Lecture 1. Notes. Notes. Notes. Introduction. Introduction digital logic February Bern University of Applied Sciences

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1 Output voltage Input voltage 3.3V Digital operation (Switch) Lecture digital logic February 26 ern University of pplied Sciences Digital vs nalog Logic =? lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b. versus nalog circuits Output voltage [V] On () Off () Input voltage [V] Transistor: Discovered in 947 by Shockley,ardeen and rattain asis for almost all digital and analog circuits nowadays pplication: nalog as amplifier Possible values unlimited Digital as switch Only 2 values: on and off Digital: 2 valued digit Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.2 We have seen: Digital: two valued digit What about: Logic Example: When I m late and take a taxi or I m not late and take the bus then I m on time for the course Each logic expression has variables Each logic expression has operators Each logic expression has one or more inputs Each logic expression has one output is based on logic expressions of two valued (binary) variables Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.3

2 asic Logic Operator: Example: Short: NOT ND OR omplement Short notation : onjunction Short notation : Disjunction Short notation : If you cannot see then you are blind If you mix flour and milk and eggs then you have pancakes If John takes money or Marry takes money then there is less money blind = see pancakes = flour milk eggs less money = John takes money Marry takes money Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.4 asic Logic (cont.) What about: If John is walking or John is biking then he is active This is a disjunction ut a special one, as John cannot walk and bike at the same time! This operator is called the Exclusive OR (XOR). Disjunction Short notation: Equivalence: = ( ) ( ) Digital vs nalog Logic =? lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.5 Each logical expression has n inputs and one output Each input can only have two states, namely True (T) or False (F) For n inputs we can therefore have 2 n different input combinations We can write the truth table of a logical expression We can compress the truth table by using the don t care symbol X or - The - denotes that the variable can be either T or F Digital vs nalog Logic? = lgebra If John takes money or Marry takes money then there is less money = F F F F T T T F T T T T F F F F T T T - T Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.6

3 ? = lgebra We have seen a transistor giving in its digital operation the two values On (3.3V) and Off (V) We have also seen logic expressions with their two values True and False Let us define that :. Off = False = and 2. On = True =. What happens than with the conjunction? We have a logic multiplication... Digital vs nalog If you have a password and a login than you can use the computer = P L P L =P L Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.7? = lgebra What happens than with the disjunction? Is it a logic addition? We cannot express 2; we only have two quantities ( and ) Hence we have only two possibilities:. +=; we call this the logic addition 2. +=; we call this the logic exclusion Digital vs nalog Logic? = lgebra O F F? =+O += += += += Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.8 = lgebra =+ Logic addition = Logic exclusion = Logic multiplication = Logic complement Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.9

4 oolean lgebra George oole (85 864) Formulated in 847 the Mathematical nalysis of Logic ased on mathematics restricted to two quantities, and Known as oolean lgebra laude Elwood Shannon (96 2) Published in 937 Symbolic nalysis of Relay and Switching ircuits Proofs that oolean lgebra could be used for digital circuit simplification asis for all circuit design and simplification tools/practices Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b. oolean lgebra in ircuits Mathematical Logic Operator: NOT ND OR XOR Schematic Equivalent and : V V V V Digital vs nalog Logic? = lgebra NSI NSI NSI NSI Logic = lgebra oolean lgebra IE IE IE IE = Exercise The NSI symbols are mainly used in the US and SI, whilst the IE ones are mainly seen in Europe Rev. f57fc2b. of oolean lgebra We have seen that digital logic forms an algebra; the boolean algebra s each algebra, the boolean algebra has properties, we will review them quickly and without proof Elementary properties:. = 2. += 3. = bsorbtion properties:. (+)= 2. +( )= onstant properties:. = 2. = 3. += 4. += 5. = 6. = Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.2

5 of oolean lgebra omplement properties:. = 2. += 3. = 4. = ommutative properties:. = 2. +=+ 3. = Distributive properties:. (+)=( )+( ) 2. +( )=(+) (+) 3. ( )=( ) ( ) ssociative properties: ( )=( ) +(+)=(+)+ ( )=( ) Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.3 Exercise Show that =, using: a) truthtable. b) oolean algebra. Solution: We know that the XOR is defined by: = a) Truthtable = =. The truthtable for the XOR. 2. We take =. 3. We see directly that =. b) oolean algebra. =. 2. We rewrite the function in it s equivalent form. 3. = We take =. 5. = = =. Digital vs nalog Logic =? lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.4 Exercise Show that ( ) ( ) ( ) using oolean algebra: Solution:. We start with the left function: ( ). 2. Using the definition: D E = D E+D E. 3. Hence: ( ) = +.. Now we take the right function: ( ) ( ). 2. Using the definition: D E = D E + D E. 3. Gives us: ( + ) ( + ) 4. Multiplying it out gives: Hence: ( ) ( ) = + nd: + + (Hint: make a truthtable). Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.5

6 of oolean lgebra The theorems of ugust de Morgan (86-87): = + NND gate NOR gate + = De Morgen postulated two theorems. We can draw the gate-equivalent. The NOT gates can be merged. nd the same for the second theorem. Digital vs nalog Logic =? lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.6 of De Morgan (cont.) The theorems of ugust de Morgan (86-87): = + + = The theorems of De Morgan are very important as they show: ny logic expression can be formulated with only OR and NOT ny logic expression can be formulated with only ND and NOT Digital vs nalog Logic =? lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.7 of De Morgan (Example) = + + Enumerating all terms for which = (minterms) leads to a sum of 7 products! etter: Enumerate all terms for which = (maxterms) and use De Morgan Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.8

7 of De Morgan (Example 2) = + + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + logic equation can be formulated in the disjunct form; this form is also called sum of products logic equation can be formulated in the conjunct form; this form is also called product of sums Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.9 of De Morgan (Example 2 gate) = + + ( ) ( ) ( ) * * ( ) ( ) ( ) ( ) ( ) ( ) Digital vs nalog Logic =? lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.2 Optimizing Logic Functions = + The disjunct form does not always provide the smallest equation For this simple example, it can be seen in the truth table, but what about equations with five inputs? We can use the graphic optimizing method of Karnaugh, the Karnaugh diagram y selecting all groups of 2 m s we can eliminate variables: = Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.2

8 The Karnaugh diagram nd we can continue for 4+ variables Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.22 Optimizing Logic Functions with three Variables = + We start selecting the biggest group of s (minterms) We continue until all minterms are selected Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.23 Valid Karnaugh groups What is a valid group? In the Karnaugh diagram below we have a group of four minterms. We have 2 m minterms with m=2. = = ( ) = ( ( + ) + ( + )) = (( + ) ( + )) = (() ()) We can put outside of the brackets; the variable is important for this group! We can also put and outside brackets. s easily can be seen: there are exactly 2 m- minterms in the area where = and the other 2 m- minterms are in the area where = Finally we play with ( + ). oth and are don t care for this group as ( + ) = and ( + ) =! Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.24

9 Valid Karnaugh groups Hence we have a valid group when:. The group has exactly 2 m minterms or 2 m maxterms. 2. The group has exactly m-variables don t care. variable E is don t care when:. There are exactly 2 m- minterms/maxterms from the group in the region where E=. 2. There are exactly 2 m- minterms/maxterms from the group in the region where E= (E-region). variable F influences the function when either:. ll 2 m minterms/maxterms from the group are in the region where F=. 2. ll 2 m minterms/maxterms from the group are in the region where F= (F-region). ll m-variables have to be checked for the above rules. ny violation of the above rules renders the group invalid! Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.25 Valid Karnaugh groups Homework Given the group in the Karnaugh diagram below. Is this a valid group? Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.26 Optimizing Logic Functions with four Variables = + D D = + D D Homework: We start again with the biggest groups of minterms This group is redundant, as it is included in the other two! We are done We can also start with the biggest groups of maxterms This group is redundant, as it is included in the other two! We are done Show with boolean algebra that both functions are identical; what can you observe? Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.27

10 Incomplete Defined Functions = + D D We start again with the biggest groups of minterms When assuming a for the don t care we can find a group of 8! We continue until all s are covered We are done In mathematics logic functions are always completely defined: for each of the input combinations the function is always either or In practice a logic function can have input combinations where we as designer do not mind the outcome: the function is defined and shows one or multiple don t cares Digital vs nalog Logic? = lgebra Logic = lgebra oolean lgebra Exercise Rev. f57fc2b.28