Transcript Chapter 1 – Introductory Concepts

Digital Systems
Presented by Prof Tim Johnson
Wentworth Institute of Technology
Department of Electrical Engineering and Tech.
Boston, MA
Text: Digital Systems by Ronald Tocci
Chapter 3: Describing Logic Circuits


Now that we understand the concept of binary
numbers, we will study ways of describing
how systems using binary logic levels make
decisions.
Boolean algebra is an important tool in
describing, analyzing, designing, and
implementing digital circuits.
3-1 Boolean Constants and Variables




Boolean algebra allows only two values; 0
and 1.
Logic 0 can be: false, off, low, no, open
switch.
Logic 1 can be: true, on, high, yes, closed
switch.
Three basic logic operations: OR, AND, and
NOT.
3-2 Truth Tables


A truth table describes the relationship
between the input and output of a logic
circuit.
The number of entries corresponds to the
number of inputs. For example a 2 input table
would have 22 = 4 entries. A 3 input table
would have 23 = 8 entries.
3-2 Truth Tables

Examples of truth tables with 2, 3, and 4 inputs.
3-3 OR Operation With OR Gates

The Boolean expression for the OR operation is
X=A+B



This is read as “x equals A or B.”
X = 1 when A = 1 or B = 1.
Truth table and circuit symbol for a two input OR
gate:
3-3 OR Operation With OR Gates


The OR operation is similar to addition but
when A = 1 and B = 1, the OR operation
produces 1 + 1 = 1.
In the Boolean expression
x=1+1+1=1
We could say in English that x is true (1) when A is true
(1) OR B is true (1) OR C is true (1).
3-4 AND Operations with AND gates

The Boolean expression for the AND operation is
X=A•B



This is read as “x equals A and B.”
x = 1 when A = 1 and B = 1.
Truth table and circuit symbol for a two input AND gate
are shown. Notice the difference between OR and AND
gates.
3-4 AND Operation With AND Gates


The AND operation is similar to
multiplication.
In the Boolean expression
X=A•B•C
X = 1 only when A = 1, B = 1, and C = 1.
3-5 NOT Operation

The Boolean expression for the NOT
operation is
XA

This is read as:



x equals NOT A, or
x equals the inverse of A, or
x equals the complement of A
3-5 NOT Operation

Truth table, symbol, and sample waveform for
the NOT circuit.
3-6 Describing Logic Circuits Algebraically


The three basic Boolean operations (OR,
AND, NOT) can describe any logic circuit.
If an expression contains both AND and OR
gates the AND operation will be performed
first, unless there is a parenthesis in the
expression.
3-6 Describing Logic Circuits Algebraically

Examples of Boolean expressions for logic
circuits:
3-6 Describing Logic Circuits Algebraically

The output of an inverter is equivalent to the
input with a bar over it. Input A through an
inverter equals A.

Examples using inverters.
3-7 Evaluating Logic Circuit Outputs

Rules for evaluating a Boolean expression:




Perform all inversions of single terms.
Perform all operations within parenthesis.
Perform AND operation before an OR operation
unless parenthesis indicate otherwise.
If an expression has a bar over it, perform the
operations inside the expression and then invert
the result.
3-7 Evaluating Logic Circuit Outputs

Evaluate Boolean expressions by substituting
values and performing the indicated
operations:
A  0, B  1, C  1, and D  1
x  ABC(A  D)
 0 11 (0  1)
 111 (0  1)
 111 (1)
 1 1 1  0
0
3-8 Implementing Circuits From Boolean
Expressions



It is important to be able to draw a logic circuit from a
Boolean expression.
The expression
x  A  BC
could be drawn as a three input AND gate.
A more complex example such as
y  AC  BC  ABC
could be drawn as two 2-input AND gates and one 3-input
AND gate feeding into a 3-input OR gate. Two of the
AND gates have inverted inputs.
3-9 NOR Gates and NAND Gates



The NOR gate is an
inverted OR gate.
An inversion “bubble” is
placed at the output of the
OR gate.
The Boolean expression is
x  A B
3-9 NOR Gates and NAND Gates



The NAND gate is an inverted AND gate.
An inversion “bubble” is placed at the output of the
AND gate.
The Boolean expression is x  AB
3-9 NOR Gates and NAND Gates


The output of NAND and NOR gates may be
found by simply determining the output of an
AND or OR gate and inverting it.
The truth tables for NOR and NAND gates
show the complement of truth tables for OR
and AND gates.
3-10 Boolean Theorems
The theorems or laws below may represent an expression containing
more than one variable.
3-10 Additional Boolean Theorems
The Boolean theorems are useful to reduce expressions to the simplest form.
(9)
Commutative law (10)
Associative law (11)
x y  yx
x y  yx
x  ( y  z)  ( x  y)  z  x  y  z
(12)
Distributive law (13a )
Distributive law (13b )
(14)
(15a)
(15b)
x( yz)  ( xy) z  xyz
x( y  z )  xy  xz
( w  x)( y  z )  wy  xy  wz  xz
x  xy  x
and (15) do not have counterparts
x  x y  x  y (14)
in the ordinary algebra. Each can
x  xy  x  y be proven by Boolean Algebra
Commutative law
Associative law
3-10 Boolean Proofs
Boolean proof of (14):
x  xy  x 1  xy x→ x∙1 by Rule 2
x 1  xy  x(1  y ) by Rule 13a
inverse application
inverse application
x(1  y )  x(1)
x(1)  x
(1+y)→(1) by Rule 6
by Rule 2
Heuristic proof of 15a & 15b
x  xy  x  y
x/ x
0/1
0/1
1/0
1/0
y
0
1
0
1
x  xy  x  y
xy
x  xy
x y
x/ x
0
1
0
0
0
1
1
1
0
1
1
1
0/1
equal
0/1
1/0
1/0
y
0
1
0
1
xy
x  xy
xy
0
0
0
1
1
1
0
1
1
1
0
1
equal
*Heuristic refers to experience-based techniques for problem solving, learning, and discovery
that give a solution.. Examples: trial & error, using a rule-of-thumb, an educated guess, an
intuitive judgment, stereotyping, or common sense.
Boolean Proof of 15a
3-10 Examples
x  AB D  AB D 
y  ( A  B)( A  B) 
z  ACD  A BCD 
w x yz 
3-11 DeMorgan’s Theorems

Two very important Boolean theorems are contributed
by a great mathematician named DeMorgan:
(16) x  y  x  y
(17) x  y  x  y

Although these two theorems are stated in single
variables of x and y, they are valid for situations where
x and/or y are expressions that contain more than one
variable.
AB  C
3-11 Practice examples

Simply the expressions to one having only single
variables inverted:
x  ( A  C )(B  D ) 
y  A  BC 
z  ( A  BC)(D  EF ) 
Practice examples
x  ( A B)  B  ( B  C ) 


y  (A  B) B  C 
z  A  B C 
w  A B  (C  D) 
Next Page
Examples
x  ( A  B )  CD 
y  AB  C 
z  ( A  B  C )( A  B C ) 
a  ( A  B  C ) D  E  ( A  B  C )(D  E ) 
b  xz  z ( x  xy) 
c  A  B (C  D  E )  A  B 
d  AB  ABC  ABCD  ABCDE  ABCDEF 
Next Page

Simplify the following Boolean expressions:
Next Page
Next Page

Write down the Boolean expression and simplify it.
3-11 DeMorgan’s Theorems

A NOR gate is equivalent to an AND gate with inverted inputs.
(16) x  y  x  y
3-11 DeMorgan’s Theorems

A NAND gate is equivalent to an OR gate with inverted inputs.
(17) x  y  x  y
3-12 Universality of NAND and NOR Gates


In the following two slides NAND or NOR
gates are used to create the three basic logic
expressions (OR, AND, and INVERT)
This characteristic provides flexibility and has
some use in logic circuit design.
Using NOR to Represent NOT, AND, OR Gates
Using NAND to Represent NOT, AND, OR Gates
3-13 Alternate Logic-Gate Representations

To convert a standard symbol to an alternate:




Invert each input and output (add an inversion bubble where
there are none on the standard symbol, and remove bubbles
where they exist on the standard symbol.
Change a standard OR gate to an AND gate, or an AND gate
to an OR gate.
In case of the inverter, the operation symbol is NOT changed.
The equivalence can be applied to gates with any number of
inputs.
3-13 Alternate Logic-Gate Representations
3-13 Alternate Logic-Gate Representations




Active high – an input or output has no
inversion bubble.
Active low – an input or output has an
inversion bubble.
An AND gate will produce an active output
when all inputs are in their active states.
An OR gate will produce an active output
when any input is in an active state.
3-13 Alternate Logic-Gate Representations
3-14 Which Gate Representation to Use


Using alternate and standard logic gate
symbols together can make circuit operation
clearer.
When possible choose gate symbols so that
bubble outputs are connected to bubble
input and the inversions are eliminated.
3-14 Which Gate Representation to Use
Original Circuit
Equivalent representation
when output is active-high
Equivalent representation
when output is active-low
Proper combination of alternate and standard logic gates can make circuit operation clearer.
3-15 IEEE/ANSI Standard Logic Symbols



Rectangular symbols represent logic gates and
circuits.
Dependency notation inside symbols show
how output depends on inputs.
A small triangle replaces the inversion
bubble.
3-15 IEEE/ANSI Standard Logic Symbols


Compare the
IEEE/ANSI symbols
to traditional
symbols.
These symbols are
not widely accepted
but may appear in
some schematics.