CS1103 Digital Logic Design CS1104: Computer Organisation http://www.comp.nus.edu.sg/~cs1104 Lecture 4: Logic Gates and Circuits

Lecture 4: Logic Gates and Circuits The Inverter The AND Gate The OR Gate The NAND Gate The NOR Gate The XOR Gate The XNOR Gate Drawing Logic Circuit Analysing Logic Circuit Propagation Delay CS1104-4 Lecture 4: Introduction to Logic Gates

Lecture 4: Logic Gates and Circuits Universal Gates: NAND and NOR NAND Gate NOR Gate Implementation using NAND Gates Implementation using NOR Gates Implementation of SOP Expressions Implementation of POS Expressions Positive and Negative Logic Integrated Circuit Logic Families CS1104-4 Lecture 4: Introduction to Logic Gates

(ANSI/IEEE Standard 91-1984) Logic Gates Gate Symbols EXCLUSIVE OR a b a.b a+b a' (a+b)' (a.b)' a  b & 1 AND 1 =1 OR NOT NAND NOR Symbol set 1 Symbol set 2 (ANSI/IEEE Standard 91-1984) CS1104-4 Logic Gates

Logic Gates: The Inverter Application of the inverter: complement. 1 Binary number 1’s Complement CS1104-4 Logic Gates: The Inverter

Logic Gates: The AND Gate (1/2) B A.B & CS1104-4 Logic Gates: The AND Gate

Logic Gates: The AND Gate (2/2) Application of the AND Gate 1 sec A Enable Counter Reset to zero between Enable pulses Register, decode and frequency display CS1104-4 Logic Gates: The AND Gate

Logic Gates: The OR Gate 1 A B A+B CS1104-4 Logic Gates: The OR Gate

Logic Gates: The NAND Gate & A B (A.B)'  NAND Negative-OR  CS1104-4 Logic Gates: The NAND Gate

Logic Gates: The NOR Gate 1 A B (A+B)'  NOR Negative-AND  CS1104-4 Logic Gates: The NOR Gate

Logic Gates: The XOR Gate =1 A B A  B CS1104-4 Logic Gates: The XOR Gate

Logic Gates: The XNOR Gate B (A  B)' A B (A  B)' =1 CS1104-4 Logic Gates: The XNOR Gate

Drawing Logic Circuit (1/2) When a Boolean expression is provided, we can easily draw the logic circuit. Examples: (i) F1 = x.y.z' (note the use of a 3-input AND gate) x y z F1 z' CS1104-4 Drawing Logic Circuit

Drawing Logic Circuit (2/2) (ii) F2 = x + y'.z (if we assume that variables and their complements are available) x y' z F2 y'.z (iii) F3 = x.y' + x'.z x' z F3 x'.z x.y' x y' CS1104-4 Drawing Logic Circuit

Quick Review Questions (1) Textbook page 77. Questions 4-1, 4-2. CS1104-4 Quick Review Questions (1)

Analysing Logic Circuit When a logic circuit is provided, we can analyse the circuit to obtain the logic expression. Example: What is the Boolean expression of F4? A'.B' A' B' C F4 A'.B'+C (A'.B'+C)' F4 = (A'.B'+C)' = (A+B).C' CS1104-4 Analysing Logic Circuit

Propagation Delay (1/3) Every logic gate experiences some delay (though very small) in propagating signals forward. This delay is called Gate (Propagation) Delay. Formally, it is the average transition time taken for the output signal of the gate to change in response to changes in the input signals. Three different propagation delay times associated with a logic gate: tPHL: output changing from the High level to Low level tPLH: output changing from the Low level to High level tPD=(tPLH + tPHL)/2 (average propagation delay) CS1104-4 Propagation Delay

Propagation Delay (2/3) Input Output Output Input H L tPHL tPLH CS1104-4 Propagation Delay

Propagation Delay (3/3) A B C In reality, output signals normally lag behind input signals: 1 time Signal for C Signal for B Signal for A Ideally, no delay: 1 time Signal for C Signal for B Signal for A CS1104-4 Propagation Delay

Calculation of Circuit Delays (1/3) Amount of propagation delay per gate depends on: (i) gate type (AND, OR, NOT, etc) (ii) transistor technology used (TTL,ECL,CMOS etc), (iii) miniaturisation (SSI, MSI, LSI, VLSI) To simplify matters, one can assume (i) an average delay time per gate, or (ii) an average delay time per gate-type. Propagation delay of logic circuit = longest time it takes for the input signal(s) to propagate to the output(s). = earliest time for output signal(s) to stabilise, given that input signals are stable at time 0. CS1104-4 Calculation of Circuit Delays

Calculation of Circuit Delays (2/3) In general, given a logic gate with delay, t. Logic Gate t1 t2 tn : max (t1, t2, ..., tn ) + t If inputs are stable at times t1,t2,..,tn, respectively; then the earliest time in which the output will be stable is: max(t1, t2, .., tn) + t To calculate the delays of all outputs of a combinational circuit, repeat above rule for all gates. CS1104-4 Calculation of Circuit Delays

Calculation of Circuit Delays (3/3) As a simple example, consider the full adder circuit where all inputs are available at time 0. (Assume each gate has delay t.) X Y S C Z max(0,0)+t = t t max(t,0)+t = 2t max(t,2t)+t = 3t 2t where outputs S and C, experience delays of 2t and 3t, respectively. CS1104-4 Calculation of Circuit Delays

Quick Review Questions (2) Textbook page 77. Questions 4-3 to 4-5. CS1104-4 Quick Review Questions (2)

Universal Gates: NAND and NOR AND/OR/NOT gates are sufficient for building any Boolean functions. We call the set {AND, OR, NOT} a complete set of logic. However, other gates are also used because: (i) usefulness (ii) economical on transistors (iii) self-sufficient NAND/NOR: economical, self-sufficient XOR: useful (e.g. parity bit generation) CS1104-4 Universal Gates: NAND and NOR

(x.x)' = x' (T1: idempotency) NAND Gate (1/2) NAND gate is self-sufficient (can build any logic circuit with it). Therefore, {NAND} is also a complete set of logic. Can be used to implement AND/OR/NOT. Implementing an inverter using NAND gate: x x' (x.x)' = x' (T1: idempotency) CS1104-4 NAND Gate

NAND Gate (2/2) Implementing AND using NAND gates: x x.y y (x.y)' ((x.y)'(x.y)')' = ((x.y)')' idempotency = (xy) involution Implementing OR using NAND gates: x x+y y x' y' ((x.x)'(y.y)')' = (x'.y')' idempotency = x''+y'' DeMorgan = x+y involution CS1104-4 NAND Gate

(x+x)' = x' (T1: idempotency) NOR Gate (1/2) NOR gate is also self-sufficient. Therefore, {NOR} is also a complete set of logic Can be used to implement AND/OR/NOT. Implementing an inverter using NOR gate: x x' (x+x)' = x' (T1: idempotency) CS1104-4 NOR Gate

NOR Gate (2/2) Implementing AND using NOR gates: x x.y y x' y' ((x+x)'+(y+y)')'=(x'+y')' idempotency = x''.y'' DeMorgan = x.y involution Implementing OR using NOR gates: x x+y y (x+y)' ((x+y)'+(x+y)')' = ((x+y)')' idempotency = (x+y) involution CS1104-4 NOR Gate

Implementation using NAND gates (1/2) Possible to implement any Boolean expression using NAND gates. Procedure: (i) Obtain sum-of-products Boolean expression: e.g. F3 = x.y'+x'.z (ii) Use DeMorgan theorem to obtain expression using 2-level NAND gates = (x.y'+x'.z)' ' involution = ((x.y')' . (x'.z)')' DeMorgan CS1104-4 Implementation using NAND gates

Implementation using NAND gates (2/2) x' z F3 (x'.z)' (x.y')' x y' F3 = ((x.y')'.(x'.z)') ' = x.y' + x'.z CS1104-4 Implementation using NAND gates

Implementation using NOR gates (1/2) Possible to implement any Boolean expression using NOR gates. Procedure: (i) Obtain product-of-sums Boolean expression: e.g. F6 = (x+y').(x'+z) (ii) Use DeMorgan theorem to obtain expression using 2-level NOR gates. = ((x+y').(x'+z))' ' involution = ((x+y')'+(x'+z)')' DeMorgan CS1104-4 Implementation using NOR gates

Implementation using NOR gates (2/2) x' z F6 (x'+z)' (x+y')' x y' F6 = ((x+y')'+(x'+z)')' = (x+y').(x'+z) CS1104-4 Implementation using NOR gates

Implementation of SOP Expressions (1/2) Sum-of-Products expressions can be implemented using: 2-level AND-OR logic circuits 2-level NAND logic circuits AND-OR logic circuit F A B D C E F = A.B + C.D + E CS1104-4 Implementation of SOP Expressions

Implementation of SOP Expressions (2/2) NAND-NAND circuit (by circuit transformation) a) add double bubbles b) change OR-with- inverted-inputs to NAND & bubbles at inputs to their complements F A B D C E E' CS1104-4 Implementation of SOP Expressions

Implementation of POS Expressions (1/2) Product-of-Sums expressions can be implemented using: 2-level OR-AND logic circuits 2-level NOR logic circuits OR-AND logic circuit G A B D C E G = (A+B).(C+D).E CS1104-4 Implementation of POS Expressions

Implementation of POS Expressions (2/2) NOR-NOR circuit (by circuit transformation): a) add double bubbles b) changed AND-with- inverted-inputs to NOR & bubbles at inputs to their complements G A B D C E E' CS1104-4 Implementation of POS Expressions

Quick Review Questions (3) Textbook page 77. Questions 4-6 to 4-8. CS1104-4 Quick Review Questions (3)

Positive & Negative Logic (1/3) In logic gates, usually: H (high voltage, 5V) = 1 L (low voltage, 0V) = 0 This convention is known as positive logic. However, the reverse convention, negative logic possible: H (high voltage) = 0 L (low voltage) = 1 Depending on convention, same gate may denote different Boolean function. CS1104-4 Positive & Negative Logic

Positive & Negative Logic (2/3) A signal that is set to logic 1 is said to be asserted, or active, or true. A signal that is set to logic 0 is said to be deasserted, or negated, or false. Active-high signal names are usually written in uncomplemented form. Active-low signal names are usually written in complemented form. CS1104-4 Positive & Negative Logic

Positive & Negative Logic (3/3) Positive logic: Enable Active High: 0: Disabled 1: Enabled Negative logic: Enable Active Low: 0: Enabled 1: Disabled CS1104-4 Positive & Negative Logic

Integrated Circuit Logic Families (1/2) Some digital integrated circuit families: TTL, CMOS, ECL. TTL: Transistor-Transistor Logic. Uses bipolar junction transistors Consists of a series of logic circuits: standard TTL, low-power TTL, Schottky TTL, low-power Schottky TTL, advanced Schottky TTL, etc. CS1104-4 Integrated Circuit Logic Families

Integrated Circuit Logic Families (2/2) CMOS: Complementary Metal-Oxide Semiconductor. Uses field-effect transistors ECL: Emitter Coupled Logic. Uses bipolar circuit technology. Has fastest switching speed but high power consumption. Performance characteristics Propagation delay time. Power dissipation. Fan-out: Fan-out of a gate is the maximum number of inputs that the gate can drive. Speed-power product (SPP): product of the propagation delay time and the power dissipation. CS1104-4 Integrated Circuit Logic Families

Summary Logic Gates Drawing Logic Circuit Analysing Logic Circuit AND, OR, NOT NAND NOR Drawing Logic Circuit Analysing Logic Circuit Given a Boolean expression, draw the circuit. Given a circuit, find the function. Implementation of a Boolean expression using these Universal gates. Implementation of SOP and POS Expressions Positive and Negative Logic Concept of Minterm and Maxterm CS1104-4 Summary

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