Digital Abstraction

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6.002

CIRCUITS AND ELECTRONICS

The Digital Abstraction

6.002 Fall 2000

Lecture 4

1

Review z Discretize matter by agreeing to

observe the lumped matter discipline

Lumped Circuit Abstraction zAnalysis tool kit: KVL/KCL, node method, superposition, Thévenin, Norton (remember superposition, Thévenin, Norton apply only for linear circuits)

6.002 Fall 2000

Lecture 4

2

Today

Discretize value

Digital abstraction

Interestingly, we will see shortly that the tools learned in the previous three lectures are sufficient to analyze simple digital circuits

Reading: Chapter 5 of Agarwal & Lang

6.002 Fall 2000

Lecture 4

3

But first, why digital? In the past … Analog signal processing R1 V0

R2

V1 + –

V1

and V2 might represent the outputs of two sensors, for example.

+ –

V2

By superposition, V0 =

R2 R1 V1 + V2 R1 + R2 R1 + R2

If R1 = R 2 , V0 =

V1 + V2 2

The above is an “adder” circuit. 6.002 Fall 2000

Lecture 4

4

Noise Problem t

add noise on this wire

Receiver: huh?



noise hampers our ability to distinguish between small differences in value — e.g. between 3.1V and 3.2V.

6.002 Fall 2000

Lecture 4

5

Value Discretization Restrict values to be one of two HIGH

LOW

5V

0V

TRUE

FALSE

1

0

…like two digits

0 and 1

Why is this discretization useful? (Remember, numbers larger than 1 can be represented using multiple binary digits and coding, much like using multiple decimal digits to represent numbers greater than 9. E.g., the binary number 101 has decimal value 5.)

6.002 Fall 2000

Lecture 4

6

Digital System sender

noise VN

VS

VR

VN = 0V

receiver

VS

VR

5V “0” “1” “0” HIGH

“0” “1” “0” 5V

t

2.5V

0V

LOW

0V

t

2.5V

With noise

VS

VN = 0.2V

“0” “1” “0” 5V

“0” “1” “0”

0.2V

t

2.5V

VS

t

t

2.5V

0V 6.002 Fall 2000

Lecture 4

7

Digital System

Better noise immunity Lots of “noise margin” For “1”: noise margin 5V to 2.5V = 2.5V For “0”: noise margin 0V to 2.5V = 2.5V

6.002 Fall 2000

Lecture 4

8

Voltage Thresholds and Logic Values

5V

1

1

sender 0

1 2.5V receiver

0

0 0V

6.002 Fall 2000

Lecture 4

9

But, but, but … What about 2.5V? Hmmm… create “no man’s land” or forbidden region For example, 5V

1 sender

3V 2V

0

1

VH

forbidden region

receiver

VL

0

0V

“1”

V

“0”

0V

6.002 Fall 2000

H

Lecture 4

5V V

L 10

But, but, but …

Where’s the noise margin? What if the sender sent 1:

VH ?

Hold the sender to tougher standards! 5V 1

V 0H

1 V IH

sender

V IL

0

receiver 0

V 0L

0V

6.002 Fall 2000

Lecture 4

11

But, but, but …

Where’s the noise margin? What if the sender sent 1:

VH ?

Hold the sender to tougher standards! 5V 1

V 0H

1

sender

Noise margins

V IH

receiver

V IL

0

0

V 0L

0V “1” noise margin: V

- V

“0” noise margin:

-

6.002 Fall 2000

Lecture 4

IH V IL

0H V 0L 12

5V V 0H V IH V IL V 0L 0V

5V V 0H V IH V IL V 0L 0V

0

1

0

1

sender

t

0

1

0

1

receiver

t

Digital systems follow static discipline: if inputs to the digital system meet valid input thresholds, then the system guarantees its outputs will meet valid output thresholds. 6.002 Fall 2000

Lecture 4

13

Processing digital signals Recall, we have only two values —

1,0

Map naturally to logic: T, F Can also represent numbers

6.002 Fall 2000

Lecture 4

14

Processing digital signals Boolean Logic If X is true and Y is true Then Z is true else Z is false. Z = X AND Y

X, Y, Z are digital signals “0” , “1”

Z = X • Y Boolean equation X Y

AND gate

Z

Truth table representation: X Y Z 0 0 1 1

0 1 0 1

0 0 0 1

Enumerate all input combinations 6.002 Fall 2000

Lecture 4

15

Combinational gate abstraction „ Adheres to static discipline „ Outputs are a function of

inputs alone.

Digital logic designers do not have to care about what is inside a gate.

6.002 Fall 2000

Lecture 4

16

Demo

X

Y

Z Noise X Y

Z

Z = X • Y 6.002 Fall 2000

Lecture 4

17

Examples for recitation X

t Y

t Z

t Z = X • Y 6.002 Fall 2000

Lecture 4

18

In recitation… Another example of a gate If (A is true) OR (B is true) then C is true else C is false C = A + B A B

Boolean equation OR C

OR gate

More gates B

B Inverter

X Y

Z NAND

Z = X • Y

6.002 Fall 2000

Lecture 4

19

Boolean Identities X X X X

• 1 = X • 0 = X + 1 = 1 +0 = X

1 = 0 0 = 1 AB + AC = A • (B + C)

Digital Circuits Implement: B C

output = A + B • C B•C output

A

6.002 Fall 2000

Lecture 4

20

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