Digital Logic Introduction

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Digital Logic Introduction

• Bryan Duggan

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Background

• Digital logic is at the core of all digital
  computers, from Microwaves to Mainframes, from
  the 1940s onwards
• Much of the pioneering work was done in Blechley
  Park, by Alan Turing, Max Newman, Freddie
  Williams etc. It came out of Turings 1936 paper
  On Computable Numbers with an application to the
  Entscheidungsproblem. Turing created the idea of
  the Universal Turing Machine
• Was used to crack the Enigma coded messages of
  the German U-Boats, using the Colossus machine,
  created by Max Newman.
• Much of the work was destroyed, but some of the
  mathematicians went on to make the Mark 1
  computer in Manchester, the first digital
  computer created by Tom Kilburn
• To learn more http//www.computer50.org/

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Background Continued

• Digital logic and the two valued number system is
  at the core of all digital computers, and yet
  it's entirely based on three fundamental
  operations, designated AND, OR, and NOT
• There are many different ways to implement
  logical functions electronically, valves, relays,
  mercury tubes, mechanical but now exclusively
  transistors and ICs (Integrated circuits).
• Applications in programming, hardware etc.
• All programming languages have a concept of true
  and false, conditional evaluations, the IF
  statement. It is useful to know digital logic as
  it will make you better programmers and thinkers.

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Background

• Everything in the digital world is based on the
  binary number system. Numerically, this involves
  only two symbols 0 and 1. Logically, we can use
  these symbols or we can equate them with others
  according to the needs of the moment. Thus, when
  dealing with digital logic, we can specify
  that0 false no1 true  yes
• Using this two-valued logic system, every
  statement or condition must be either "true" or
  "false" it cannot be partly true and partly
  false
• The essential reason for basing logical
  operations on the binary number system is that it
  is easy to design simple, stable electronic
  circuits that can switch back and forth between
  two clearly-defined states, with no ambiguity
  attached. It is also possible to design and build
  circuits that will remain indefinitely in one
  state unless and until they are deliberately
  switched to the other state. This makes it
  possible to construct a machine (the computer)
  which can remember sequences of events and adjust
  its behavior accordingly. It would be difficult
  to design circuits with 10 or more states
  required to represent 0 -9 or A- Z

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Introduction

• At an abstract level, any digital logic system
  may be regarded as a black box, with inputs and
  outputs, (GIGO). Each input and output may be
  either 1 or 0. Voltages are used to represent
  these values. A low voltage (0V) for false and a
  high voltage for true (5V). This is because of
  the way transistors work

Digital Logic System
Outputs
Inputs
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Introduction

• Digital logic may be divided into two classes
  combinational logic, in which the logical outputs
  are determined by the logical function being
  performed and the logical input states at that
  particular moment and sequential logic, in which
  the outputs also depend on the prior states of
  those outputs. In other words the history of the
  inputs. Sequential logic systems are also known
  as Finite State Machines.
• Both classes of logic are used extensively in all
  digital computers. Since both types of logic
  circuits begin with logic gates to combine
  logical input signals in various ways to produce
  the desired outputs, we will be seeing how the
  basic logic gates work.

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Sequential Logic circuit
Combinatorial Logic
Inputs
Outputs
Memory
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1. The Buffer Gate

• Input of A and an output of Z
• Although the use of the circuit is not
  immediately apparent, it is in fact widely
  used.Truth Table

Z A
A
Z
A Z 0 0 1 1
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2. The Not Gate

• With a not gate, the value of the input is the
  inverse or complement of the value of the input
• It always has exactly one input as well as one
  output. Whatever logical state is applied to the
  input, the opposite state will appear at the
  output.The NOT function, as it is called, is
  necessary in many applications and highly useful
  in others. A practical verbal application might
  beThe door is NOT locked You may enter
• The NOT function is denoted by a horizontal bar
  over the value to be invertedZ A

A
Z
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2. The NOT Gate Continued

• Truth TableA Z 0 1 1 0
• The logical symbol for the NOT gate consists of
  two parts, a triangle and a circle, or bubble as
  it is called. The triangle means the element
  consists of a buffer the bubble indicates it is
  followed by an inverter. The simplicity of the
  NOT gate hides its importance, without it many
  digital logic systems would be impossible.

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3. The And Gate

• The AND gate implements the AND function. With
  the gate shown, both inputs must have logic 1
  signals applied to them in order for the output
  to be a logic 1. With either input at logic 0,
  the output will be held to logic 0.
• The equation is read Z A AND B. There is no
  limit to the number of inputs that may be applied
  to an AND function, so there is no functional
  limit to the number of inputs an AND gate may
  have. The number of inputs can be increased to n
  to give an n input AND gate. The generalised
  equation then becomesZ A.B.C...

A
Z
B
Z A.B
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3. The AND gate Continued

• The Truth table for an AND gate looks like this
• It has 3 columns, one for each input A and B and
  one for the output Z. There are 4 rows, since
  there are 22 4 combinations of the inputs A and
  B. The output Z will only be 1 when all the
  inputs are 1
• Question 1 How many rows would there be for a 3
  input AND gate?
• Question 2 How can you implement a NOT gate with
  an AND gate? (2 answers)

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4. The OR Gate

• The output of the OR gate is true, when either of
  the inputs A OR B is true, or when both are true.
  It looks like this
• The OR gate is sort of the reverse of the AND
  gate. The OR function, like its verbal
  counterpart, allows the output to be true (logic
  1) if any one or more of its inputs are true.
  Verbally, we might say, "If it is raining OR if I
  turn on the sprinkler, the lawn will be wet."
  Note that the lawn will still be wet if the
  sprinkler is on and it is also raining. This is
  correctly reflected by the basic OR function.
• As with the AND function, the OR function can
  have any number of inputs.

A
Z
B
Z A B
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4. The OR Gate Continued

• The truth table for an OR gate looks like this
• The OR gate can also be generalised into the
  formZ A B CIn this case, the output Z
  is true when any of the inputs is true

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• Any logic circuit, no matter how complex may be
  completely described using the Boolean operations
  previously defined, because the OR gate, the AND
  gate and the NOT circuit are the basic building
  blocks of digital systems. The above operations
  are useful on their own, but are usually combined
  to make memory units, calculation units,
  registers etc
• Question Derive the truth table for this
  circuit.

A
A.B
B
Z A.B C
C
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Truth Table for A.B C
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Circuits containing invertors

• Whenever an inverter is present in a logic
  -circuit diagram, its output is simply the input
  expression with a bar over it. Question Draw
  the truth table for the following

A
Z A B
B
A
B
Z A B
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Evaluating a Boolean Expression

• You can evaluate a Boolean expression by
  substitution. For example, A 0, B 0, C 1,
  D 1 E 1Z D (A B)C . E
• (72)
• General rule for substitutions are1. Perform
  all inversions for single terms first2. Then
  perform all operations within parenthesis3.
  Perform ANDs before Ors, unless parenthesis
  dictate otherwise4. If an expression has a bar
  over it, perform the operations of the expression
  first and then invert the result.

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Implementing a circuit from a Boolean expression

• If the operation of a circuit is defined by a
  Boolean expression, a logic circuit diagram can
  be implemented directly from the expression. For
  example if we needed a circuit which was defined
  by the expression Z A . B . C, we would know
  that all we needed was a three input AND
  gate.What would we need for thisZ A B
• Question Construct a circuit whose output is Z
  AC BC ABC
• Hint This Boolean expression contains 3 terms
  which are ORed together, so we need a 3 input OR
  gate, with inputs which equal to AC, BC and ABC
  (74)

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5. The NAND Gate

• The NAND gate implements the NAND function, which
  is exactly inverted from the AND function you
  already examined. Both inputs must have logic 1
  signals applied to them in order for the output
  to be a logic 0. With either input at logic 0,
  the output will be held to logic 1.
• The circle at the output of the NAND gate denotes
  the logical inversion, just as it did at the
  output of the inverter. Note that the overbar is
  a solid bar over both input values at once. This
  shows that it is the AND function itself that is
  inverted, rather than each separate input.

A
Z A . B
B
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5. The NAND Gate Continued
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6. The NOR Gate

• The NOR gate is an OR gate with the output
  inverted. Where the OR gate allows the output to
  be true (logic 1) if any one or more of its
  inputs are true, the NOR gate inverts this and
  forces the output to logic 0 when any input is
  true.
• In symbols, the NOR function is designated with a
  plus sign (), with an overbar over the entire
  expression to indicate the inversion. In logical
  diagrams, the symbol to the left designates the
  NOR gate. As expected, this is an OR gate with a
  circle to designate the inversion

A
Z A B
B
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6 . The NOR Gate Continued
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6. And finally...the XOR Gate

• The Exclusive-OR, or XOR function is an
  interesting and useful variation on the basic OR
  function. Verbally, it can be stated as, "Either
  A or B, but not both." The XOR gate produces a
  logic 1 output only if its two inputs are
  different. If the inputs are the same, the output
  is a logic 0.
• The XOR symbol is a variation on the standard OR
  symbol. It consists of a plus () sign with a
  circle around it. The logic symbol, as shown
  here, is a variation on the standard OR symbol.
• Unlike standard OR/NOR and AND/NAND functions,
  the XOR function always has exactly two inputs,
  and commercially manufactured XOR gates are the
  same

A
Z A B
B
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6. The XOR Gate
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The Universality of NAND gates
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The Universality of NOR gates