Logical Circuits Part 2 Review

To Study

1. K Maps
2. Don’t Cares
3. MSI Functions

K Maps

TODO

1. Watch Lecture
2. View Slides
3. View and Practice Drill Problems

Rules

1. The number of inputs is $n$, while the number of squares is $2^n$
2. To find the minimum expression, group the adjacent cells
   1. Diagonal cells are not adjacent
   2. Cells at the extremities are adjacent
3. Groups should have either a square or a rectangular shape
4. The number of elements in a group is restricted to the power of 2. Even numbers that are not a power of 2 do not count
5. Smaller number of groups with the most amount of elements is preferred
6. We group the 1’s to get the minimal SOP
7. We group the 0’s to get the minimal POS
8. Although cells can be repeatedly grouped, each group should maintain at least 1 unique member.

Don’t Cares

TODO

1. Watch Lecture
2. View Slides
3. View and Practice Drill Problems

Rules

• Occurs when a certain input combinations are impossible in a given system
• Also occurs when an input(s) is unnecessary when determining the output—you can predict the output even without knowing a particular input’s value
• Is symbolized by $x$
• It can be treated like 1 when grouping minterms and treated as 0 when grouping maxterms. Nonetheless, only utilize them in grouping when they can further simplify functions.

MSI Functions

TODO

1. Adder and Decoder Lecture
   1. Adder
   2. Decoder
2. Adder and Decoder Slides
   1. Adder
   2. Decoder
3. Multiplexer
   1. Lecture
   2. Slides
4. Drill Problems
   1. MSI Drill Problems Lecture
   2. MSI Drill Problems Powerpoint
   3. Re-Answer all Exercises

Introduction

MSI Functions covers the readily available components you can buy without manually designing them. Using traditional steps such as starting with a truth table, then doing a K-map, etc. would sometimes be too laborious, thereby giving rise to the need for a simpler approach—MSI functions.

Integrated Circuit (IC) Density

IC Density	Approximated Number of Components Inside the IC
Small Scale Integration (SSI)	1 to 50
Medium Scale Integration (MSI)	50 to 1000
Large Scale Integration (LSI)	1000 to 100,000
Very Large Scale Integration (VLSI)	100,000 to 1M
Ultra Large Scale Integration (ULSI)	ABOVE 1M

EXOR Gates

Input	Output
Both same	0
Both different	1

The exclusive or gate can function as either an inverter or a buffer.

• When high, it is an inverter
• When low, it is a buffer

Adder

Adders are used to add bits together. They consist of input values, output, carry in, and carry out

Decoder

• Input is $n$ while output is $2^n$
• Only one output will stand out, the rest will posses the same value/logic
• Every D output corresponds to a complete combination of ABC

Active HIGH	Active LOW
Output is not inverted	Output is inverted
Activated component is high	Activated component is low

TIP

• A nor gate with all inputs inverted can be converted into an a nand gate, where everything is inverted at the output instead

Adder and Decoder Summary

1. Active HIGH
   1. Decoder + OR gate (SOP only)
2. Active LOW
   1. Decoder + NAND gate (SOP only)
   2. Decoder + AND gate (POS only)

Multiplexer

To implement a function using MUX:

1. Only choose the function inputs connected to the MUX select lines
2. The others will serve as reference inputs of the MUX
3. Line up the input values and determine the equivalent values based on both the input values and the value of the reference input
4. Mark the given numbers from the problem
5. Add the marked numbers based on the reference value

TIP

• Inputs is $2^n$, while select lines is $n$

Notes

• Decoder/demux means that there is an enabler to the decoder. Active low means that the enabler has an inverter
• For BCD, add 6 to rectify incorrect sum due to it exceeding 9