School of Electrical & Computer Engineering ECE 270 Lecture Module 2 Purdue University, College of Engineering Spring 2019 Edition Purdue IM:PACT* Spring 2019 Edition *Instruction Matters: Purdue Academic Glossary of Common Terms Course Transformation DISCRETE LOGIC – a circuit constructed using small-scale integrated (SSI) and medium-scale integrated (MSI) logic devices (NAND gates, decoders, Introduction to Digital System Design multiplexers, etc.) PROGRAMMABLE LOGIC DEVICE (PLD) – an integrated circuit onto which a generic logic circuit can be programmed (and subsequently erased and re-programmed) GENERIC ARRAY LOGIC (GAL) –a (legacy) flash memory based PLD, which is typically erased and re-programmed out- Module 2 of-circuit Combinational Logic Circuits COMPLEX PLD (CPLD) – large flash memory based PLD that is programmable in-circuit 2 Glossary of Common Terms Module 2 isp (IN-SYSTEM PROGRAMMING) –prefix Learning Outcome: “An ability to analyze and design used on CPLDs that can be erased and re- combinational logic circuits” programmed in-circuit A. Combinational Circuit Analysis and Synthesis FIELD PROGRAMMABLE GATE ARRAY B. Mapping and Minimization (FPGA) – an SRAM-based PLD that can be C. Timing Hazards programmed in-circuit (no need to “erase” D. XOR/XNOR Functions since SRAM-based) E. Programmable Logic Devices ADVANCED BOOLEAN EXPRESSION F. Hardware Description Languages LANGUAGE (ABEL) – a “classic” hardware G. Combinational Building Blocks: Decoders description language (HDL) for specifying the behavior of PLDs H. Combinational Building Blocks: Encoders and Tri-State Outputs I. Combinational Building Blocks: Multiplexers VHDL and VERILOG – advanced hardware J. Top Level Modules simulation and description languages 3 4 Purdue IM:PACT* Spring 2019 Edition Reading Assignment: *Instruction Matters: Purdue Academic th th Course Transformation DDPP 4 Ed. pp. 196-210, 5 Ed, pp. 100-117 Learning Objectives: Identify minterms (product terms) and maxterms (sum terms) Introduction to Digital System Design List the standard forms for expressing a logic function and give an example of each: sum-of-products (SoP), product-of-sums (PoS), ON set, OFF set Analyze the functional behavior of a logic circuit by constructing a truth table that lists the relationship between input variable combinations and the output variable Module 2-A Transform a logic circuit from one set of symbols to another through graphical application of DeMorgan’s Law Combinational Circuit Analysis and Synthesis Realize a combinational function directly using basic gates (NOT, AND, OR, NAND, NOR) 6 © 2019 by D. G. Meyer 1 School of Electrical & Computer Engineering ECE 270 Lecture Module 2 Purdue University, College of Engineering Spring 2019 Edition Outline Overview We analyze a combinational logic circuit by obtaining a Overview formal description of its logic function Definitions Once we have a description of the logic function, we can: – determine the behavior of the circuit for various input Minterm identification combinations Maxterm identification – manipulate an algebraic description to suggest different circuit structures ON Sets and OFF sets – transform an algebraic description into a standard form Combinational circuit analysis (e.g., sum-of-products for PLD implementation) – use an algebraic description of the circuit’s functional Equivalent symbols behavior in the analysis of a larger system that includes the circuit Combinational circuit synthesis 7 8 Definitions Definitions Definition: A combinational logic circuit is one whose Definition: A literal is a variable or the complement of output depend only on its current combination of input a variable values (or “input combination”) Definition: A product term is a single literal or a logical Definition: A logic function is the assignment of “0” or product of two or more literals “1” to each possible combination of its input variables Definition: A sum-of-products expression is a logical X1 sum of product terms X2 Definition: A sum term is a single literal or a logical . sum of two or more literals . f f (X1,X2, … , Xn) . Definition: A product-of-sums expression is a logical Xn No Feedback product of sum terms 9 10 Examples Definitions Definition: A normal term is a product or sum term in which W, X, Y Literals no variable appears more than once Definition: An n-variable minterm is a normal product term W X Z Product Term with n literals Definition: An n-variable maxterm is a normal sum term X Y + W Z Sum of Products Expression with n literals Definition: The canonical sum of a logic function is a sum X + Y + Z Sum Term of minterms corresponding to input combinations for which the function produces a “1” output (X + Y) (W + Z ) Product of Sums Expression Definition: The canonical product of a logic function is a product of maxterms corresponding to input combinations for which the function produces a “0” output 11 12 © 2019 by D. G. Meyer 2 School of Electrical & Computer Engineering ECE 270 Lecture Module 2 Purdue University, College of Engineering Spring 2019 Edition Minterm Identification Maxterm Identification 0 → complemented 0 → true 1 → true 1 → complemented 13 14 X Y Z F(X,Y,Z) ON Sets and OFF Sets Example 000 1 Definition: The minterm list that “turns on” an output 001 0 function is called the on set Based on the truth table, 010 0 determine the following 011 1 Example: X,Y,Z(0,1,2,3) 100 0 Indicates “sum” (of products) 101 0 Rows of truth table that are “1” F(X,Y,Z) expressed as: 110 1 an on-set: ____________________ Definition: The maxterm list that “turns off” an output 111 1 function is called the off set an off-set: ____________________ Example: X,Y,Z(4,5,6,7) a sum of minterms: _____________________________ Indicates “product” (of sums) a product of maxterms: __________________________ Rows of truth table that are “0” _______________________________________________ 15 16 X Y Z F(X,Y,Z) X Y Z F(X,Y,Z) Example 000 1 Example 000 1 001 0 001 0 Based on the truth table, 010 0 Based on the truth table, 010 0 determine the following 011 1 determine the following 011 1 100 0 100 0 101 0 101 0 F(X,Y,Z) expressed as: 110 1 F(X,Y,Z) expressed as: 110 1 an on-set: ____________________X,Y,Z(0,3,6,7) 111 1 an on-set: ____________________X,Y,Z(0,3,6,7) 111 1 an off-set: ____________________ an off-set: ____________________X,Y,Z(1,2,4,5) a sum of minterms: _____________________________ a sum of minterms: _____________________________ a product of maxterms: __________________________ a product of maxterms: __________________________ _______________________________________________ _______________________________________________ 17 18 © 2019 by D. G. Meyer 3 School of Electrical & Computer Engineering ECE 270 Lecture Module 2 Purdue University, College of Engineering Spring 2019 Edition X Y Z F(X,Y,Z) X Y Z F(X,Y,Z) Example 000 1 Example 000 1 001 0 001 0 Based on the truth table, 010 0 Based on the truth table, 010 0 determine the following 011 1 determine the following 011 1 100 0 100 0 101 0 101 0 F(X,Y,Z) expressed as: 110 1 F(X,Y,Z) expressed as: 110 1 an on-set: ____________________X,Y,Z(0,3,6,7) 111 1 an on-set: ____________________X,Y,Z(0,3,6,7) 111 1 an off-set: ____________________X,Y,Z(1,2,4,5) an off-set: ____________________X,Y,Z(1,2,4,5) a sum of minterms: X’•Y’•Z’_____________________________ + X’•Y•Z + X•Y•Z’ + X•Y•Z a sum of minterms: X’•Y’•Z’_____________________________ + X’•Y•Z + X•Y•Z’ + X•Y•Z a product of maxterms: __________________________ a product of maxterms: __________________________ (X+Y+Z’)•(X+Y’+Z)• (X’+Y+Z)•(X’+Y+Z’) _______________________________________________ _______________________________________________ 19 20 1. The ON set for a 3-input NAND gate XYZF (X,Y,Z) (with inputs X, Y, and Z) is: NAND 000 1 A. (7) X,Y,Z 001 1 B. X,Y,Z(0) 010 1 C. X,Y,Z(0,1,2,3,4,5,6) 011 1 D. X,Y,Z(1,2,3,4,5,6,7) 100 1 Clicker Quiz E. none of the above 101 1 110 1 111 0 21 22 2. The OFF set for a 3-input NOR gate XYZ F (X,Y,Z) 3. If the function F(X,Y,Z) is represented by the (with inputs X, Y, and Z) is: NOR 000 1 ON SET X,Y,Z(0,3,5,6), then the complement A. X,Y,Z(7) 001 0 of this function F(X,Y,Z) is represented by the B. (0) ON SET: X,Y,Z 010 0 XYZ F(X,Y,Z) XYZ F (X,Y,Z) A. (0,3,5,6) C. X,Y,Z(0,1,2,3,4,5,6) 011 0 X,Y,Z 000 1 000 0 B. (1,2,4,7) D. X,Y,Z(1,2,3,4,5,6,7) 100 0 X,Y,Z 001 0 001 1 E. none of the above 101 0 C. X,Y,Z(1,2,4,6) 010 0 010 1 110 0 D. X,Y,Z(1,3,5,7) 011 1 011 0 111 0 E. none of the above 100 0 100 1 101 1 101 0 110 1 110 0 111 0 111 1 23 24 © 2019 by D. G. Meyer 4 School of Electrical & Computer Engineering ECE 270 Lecture Module 2 Purdue University, College of Engineering Spring 2019 Edition 4. If the function F(X,Y,Z) is represented by the Example - Combinational Analysis ON SET X,Y,Z(0,3,5,6), then the dual of this function FD(X,Y,Z) is represented by the ON SET: A.