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Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Digital Logic Circuits Digital circuits make up all computers and computer systems. The operation of digital circuits is based on Boolean algebra, algebra the mathematics of binary numbers. numbers Boolean algebra is very simple, having only three basic functions, AND, OR, and NOT. These basic functions can be combined in many ways to provide all the functions required in the central processor of a digital computer. computer Digital circuits operate by performing Boolean operations on binaryy numbers ((more about binaryy numbers in EE 2310). ) 1 EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Erik Jonsson School of Engineering g g and Computer Science Th U The University i it off T Texas att D Dallas ll First Boolean Function: NOT NOT is the simplest logical function: 1 input and 1 output. NOT is defined as follows: “The output f of NOT, given an input a is the complement or opposite of the input a, input.” Or : f a Since NOT can have only a 0 or 1 input, the output of NOT is the reverse, or complement, of the input. – If th the iinputt off NOT is i 1, 1 the th output t t iis 0. 0 – If the input of NOT is 0, the output is 1. The NOT function is called inversion, and the digital circuit which iinverts t iis an iinverter. t Th l t i circuit i it symbol b l ffor NOT iis: The electronic a 2 a EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science The Truth Table The inverter input/output relationship, with one input Boolean Function Truth Table and output output, is easy to show show. I Input x Input I y Output O f For complex functions, an 0 0 0 I/O table is helpful. 0 1 1 We W call ll this thi a truth t th ttable, bl 1 0 1 since it indicates the 1 1 1 0 (“true”) outputs, although it normally shows outputs for Note: This 2-input 2 input truth table all input combinations. shows the output f for all possible combinations of the binary inputs We will demonstrate some Boolean functions using truth x and y. tables. 3 EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Second Boolean Function: AND 4 AND Truth Table AND has two or more inputs. a b a AND b The truth table for a two-input AND with 0 0 0 inputs a and b is shown in the chart. 0 1 0 AND is defined as follows: a AND b 1 if and only if (iff) a 1 and b 1. 1 0 0 Mathematically, we represent “a AND b” 1 1 1 as a·b (an unfortunate choice). AND may have more than two inputs, i. 22-Input Input AND a aa·bb e : a AND b AND c AND d. e.: d b The electronic circuit symbols for 2- and a 4-input ANDs are shown at the right. b a·b·c·d 4-Input p AND c Regardless of the number of inputs, the d output of AND is 1 iff all inputs are 1. EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Third Boolean Function: 5 OR OR has two or more inputs. OR Truth Table a b a OR b The OR truth table for two inputs a, b is 0 0 0 shown h iin th the adjacent dj t chart. h t 0 1 1 OR is defined as follows: a OR b 1 if 1 0 1 either a or b or both a and b 1. 1 1 1 Mathematically we represent “a OR b” Mathematically, as a b (another bad choice). OR may have more than two inputs, i. e.: a 2-Input OR b a OR b OR c OR d. The electronic circuit symbols for 2- and a 4- input ORs are shown at the right. b 4-Input OR c Regardless of the number of inputs, the d output of OR is 0 iff all inputs are 0. EE 1202 Lab Briefing #3 a b a b c d N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Logic “1” and “0” 6 Electronic circuits don’t manipulate logic 1 and 0 literally. In digital circuits, the values “1” and “0” are levels of voltage, and the logic circuits that we use are technically “inverting amplifiers with saturated outputs.” In the circuits we will use, logic 0 is 0 volts, and logic 1 is 5 volts. Logic L i 0/1 inputs (0V, 5 V) EE 1202 Lab Briefing #3 Logic 0/1 outputs (0V, 5 V) N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Making More Complex Boolean Functions The three Boolean functions discussed above can be used to form more complex p functions. ANY computer function can be performed using combinations of AND, OR, and NOT. To simplify the definition of combinational logic (the logic of the computer CPU), any logic function can be composed p of a level of AND ggates followed byy a single g OR gate. There are a few other ways to form Boolean circuits, b t we will but ill cover only l this thi one method th d in i Lab L b 3. 3 7 EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Uniqueness q of AND The uniqueness of the AND function is that the output p of AND is 0 except when EVERY input 1. In I th the 4 gates t tto th the right, i ht a SINGLE 0 input into each gate forces the output to 0. The output of AND is 1 only when ALL inputs 1 (8input gate to right). right) 8 EE 1202 Lab Briefing #3 1 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science The “Any 1” Quality of OR The output of OR 1 if ANY input i 1. OR outputs a 0 iff ALL inputs p 0. We can use the ability of OR to “pass” any 1 and the unique 1- outputs of the AND to create Boolean functions. 9 EE 1202 Lab Briefing #3 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Digital Design In circuit design, inputs and outputs are defined by a “spec.” Since computer circuits use only binary numbers, inputs are always 0 and 1, and the output is always 0 and 1. The engineer designs the circuit between input and output by: – – – Makingg a truth table to represent p the input/output p p relationship. p Defining a Boolean expression that satisfies the truth table. Constructing a circuit that represents the Boolean function. Establish inputs and outputs 10 Construct Truth Table Define Boolean expression in SOP or POS form EE 1202 Lab Briefing #3 Design digital circuit based on Boolean expression N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Creating a Computer (“Boolean”) Function A “spec” for a function f of two variables x and y is that f 1 when x and y are different, and 0 otherwise. The truth table charts f per the “spec.” How can we describe this behavior with a Boolean expression? For the first 1, we can create an AND function: x y. Note that this expression is 1 ONLY when x 0, y 1. For the second 1, we create x y , which is only 1 for x 1, y 0. 11 EE 1202 Lab Briefing #3 x 0 0 1 1 y 0 1 0 1 x y x y f 0 1 1 0 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Boolean Function (2) First 1 AND function: x y . Second 1 AND function: x y . The two Boolean AND functions each describe one of the two conditions in which f is 1. How do we create a Boolean function that describes BOTH conditions of f 1? Recall that any 1 is passed through OR. OR Then if we send both ones through a single OR, its output will match the specified performance of f. 12 EE 1202 Lab Briefing #3 x 0 0 1 1 y 0 1 0 1 f 0 1 1 0 x y x y N. B. Dodge 01/12

Erik Jonsson School of Engineering g g and Computer Science Th U The University i it off T Texas att D Dallas ll Boolean Function (3) We OR the two AND functions: f x y x y We now have a complete description (Boolean expression) for the function f. Since Si we know k what h t AND and d OR circuits look like, we can build a digital circuit that produces f. x x 0 0 1 1 y 0 1 0 1 f 0 1 1 0 x y x y f y 13 EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Erik Jonsson School of Engineering g g and Computer Science Th U The University i it off T Texas att D Dallas ll Building Boolean Functions As we have just seen, if we have Boolean functions that result from a truth table and “spec,” we can convert the Boolean functions to computer circuits. C id th Consider these ffunctions: ti 1: 2: a b c f a b f (a b) c f 14 3: EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Quick Exercise Designing a circuit from the Boolean expressions: ( a·b ) ( c·d ) f a b c d f 15 EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Design: “Spec” to Truth Table to Circuit Say you have the following “design spec” for function f: – There are three inputs x, y, and z, to a digital circuit. – The circuits must have an output of 1 when y z 1 and x 0; and when x z 1 and y 0. 0 – Design the circuit using AND, OR, and NOT gates. 16 x 0 0 0 0 1 1 1 1 y 0 0 1 1 0 0 1 1 z 0 1 0 1 0 1 0 1 f(x, y, z) 0 0 0 1 0 1 0 0 AND’s f xyz f x yz The truth table above shows the desired output: f 1 when x 0, y, z 1, f 1 when yy 00, x, x z 1. 1 EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Getting the Boolean Function We can create a Boolean function for each of the two “1” conditions: – Inverting x and ANDing it with y and z creates a 1 for the first condition. – Invertingg y and ANDingg it with x and z creates the other 1. – Notice that each AND function produces a 1 ONLY for that combination of variables. According to the definition of OR, any 1 goes through that gate. x 0 0 0 0 1 1 1 1 y 0 0 1 1 0 0 1 1 z 0 1 0 1 0 1 0 1 This is a 1 ONLY for (0,1,1) f(x, y, z) 0 0 0 1 0 1 0 0 AND’s f xyz f x yz This is a 1 ONLY for (1,0,1) – Therefore OR the two AND functions together to get a function that is 1 for both cases! 17 EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Boolean Function and Circuit x 0 0 0 0 1 1 1 1 x y z y 0 0 1 1 0 0 1 1 z 0 1 0 1 0 1 0 1 f(x, y, z) 0 0 0 1 0 1 0 0 AND’s x y z x y z x y z The OR function (f) completely satisfies the spec and truth table! 18 EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Lab Instrument: IDL-800 Key parts: – – – – – 19 LED indicators Circuit board 5V power Momentary 0-1 switches 0-1 input switches EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Digital Circuit Kit The digital circuit kits are also used in EE 2310. You will only be using AND, OR, and NOT circuits (see below). NEVER replace a bad or broken circuit in the kit. Give to the TA to be replaced. ALWAYS put up the kit when you are done. 20 EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Th U The University i it off T Texas att D Dallas ll Erik Jonsson School of Engineering g g and Computer Science Plugging in a Digital IC and Wiring Up a Circuit Remember: The circuit is ALWAYS plugged in so that h iit spans a channel in the circuit board. Therefore a wire plugged into any of the parallel holes into which a chip leg is plugged is connected to that leg of the chip. 21 EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Erik Jonsson School of Engineering g g and Computer Science Th U The University i it off T Texas att D Dallas ll Digital IC Circuit Diagrams Circular dot 1 2 3 4 5 Ground (0V.) 6 connection 7 Notch 14 13 Power ( 5V.) connection 12 11 10 9 8 32 08 04 13 12 11 10 9 8 5 6 3 4 1 2 1 2 3 1 2 4 5 6 4 5 6 9 10 8 9 10 8 12 13 11 12 13 11 3 74 LS XXX Chip Outline SN 74LS04 Hex inverter gate 22 SN 74LS08 Quad 2‐input AND gate SN 74LS32 Quad 2‐input OR gate You will be using the 74LS04 74LS04, 74LS08 74LS08, and 74LS32 digital integrated circuits circuits. The diagram above (also in your manual) shows the outline of the chip, with power/ground inputs. The three chip schematics show how the circuits in each chip connect to the input pins. EE 1202 Lab Briefing #3 N. B. Dodge 01/12

Erik Jonsson School of Engineering and Th U i it f T t D ll gg Computer Science The University of Texas at Dallas First Boolean Function: NOT NOT is the simplest logical function: 1 input and 1 output. NOT is defined as follows: "The output f of NOT, given an input a is the complement or opposite of the inputa, is the complement or opposite of the input .

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