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A Simple Circuit
● Design a garage close circuit that activates the garage door motor when the door is open, the object sensor is off, and Close button is pressed.
Boolean Algebra
Symbol Name Description Priority
( ) Parenthesis Evaluate expression inside parenthesis first
Highest
' NOT Evaluate left to right
* AND Evaluate left to right
+ OR Evaluate left to right Lowest
Boolean Algebra (Terms)
● Variable – A term that represents a value● Literal – The usage of a variable in a
formula● Sum-Of-Products (SOP)
F = XYZ + X'Y'Z + XY'Z'● Product-Of-Sums (POS)
G = (X + Y + Z)(X' + Y' + Z)(X + Y' + Z')
Boolean Algebra (Axioms)
(A1) X = 0 if X != 1 (A1') X = 1 if X != 0
(A2) If X = 0, then X' = 1 (A2') if X = 1, then X' = 0
(A3) 0 * 0 = 0 (A3') 1 + 1 = 1
(A4) 1 * 1 = 1 (A4') 0 + 0 = 0
(A5) 0 * 1 = 1 * 0 = 0 (A5') 1 + 0 = 0 + 1 = 1
Boolean Algebra (Postulates)● Commutative
A + B = B + A
A * B = B * A● Distributive
A * (B + C) = A * B + A * C
A + (B * C) = (A + B) * (A + C)
Pay attention to the second distributive
Boolean Algebra (Postulates)● Associative
(A + B) + C = A + (B + C)
(A * B) * C = A * (B * C)● Identity
0 + A = A + 0 = A
1 * A = A * 1 = A
Boolean Algebra (Theorems)● Involution Law
(A')' = A● DeMorgan's Law
(A + B)' = A'B'
(AB)' = A' + B'
DeMorgan's can be extended to more variables
Boolean Algebra (Theorems)● Covering
X + X * Y = X
X * (X + Y) = X● Combining
X * Y + X * Y' = X
(X + Y) * (X + Y') = X
Boolean Algebra (Theorems)● Consensus
X * Y + X' * Z + Y * Z = X * Y + X' * Z
(X + Y) * (X' + Z) * (Y + Z) = (X + Y) * (X' + Z)
Boolean Algebra
● Prove the following using algebraic manipulation
X'Y' + X'Y + XY = X' + Y
A'B + B'C' + AB + B'C = 1
Y + X'Z + XY' = X + Y + Z
X'Y' + Y'Z + XZ + XY + YZ' = X'Y' + XZ + YZ'
Boolean Algebra
● Prove the following using algebraic manipulation
AB + BC'D' + A'BC + C'D = B + C'D
Boolean Algebra
Given that A * B = 0 and A + B = 1, use algebraic manipulation to prove that
(A + C) * (A' + B) * (B + C) = B * C
Principle of Duality
● Any theorem or identity in Boolean Algebra remains true if 0 and 1 are swapped and the AND and OR operations are swapped throughout
Truth Tables
● A truth table may be expressed by many different equations.
● Prove two functions are equal by induction.● Optimizing a function usually requires
creating a truth table.
Standard Forms
A B C MinTerm MaxTerm
0 0 0 A'B'C' m0 A+B+C M00 0 1 A'B'C m1 A+B+C' M10 1 0 A'BC' m2 A+B'+C M20 1 1 A'BC m3 A+B'+C' M31 0 0 AB'C' m4 A'+B+C M41 0 1 AB'C m5 A'+B+C' M51 1 0 ABC' m6 A'+B'+C M61 1 1 ABC m7 A'+B'+C' M7
Standard Forms
● Sum of Minterms– Product terms
– Each term contains each variable.
– A term is one line or element on a truth table.
– For each line in a truth table that is 1, that term is part of the final equation.
– Write the Sum of Minterms for table 2.3 on page 67.
– Can be written as Σm(mx, my, ...)
Standard Forms
● Product of Maxterms– Sum terms.
– Each term contains each variable.
– A term is one line or element on a truth table.– For each line in a truth table that is 0, that term is part
of the final equation.
– Write the Product of Maxterms for table 2.3 on page 67.
– Can be written as ∏M(Mx, My, ...)
Combinational Logic Design Process● Create a truth table.● Write optimized equations. (We still need to
cover optimization in 6.2).● Draw schematic or create hardware
description from optimized equations.
Timing
● Real logic gates take time to react to an input change.
● The delay is called the propagation delay.● A complicated circuit may have “glitches” in its
output signals when the inputs change from one state to another.
● Reducing “glitches” often requires extra logic gates.
Optimization
● The second part of our design process.● Optimization criteria:
– Performance
– Size
– Power
Two-level Optimization
● Manipulating a function until it is in a minimized SOP or POS form.
● Minimizes the number of literals in an equation and results in a smaller circuit than a minimized sum-of-minterms or product of maxterms.
● Use Boolean Algebra● Use a Karnaugh Map
Karnaugh Maps (K-maps)
● A visual method.● Good for up to 4 variables. Difficult for 5 or 6
variables. Very difficult for more than 6 variables.
● Start with a truth table for your function.● A rectangular grid.● Number of squares is equal to the number of
lines in truth table.
K-Maps
● A two variable K-map.● Each square is a
minterm● Only one bit may
change between adjacent squares.
● Each side of map is a power of 2.
K-Maps
● 3 variable K-map● Map the equation
Σm(2,3,5,6)● Implicant – any
rectangular group of 1's where rectangle is a power of 2 on each side.
● Prime Implicant – An implicant that can't “grow larger”.
K-Maps
● Essential Prime Implicant – A prime implicant that has one square that is not part of another prime implicant.
● Non-essential Prime Implicant – A prime implicant where every one of its squares is part of another prime implicant.
K-Maps
● An optimized SOP has all of the essential prime implicants. It may have non-essential prime implicants only if the essential prime implicants don't cover all squares.
● There can be more than one simplified SOP due to the selection of non-essential prime implicants.
K-Maps
● Determine essential and non-essential prime implicants:– Σm(1,2,3,5,8,9,11,15)
– Σm(5,7,9,13,14,15)
– Σm(0,1,3,4,6,9,11,14,15)
– Σm(1,5,13,14,15)
– Σm(2,3,5,7,8,10,12,13)
K-Maps
● Determine essential and non-essential prime implicants:– Σm(0,1,3,5,6,7,8,9,10,12,14,15)
– Σm(1,2,3,5,6,13,14,15)
K-Maps
● To create a simplified POS, select rectangular regions of 0's.
● These are called Implicates.● A simplified POS is made up of essential
prime implicates and maybe non-essential prime implicates.
● Map the equation ПM(0,1,4,6)
K-Maps
● Selecting the 0's as a minimized SOP gives the inverse of the function.
● Use Demorgan's Theorem to get the function and at the same time minimized POS form.
● Usually go straight from K-Map to minimized POS form.
K-Maps
● Determine essential and non-essential prime implicates:– ∏M(1,2,3,5,8,9,11,15)
– ∏M(5,7,9,13,14,15)
– ∏M(0,1,3,4,6,9,11,14,15)
– ∏M(1,5,13,14,15)
– ∏M(2,3,5,7,8,10,12,13)
K-Maps
● Determine essential and non-essential prime implicates:– ∏M(0,1,3,5,6,7,8,9,10,12,14,15)
– ∏M(1,2,3,5,6,13,14,15)