[CSCI150] Lecture 2: Combinational Logic Circuits

Starting from Lecture 2, we will have some separate videos focusing on Tutorials. I will post videos on how to use the software, but sometimes if you do encounter more difficult questions, please contact me for an office hour, or just attend the regular one on Wednesday.

LS4 Part 1 (Lecture): Gates, Operators, Timing Diagram, Truth Table, Gating Delay

Highlight:

  • Gates
    • AND, OR, NOT gates
    • Physical components
  • Operators
    • Differences with physical gates: conceptually speaking, one is mathematical, one is physical simulation
  • Timing Diagram
  • Truth Table
  • Gating Delay
    • Important to know what it is, that it exists

LS4 Part 2 (Tutorial): LogicWorks, Basic circuit drawing and Simulation

Highlight:

  • Basic Interface
    • Simulation I/O, binary switches for inputs, probs for outputs
    • Simulation Gates
  • Timing Diagram control
    • Clock units

LS5 Part 1 (Lecture): Boolean Algebra, Binary Identities

This is one of the slightly longer lecture recordings, it covers some very important things as well as providing some important proofs that you should attempt on your own first.

Highlights:

  • Binary Identities: HS401 handout
    • 17 basic rules, prove using truth table
    • 6 extended rules, prove using the 17 basic rules.
  • Complement using DeMorgan’s Law
    • One should practice the application of DeMorgan’s Law very carefully, as it is quite common to make mistakes
  • Duality
    • The dual of a boolean expression: AND replaced with OR, OR replaced with AND, 0 with 1, 1 with 0
    • Duality principle.
  • Algebraic Manipulation
    • Trial and Error: try all possible binary identities, see if they make the expressions simpler
    • Utilise known conditions wisely, try to compute the values for as much expressions as possible
    • Practice, practice, practice

LS5 Part 2 (Exercise): Boolean Algebra

Here’s what you need before watching this video: bring out a pen/pencil and a few pieces of paper, and do the exercises. Each question is already included in LS5, and will be displayed for 10seconds, during which you should pause the video, attempt to solve it, and only when you have tried sufficient amount of times or have solved it successfully should you proceed to watch my solutions.

The answers will NOT be available in the PDF version of LS5.

LS6 Part 1 (Lecture): Minterms/Maxterms

Highlights:

  • Standard Forms
    • A particular way of writing boolean expressions, so they can have special properties
    • Minterms
      • Product Term, includes all input variables (or inverted ones)
      • Each minterm for any given input variables, can have an output value of 1 with only 1 input combination
      • Represented using lower-case m with subscript
      • Correspond to positive output rows on the truth table
      • Sum-of-Minterm can be directly taken from any truth table, so you get easy boolean expression from any truth table
    • Maxterms
      • Exact opposite of minterm
      • Sum Term, includes all input variables (or inverted ones)
      • Each maxterm for any given input variables, can have an output value of 0 with only 1 input combination
      • Represented using UPPER-case M with subscript
      • Correspond to negative output rows on the truth table
      • Product-of-Maxterm can be directly taken from any truth table, so you get easy boolean expression from any truth table
    • For any F(…),

          \[m_i = \overline{M_i}\]

LS6 Part 2 (Exercise): Standard Forms

Here’s what you need before watching this video: bring out a pen/pencil and a few pieces of paper, and do the exercises. Each question is already included in LS6, and will be displayed for 10seconds, during which you should pause the video, attempt to solve it, and only when you have tried sufficient amount of times or have solved it successfully should you proceed to watch my solutions.

The answers will NOT be available in the PDF version of LS5.

LS7 Part 1: How to draw the K-Maps

Highlights:

  • K-Maps are used for simplifying Sum-of-Minterms, will discuss how it works in LS7 Part 2. In this video, we only introduce how to draw them, in the next one (LS7 P2) we’ll teach you how to use them.
  • We only cover 2, 3, 4 variable K-Maps
    • 2 variable: 2×2 table, 4 minterms
    • 3 variable: 2×4 roll, 8 minterms
    • 4 variables: 4×4 doughnut
  • K-Map properties
    • Same number of regions as minterms (for their respective variables)
    • Two minterms are adjacent if and only if they differ in one literal, both vertically and horizontally
      • Say, 01 and 10 differ in 2 variables, 01 and 11 differ in 1 variable.
    • Each literal’s positive rows/columns are indicated by brackets on the side

LS7 Part 2: How to use K-Map to Optimise

Highlights:

  • You can use K-Maps to optimise truth tables, or Sum-of-Minterms
  • 3 steps
    • Enter the 1 values into minterm positions in the K-Map
    • Select the largest rectangles (with edges of 1, 2, or 4) to cover all 1s, each rectangle will give you a product term, which is equal to all minterms in that rectangle connected with OR operator
    • Read off for rectangles
      • Look at the boolean expression for the rectangle’s rows
      • Look at the boolean expression for the rectangle’s columns
      • Combine with AND, and you’ll have your product term
      • When you are done with all rectangles, connect all these product terms with OR, this is your final answer.

LS7 Part 3: Don’t Care Conditions, Exercises

Highlights:

  • Don’t Care Conditions
  • Optimising for Don’t Care Conditions
  • Exercises:
    • Here’s what you need before watching this video: bring out a pen/pencil and a few pieces of paper, and do the exercises. Each question is already included in LS7, and will be displayed for 10seconds, during which you should pause the video, attempt to solve it, and only when you have tried sufficient amount of times or have solved it successfully should you proceed to watch my solutions.

LS8: Delays, and Other Gates

Highlights:

  • Propagation/Transport delay: pretty much the same in our context, extends Gating Delay to entire circuit
  • Inertial Delay: Transport delay with Rejection time window that increases robustness
  • Standard Load: how many other gates, can the output of a single gate drive (Computation not required). SL is also used in estimating gating delay, but we don’t ask you to know how.
  • Buffer, 3-state Buffer: basically, boosting voltage while sacrificing delay, so that you can drive more gates through longer wires.
  • XOR gate: remember its truth table, and boolean expression!
  • N-Gates

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