Slides for Lecture 8 ENEL 353: Digital Circuits — Fall 2013 Term Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary 25 September, 2013
slide 2/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Previous Lecture 2-input NAND, NOR, and XNOR gates. Gates with 3 or more inputs. Electrical signalling for gate inputs and outputs, voltage levels and noise margins.
slide 3/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Today’s Lecture Completion of material on voltage levels and noise margins. Introduction to combinational logic design. Introduction to Boolean algebra. Related reading in Harris & Harris: Sections 1.6, 2.1 and 2.2.
slide 4/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Noise margin example: Low-voltage CMOS (This is pretty much a repeat of a slide from Lecture 7.) What are NM L and NM H for this logic Here are family? numbers for the low-voltage Suppose the input to NOT gate 1 is 3 . 1 V. How much noise can there be on CMOS logic family . . . the wire from A to B before NOT gate 2 might produce wrong output? What if item voltage the input to NOT gate 1 is 0 . 2 V? 3 . 3 V V DD added noise V IL 0 . 9 V 1 . 8 V V IH A B V OL 0 . 36 V V OH 2 . 7 V NOT NOT gate 1 gate 2
slide 5/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Inputs, outputs, elements and nodes Here is an example logic circuit: n1 A E1 B E3 Y C E2 Z ◮ The inputs are A , B and C . ◮ The outputs are Y and Z . Image is Figure 2.2 from Harris D. M. and Harris S. L., Digital Design and Computer Architecture, 2nd ed. , c � 2013, Elsevier, Inc.
n1 A E1 B E3 Y C E2 Z ◮ E1, E2 and E3 are called elements —each element is itself a logic circuit , simpler than the overall circuit. ◮ A node is a wire whose voltage communicates a bit value between elements. ◮ A , B and C are called input nodes ; Y and Z are called output nodes . ◮ n1 is an example of an internal node —a node that is neither an input nor an output of the overall circuit.
slide 7/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Review: Combinational versus Sequential Logic The outputs of a combinational logic circuit depend only the current values of its inputs. The outputs of a sequential logic circuit depend on the history of its input values. A logic gate is a combinational logic circuit that has a single output, not multiple outputs.
slide 8/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Generic symbol for combinational logic The symbol C L is often used as a label to indicate that a logic element is combinational, without actually specifying the function of the logic element. Let’s draw a diagram for a generic 3-input, 2-output combinational logic element, then let’s draw it another way. Let’s draw a diagram for a generic combinational logic element, with unspecified numbers of inputs and outputs.
slide 9/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Combinational composition Combinational composition is a term for a method of building complex combinational circuits out of simpler elements. The rules for combinational composition are: ◮ Each of the elements must itself be combinational. ◮ Each node in the circuit is either an input node or connects to exactly one output of an element . ◮ There must be no cyclic paths —no way to make a path through the circuit that goes more than once through a single element. If a circuit satifies all three of these rules, it is guaranteed to be combinational.
slide 10/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Examples with logic gates: Which ones satisfy the combinational composition rules? R A X B (a) (b) Y Y C S A (c) Y B
slide 11/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Examples with generic combinational elements Does either circuit satisfy the combinational composition rules? (a) A C C Y L L B C C Z L L C Y (b) A C C Z L L B
slide 12/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Ordinary Algebra We’re very accustomed to seeing things like ( a + b )( c + d ) = ac + ad + bc + bd and If x 2 + x = 0 and x � = 0 , then x + 1 = 0 . Symbols such as a , b , c , d , and x are variables that could have any value taken from a set such as ◮ the set of all real numbers, or ◮ the set of all complex numbers.
slide 13/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Boolean Algebra In Boolean algebra , variables have values taken from this set: { 0 , 1 } . Ordinary algebra has many operators : addition, subtraction, multiplication, division, and others. Boolean algebra has three operators : NOT, AND, and OR. Boolean algebra is very useful for description, analysis and design of combinational logic systems. Note: Past years’ textbooks for ENEL 353 use the term “switching algebra” instead of “Boolean algebra”. Watch out for that if you are looking at previous years’ course materials.
slide 14/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Terminology for Boolean algebra There are many words that have special meanings in discussion of Boolean algebra. Here are some of them: complement , literal , product , minterm , maxterm . There are several others. Make sure you learn exact meanings for all of them! Having only a rough idea what they mean is not good enough —that will lead to confusion and errors .
slide 15/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Variables, complements and literals As stated two slides back, variables in Boolean algebra are things that may have one of two values: 0 (FALSE), or 1 (TRUE). The complement of a variable is the NOT of that variable. So the complement of A is A , the complement of B is B , and so on. A literal is either a variable or the complement of a variable. So, if A and B are Boolean variables, which of the following are literals? ◮ A ◮ AB ◮ A ◮ A + B ◮ B ◮ A + A ◮ B
slide 16/18 ENEL 353 F13 Section 02 Slides for Lecture 8 True form and complementary form These are terms that distinguish the two kinds of literals. The true form of a variable is just the “plain” version of that variable. For example, the true forms of A , B , etc., are simply A , B , etc. The complementary form of a variable is the NOT of that variable. For example, the complementary forms of A , B , and so on, are A , B , and so on.
slide 17/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Products In Boolean algebra, a product is defined to be either ◮ a literal; or ◮ the AND of two or more literals. (Our textbook defines a product as “the AND of one or more literals”, which is correct, but makes you think harder than necessary about what the AND of one literal is.) Suppose A , B , and C are Boolean variables. Which of the following are products? ◮ C ◮ A ◮ AB ◮ A + B ◮ ABC ◮ AB + BC
slide 18/18 ENEL 353 F13 Section 02 Slides for Lecture 8 Next Lecture More about Boolean algebra. Related reading in Harris & Harris: Sections 2.2 and 2.3
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