ThisLecture will cover the following points: Canonical and Standard Forms MinTerms and MaxTerms Digital Logic Families 24 March 2010 Standard Expression Forms Two standard (canonical) expression forms Canonical sum form Means that sum of products (MinTerms) OR of AND terms Canonical product form Means that product of sums (MaxTerms) AND of OR terms 24 March 2010 Page 2 1
Standard Forms (cont.) Standard Sum Form Sum of Products (OR of AND terms) ( A B C ) + ( A B C ) + ( A B C ) + ( A B C ) F[A, B,C] = + Minterms Standard Product Form Product of Sums (AND of OR terms) ( A + B + C) ( A + B + C) ( A + B + C) ( A + B C) F[A, B,C] = + Maxterms 24 March 2010 Page 3 Standard forms of expressions We can write expressions in many ways, but some ways are more useful than others A sum of products (SOP) expression contains: Only OR (sum) operations at the outermost level Each term that is summed must be a product of literals f(x,y,z) = y + x yz + xz The advantage is that any sum of products expression can be implemented using a two level circuit literals and their complements at the 0th level AND gates at the first level a single OR gate at the second level This diagram uses some short hands NOT gates are implicit literals are reused this is not okay in Logic Works! 24 March 2010 Page 4 2
MinTerms A minterm is a special product of literals, in which each input variable appears exactly once. A function with n variables has 2 n minterms (since each variable can appear complemented or not) If the function has two variables x and y so the minterms can be calculated as follows: Each minterm is true for exactly one combination of inputs x y, xy, x y, xy x y MinTerms Term shorthand 24 March 2010 0 0 x y m 0 0 1 x y m 1 1 0 x y m 2 1 1 x y m 3 Page 5 Minterms A three variable function, such as f(x,y,z), has 2 3 = 8 minterms Each minterm is true for exactly one combination of inputs: x y z x y z x yz x yz xy z xy z xyz xyz Minterm Is true when Shorthand x y z x=0, y=0, z=0 m 0 x y z x=0, y=0, z=1 m 1 x yz x=0, y=1, z=0 m 2 x yz x=0, y=1, z=1 m 3 xy z x=1, y=0, z=0 m 4 xy z x=1, y=0, z=1 m 5 xyz x=1, y=1, z=0 m 6 xyz x=1, y=1, z=1 m 7 24 March 2010 Page 6 3
Sum of minterms form Every function can be written as a sum of minterms, which is a special kind of sum of products form (SOP) The sum of minterms form for any function is unique If you have a truth table for a function, you can write a sum of minterms expression just by picking out the rows of the table where the function output is 1. x y z f(x,y,z) f (x,y,z) 0 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 1 f = x y z + x y z + x yz + x yz + xyz = m 0 + m 1 + m 2 + m 3 + m 6 = Σm(0,1,2,3,6) f = xy z + xy z + xyz = m 4 + m 5 + m 7 = Σm(4,5,7) f contains all the minterms not in f 24 March 2010 Page 7 Example: SOP 1)Example: Express the Boolean Function F=x+yz by two different ways as a sum of minterms. 1 st method by using the rules of Boolean Algebra 111 110 101 100 011 24 March 2010 Page 8 4
2 nd method by using the truth table Example: SOP F=x+yz cont. x y z yz F minterms 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 1 1 m3 1 0 0 0 1 m4 1 0 1 0 1 m5 1 1 0 0 1 m6 1 1 1 1 1 m7 F = ( m3 + m4 + m5 + m6 + m7 ) F = m(3,4,5,6,7) 24 March 2010 Page 9 The dual idea: products of sums Just to keep you on your toes... A product of sums (POS) expression contains: Only AND (product) operations at the outermost level Each term must be a sum of literals f(x,y,z) = y (x + y + z ) (x + z) Product of sums expressions can be implemented with two level circuits literals and their complements at the 0th level OR gates at the first level a single AND gate at the second level Compare this with sums of products 24 March 2010 Page 10 5
MaxTerms A maxterm is a sum of literals, in which each input variable appears exactly once. A function with n variables has 2 n maxterms Ifthe function has two variables x and y so the maxterms can be calculated as follows: Each maxterm is false for exactly one combination of inputs x + y, x + y, x + y, x + y x y MaxTerms Term shorthand 24 March 2010 0 0 x + y M 0 0 1 x + y M 1 1 0 x + y M 2 1 1 x + y M 3 Page 11 Maxterms The maxterms for a three variable function f(x,y,z): Each maxterm is false for exactly one combination of inputs: x + y + z x + y + z x + y + z x + y + z x + y + z x + y + z x + y + z x + y + z Maxterm Is false when Shorthand x + y + z x=0, y=0, z=0 M 0 x + y + z x=0, y=0, z=1 M 1 x + y + z x=0, y=1, z=0 M 2 x + y + z x=0, y=1, z=1 M 3 x + y + z x=1, y=0, z=0 M 4 x + y + z x=1, y=0, z=1 M 5 x + y + z x=1, y=1, z=0 M 6 x + y + z x=1, y=1, z=1 M 7 24 March 2010 Page 12 6
Product of maxterms form Every function can be written as a unique product of maxterms If you have a truth table for a function, you can write a product of maxterms expression by picking out the rows of the table where the function output is 0. (Be careful if you re writing the actual literals!) x y z f(x,y,z) f (x,y,z) 0 0 0 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 1 f = (x + y + z)(x + y + z )(x + y + z ) = M 4 M 5 M 7 = M(4,5,7) f = (x + y + z)(x + y + z )(x + y + z) (x + y + z )(x + y + z) = M 0 M 1 M 2 M 3 M 6 = M(0,1,2,3,6) f contains all the maxterms not in f 24 March 2010 Page 13 Minterms and maxterms are related Any minterm m i is the complement of the corresponding maxterm M i Minterm Shorthand Maxterm Shorthand x y z m 0 x + y + z M 0 x y z m 1 x + y + z M 1 x yz m 2 x + y + z M 2 x yz m 3 x + y + z M 3 xy z m 4 x + y + z M 4 xy z m 5 x + y + z M 5 xyz m 6 x + y + z M 6 xyz m 7 x + y + z M 7 For example, m 4 = M 4 because (xy z ) = x + y + z 24 March 2010 Page 14 7
Converting between standard forms We can convert a sum of minterms to a product of maxterms From before f = Σm(0,1,2,3,6) and f = Σm(4,5,7) = m 4 + m 5 + m 7 complementing (f ) = (m 4 + m 5 + m 7 ) so f = m 4 m 5 m 7 [ DeMorgan s law ] = M 4 M 5 M 7 [ By the previous page ] = M(4,5,7) In general, just replace the minterms with maxterms, using maxterm numbers that don t appear in the sum of minterms: f = Σm(0,1,2,3,6) = M(4,5,7) The same thing works for converting from a product of maxterms to a sum of minterms 24 March 2010 Page 15 Digital Logic Families Digital integrated circuits are not classified not only by their complexity or logical operation, but also by the specific circuit technology to which they belong. The circuit technology is referred to as a digital logic family. Each logic family has its own basic electronic circuit upon which more complex digital circuits and components are developed. Many different logic families of digital integrated circuit have been introduced commercially. The following are the most popular: Type TTL ECL MOS CMOS The abbreviation Transistor Transistor Logic Emitter Coupled Logic Metal Oxide Semiconductor Complementary Metal Oxide Semiconductor Advantages The standard one High speed operation High component density Low power consumption 24 March 2010 Page 16 8
Exercises 1) Express the Boolean Function F = A + BC by two different ways as a sum of minterms. 2) Express the Boolean Function by two different ways as a sum of minterms. 3) Express the Boolean Function by two different ways as a product of maxterms. F = x y + xz 4) Express the Boolean Function by two different ways as a product of maxterms. 5) Find the Maxterms from the following Function: 6) Find the minterms from the following Function: F( A, B, C ) = (1,4,5,6,7) F( A, B, C ) = (0,2,4,6) 7) Implement the following function in two level gating structure: F = AB + C ( D + E ) 8) What is the difference between the positive and negative logic (search on the Internet) 24 March 2010 Page 17 Simplify By Boolean Rules 1 F1=(A + C)(AD + AD ) + AC + C 2 F2=A F2=A (A + B) + (B + AA)(A + B ) 3 F3=(AB) (A + B)(B + B) 24 March 2010 Page 18 9