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Contoh Aplikasi K-MAP pada Perancangan  level suhu AC otomatis

Oleh : Arif Johar Taufiq

The Karnaugh map (K-map for short), Maurice Karnaugh's 1953 refinement of Edward Veitch's 1952 Veitch diagram, is a method to simplify Boolean algebra expressions. The Karnaugh map reduces the need for extensive calculations by taking advantage of humans' pattern-recognition capability, permitting the rapid identification and elimination of potential race conditions. In a Karnaugh map the boolean variables are transferred (generally from a truth table) and ordered according to the principles of Gray code in which only one variable changes in between squares. Once the table is generated and the output possibilities are transcribed, the data is arranged into the largest possible groups containing 2ⁿ cells (n=0,1,2,3...) and the minterm is generated through the axiom laws of boolean algebra. The Karnaugh map was invented in 1952 by Edward W. Veitch and developed further in 1953 by Maurice Karnaugh, a telecommunications engineer at Bell Labs.

Problems

Karnaugh maps generally become more cluttered and hard to interpret when adding more variables. A general rule is that Karnaugh maps work well for up to four variables, and shouldn't be used at all for more than six variables. For expressions with larger numbers of variables, the Quine–McCluskey algorithm can be used. Nowadays in general the minimization process is carried out by computer, for which the Espresso heuristic logic minimizer has become the standard minimization program.

Procedures

The K-Map method may theoretically be applied for the simplification of any boolean expression regardless of its number of variables, but is most often used when there are fewer than six variables because K-Maps of expressions with more than six variables are complex and tedious to simplify. Each variable contributes two possibilities: the initial value, and its inverse; it therefore organizes all possibilities of the system. The variables are arranged in Gray code in which only one variable changes between two adjacent grid boxes.

Once the variables have been defined, the output possibilities are transcribed according to the grid location provided by the variables. Thus for every possibility of a boolean input or variable the output possibility is defined. When the Karnaugh map has been completed, to derive a minimized function the "1s" or desired outputs are grouped into the largest possible rectangular groups in which the number of grid boxes (output possibilities) in the groups must be equal to a power of 2. For example, the groups may be 4 boxes in a line, 2 boxes high by 4 boxes long, 2 boxes by 2 boxes, and so on. "Don't care(s)" possibilities (generally represented by a "X") are grouped only if the group created is larger than the group with "Don't care" is excluded. The boxes can be used more than once only if it generates the least number of groups. Each "1" or desired output possibilities must be contained within at least one grouping.

The groups generated are then converted to a boolean expression by: locating and transcribing the variable possibility attributed to the box, and by the axiom laws of boolean algebra—in which if the (initial) variable possibility and its inverse are contained within the same group the variable term is removed. Each group provides a "product" to create a "sum-of-products" in the boolean expression. To determine the inverse of the Karnaugh map, the "0s" are grouped instead of the "1s". The two expressions are non-complementary.

Contoh Kasus:

Suatu ruang dengan 1 pintu masuk dan 1 pintu keluar. Kapasitas maksimum ruang tersebut adalah untuk 12 orang. Di dalam ruang terdapat AC yang secara otomatis akan menyesuaikan level suhunya tergantung dari jumlah orang di dalam ruang tersebut. Penyesuaiannya adalah sbb:

Atau dalam hal ini kasus di atas kalau mau dianalogikakan dengan kasus lain sama seperti kasus persnelling otomatis pada motor bebek (MIO) yaitu saat awal nol -> netral, saat RPM mesin naik dari 10-40 maka gigi persnelling pindah ke-1, saat RPM mesin naik dari 50-80 gigi pindah ke-2 dan saat RPM menjadi 90-12 maka gigi pindah ke gigi 4.

maka penyelesaian sintesa rangkaian kombinational untuk pengatur level suhu AC otomatis tersebut adalah sbb:

• Input menunjukkan jumlah orang dari 0 s/d 12. Diperlukan input 4 bit untuk menyatakan

besaran 0 (0000) s/d 12 (1100)

• Output menunjukkan output 2 bit untuk menyatakan besaran 0(00) s/d 3 (11)

d= don't care

K-Map penyelesaian

Jika kita simulasikan dengan Workbench telah sesuai rancangan yang diinginkan sbb:

File workbench untuk simulasi download disini. Download rekaman simulasi disini