4. 4
Boolean Expression for Logic
Gate Network
ā¢ Similar to finding the expression for a
single gate.
ā¢ Inputs may be compound expressions
that represent outputs from previous
gates.
5. 5
Boolean Expression for Logic
Gate Network
= +
Here, we just
work our way
through from
left to right and
write out the
intermediate
terms...
6. 6
Bubble-to-Bubble Convention
ā¢ Choose gate symbols so that outputs
with bubbles connect to inputs with
bubbles. (If itās practicalā¦)
ā¢ Results in a cleaner notation and a
clearer idea of the circuit function.
8. 8
Letās redraw this removing as many bubble to bubble
connections then see how much easier it is to derive the
Boolean expression for S
Example 3.3
Figure 3.5 b
9. 9
Order of Precedence
ā¢ Unless otherwise specified, in Boolean
expressions AND functions are
performed first, followed by ORs.
ā¢ To change the order of precedence, use
parentheses.
12. 12
Logic Diagrams from Boolean
Expressions
ā¢ Use order of precedence.
ā¢ A bar over a group of variables is the
same as having those variables in
parentheses.
14. 14
Truth Tables from Logic Diagrams
or Boolean Expressions
ā¢ Two methods:
ā Combine individual truth tables from each
gate into a final output truth table.
ā Develop a Boolean expression and use it
to fill in the truth table.
16. 16
Truth Tables from Logic Diagrams
or Boolean Expressions
So here weāve just
combined the T/T columns
from the individual gates in
the diagram
17. 17
Circuit Description Using Boolean
Expressions
ā¢ Product term:
ā Part of a Boolean expression where one or
more true or complement variables are
ANDed.
ā¢ Sum term:
ā Part of a Boolean expression where one or
more true or complement variables are
ORed.
18. 18
Circuit Description Using Boolean
Expressions
ā¢ Sum-of-products (SOP):
ā A Boolean expression where several
product terms are summed (ORed)
together.
ā¢ Product-of-sum (POS):
ā A Boolean expression where several sum
terms are multiplied (ANDed) together.
19. 19
Examples of SOP and POS
Expressions
)()()(:POS
:SOP
CACBBAY
DACBABY
ļ«ļ·ļ«ļ·ļ«ļ½
ļ«ļ«ļ½
20. 20
SOP and POS Utility
ā¢ SOP and POS formats are used to
present a summary of the circuit
operation.
21. 21
Bus Form
ā¢ A schematic convention in which each
variable is available, in true or
complement form, at any point along a
conductor.
23. 23
Deriving a SOP Expression from
a Truth Table
ā¢ Each line of the truth table with a 1 (HIGH) output
represents a product term .
ā¢ Each product term is summed (ORed).
ā¢ Minterm = Boolean product term that contains
EACH variable ONCE in true or complemented form
ā¢ Maxterm = Boolean sum term that contains EACH
variable ONCE in true or complemented form
From T/T
To gatesā¦
24. 24
The Big Picture
A significant goal of this class is synthesis
How do we get from a specification to hardware?
T/T ā minterms ā Boolean expression ā
synthesis ā gates
Do Fig 3.16, generate SOP from T/T then gatesā¦
ā¢Pg16r2
25. 25
Digital
Circuit
A
B
C
Y
Figure 3.16 Digital Circuit with Unknown Function
A B C Y
0 0 0 1
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 0
1 1 1 0
First, find the minterms and the
SOP solution
Then lets find the maxterms and
the POS solution.
27. 27
Now ā the other way - Deriving a POS
Expression from a Truth Table
ā¢ Each line of the truth table with a 0 (LOW)
output represents a sum term.
ā¢ The sum terms are multiplied (ANDed).
ā¢ Letās revisit the last example ā this time do
the Product of Sums solutionā¦
ā Generate the maxterms then OR inputs together
ā then AND to get the final output function
28. 28
Deriving a POS Expression from
a Truth Table
This is kind of a worst
case T/T ā eight
terms for either SOP
or POS
This would require
eight ā 4 input ANDās
and one 8 input ORā¦
29. 29
Theorems of Boolean Algebra
ā¢ Used to minimize a Boolean expression
to reduce the number of logic gates in a
network.
ā¢ 24 theorems.
30. 30
Commutative Property
ā¢ The operation can be applied in any
order with no effect on the result.
ā Theorem 1: xy = yx
ā Theorem 2: x + y = y + x
lets do the truth table on
the board...
31. 31
Associative Property
ā¢ The operands can be grouped in any
order with no effect on the result.
ā Theorem 3: (xy)z = x(yz) = (xz)y
ā Theorem 4: (x + y) + z = x + (y + z) = (x + z) + y
32. 32
Distributive Property
ā¢ Allows distribution (multiplying through)
of AND across several OR functions.
ā¢ Theorem 5: x(y + z) = xy + xz
ā¢ Theorem 6: (x + y)(w + z) = xw + xz + yw + yz
ā Theorem 6 effectively changes a POS representation to SOP
Letās do theorem 5 on the boardā¦
Also do example 3.9 from the textā¦
34. 34
Operations with Logic 0
ā¢ Theorem 7: x ļ· 0 = 0
ā¢ Theorem 8: x + 0 = x
ā¢ Theorem 9: x ļ 0 = x
Letās do theorem 7 on the boardā¦
35. 35
Operations with Logic 1
ā¢ Theorem 10: x ļ· 1 = x
ā¢ Theorem 11: x + 1 = 1
ā¢ xx 1:12Theorem ļ½ļ
These mostly make sense, but lets check theorem
12ā¦
42. 42
Multivariable Theorems
ā¢ Theorem 22: x + xy = x
ā¢ Theorem 23: (x + y)(x + z) = x + yz
ā¢ yxyxx:24Theorem ļ«ļ½ļ«
Letās check these on the boardā¦
44. 44
Simplifying SOP and POS
Expressions
ā¢ A Boolean expression can be simplified
by:
ā Applying the Boolean Theorems to the
expression
ā Application of graphical tool called a
Karnaugh map (K-map) to the expression
47. 47
Simplification By Karnaugh
Mapping
ā¢ A Karnaugh map, called a K-map, is a
graphical tool used for simplifying
Boolean expressions.
ā¢ Makes use of humans āpattern
recognitionā skills
ā¢ K-map works well for 2,3,4 variables,
more difficult for 5 or 6ā¦
48. 48
Construction of a Karnaugh Map
ā¢ Square or rectangle divided into cells.
ā¢ Each cell represents a line in the truth
table.
ā¢ Cell contents are the value of the
output variable on that line of the
truth table.
51. 51
Construction of a Karnaugh Map
Letās draw the truth
table for this on the
board to see how the
K-map is loadedā¦
52. 52
K-map Cell Coordinates
ā¢ Adjacent cells differ by only one
variable.
ā¢ Grouping adjacent cells allows
canceling variables in their true and
complement forms.
53. 53
Grouping Cells
ā¢ Cells can be grouped as pairs, quads,
and octets.
ā¢ A pair cancels one variable.
ā¢ A quad cancels two variables.
ā¢ An octet cancels three variables.
57. 57
Grouping Cells along the Outside
Edge
ā¢ The cells along an outside edge are
adjacent to cells along the opposite
edge.
ā¢ In a four-variable map, the four corner
cells are adjacent.
59. 59
Loading a K-Map from a Truth
Table
ā¢ Each cell of the K-map represents one
line from the truth table.
ā¢ The K-map is not laid out in the same
order as the truth table.
62. 62
Multiple Groups
ā¢ Each group is a term in the maximum
SOP expression.
ā¢ A cell may be grouped more than once
as long as every group has at least one
cell that does not belong to any other
group. Otherwise, redundant terms will
result.
66. 66
Using K-Maps for Partially
Simplified Circuits
ā¢ Fill in the K-map from the existing
product terms.
ā¢ Each product term that is not a minterm
will represent more than one cell.
ā¢ Once completed, regroup the K-map for
maximum simplification.
72. 72
POS Simplification Using
Karnaugh Mapping
ā¢ Group those cells with values of 0.
ā¢ Use the complements of the cell
coordinates as the sum term.
74. 74
Simplification by DeMorganās Equivalent Gates
Bubble-to-Bubble Convention ā Step 1
ā¢ Start at the output and work towards the
input.
ā¢ Select the OR gate as the last gate for an
SOP solution.
ā¢ Select the AND gate as the last gate for an
POS solution.
75. 75
Simplification by DeMorganās Equivalent Gates
Bubble-to-Bubble Convention ā Step 2
ā¢ Choose the active level of the output if
necessary.
ā¢ Go back to the circuits inputs to the next level
of gating.
76. 76
Simplification by DeMorganās Equivalent Gates
Bubble-to-Bubble Convention ā Step 3
ā¢ Match the output of these gates to the input of
the final gate. This may require converting the
gate to its DeMorganās equivalent.
ā¢ Repeat Step 2 until you reach the circuits
input
77. 77
Simplification by DeMorganās Equivalent Gates
Bubble-to-Bubble Convention
=
See Pg 120 of the text, there is an error in Fig 3.63
L7
78. 78
Simplification by DeMorganās Equivalent Gates
Bubble-to-Bubble Convention
=
=
So here by using the DeMorganās equivalent circuit we
are able to remove the bubble to bubble connection
94. 94
Practical Circuit Implementation
in SSI
ā¢ Not all gates are available in TTL.
ā¢ TTL components are becoming more
difficult to find.
ā¢ In a circuit design, it may be necessary
to replace gates with other types of
gates in order to achieve the final
design.
95. Practical Implementation of Small Scale Integration Devices
ā¢ So far, we have focused
on simplification of
designs to save time,
space, gates, power, etc.
ā¢ Now, letās examine some
practical information
related to devices you will
use in the lab
Most device available in several physical
Packages e.g. DIP, Small Outline IC etc.
Gate packages generally have 1,2,3,4 or
6 gates in a pkg ā limited nbr of pins! 95
96. A few important design notes
ā¢ All devices, regardless of the physical package require
Vcc and Gnd
ā¢ Un-used devices need to be pulled Hi or Lo to prevent
noise or oscillation
ā¢ Unused TTL ā configure for o/p = HIGH
ā¢ Unused CMOS ā Hi or Lo ā either is OK
ā¢ Every circuit requires a schematic and / or block
diagram with pin numbers, Vcc and Gnd labeled etc. ā
this really speeds up troubleshooting
96
98. ā¢ Donāt mix logic families (if you have a choice)
ā¢ Donāt connect outputs together
ā Outputs go to inputs (unless tristateā¦)
ā¢ Ensure every package has Vcc and ground
ā¢ Configure unused gates
ā¢ Letās do example 3.27
98
More on Design
99. Here is the authors solution ā
he has used DME to create a
four i/p OR with
Inverting inputs ā perhaps a
bit messy.
Another way would be to
make a 4 i/p OR from
3 ā two i/p OR gates.
His result is pin accurate, but
has quite a bit of negative
logic ā this makes
troubleshooting more difficult
99
100. More on Positive Logic
ā¢ Easier for others to understand
ā¢ Permits faster integration into larger
projects
ā¢ In industry, the cost related to
troubleshooting, integration, regression
testing and documentation swamp the
cost of a few extra gates on the pcb
100
101. 101
Pulsed Operation
ā¢ The enabling and inhibiting properties of the
basic gates are used to pass or block
pulsed digital signals.
ā¢ The pulsed signal is applied to one input.
ā¢ One input is used to control (enable/inhibit)
the pulsed digital signal.
104. 104
General Approach to Logic
Circuit Design ā 1
ā¢ Have an accurate description of the
problem.
ā¢ Understand the effects of all inputs on
all outputs.
ā¢ Make sure all combinations have been
accounted for.
105. 105
General Approach to Logic
Circuit Design ā 2
ā¢ Active levels as well as the
constraints on all inputs and outputs
should be specified.
ā¢ Each output of the circuit should be
described either verbally or with a
truth table.
106. 106
General Approach to Logic
Circuit Design ā 3
ā¢ Look for keywords AND, OR, NOT that can
be translated into a Boolean expression.
ā¢ Use Boolean algebra or K-maps to simplify
expressions or truth tables.
ā¢ Generate gate schematics from simplified
expressions or truth tables
Lets do Example 3.29