🗊Презентация Combinational logic design

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Combinational logic design, слайд №1
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Introduction Introduction Boolean Equations Boolean Algebra From Logic to Gates Multilevel Combinational Logic X’s and Z’s, Oh My Karnaugh Maps Combinational Building Blocks Timing
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Introduction Introduction Boolean Equations Boolean Algebra From Logic to Gates Multilevel Combinational Logic X’s and Z’s, Oh My Karnaugh Maps Combinational Building Blocks Timing

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A logic circuit is composed of: A logic circuit is composed of: Inputs Outputs Functional specification Timing specification
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A logic circuit is composed of: A logic circuit is composed of: Inputs Outputs Functional specification Timing specification

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Nodes Nodes Inputs: A, B, C Outputs: Y, Z Internal: n1 Circuit elements E1, E2, E3 Each a circuit
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Nodes Nodes Inputs: A, B, C Outputs: Y, Z Internal: n1 Circuit elements E1, E2, E3 Each a circuit

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Combinational Logic Combinational Logic Memoryless Outputs determined by current values of inputs Sequential Logic Has memory Outputs determined by previous and current values of inputs
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Combinational Logic Combinational Logic Memoryless Outputs determined by current values of inputs Sequential Logic Has memory Outputs determined by previous and current values of inputs

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Every element is combinational Every element is combinational Every node is either an input or connects to exactly one output The circuit contains no cyclic paths Example:
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Every element is combinational Every element is combinational Every node is either an input or connects to exactly one output The circuit contains no cyclic paths Example:

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Functional specification of outputs in terms of inputs Functional specification of outputs in terms of inputs Example: S = F(A, B, Cin) Cout = F(A, B, Cin)
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Functional specification of outputs in terms of inputs Functional specification of outputs in terms of inputs Example: S = F(A, B, Cin) Cout = F(A, B, Cin)

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Complement: variable with a bar over it Complement: variable with a bar over it A, B, C Literal: variable or its complement A, A, B, B, C, C Implicant: product of literals ABC, AC, BC Minterm: product that includes all input variables ABC, ABC, ABC Maxterm: sum that includes all input variables (A+B+C), (A+B+C), (A+B+C)
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Complement: variable with a bar over it Complement: variable with a bar over it A, B, C Literal: variable or its complement A, A, B, B, C, C Implicant: product of literals ABC, AC, BC Minterm: product that includes all input variables ABC, ABC, ABC Maxterm: sum that includes all input variables (A+B+C), (A+B+C), (A+B+C)

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Sum-of-Products Form
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Sum-of-Products Form

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Sum-of-Products Form
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Sum-of-Products Form

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You are going to the cafeteria for lunch You are going to the cafeteria for lunch You won’t eat lunch (E) If it’s not open (O) or If they only serve corndogs (C) Write a truth table for determining if you will eat lunch (E).
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You are going to the cafeteria for lunch You are going to the cafeteria for lunch You won’t eat lunch (E) If it’s not open (O) or If they only serve corndogs (C) Write a truth table for determining if you will eat lunch (E).

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You are going to the cafeteria for lunch You are going to the cafeteria for lunch You won’t eat lunch (E) If it’s not open (O) or If they only serve corndogs (C) Write a truth table for determining if you will eat lunch (E).
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You are going to the cafeteria for lunch You are going to the cafeteria for lunch You won’t eat lunch (E) If it’s not open (O) or If they only serve corndogs (C) Write a truth table for determining if you will eat lunch (E).

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SOP & POS Form SOP – sum-of-products POS – product-of-sums
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SOP & POS Form SOP – sum-of-products POS – product-of-sums

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SOP – sum-of-products SOP – sum-of-products POS – product-of-sums
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SOP – sum-of-products SOP – sum-of-products POS – product-of-sums

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Axioms and theorems to simplify Boolean equations Axioms and theorems to simplify Boolean equations Like regular algebra, but simpler: variables have only two values (1 or 0) Duality in axioms and theorems: ANDs and ORs, 0’s and 1’s interchanged
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Axioms and theorems to simplify Boolean equations Axioms and theorems to simplify Boolean equations Like regular algebra, but simpler: variables have only two values (1 or 0) Duality in axioms and theorems: ANDs and ORs, 0’s and 1’s interchanged

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Y = AB + AB Y = AB + AB
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Y = AB + AB Y = AB + AB

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Y = AB + AB Y = AB + AB = B(A + A) T8 = B(1) T5’ = B T1
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Y = AB + AB Y = AB + AB = B(A + A) T8 = B(1) T5’ = B T1

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Y = A(AB + ABC) Y = A(AB + ABC)
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Y = A(AB + ABC) Y = A(AB + ABC)

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Y = A(AB + ABC) Y = A(AB + ABC) = A(AB(1 + C)) T8 = A(AB(1)) T2’ = A(AB) T1 = (AA)B T7 = AB T3
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Y = A(AB + ABC) Y = A(AB + ABC) = A(AB(1 + C)) T8 = A(AB(1)) T2’ = A(AB) T1 = (AA)B T7 = AB T3

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Y = AB = A + B Y = AB = A + B Y = A + B = A B
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Y = AB = A + B Y = AB = A + B Y = A + B = A B

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Backward: Backward: Body changes Adds bubbles to inputs Forward: Body changes Adds bubble to output
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Backward: Backward: Body changes Adds bubbles to inputs Forward: Body changes Adds bubble to output

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Two-level logic: ANDs followed by ORs Two-level logic: ANDs followed by ORs Example: Y = ABC + ABC + ABC
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Two-level logic: ANDs followed by ORs Two-level logic: ANDs followed by ORs Example: Y = ABC + ABC + ABC

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Inputs on the left (or top) Inputs on the left (or top) Outputs on right (or bottom) Gates flow from left to right Straight wires are best
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Inputs on the left (or top) Inputs on the left (or top) Outputs on right (or bottom) Gates flow from left to right Straight wires are best

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Wires always connect at a T junction Wires always connect at a T junction A dot where wires cross indicates a connection between the wires Wires crossing without a dot make no connection
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Wires always connect at a T junction Wires always connect at a T junction A dot where wires cross indicates a connection between the wires Wires crossing without a dot make no connection

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Example: Priority Circuit Example: Priority Circuit Output asserted corresponding to most significant TRUE input
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Example: Priority Circuit Example: Priority Circuit Output asserted corresponding to most significant TRUE input

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Example: Priority Circuit Example: Priority Circuit Output asserted corresponding to most significant TRUE input
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Example: Priority Circuit Example: Priority Circuit Output asserted corresponding to most significant TRUE input

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Contention: circuit tries to drive output to 1 and 0 Contention: circuit tries to drive output to 1 and 0 Actual value somewhere in between Could be 0, 1, or in forbidden zone Might change with voltage, temperature, time, noise Often causes excessive power dissipation Warnings: Contention usually indicates a bug. X is used for “don’t care” and contention - look at the context to tell them apart
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Contention: circuit tries to drive output to 1 and 0 Contention: circuit tries to drive output to 1 and 0 Actual value somewhere in between Could be 0, 1, or in forbidden zone Might change with voltage, temperature, time, noise Often causes excessive power dissipation Warnings: Contention usually indicates a bug. X is used for “don’t care” and contention - look at the context to tell them apart

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Floating, high impedance, open, high Z Floating, high impedance, open, high Z Floating output might be 0, 1, or somewhere in between A voltmeter won’t indicate whether a node is floating Tristate Buffer
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Floating, high impedance, open, high Z Floating, high impedance, open, high Z Floating output might be 0, 1, or somewhere in between A voltmeter won’t indicate whether a node is floating Tristate Buffer

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Floating nodes are used in tristate busses Floating nodes are used in tristate busses Many different drivers Exactly one is active at once
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Floating nodes are used in tristate busses Floating nodes are used in tristate busses Many different drivers Exactly one is active at once

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Boolean expressions can be minimized by combining terms Boolean expressions can be minimized by combining terms K-maps minimize equations graphically PA + PA = P
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Boolean expressions can be minimized by combining terms Boolean expressions can be minimized by combining terms K-maps minimize equations graphically PA + PA = P

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Complement: variable with a bar over it Complement: variable with a bar over it A, B, C Literal: variable or its complement A, A, B, B, C, C Implicant: product of literals ABC, AC, BC Prime implicant: implicant corresponding to the largest circle in a K-map
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Complement: variable with a bar over it Complement: variable with a bar over it A, B, C Literal: variable or its complement A, A, B, B, C, C Implicant: product of literals ABC, AC, BC Prime implicant: implicant corresponding to the largest circle in a K-map

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Every 1 must be circled at least once Every 1 must be circled at least once Each circle must span a power of 2 (i.e. 1, 2, 4) squares in each direction Each circle must be as large as possible A circle may wrap around the edges A “don't care” (X) is circled only if it helps minimize the equation
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Every 1 must be circled at least once Every 1 must be circled at least once Each circle must span a power of 2 (i.e. 1, 2, 4) squares in each direction Each circle must be as large as possible A circle may wrap around the edges A “don't care” (X) is circled only if it helps minimize the equation

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Multiplexers Multiplexers Decoders
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Multiplexers Multiplexers Decoders

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Selects between one of N inputs to connect to output Selects between one of N inputs to connect to output log2N-bit select input – control input Example: 2:1 Mux
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Selects between one of N inputs to connect to output Selects between one of N inputs to connect to output log2N-bit select input – control input Example: 2:1 Mux

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Logic gates Logic gates Sum-of-products form
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Logic gates Logic gates Sum-of-products form

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