6.002
CIRCUITS AND ELECTRONICS
The Digital Abstraction
6.002 Fall 2000
Lecture 4
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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)
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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
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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
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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.
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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.)
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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
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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
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Voltage Thresholds and Logic Values
5V
1
1
sender 0
1 2.5V receiver
0
0 0V
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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
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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:
-
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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
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Processing digital signals Recall, we have only two values —
1,0
Map naturally to logic: T, F Can also represent numbers
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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
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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.
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Demo
X
Y
Z Noise X Y
Z
Z = X • Y 6.002 Fall 2000
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Examples for recitation X
t Y
t Z
t Z = X • Y 6.002 Fall 2000
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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
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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
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