Digital Logic Circuits Winter 2012 COMP 2130 Intro Computer Systems Computing Science Thompson Rivers University

Digitial Logic Circuits Course Contents Part I – Review Mumber Systems Boolean Logic Digital Logic Circuits Part II – C Programming Language Part III – Introduction to Computer Systems Not the replacement of IPv6; Inter-transition mechanism; But if IPv6 would fail TRU-COMP2130 Digitial Logic Circuits

Unit Learning Objectives Find a Boolean expression corresponding to circuit. Find a circuit corresponding to a Boolean expression. Find a Boolean expression corresponding to an i/o table. Create an i/o table corresponding to given requirements. Design AND, OR, and NOT gates using NOR gates. Understand how multiplexers and decoders work. TRU-COMP2130 Digitial Logic Circuits

Digitial Logic Circuits Unit Contents Gates Input/Output Table for a Circuit Boolean Expression Corresponding to a Circuit Circuit Corresponding to a Boolean Expression Finding a Circuit Corresponding to a Given Input/Output Table Simplifying Combinational Circuits NAND and NOR Gates Combinatorial Circuits Sequential Circuits TRU-COMP2130 Digitial Logic Circuits

Digitial Logic Circuits Analogy between the operations of switching devices and the operations of logical connectives P Q Switches in series P Q P  Q 1 ? P Q Switches in parallel P Q P  Q 1 ? TRU-COMP2130 Digitial Logic Circuits

Digitial Logic Circuits Gates AND (conjunction) OR (disjuction) NOT (negation) TRU-COMP2130 Digitial Logic Circuits

Input/Output Table for a Circuit P = 1 an Q = 0, then R = ??? I/O table ??? P = 0, Q = 1, and R = 1. Then S = ??? I/O table ??? TRU-COMP2130 Digitial Logic Circuits

Digitial Logic Circuits I/O tables for circuits??? TRU-COMP2130 Digitial Logic Circuits

Boolean Expression for a Circuit TRU-COMP2130 Digitial Logic Circuits

Circuit Corresponding to a Boolean Expression (~P  Q)  ~Q ((P  Q)  (R  S))  T TRU-COMP2130 Digitial Logic Circuits

Finding a Circuit Corresponding to a Given I/O Table How to find a circuit that corresponds to the above I/O table? Input Output P Q R S 1 TRU-COMP2130 Digitial Logic Circuits

Digitial Logic Circuits Input Output P Q R S 1 For the 1st, the 2nd and the 3rd rows, P  Q  R P  ~Q  R P  ~Q  ~R Then ??? TRU-COMP2130 Digitial Logic Circuits

Digitial Logic Circuits For the 1st, the 2nd and the 3rd rows, P  Q  R P  ~Q  R P  ~Q  ~R All the other cases will produce 0. Therefore, S = (P  Q  R)  (P  ~Q  R)  (P  ~Q  ~R) This is a disjunction of conjunctions (called disjunctive normal form). This boolean expression might be simplified by using equivalences. How to find a circuit that corresponds to the above boolean expression? = (P  R  Q)  (P  R  ~Q)  (P  ~Q  ~R) = ((P  R)  (Q  ~Q ))  (P  ~Q  ~R) = ((P  R)  T)  (P  ~Q  ~R) = (P  R)  (P  ~Q  ~R) = P  (R  (~Q  ~R)) = P  (R  ~Q) TRU-COMP1380 Digitial Logic Circuits

Digitial Logic Circuits 1 bit full adder 01 + 11 = ??? S = (P  Q  Cin)  (P  ~Q  ~Cin)  (~P  Q  ~ Cin)  (~P  ~Q  Cin) Cout = ... Corresponding circuit ??? Input Output P Q Cin S Cout 1 TRU-COMP2130 Digitial Logic Circuits

Digitial Logic Circuits 2-to-1 multiplexer When C = 0; A is selected, and when C = 1, B is selected. Corresponding i/o table? S = ??? A B C S Input Output A B C S 1 TRU-COMP1380 Digitial Logic Circuits

Simplifying Combinational Circuits Any good idea? Convert to a boolean expression, and then Simplify the boolean expression by using equivalences R = ((P  ~Q)  (P  Q))  Q = ... by distributive law = ... TRU-COMP2130 Digitial Logic Circuits

Digitial Logic Circuits NAND and NOR Gates AND and NAND gates NAND = NOT (AND) OR and NOR gates NOR = NOT (OR) TRU-COMP2130 Digitial Logic Circuits

NOT AND OR XOR NAND NOR TRU-COMP2130 Digital Logic

Can you make NAND, NOR and XOR, by using AND, OR and NOT? A  B = (~A  B) + (A  ~B) Can you make AND by using OR and NOT? (Using DeMorgan’s Law) A  B = ~(~A + ~B) Can you make OR by using AND and NOT? A + B = ~(~A  ~B) TRU-COMP2130 Digital Logic

Can you make AND, OR and NOT, by using NOR? ~A = A NOR A A + B = (A NOR B) NOR (A NOR B) A  B = (A NOR A) NOR (B NOR B) Can you make AND, OR and NOT, by using NAND? Digital circuits can be, and frequently are, implemented solely with NOR gates or solely with NAND gates. TRU-COMP2130 Digital Logic

Combinational Circuits A combinational circuit is an interconnected set of gates whose output at any time is a function only of the input at that time. E.g., a boolean function of three variables? F(A,B,C) = ~A  B  ~C + ~A  B  C + A  B  ~C = (A+B+C)(A+B+~C)(~A+B+C)(~A+B+~C)(~A+~B+~C) The second form is easier to implement. TRU-COMP2130 Digital Logic

Multiplexer (E.g., to control signal and data routing) Two + four inputs, one output 0 0 => value from input D0 is selected from D0, D1, D2, D3 0 1 => value from input D1 is selected 1 0 => value from input D2 is selected 1 1 => value from input D3 is selected Decoder (E.g., address decoding) Two inputs, four outputs 0 0 => output D0 selected 0 1 => output D1 selected 1 0 => output D2 selected 1 1 => output D3 selected TRU-COMP2130 Digital Logic

TRU-COMP2130 Digital Logic

Adder A B Cin S Cout 1 ... 1 bit full adder 4 bit adder TRU-COMP2130 1 ... Adder 1 bit full adder 4 bit adder TRU-COMP2130 Digital Logic

Sequential Circuits Combinational circuits implement the essential functions of a digital computer. However, they provide no memory or state information. The current output of a sequential circuit depends not only on the current input but also on the past history of inputs TRU-COMP2130 Digital Logic

Flip-flop Register Counter ... The simplest form of sequential circuit The S-R latch: 1 bit memory 1 0 -> Q = 1 0 1 -> Q = 0 0 0 -> no change Register Counter ... TRU-COMP2130 Digital Logic

Programmable Logic Devices TRU-COMP2130 Digital Logic