ELEC1303 - Lecture 3

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ELEC1303 - Lecture 2
Describing Logic Circuits - Slides
Read: Combinational vs Sequential Logic

Digital Circuits/Logic Circuits

• Digital circuits are also known as logic circuits
  • This is because they obey a set of logic rules
• The manner in which a digital circuit responds to an input is referred to as the circuit’s logic

Combinational Device

• A combinational device is a circuit element that has:
  • One or more digital inputs $0\le {\text{V}}_{\text{IL}}$, $1\ge {\text{V}}_{\text{IH}}$
  • One or more digital outputs $0\le {\text{V}}_{\text{OL}}$, $1\ge {\text{V}}_{\text{OH}}$
  • Functional specification
    i.e. specific output for every possible input combination
  • Timing specification (at minimum, propagation delay (${\text{t}}_{\text{PD}}$))
    ${\text{t}}_{\text{PD}}$ is the time for a signal to travel from the input of a system to its output

Functional Specifications

• Besides language, there are other ways to specify and represent the function of a combinational device
• These are three examples:
  1. Truth table
  2. Boolean expression
  3. Schematic
• You can convert between these three representations
  
• Truth table is the only unique representation
  • i.e. there are many ways to make an expression or schematic for a given circuit that all mean the same thing
• See slides for more information

Boolean Expressions and Logic Gates

x equals A and B → $X=A\cdot B$

• The dot is implicit and may be left out

x equals A or B → $X=A+B$

x equals NOT A → $X=\bar{A}$ or $X=A^{\prime }$

• NOT circuit commonly known as inverter

Duality Principle

• The dual of a Boolean expression can be obtained by:
  • Interchanging AND ($\cdot$) and OR ($+$) operators
  • Interchanging constants 0’s and 1’s
  • The complement operator does not change
• The properties of Boolean algebra appear in dual pairs
  • If a property is proven true then its dual is also true
  • The “dual” is NOT equal to the original expression
• This helps us derive new theorems quickly from existing ones
  • We do not require additional proofs

Example: the dual of $x(y+z^{\prime })$ is $x+y{z}^{\prime }$

De Morgan’s Theorem

• The complement of the logical AND of two or more variables is equal to the logical OR of the complements of those variables
• The complement of the logical OR of two or more variables is equal to the logical AND of the complements of those variables

(x + y)' = x'y'$$$$(x\ y)' = x' + y'

• Complement $f=(x^{\prime }y{z}^{\prime })+(x{y}^{\prime }{z}^{\prime })$?
  Answer: $f^{\prime }=(x+y^{\prime }+z)\cdot (x^{\prime }+y+z)$

Boolean algebra allows us to simplify a function so that it contains the smallest number of literals

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ELEC1303 - Lecture 4