Chapter 5 Boolean Algebra and Reduction Techniques William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Combinational Logic Using two or more logic gates to form a more useful, complex function A combination of logic functions B = KD + HD Boolean Reduction B = D(K+H) William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Boolean Laws and Rules Commutative law of addition A + B = B + A Associative law of addition A + (B + C) = (A + B) + C Distributive law A(B + C) = AB + AC Equivalent circuits –See Figures 5-9 to 5-14 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Figure 5-9 Figure 5-10 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Figure 5-11 Figure 5-12 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Figure 5-13 Figure 5-14 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Boolean Laws and Rules Rule 1: Anything ANDed with a 0 is equal to 0 Rule 2: Anything ANDed with a 1 is equal to itself - Figure 5-16 Rule 3: Anything ORed with a 0 is equal to itself - Figure 5-17 Rule 4: Anything ORed with a 1 is equal to 1 - Figure 5-18 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Figure 5-16 Figure 5-17 Figure 5-18 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Boolean Laws and Rules Rule 5: Anything ANDed with itself is equal to itself - Figure 5-19 Rule 6: Anything ORed with itself is equal to itself - Figure 5-20 Rule 7: Anything ANDed with its own complement equals 0 - Figure 5-21 Rule 8: Anything ORed with its own complement equals 1 - Figure 5-22 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Figure 5-19 Figure 5-20 Figure 5-21 Figure 5-22 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Boolean Laws and Rules Rule 9: Anything complemented twice will return to its original logic level - Figure Rule 10: –A + AB = A + B –See Table 5-1 Law and Rule Summary - See Table 5-2 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Figure 5-23 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Simplification of Combinational Logic Circuits Using Boolean Algebra Equivalent circuits can be formed with fewer gates Cost is reduced Reliability is improved Use laws and rules of Boolean Algebra William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Using Quartus II To Determine Simplified Equations The software will determine the simplest form before synthesizing –eliminates unnecessary inputs –minimizes gates used in CPLD Quartus II solution to example 5-9 –see figures 5-32 (a) and (b) William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Figures 5-32 (a) and (b) William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Using Quartus II To Determine Simplified Equations If the program determines an input is not necessary, a warning message will be displayed when the program is compiled –See figure 5-33 The floor plan view provides implementation details –The Equations Section, see figure 5-35 –Arithmetic operators: & AND ! NOT # OR $ Exclusive OR William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Using Quartus II To Determine Simplified Equations Floorplan Editor Display –alternate way to view reduced logic equation –useful for viewing and modifying pin and macrocell assignments Logic Array Block (LAB) view shows interconnections and equations See figure 5-36(b) William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Figure 5-36(b) William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

DeMorgan’s Theorem To simplify circuits containing NAND and NOR gates A B = A + B A + B = A B William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

DeMorgan’s Theorem Break the bar over the variables and change the sign between them Inversion bubbles - used to show inversion rather than inverters Use parentheses to maintain proper groupings End up with Sum-of-Products (SOP) form William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

DeMorgan’s Theorem Bubble Pushing –See Figure 5-60 –shortcut method of forming equivalent gates –change the logic gate (AND to OR or OR to AND) –Add bubbles to the inputs and outputs where there were none and remove original bubbles William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Figure 5-60 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Entering a Truth Table in VHDL Using a Vector Signal Truth table to be entered, table 5-5 –define internal signal to represent the three inputs –the three inputs will be grouped as a 3 bit vector –the use of comments to document a program –use selected signal assignments, built to look like truth table entries see figure 5-62 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Figure 5-62 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

The Universal Capability of NAND and NOR Gates NANDs can be combined to form all other gates –See Figures 5-68, -69, -71, -73, and -74 NORs can be combined to form all other gates –See Figure 5-75 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Figure 5-68 Figure 5-69 Figure 5-73 Figure 5-71 Figure 5-74 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Figure 5-75 William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

AND-OR-INVERT Gates for Implementing Sum-of-Products Expressions Product-of-sums (POS) form Sum-of-products (SOP) form –can easily be implemented using an AOI gate TTL and CMOS AOI devices are available William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Karnaugh Mapping To minimize the number of gates Reduce circuit cost Reduce physical size Reduce gate failures Requires SOP form William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Karnaugh Mapping Graphically shows output level for all possible input combinations Moving from one cell to an adjacent cell, only one variable changes William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Karnaugh Mapping Transform the Boolean equation into SOP form Fill in the appropriate cells of the K-map Encircle adjacent cells in groups of 2, 4 or 8 –watch for wraparound Find terms by which variables remain constant within circles William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

System Design Applications Use Karnaugh Mapping to reduce equations Use AND-OR-INVERT gates to implement logic William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Summary Several logic gates can be connected together to form combinational logic. There are several Boolean laws and rules that provide the means to form equivalent circuits. Boolean algebra is used to reduce logic circuits to simpler equivalent circuits that function identically to the original circuit. William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Summary DeMorgan’s theorem is required in the reduction process whenever inversion bars cover more than one variable in the original Boolean equation. NAND and NOR gates are sometimes referred to as universal gates, because they can be used to form any of the other gates. William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Summary AND-OR-INVERT (AOI) gates are often used to implement sum-of-products (SOP) equations Karnaugh mapping provides a systematic method of reducing logic circuits. William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Summary Combinational logic designs can be entered into a computer using graphic design software or VHDL. Using vectors in VHDL is a convenient way to group like signals together similar to an array. William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

Summary Truth tables can be implemented in VHDL using vector signals with the selected signal assignment statement. Quartus II can be used to determine the simplified equation of combinational circuits. William Kleitz Digital Electronics with VHDL, Quartus® II Version Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.