented by
• Kristina Miguel

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Introduction

• The CPU is constructed from logic gates.
• The basic activity of the control unit is
  decoding instructions.
• Decoder circuits use an input binary number to
  select an output line, or several lines.
• Logic circuits can be implemented directly from
  truth table information.

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Review Basic Logic Gates with Truth Tables
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Traffic Light Controllers Impossible to avoid!

• Traffic controllers are an example of decoder
  circuits.
• Integrated urban traffic management schemes
  depend on communication links between the traffic
  light controllers and a central control computer.

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Example System

• Traffic light controllers found at British road
  junctions have the sequence Red, Red/Amber,
  Green, Amber, and then Red again.
• We will analyze the corresponding truth tables.

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Circuit Implementation from Truth Tables some
practical tips
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Tip 1 Identical Columns

• The input is directly connected to the output if
  an output column is identical to an input column.
  No logic is required.
• The level crossing Amber output and Y input are
  identical and can be expressed as

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Tip 2 Nearly Identical Columns

• An output can be generated by a simple logic
  function from only some of the inputs.
• The level crossing Red is the inverse of input X.

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Tip 3 Solo Row

• Use an AND gate to detect the input row if an
  output column only has a single 1.
• The level crossing Green can be expressed as

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Tip 3 Solo Row (cont.)

• The crossroads Green can be expressed as

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Tip 4 Inverted Solo Row

• Use an AND gate to detect the input row pattern,
  and then a NOT inverter when an output column
  only has a single 0.
• Crossroads Red can be expressed as

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Tip 5 Standard Patterns

• Sometimes it is possible to utilize an existing
  logic circuit, with minimum modification.
• Let us consider a simple washing machine
  controller.
• XOR can be used as a difference detector as
  seen in the motor control output from the
  following washing machine example.

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Tip 5 Standard Patterns (cont.)

• Motor can be expressed as
• Motor (X XOR Y) AND Z

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Tip 6 Elimination

• A shortcut can be taken when considering the
  Crossroads truth table.
• There are two rows where both X and Z contains
  1 and the value of Y has no effect in
  determining the output.
• Crossroads Amber can be expressed as

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Tip 7 Sum-of-Products

• When these short cuts do not work out, the brute
  force method remains
• Run down each output column and mark every row
  which contributes a 1.
• Next set up an AND gate pattern detector for each
  marked row, using NOT gates on inputs if 0 is
  detected.
• Then allocate an OR gate to each output column.

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Decoder Logic essential for control units and
memories

• The essential purpose of a decoder is to
  recognize a code number and invoke the
  corresponding action.
• The schematic diagram for a decoder is a box with
  more output than input lines.
• Proper decoders select a single output line at
  a time.
• Example 74F138 Decoder
• 3 to 8 line decoder
• Accepts three binary weighted inputs.
• The output selected depends on the input number.
• Ideal for memory chip select decoding.

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Example 74F138 Decoder
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Another Example Seven-Segment Converter

• The seven-segment decoder has three inputs and
  eight outputs and could be modified to implement
  a binary to seven-segment display converter.

• The unit is constructed from seven LEDs
  (Light-Emitting Diodes).

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Applying the Brute Force Method

• Using logic expressions you can design a
  seven-segment display converter driver circuit.
• The truth table logic terms for each output
  column can be directly implemented by the brute
  force method.

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Applying the Brute Force Method (cont.)
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Applying the Brute Force Method (cont.)
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Using Logic Expressions to Design a Logic Circuit

• Logic circuit for the a segment of a
  seven-segment LED driver.
• a can be derived by taking the complement of
  inverse a

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Summary

• Truth tables can clearly express logic circuit
  functionality by displaying input-output
  relationships.
• Some short cuts can be used to speed up the
  process of deriving logic circuits from truth
  tables.
• Logic minimization may not be the priority when
  building circuits.