Digital Logic: Understanding Switches, Functions, and Gates, Slides of Computer Aided Design (CAD)

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Elementary Digital Logic

Apps

O/S

Arch

mArch

Logic

Digital

Analog

Devices

Physics

Variables and Transistors (Switches)

• Digital systems use binary logic
  • Two values: 0 and 1 (or false and true if you like)
• Transistors are essentially a switch controlled by a

single binary variable x

• A generic symbol for a switch is

x = 0 x = 1

S

x

Switches and Functions (2)

• Can do more complex logic, too: L(x1,x2,x3) = (x1 + x2) x
• Don’t get the idea that this is the actual physical way it is done
  • This is just an abstraction

S

x Power supply S

x

S

x3 L

Seven Common Logic Primitives

• AND
• OR
• NOT (complement, inversion)
• NAND
• NOR
• XOR
• XNOR

Universal Logic Families

Universal Logic Families:

• Any logic function can be designed using one of these sets of gates:
  • {AND, OR, NOT}
  • {NAND}
  • {NOR}
• These are called “universal logic families”
• Actual synthesized circuits are usually designed using either NAND or NOR gates only - Remember Regularity
• We’ll use {AND, OR, NOT} since it is easier to do “on paper”

Digital Design Process

• How do you build a digital logic circuit?
  • Represent all inputs and outputs in binary
    • Create a model / representation for the system
  • Formalize the design in a truth table or in an algebraic expression - Defines the function of the system
  • Simplify the description  simplify the algebraic expression - Ex: X ( X + Y ) = X
  • Implement the system with available components (gates)

Apps

O/S

Arch

mArch

Logic

Digital

Analog

Devices

Physics

Converting Boolean Algebra into

Gates

• C = SA + SB

• What does this device do?
  • When S = 0  C = A
  • When S = 1  C = B
• A steering device: S steers/switches A or B onto the output C

A

B

C

S

2x1 MUX (Multiplexor)

• C = SA + SB
• This is a 2x1 MUX
• “Black box” version is
• 2 input by 1 bit of data multiplexor (steering device)
  • 2 inputs requires a 1 bit selector S

A

B

C

S

S

B

A C

0 1 S

B

A C

0 1 S

B

A C

0 1

Combinational Logic

• Changes in input values are reflected immediately (subject to the speed of light and electrical delays) on the outputs
• Each gate has an associated “electrical delay”
• Delays are often ignored for the purpose of the logic design (but not for the real implementation!)
• As soon as inputs change, the outputs change – no memory of what happened before - (at least conceptually)