Chapter 3

Chapter 3Penyederhanaan Fungsi/Gerbang Logika1Penyederhanaan fungsi/gerbang logika dapat dilakukan dengan :Penyederhanaan langsung dari ungkapan BooleKarnaugh Map (K-Map)2Review Gerbang Logika3Penyederhanaan LangsungTeorema Boole4Karnaugh Maps (K-Map)sebuah representasi grafik dari sebuah fungsi logika tabel kebenaran yang dapat digunakan untuk menyederhanakan fungsi tersebutK-map berisi semua kemungkinan logika dari sistem logika yang dirangkai dalam bentuk tabel

5Hubungan Tabel kebenaran dengan K-map (dua variabel)

posABF000F0101F1210F2311F3Tabel kebenaran K-map6Contoh :Buatlah penyederhanaan u/ fungsi F=A.B+A.B dengan menggunakan cara K-map

7Penyelesaian1. Buatlah kotak dgn 4 keluaran (22 =4) 2. Tentukan sel keluaran yang bernilai logika 13. Sel bernilai 1 adalah sel pada koordinat A.B dan koordinat A.B spt gambar disamping sedangkan (sel yang lain boleh dibiarkan kosong atau diberi nilai 0)

8Penyelesaian (2)

4. Kelompokkan sel yang bernilai 1 yang berdekatan spt gambar disamping5. Dlm kelompok tersebut yaitu A tidak berubah, tetap benilai 1, tetapi B dapat bernilai 0 atau 1Sehingga keluaran diatas dapat disederhanakan menjadi F=A.B+A.B = A6. Uji jawaban dengan tebel kebenaran.

1

19Tabel Kebenaran F = A.B + A.BposABBA.BA.BF=A.B+A.B0001000101000021011013110011F = A.B + A.BF = A10K-Map tiga-masukan 8 kombinasi

posABC Fm0000Xm1001Xm2010Xm3011XM4100Xm5101Xm6110Xm7111x

11Misal suatu ungkapan Boole sbb:F = ABC+ABC+ABC+ABCMaka pada K-Map logika 1 ditempatkan pada sel-sel sesuai ungkapan berikut :ABC A=0, B=1, C=0 m2, ABC A=0, B=1, C=0 m3ABC A=1, B=0, C=0 m4, ABC A=1, B=0, C=1 m5

12SoalBuatlah penyederhanaan u/ fungsi F=ABC+ABC+ABC+ABC dengan menggunakan cara K-map

13Penyelesaian1. Dari F=ABC+ABC+ABC+ABC tentukan sel-sel yang berlogika 1 pada K-Map

14Penyelesaian2. Kelompokkan

1

11

1F = AB+AB15Penyelesaian3. Perluas pengelompokan

1 1 1

Nilai A dan C akan berubah jika kita bergerak dari dari satu sel ke sel lain dengan masukan B tetap.Nilai A dan C tidak berpengaruh pada keluaran yang hanya merufakan fungsi dari BSehingga F = B16Buktikan dengan Tabel Kebenaran!17K-Map dengan 4 masukan

18Four-variable K-MapWXYZFWXYZ00000X10001X20010X30011X40100X50101X60110X70111X81000X91001X101010X111011X121100X131101X141110X151111XV0132457612131514891110

192021Four-variable K-Map

Edges are adjacentEdges are adjacent22Plotting Functions on the K-mapSOP Form23Canonical SOP FormThree Variable Example

using shorthand notation

24Three-Variable K-Map Example

Plot 1s (minterms) of switching function111125Three-Variable K-Map Example

Plot 1s (minterms) of switching function1111

26Four-variable K-Map Example

1111127Karnaugh Maps (K-Map)Simplification of Switching Functionsusing K-MAPS28Terminology/DefinitionLiteralA variable or its complementLogically adjacent termsTwo minterms are logically adjacent if they differ in only one variable positionEx:

andm6 and m2 are logically adjacent Note:

Or, logically adjacent terms can be combined 29Terminology/DefinitionImplicantProduct term that could be used to cover minterms of a functionPrime ImplicantAn implicant that is not part of another implicantEssential Prime ImplicantAn implicant that covers at least one minterm that is not contained in another prime implicantCoverA minterm that has been used in at least one group

30Guidelines for Simplifying FunctionsEach square on a K-map of n variables has n logically adjacent squares. (i.e. differing in exactly one variable)When combing squares, always group in powers of 2m , where m=0,1,2,. In general, grouping 2m variables eliminates m variables.31Guidelines for Simplifying FunctionsGroup as many squares as possible. This eliminates the most variables.Make as few groups as possible. Each group represents a separate product term.You must cover each minterm at least once. However, it may be covered more than once.32K-map Simplification Procedure Plot the K-mapCircle all prime implicants on the K-mapIdentify and select all essential prime implicants for the cover.Select a minimum subset of the remaining prime implicants to complete the cover. Read the K-map33ExampleUse a K-Map to simplify the following Boolean expression

34Three-Variable K-Map Example

Step 1: Plot the K-map1111

135Three-Variable K-Map Example

Step 2: Circle ALL Prime Implicants 1111

136Three-Variable K-Map Example

Step 3: Identify Essential Prime Implicants1111

1EPIEPIPIPI37Three-Variable K-Map Example

Step 4: Select minimum subset of remaining Prime Implicants to complete the cover.1111

1EPIPIEPI38Three-Variable K-Map Example

Step 5: Read the map.1111

1

39Solution

40ExampleUse a K-Map to simplify the following Boolean expression

41Three-Variable K-Map Example

Step 1: Plot the K-map1111

42Three-Variable K-Map Example

Step 2: Circle Prime Implicants 1111

Wrong!!We reallyshould drawA circle aroundall four 1s 43Three-Variable K-Map Example

Step 3: Identify Essential Prime ImplicantsEPIEPI

1111Wrong!!We reallyshould drawA circle aroundall four 1s 44Three-Variable K-Map Example

Step 4: Select Remaining Prime Implicants to complete the cover.EPIEPI1111

45Three-Variable K-Map Example

Step 5: Read the map.

1111

46Solution

Since we can still simplify the functionthis means we did not use the largestpossible groupings.47Three-Variable K-Map Example

Step 2: Circle Prime Implicants 1111

Right!48Three-Variable K-Map Example

Step 3: Identify Essential Prime ImplicantsEPI

111149Three-Variable K-Map Example

Step 5: Read the map.

1111

50Solution

51Special Cases52Three-Variable K-Map Example

1111

111153Three-Variable K-Map Example

54Three-Variable K-Map Example

1

11155Four Variable Examples56ExampleUse a K-Map to simplify the following Boolean expression

57Four-variable K-Map

1111111158Four-variable K-Map

1111111159Four-variable K-Map

1111111160ExampleUse a K-Map to simplify the following Boolean expression

D=Dont care (i.e. either 1 or 0)61Four-variable K-Map

11d11111

dd62Four-variable K-Map

11d11111

dd63Five Variable K-Maps

64Five variable K-mapA=1A=0Use two four variable K-maps65Use Two Four-variable K-Maps

A=0 mapA=1 map66Five variable example

67Use Two Four-variable K-Maps

A=0 mapA=1 map

1111111168Use Two Four-variable K-Maps

A=0 mapA=1 map11111111

69Five variable example

70Plotting POS Functions71K-map Simplification Procedure Plot the K-map for the function FCircle all prime implicants on the K-mapIdentify and select all essential prime implicants for the cover.Select a minimum subset of the remaining prime implicants to complete the cover. Read the K-mapUse DeMorgans theorem to convert F to F in POS form72ExampleUse a K-Map to simplify the following Boolean expression

73Three-Variable K-Map Example

Step 1: Plot the K-map of F11111

74Three-Variable K-Map Example

Step 2: Circle ALL Prime Implicants 1111175Three-Variable K-Map Example

Step 3: Identify Essential Prime Implicants11111EPIEPIPIPI76Three-Variable K-Map Example

Step 4: Select minimum subset of remaining Prime Implicants to complete the cover.11111EPIPIEPI77Three-Variable K-Map Example

Step 5: Read the map.11111

78Solution

79TPS Quiz80A

A

B

B

A

B

A

A

B

B

A

B

A

A

B

B

A

B

A A A A

C

C

B B B B

A 0 0 1 1B 0 1 1 0

C=0

C=1

A 0 0 1 1B 0 1 1 0

C=0

C=1

A 0 0 1 1B 0 1 1 0

C=0

C=1

A 0 0 1 1B 0 1 1 0

C=0

C=1

A 0 0 1 1B 0 1 1 0

C=0

C=1

AB

CD

00

01

11

10

00

01

11

10

abcd0001111000011110abc0001111001abc0001111001abcd0001111000011110abc0001111001abc0001111001abc0001111001abc0001111001abc0001111001abc0001111001abc0001111001abc0001111001abc0001111001abc0001111001abc0001111001abc0001111001abc0001111001abc0001111001abc0001111001abc0001111001abcd0001111000011110abcd0001111000011110abcd0001111000011110abcd0001111000011110abcd0001111000011110bcde0001111000011110bcde0001111000011110bcde0001111000011110bcde0001111000011110bcde0001111000011110bcde0001111000011110abc0001111001abc0001111001abc0001111001abc0001111001abc0001111001