Logic Circuits and Computers

1
Logic Circuits and Computers

• Switches provide a simple way of storing
  information

The light turns on if and only if current flows
through, that is when the switch is closed (on).
2
The light bulb turns on if and only if at least
one of the switches are closed
The light bulb turns on if and only if the both
switches are closed
3
Transistors

• Storing information using light switches would
  be very slow
• Transistor is a solid state device that uses
  electricity to turn on and off

4
Transistors

• Voltage is connected to current
• In controls a switch
• Supply voltage to the wire in then no current at
  out
• No voltage at in then current at out
• This represents a NOT gate

5
Transistors

• Only three gates are needed to build the computer
• NOT, AND and the OR gate
• AND gate has two input wires and one output

6
AND gate
If both of the wires have voltage, then the AND
gate must produce a voltage on the output wire
7
OR Gate
If either or both input wires have voltage then
the gate must produce voltage on the output. If
neither of the wires have voltage, then no
voltage is produced on the output
8
Logic circuits

• Three gates have been repeatedly used by
  electronic engineers and have their own symbols

AND
OR
NOT
9
Logic circuits

• Rules for combinational circuits
• Never combine two input wires
• A single input wire can be split partway and used
  as input for separate gates
• An output wire can be used as input

10
Determining output for a given Input
Indicate the output of the circuit below for the
given input signals. P1, Q0, R1
Solution
11
Constructing the input/output table for a Circuit
12
Finding a Boolean expression for a circuit

• Boolean expressions can be obtained by tracing
  the actions of the gates of input variables

P?Q
(P ? Q) ??(P?Q)
?(P?Q)
P?Q
13
Circuit corresponding to Boolean expression

• Write the input variables on the left hand side
  of the diagram
• Go from the right side of the diagram to the
  left, working from the outermost part of the
  expression to the innermost.
• Example
• Construct a circuit for the Boolean expression
• (?P?Q) ??Q

14
(?P?Q) ??Q
15
Circuit from a given I/O Table

• Identify each row for which the output is 1
• For each such row, construct an AND expression
  that produces a 1(Clause) for the exact
  combination of values for that row and a 0 for
  all other combinations of input values.
• Join the clauses that are true together using the
  OR, this gives the disjunctive normal form or
  sum-of-products

16
Circuit from a given I/O Table
17
Circuit from a given I/O Table

• P?Q?R is 1 if P1 and Q1 and R1
• P??Q?R is 1 if P1 and Q0 and R1
• P??Q??R is 1 if P1 and Q0 and R0

18
(P?Q?R)?(P??Q?R)?(P??Q??R)
19
Simplifying the Normal Form

• With simpler Boolean expressions, fewer gates are
  needed to implement the function
• Three methods that can be used to achieve
  simplification are
• Algebraic simplification
• Karnaugh Maps
• Quine-Mckluskey tables

20
Karnaugh Maps

• Karnaugh Maps are a convenient way of
  representing a Boolean function of small number
  of variables
• Among true clauses, find those that belong to a
  unique largest block of either 1,2,4, or 8 and
  circle those blocks
• (P?Q?R)?(P??Q?R)?(P??Q??R)

21
Karnaugh Maps

• (P?Q?R)?(P??Q?R)?(P??Q??R)

(P?R)?(P??Q)
22
(P?R)?(P??Q)