COMPUTER ORGANIZATION & ARCHITECTURE DIGITAL LOGIC CSCD211- DEPARTMENT OF COMPUTER SCIENCE, UNIVERSITY OF GHANA

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1 COMPUTER ORGANIZATION & ARCHITECTURE DIGITAL LOGIC

2 LOGIC Logic is a branch of math that tries to look at problems in terms of being either true or false. It will use a set of statements to derive new true statements. Computers rely heavily on logic to run programs. If and else if statements in programming language are a form of logic that computers use If x is equal to 5 then do something if (x == 5) { do something; } 2

3 Computers must simulate logic in hardware. To do this, it must use special pieces that can act like logic functions Example: CPUs use transistors Transistors are switches that are on or off On means there is an electrical current Off means there is no electrical current We can use these to mean either true or false. On = true = = electricity detected Off = false = = no electricity detected 3

4 BOOLEAN ALGEBRA Boolean algebra uses logic to analyze and perform operations on variables. Boolean means the math uses just true and false Boolean functions can always be represented with a truth table. Boolean algebra turns out to be a convenient tool in two areas: Analysis: It is an economical way of describing the function of digital circuitry. Design: Given a desired function, Boolean algebra can be applied to develop a simplified implementation of that function 4

5 The basic logical operations are AND, OR, and NOT, which are symbolically represented by dot, plus sign, and overbar: The operation AND yields true (binary value ) if and only if both of its operands are true. The operation OR yields true if either or both of its operands are true. The unary operation NOT inverts the value of its operand. 5

6 There are also three other useful operators: XOR, NAND, and NOR. The exclusive-or (XOR) of two logical operands is if and only if exactly one of the operands has the value. The NAND function is the complement (NOT) of the AND function, and the NOR is the complement of OR The table above shows basic logical operations in a form known as a truth table, which lists the value of an operation for every possible combination ofvalues of operands. 6

7 The table below summarizes key identities of Boolean algebra. The equations have been arranged in two columns to show the complementary, or dual, nature of the AND and OR operations. There are two classes of identities: basic rules (or postulates), which are stated without proof, and other identities that can be derived from the basic postulates. The postulates define the way in which Boolean expressions are interpreted. 7

8 Verify using truth tables. X + XY = X 2. (X + Y) = X Y 3. (A + B)(A + C)(A + B ) Draw a Truth Table for the following;. M=N(P + R) 2. M=N + P + NP Prove that;. (X + Y)(X + Z) = X + YZ 2. X + X Y = X + y 8

9 Simplify the following. AB + B A + CDE + C DE + EC D 2. AB + AC + BA 3. (PQ + R + ST)TS 9

10 LOGIC GATES The fundamental building block of all digital logic circuits is the gate. Logical functions are implemented by the interconnection of gates. A gate is an electronic circuit that produces an output signal that is a simple Boolean operation on its input signals. The basic gates used in digital logic are AND, OR, NOT, NAND, NOR, and XOR. Each gate is defined in three ways: graphic symbol, algebraic notation, and truth table.

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12 TRANSISTORS A transistor is a device that acts, depending on the voltage level of an input signal, either as a wire that conducts electricity or as a resistor that blocks the flow of electricity A transistor has no moving parts, yet acts like a switch It is made of a semiconductor material, which is neither a particularly good conductor of electricity, such as copper, nor a particularly good insulator, such as rubber 2

13 Constructing Gates A transistor has three terminals - A source - A base - An emitter, typically connected to a ground wire If the electrical signal is grounded, it is allowed to flow through an alternative route to the ground (literally) where it can do no harm 3

14 Constructing gates using transistors It turns out that, because the way a transistor works, the easiest gates to create are the NOT, NAND, and NOR gates 4

15 Circuits Two general categories In a combinational circuit, the input values explicitly determine the output In a sequential circuit, the output is a function of the input values as well as the existing state of the circuit As with gates, we can describe the operations of entire circuits using three notations Boolean expressions Logic diagrams Truth tables 5

16 Combinational Circuits Combinational circuit is one that combines different gates in the circuit. Some characteristics of combinational circuits are listed below The output of the circuit depends only on the present input terminals. They do not use any memory. Previous state of the inputs do not have effect on the present state of the circuit. It can have n number of input and m number of outputs. 6

17 Gates are combined into circuits by using the output of one gate as the input for another 7

18 Consider the following Boolean expression: A(B + C) Now compare the final result column in this truth table to the truth table for the previous example 8

19 Exercise. Consider the function: Y = (A B) + (A C) B (a) Draw a combinational logic circuit that implements this function. (b) Draw a truth table for this function. 2. Y = A + (B C) (a) Draw a combinational logic circuit that implements this function. (b) Draw a truth table for this function. 3.Using only NAND gates, draw a combinational logic circuit for the exclusive or function Y = A B + A B 9

20 ADDERS At the digital logic level, addition is performed in binary Addition operations are carried out by special circuits called Adders There are basically two kinds of adders Half Adder Full Adder 2

21 HALF ADDER The result of adding two binary digits could produce a carry value Half Adder is a combinational circuit with two inputs and two outputs It s a circuit that computes the sum of two bits and produces the correct carry bit It is the basic building block for addition of two single bit numbers 2

22 HALF ADDER Cont The sum aspect of the half adder uses XOR gates while the carry bit uses the AND gate. Below is a truth table of a half adder 22

23 Circuit diagram representing a half adder Two expressions: Boolean Sum = A B Carry = AB 23

24 FULL ADDER Full Adder is a combinational circuit that performs the addition of three bits (two significant bits and a carryin bit) It consists of three inputs and two outputs, two inputs are the bits to be added, the third input represents the carry in bit. The full adder is usually a component in a cascade of adders, which add 8, 6, etc, binary numbers 24

25 FULL ADDER Cont It can be implemented using two half adders (thus two XOR gates and two AND gates) and an OR gate. A Circuit diagram is shown below; 25

26 The sum output is equal to when only one input is equal to or when all three inputs are equal to. The carry-out output has a carry of if two or three inputs are equal to The truth table is shown below 26

27 MULTIPLEXER Multiplexer is a general circuit that produces a single output signal. It is a special type of combinational circuit with n- data inputs, one output and m select inputs. The output is equal to one of several input signals to the circuit 27

28 MULTIPLEXER Cont The multiplexer selects which input signal is used as an output signal based on the value represented by a few more input signals, called select signals or select control lines Depending on the digital code applied at the selected inputs, one out of the n data sources is selected and transmitted to the single output. 28

29 The control lines S, S, and S2 determine which of the eight other input lines (D through D7) are routed to the output (F) 29

30 CSCD2-DEPARTMENT OF COMPUTER SCIENCE, 3

31 SEQUENTIAL CIRCUITS Sequential circuits are digital circuits in which the outputs depend not only on the current inputs, but also on the previous state of the output. They basic sequential circuit elements can be divided in two categories: Level-sensitive (Latches) High-level sensitive Low-level sensitive Edge-triggered (Flip-flops) Rising (positive) edge triggered Falling (negative) edge triggered Dual-edge triggered CSCD2-DEPARTMENT OF COMPUTER SCIENCE, 3

32 The Set/Reset (SR) Latch The Set/Reset latch is the most basic unit of sequential digital circuits. It has two inputs (S and R) and two outputs Q and Q. The two outputs must always be complementary, i.e if Q is then Q must be, and vice-versa. The S input sets the Q output to a logic. The R input resets the Q output to a logic. CSCD2-DEPARTMENT OF COMPUTER SCIENCE, 32

33 The design of this circuit guarantees that the two outputs X and Y are always complements of each other The value of X at any point in time is considered to be the current state of the circuit Therefore, if X is, the circuit is storing a ; if X is, the circuit is storing a CSCD2-DEPARTMENT OF COMPUTER SCIENCE, 33

34 Truth Table S R Q+ Q + Function Q Q Latch Reset Set Logic Symbol S Q Illegal R Q CSCD2-DEPARTMENT OF COMPUTER SCIENCE, 34

35 The Gated Set/Reset (SR) Latch To be able to control when the S and R inputs of the SR latch can be applied to the latch and thus change the outputs, an extra input is used. This input is called the Enable or clock. If the Enable is then the S and R inputs have no effect on the outputs of the SR latch. If the Enable is then the Gated SR latch behaves as a normal SR latch. CSCD2-DEPARTMENT OF COMPUTER SCIENCE, 35

36 Circuit Diagram Truth Table Truth Table S EN R S Q Q R Q Q EN S R Q+ Q Q Q EN S R Q+ X X Function Logic Symbol Q Q S Q EN R Q U CSCD2-DEPARTMENT OF COMPUTER SCIENCE, 36

37 Edge Detection Circuits Edge detection circuits are used to detect the transition of the Enable from logic to logic (positive edge) or from logic to logic (negative edge). The operation of the edge detection circuits shown below is based on the fact that there is a time delay between the change of the input of a gate and the change at the output. This delay is in the order of a few nanoseconds. The Enable in this case is called the Clock (CLK) Positive Edge Detection Negative Edge Detection EN EN' EN EN' EN EN EN EN EN EN EN EN EN' EN' CSCD2-DEPARTMENT OF COMPUTER SCIENCE, 37

38 The JK Edge Triggered Flip Flop The JK edge triggered flip flop can be obtained by inserting an edge detection circuit at the Enable (CLK) input of a JK latch. This ensures that the outputs of the flip flop will change only when the CLK changes ( to for +ve edge or to for ve edge) Positive Edge JK Flip Flop Negative Edge JK Flip Flop J J S Q Q S Q Q CLK CLK R Q Q R Q Q K K Logic Symbol J Q CLK K Q CLK J X K X QN+ Q Q Function Logic Symbol J Q CLK K Q CLK J X K X QN+ Function Q CS4HST- DEPARTMENT OF COMPUTER SCIENCE, 38

39 Flip Flops with asynchronous inputs (Preset and Clear) Two extra inputs are often found on flip flops, that either clear or preset the output. These inputs are effective at any time, thus are called asynchronous. If the Clear is at logic then the output is forced to, irrespective of the other normal inputs. If the Preset is at logic then the output is forced to, irrespective of the other normal inputs. The preset and the clear inputs can not be simultaneously. In the Preset and Clear are both then the flip flop behaves according to its normal truth table. Positive Edge JK Flip Flop with Preset and Clear PRESET CLEAR PR J Q K Q CLR CLK PR CLR J K QN+ Function X X X X X X Q Q 39