Gerbang Logika dan Aljabar
Boolean*Sekarang kita telah mengetahui konsep
bilangan biner, dan kita akan mempelajari
cara menggambarkan bagaimana sistem
menggunakan menggunakan level logika biner dalam membuat keputusan.
* Aljabar Boolean adalah alat yang penting
dalam menggambarkan, menganalisa,
merancang, dan mengimplementasikan
rangkaian digital.
Konstanta Boolean dan
Variabel.
Aljabar Boolean dibawah ini hanya
mempunyai dua nilai : 0 dan 1.
Logika 0 dapat dikatakan : false, off, low, no,
saklar terbuka.
Logika 1 dapat dikatakan: true, on, high, yes,
saklar tertutup.
Tiga operasi logika dasar: OR, AND, dan
NOT.
Tabel Kebenaran
Sebuah tabel kebenaran menggambarkan
hubungan antara input dan ouput sebuah
rangkaian logika.
Jumlah The number of entries corresponds to
the number of inputs. For example a 2 input
table would have 2
2 = 4 entries. A 3 input
table would have 2
3 = 8 entries.
Contoh tabel kebenaran dengan masukan 2, 3 dan
4 buah.
Operasional OR dengan gerbang ORŸ The
Boolean expression for the OR operation is
X = A + B
Ÿ This
is read as “x equals A or B.”
Ÿ X
= 1 when A = 1 or B = 1.
Ÿ Truth
table and circuit symbol for a two input OR
gate:
OR Operation With OR Gates
Ÿ The
OR operation is similar to addition but
when A = 1 and B = 1, the OR operation
produces 1 + 1 = 1.
Ÿ 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).
Ÿ There
are many examples of applications where
an output function is desired when one
of multiple inputs is activated.
AND Operations with AND gates
Ÿ The
Boolean expression for the AND operation is
X = A • B
Ÿ This
is read as “x equals A and B.”
Ÿ 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
Ÿ The
AND operation is similar to multiplication.
Ÿ In
the Boolean expression
X = A • B • C
X = 1 only
when A = 1, B = 1, and C = 1.
NOT Operation
Ÿ The
Boolean expression for the NOT operation
is
X = A
Ÿ This
is read as:
Ÿ x
equals NOT A, or
Ÿ x
equals the inverse of A, or
Ÿ x
equals the complement of A
Ÿ Truth
table, symbol, and sample waveform for
the NOT circuit.
Describing Logic Circuits
Algebraically
Ÿ The
three basic Boolean operations (OR, AND,
NOT) can describe any logic circuit.
Ÿ If
an expression contains both AND and OR
gates the AND operation will be performed first, unless there is a parenthesis in the expression.
Ÿ Examples
of Boolean expressions for logic circuits:
Ÿ The
output of an inverter is equivalent to the
input with a bar over it. Input
A through an inverter equals A.
Ÿ Examples
using inverters.
Evaluating Logic Circuit Outputs
Ÿ Rules
for evaluating a Boolean expression:
Ÿ Perform
all inversions of single terms.
Ÿ Perform
all operations within parenthesis.
Ÿ Perform
AND operation before an OR operation
unless parenthesis indicate otherwise.
Ÿ If
an expression has a bar over it, perform the
operations inside the expression and then invert the result.
Ÿ 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
Ÿ Output
logic levels can be determined directly
from a circuit diagram.
Ÿ The
output of each gate is noted until a final
output is found.
Implementing Circuits From Boolean Expressions
Ÿ It
is important to be able to draw a logic circuit from a Boolean expression.
Ÿ The
expression
x = A ×B×C
could be drawn as a three input AND gate.
Ÿ 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.
NOR Gates and NAND Gates
Ÿ Combine
basic AND, OR, and NOT operations.
Ÿ The
NOR gate is an inverted OR gate. An inversion “bubble” is placed at the
output of the OR gate.
The Boolean expression is,
Ÿ The
NAND gate is an inverted AND gate. An inversion
“bubble” is placed at the output of the AND
gate.
Ÿ The
Boolean expression is
x = AB
Ÿ The
output of NAND and NOR gates may be
found by simply determining the output of an AND or OR gate and inverting it.
Ÿ The
truth tables for NOR and NAND gates show
the complement of truth tables for OR
and AND gates.
Universality of NAND and NOR Gates
Ÿ NAND
or NOR gates can be used to create the
three basic logic expressions (OR, AND, and
INVERT)
Ÿ This
characteristic provides flexibility and is
very useful in logic circuit design.
IEEE/ANSI Standard Logic
Symbols
Ÿ Compare
the IEEE/ANSI symbols to traditional symbols.
Ÿ These
symbols are not widely accepted but may appear in some schematics.
Application
Summary of Methods to Describe Logic Circuits
Ÿ The
three basic logic functions are AND, OR,
and NOT.
Ÿ Logic
functions allow us to represent a decision
process.
Ÿ If
it is raining OR it looks like rain I will take an umbrella.
Ÿ If
I get paid AND I go to the bank I will have
money to spend.
sumber : https://onlinelearning.uhamka.ac.id
