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CONTOH PERANCANGAN DGN KAIDAH SOP

(SUM OF PRODUCT) / MINTERM

Oleh : Arif Johar Taufiq

 

PENGANTAR

In Boolean algebra, any Boolean function can be expressed in a canonical form using the dual concepts of minterms and maxterms. Minterms are called products because they are the logical AND of a set of variables, and maxterms are called sums because they are the logical OR of a set of variables (further definition appears in the sections headed Minterms andMaxterms below). These concepts are called duals because of their complementary-symmetry relationship as expressed by De Morgan's laws, which state that AND(x,y,z,...) = NOR(x',y',z',...) and OR(x,y,z,...) = NAND(x',y',z',...) (the apostrophe ' is an abbreviation for logical NOT, thus " x' " represents " NOT x ", the Boolean usage " x'y + xy' " represents the logical equation " (NOT(x) AND y) OR (x AND NOT(y)) ").

The dual canonical forms of any Boolean function are a "sum of minterms" and a "product of maxterms." The term "Sum of Products" or "SoP" is widely used for the canonical form that is a disjunction (OR) of minterms. Its De Morgan dual is a "Product of Sums" or "PoS" for the canonical form that is a conjunction (AND) of maxterms. These forms allow for greater analysis into the simplification of these functions, which is of great importance in the minimization or other optimization of digital circuits.

The usual purpose of doing Boolean algebra is to simplify the design of a digital circuit that performs a function, either to minimize the number of gates, or to minimize the time for the value of the function to settle down after a change in its input(s), or some other practical criterion.

There are sixteen possible functions of two variables, but in digital logic hardware, the simplest gate circuits implement only four of them: conjunction (AND), disjunction (inclusive OR), and the complements of those (NAND and NOR).

Most gate circuits accept more than 2 input variables; for example, the spaceborne Apollo Guidance Computer, which pioneered the application of integrated circuits in the 1960s, was built with only one type of gate, a 3-input NOR, whose output is true only when all 3 inputs are false.

 

Functional equivalence

It is apparent that minterm n gives a true value (i.e., 1) for just one combination of the input variables. For example, minterm 5, a b' c, is true only when a and c both are true and b is false--the input arrangement where a = 1, b = 0, c = 1 results in 1.

If one is given a truth table of a logical function, it is possible to write the function as a "sum of products". This is a special form of disjunctive normal form. For example, if given the truth table for the arithmetic sum bit u of one bit position's logic of an adder circuit, as a function of x and y from the addends and the carry in, ci:

ci

x

y

u(ci,x,y)

0

0

0

0

0

0

1

1

0

1

0

1

0

1

1

0

1

0

0

1

1

0

1

0

1

1

0

0

1

1

1

1

Observing that the rows that have an output of 1 are the 2nd, 3rd, 5th, and 8th, we can write u as a sum of minterms m1,m2,m4, and m7. If we wish to verify this: u(ci, x, y) =m1 + m2 + m4 + m7 = (ci' x' y) + (ci' x y') + (ci x' y') + (ci x y) evaluated for all 8 combinations of the three variables will match the table.

 

 

INTI ARTIKEL

Perancangan sederhanan menggunakan gerbang logika dasar seperti AND, OR dan NOT sebenarnya sudah mengakomodir permasalahan sehari-hari kita.

Contoh kasus:

Suatu Bank HAFINA menerapkan sistem keamanan untuk membuka brangkas penyimpan uang dengan sistem tiga kunci. Pintu brangkas dapat dibuka jika paling sedikit ada dua orang yang memasukkan kunci. Kunci dipegang oleh tiga orang yaitu Kepala Bank (A), Manager Keuangan (B) dan Manager Perkreditan (C). Pintu brangkas tidak akan terbuka jika hanya satu orang yang memasukkan kunci.

Bagaimana penyelesaian masalah ini?

Dengan menerapkan sistem digital maka perancangan dapat kita diskripsikan sbb:

1. Ada tiga masukan (A, B, C)

2. Kunci masuk = "1", kunci tdk masuk = "0"

3. Pintu brangkas membuka = "1", pintu brangkas tertutup = "0"

 

 

Selanjutnya dapat disusun tabel kebenaran sbb:

Tabel 1. Tabel Kebenaran Masalah

 

Baris warna kuning menunjukkan bahwa paling tidak ada 2 orang yang mambawa kunci dan memasukkannya sehingga pintu brangkas terbuka (F=1), kaidah penyelesaian  logika yang kita pakai adalah, kita fokus pada baris yang manghasilkan output F=1, yaitu jika untuk masukan A, B, C yang kondisinya adalah:

 

                  

sehingga dapat disusun rangkaian logika

 

Gambar 1. Rangkaian logika hasil disain

 

Coba Anda uji masukan A, B, C maka keluaran F sudah sesuai dengan tabel kebenaran. Rangkaian di atas dalam dunia elektronika digital disebut kaidah SOP (Sum Of Product) atau penjumlahan dari beberapa perkalian. Istilah lain ada yang menyebut dengan Minterm.

 

Dalam Gambar 1 rangkaian logika dibutuhkan: 3 gerbang NOT, 4 gerbang AND tiga input dan 1 gerbang OR tiga input. Nah nantinya dengan aljabar boolean atau K-MAP rangkaian yg rumit itu dapat lebih disederhanakan, sehingga cuma dibutuhkan 3 gerbang AND dua input dan 1 gerbang or tiga input. Mau tahu caranya ikuti kuliah berikutnya~.

 

 

 

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