1. Digital Logic Circuits
Winter 2012
COMP 2130 Intro Computer Systems
Computing Science
Thompson Rivers University

2. Digitial Logic Circuits
Course Contents
Part I – Review
Mumber Systems
Boolean Logic
Digital Logic Circuits
Part II – C Programming Language
Part III – Introduction to Computer Systems
Not the replacement of IPv6; Inter-transition mechanism; But if IPv6 would fail
TRU-COMP2130
Digitial Logic Circuits

3. Unit Learning Objectives
Find a Boolean expression corresponding to circuit.
Find a circuit corresponding to a Boolean expression.
Find a Boolean expression corresponding to an i/o table.
Create an i/o table corresponding to given requirements.
Design AND, OR, and NOT gates using NOR gates.
Understand how multiplexers and decoders work.
TRU-COMP2130
Digitial Logic Circuits

4. Digitial Logic Circuits
Unit Contents
Gates
Input/Output Table for a Circuit
Boolean Expression Corresponding to a Circuit
Circuit Corresponding to a Boolean Expression
Finding a Circuit Corresponding to a Given Input/Output Table
Simplifying Combinational Circuits
NAND and NOR Gates
Combinatorial Circuits
Sequential Circuits
TRU-COMP2130
Digitial Logic Circuits

5. Digitial Logic Circuits
Analogy between the operations of switching devices and the operations of logical connectives P Q
Switches in series P Q
P  Q 1 ? P Q
Switches in parallel P Q
P  Q 1 ?
TRU-COMP2130
Digitial Logic Circuits

6. Digitial Logic Circuits
Gates
AND (conjunction)
OR (disjuction)
NOT (negation)
TRU-COMP2130
Digitial Logic Circuits

7. Input/Output Table for a Circuit
P = 1 an Q = 0, then R = ??? I/O table ???
P = 0, Q = 1, and R = 1. Then S = ??? I/O table ???
TRU-COMP2130
Digitial Logic Circuits

8. Digitial Logic Circuits
I/O tables for circuits???
TRU-COMP2130
Digitial Logic Circuits

9. Boolean Expression for a Circuit
TRU-COMP2130
Digitial Logic Circuits

10. Circuit Corresponding to a Boolean Expression
(~P  Q)  ~Q
((P  Q)  (R  S))  T
TRU-COMP2130
Digitial Logic Circuits

11. Finding a Circuit Corresponding to a Given I/O Table
How to find a circuit that corresponds to the above I/O table?
Input
Output P Q R S 1
TRU-COMP2130
Digitial Logic Circuits

12. Digitial Logic Circuits
Input
Output P Q R S 1
For the 1st, the 2nd and the 3rd rows,
P  Q  R
P  ~Q  R
P  ~Q  ~R
Then ???
TRU-COMP2130
Digitial Logic Circuits

13. Digitial Logic Circuits
For the 1st, the 2nd and the 3rd rows,
P  Q  R
P  ~Q  R
P  ~Q  ~R
All the other cases will produce 0.
Therefore,
S = (P  Q  R)  (P  ~Q  R)  (P  ~Q  ~R)
This is a disjunction of conjunctions (called disjunctive normal form).
This boolean expression might be simplified by using equivalences.
How to find a circuit that corresponds to the above boolean expression?
= (P  R  Q)  (P  R  ~Q)  (P  ~Q  ~R)
= ((P  R)  (Q  ~Q ))  (P  ~Q  ~R) = ((P  R)  T)  (P  ~Q  ~R)
= (P  R)  (P  ~Q  ~R)
= P  (R  (~Q  ~R))
= P  (R  ~Q)
TRU-COMP1380
Digitial Logic Circuits

14. Digitial Logic Circuits
1 bit full adder
= ???
S = (P  Q  Cin)  (P  ~Q  ~Cin)  (~P  Q  ~ Cin)  (~P  ~Q  Cin)
Cout = ...
Corresponding circuit ???
Input
Output P Q
Cin S
Cout 1
TRU-COMP2130
Digitial Logic Circuits

15. Digitial Logic Circuits
2-to-1 multiplexer
When C = 0; A is selected, and
when C = 1, B is selected.
Corresponding i/o table?
S = ??? A B C S
Input
Output A B C S 1
TRU-COMP1380
Digitial Logic Circuits

16. Simplifying Combinational Circuits
Any good idea?
Convert to a boolean expression, and then
Simplify the boolean expression by using equivalences
R = ((P  ~Q)  (P  Q))  Q
= ... by distributive law
= ...
TRU-COMP2130
Digitial Logic Circuits

17. Digitial Logic Circuits
NAND and NOR Gates
AND and NAND gates
NAND = NOT (AND)
OR and NOR gates
NOR = NOT (OR)
TRU-COMP2130
Digitial Logic Circuits

18. NOT
AND OR
XOR
NAND
NOR
TRU-COMP2130
Digital Logic

19. Can you make NAND, NOR and XOR, by using AND, OR and NOT?
A  B = (~A  B) + (A  ~B)
Can you make AND by using OR and NOT? (Using DeMorgan’s Law)
A  B = ~(~A + ~B)
Can you make OR by using AND and NOT?
A + B = ~(~A  ~B)
TRU-COMP2130
Digital Logic

20. Can you make AND, OR and NOT, by using NOR?
~A = A NOR A
A + B = (A NOR B) NOR (A NOR B)
A  B = (A NOR A) NOR (B NOR B)
Can you make AND, OR and NOT, by using NAND?
Digital circuits can be, and frequently are, implemented solely with NOR gates or solely with NAND gates.
TRU-COMP2130
Digital Logic

21. Combinational Circuits
A combinational circuit is an interconnected set of gates whose output at any time is a function only of the input at that time.
E.g., a boolean function of three variables?
F(A,B,C) = ~A  B  ~C + ~A  B  C + A  B  ~C
= (A+B+C)(A+B+~C)(~A+B+C)(~A+B+~C)(~A+~B+~C)
The second form is easier to implement.
TRU-COMP2130
Digital Logic

22. Multiplexer (E.g., to control signal and data routing)
Two + four inputs, one output
0 0 => value from input D0 is selected from D0, D1, D2, D3
0 1 => value from input D1 is selected
1 0 => value from input D2 is selected
1 1 => value from input D3 is selected
Decoder (E.g., address decoding)
Two inputs, four outputs
0 0 => output D0 selected
0 1 => output D1 selected
1 0 => output D2 selected
1 1 => output D3 selected
TRU-COMP2130
Digital Logic

23. TRU-COMP2130
Digital Logic

24. Adder A B Cin S Cout 1 ... 1 bit full adder 4 bit adder TRU-COMP21301
...
Adder
1 bit full adder
4 bit adder
TRU-COMP2130
Digital Logic

25. Sequential Circuits
Combinational circuits implement the essential functions of a digital computer. However, they provide no memory or state information.
The current output of a sequential circuit depends not only on the current input but also on the past history of inputs
TRU-COMP2130
Digital Logic

26. Flip-flop Register Counter ... The simplest form of sequential circuit
The S-R latch: 1 bit memory
> Q = 1
> Q = 0
> no change
Register
Counter
...
TRU-COMP2130
Digital Logic

27. Programmable Logic Devices
TRU-COMP2130
Digital Logic