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# Lecture 5

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### Lecture 5

1. 1. Computer & Network Technology Chamila Fernando 02/03/14 Information Representation BSc(Eng) Hons,MBA,MIEEE 1
2. 2. Lecture 5: Logic Gates and Circuits  Logic Gates        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 02/03/14 Logic Gates 2
3. 3. Lecture 4: Logic Gates and Circuits  Universal Gates: NAND and NOR  NAND Gate  NOR Gate       02/03/14 Implementation using NAND Gates Implementation using NOR Gates Implementation of SOP Expressions Implementation of POS Expressions Positive and Negative Logic Integrated Circuit Logic Families Logic Gates 3
4. 4. Logic Gates  Gate Symbols AND OR NOT a b a b a a NAND NOR EXCLUSIVE OR 02/03/14 Symbol set 2 Symbol set 1 b a b a b Logic Gates a.b a+b a' (a.b)' (a+b)' a⊕ b (ANSI/IEEE Standard 91-1984) a & a.b b a b a a b a b a b ≥1 a+b 1 a' & (a.b)' ≥1 (a+b)' =1 a⊕b 4
5. 5. Logic Gates: The Inverter  The Inverter A A' A A' A 0 1 A' 1 0  Application of the inverter: complement. 1 1 Binary number 0 1 0 0 0 0 1 0 1 1 0 1 1 0 1’s Complement 02/03/14 Logic Gates 5
6. 6. Logic Gates: The AND Gate  The AND Gate A A.B B A 0 0 1 1 02/03/14 B 0 1 0 1 A B & A.B A.B 0 0 0 1 Logic Gates 6
7. 7. Logic Gates: The AND Gate  Application of the AND Gate 1 sec A A Enable Counter Enable 1 sec Reset to zero between Enable pulses 02/03/14 Logic Gates Register, decode and frequency display 7
8. 8. Logic Gates: The OR Gate  The OR Gate A A+B B A 0 0 1 1 02/03/14 B 0 1 0 1 A B ≥ 1 A+B A+B 0 1 1 1 Logic Gates 8
9. 9. Logic Gates: The NAND Gate  The NAND Gate A (A.B)' B A 0 0 1 1 02/03/14 B 0 1 0 1 ≡ A (A.B)' B (A.B)' 1 1 1 0 A B & (A.B)' ≡ NAND Logic Gates Negative-OR 9
10. 10. Logic Gates: The NOR Gate  The NOR Gate A (A+B)' B A 0 0 1 1 02/03/14 B 0 1 0 1 ≡ A (A+B)' B (A+B)' 1 0 0 0 A B ≥ 1 (A+B)' ≡ NOR Logic Gates Negative-AND 10
11. 11. Logic Gates: The XOR Gate  The XOR Gate A A⊕B B A 0 0 1 1 02/03/14 B 0 1 0 1 A B =1 A⊕B A ⊕B 0 1 1 0 Logic Gates 11
12. 12. Logic Gates: The XNOR Gate  The XNOR Gate A (A ⊕ B)' B A 0 0 1 1 02/03/14 A B =1 (A ⊕ B)' B (A ⊕ B) ' 0 1 1 0 0 0 1 1 Logic Gates 12
13. 13. Drawing Logic Circuit  When a Boolean expression is provided, we can easily draw the logic circuit.  Examples: (i) F1 = xyz' (note the use of a 3-input AND gate) x y z 02/03/14 F1 z' Logic Gates 13
14. 14. Drawing Logic Circuit (ii) F2 = x + y'z (can assume that variables and their complements are available) x F2 y' z (iii) F3 = xy' + x'z y'z x y' xy' F3 x' z 02/03/14 Logic Gates x'z 14
15. 15. 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' A'B' B' A'B'+C C (A'B'+C)' F4 F4 = (A'B'+C)' = (A+B).C' 02/03/14 Logic Gates 16
16. 16. Propagation Delay  Every logic gate experiences some delay (though very small) in propagating signals forward.  This delay is called Gate (Propagation) Delay. 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 02/03/14 (average propagation delay) Logic Gates 17
17. 17. Propagation Delay Input Output H Input L Output H L tPHL 02/03/14 Logic Gates tPLH 18
18. 18. Propagation Delay A B C  In reality, output signals  Ideally, no normally lag behind input signals: delay: 1 0 1 Signal for A 0 1 0 1 0 1 Signal for B 0 1 Signal for C 0 time 02/03/14 Signal for A Signal for B Signal for C time Logic Gates 19
19. 19. Calculation of Circuit Delays  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. 02/03/14 Logic Gates 20
20. 20. Calculation of Circuit Delays  In general, given a logic gate with delay, t. t1 t2 : tn : Logic Gate 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. 02/03/14 Logic Gates 21
21. 21. Calculation of Circuit Delays  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 0 0 max(0,0)+t = t max(t,0)+t = 2t S t 2t max(t,2t)+t = 3t C Z 0 where outputs S and C, experience delays of 2t and 3t, respectively. 02/03/14 Logic Gates 22
22. 22. 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) 02/03/14 Logic Gates 24
23. 23. NAND Gate  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' 02/03/14 x' (T1: idempotency) Logic Gates 25
24. 24. NAND Gate  Implementing AND using NAND gates: x y (x.y)' x.y ((xy)'(xy)')' = ((xy)')' idempotency = (xy) involution  Implementing OR using NAND gates: x y 02/03/14 x' ((xx)'(yy)')' = (x'y')' idempotency = x''+y'' DeMorgan x+y = x+y involution y' Logic Gates 26
25. 25. NOR Gate     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' 02/03/14 x' (T1: idempotency) Logic Gates 27
26. 26. NOR Gate  Implementing AND using NOR gates: x' x y y' x.y ((x+x)'+(y+y)')'=(x'+y')' = x''.y'' = x.y idempotency DeMorgan involution  Implementing OR using NOR gates: x y 02/03/14 (x+y)' x+y ((x+y)'+(x+y)')' = ((x+y)')' = (x+y) Logic Gates idempotency involution 28
27. 27. Implementation using NAND gates  Possible to implement any Boolean expression using NAND gates. Procedure: (i) Obtain sum-of-products Boolean expression: e.g. F3 = xy'+x'z (ii) Use DeMorgan theorem to obtain expression using 2-level NAND gates e.g. F3 = xy'+x'z = (xy'+x'z)' ' involution = ((xy')' . (x'z)')' DeMorgan 02/03/14 Logic Gates 29
28. 28. Implementation using NAND gates x y' (xy')' x' z (x'z)' F3 F3 = ((xy')'.(x'z)') ' = xy' + x'z 02/03/14 Logic Gates 30
29. 29. Implementation using NOR gates  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. e.g. F6 = (x+y').(x'+z) = ((x+y').(x'+z))' ' involution = ((x+y')'+(x'+z)')' DeMorgan 02/03/14 Logic Gates 31
30. 30. Implementation using NOR gates x y' (x+y')' x' z (x'+z)' F6 F6 = ((x+y')'+(x'+z)')' = (x+y').(x'+z) 02/03/14 Logic Gates 32
31. 31. Implementation of SOP Expressions  Sum-of-Products expressions can be implemented using:  2-level AND-OR logic circuits  2-level NAND logic circuits  AND-OR logic circuit A B F = AB + CD + E C D F E 02/03/14 Logic Gates 33
32. 32. Implementation of SOP Expressions  NAND-NAND circuit (by circuit transformation) a) add double bubbles b) change OR-withinverted-inputs to NAND & bubbles at inputs to their complements A B C D F E A B C D F E' 02/03/14 Logic Gates 34
33. 33. Implementation of POS Expressions  Product-of-Sums expressions can be implemented using:  2-level OR-AND logic circuits  2-level NOR logic circuits  OR-AND logic circuit A B G = (A+B).(C+D).E C D G E 02/03/14 Logic Gates 35
34. 34. Implementation of POS Expressions  NOR-NOR circuit (by circuit transformation): A B a) add double bubbles b) changed AND-withinverted-inputs to NOR & bubbles at inputs to their complements C D G E A B C D G E' 02/03/14 Logic Gates 36
35. 35. Positive & Negative Logic  In logic gates, usually:  H (high voltage, 5V) = 1  L (low voltage, 0V) = 0  This convention – positive logic. 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. 02/03/14 Logic Gates 38
36. 36. Positive & Negative Logic  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. 02/03/14 Logic Gates 39
37. 37. Positive & Negative Logic Positive logic: Enable Active High: 0: Disabled 1: Enabled Negative logic: Enable 02/03/14 Logic Gates Active Low: 0: Enabled 1: Disabled 40
38. 38. Integrated Circuit Logic Families  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. 02/03/14 Logic Gates 41
39. 39. Integrated Circuit Logic Families TTL Series Standard TTL 54 or 74 7400 (quad NAND gates) Low-power TTL 54L or 74L 74L00 (quad NAND gates) Schottky TTL 54S or 74S 74S00 (quad NAND gates) Low-power Schottky TTL 02/03/14 Prefix Designation Example of Device 54LS or 74LS 74LS00 (quad NAND gates) Logic Gates 42
40. 40. Integrated Circuit Logic Families  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. 02/03/14 Logic Gates 43
41. 41. Integrated Circuit Logic Families  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. 02/03/14 Logic Gates 44
42. 42. Summary Logic Gates AND, OR, NOT NAND NOR Implementation of a Boolean expression using these Universal gates. 02/03/14 Drawing Logic Circuit Analysing Logic Circuit Given a Boolean expression, draw the circuit. Given a circuit, find the function. Implementation of SOP and POS Expressions Positive and Negative Logic Concept of Minterm and Maxterm Logic Gates 45
43. 43. End of file 02/03/14 Logic Gates 46