Transcript CIS21JA Basic HW symbols and Adders

Hardware Logic Diagrams
- The Basics Marshall Thomas
CIS 21JA – Fall Quarter 2012
Objectives
• Explain some hardware concepts at a high level
– appropriate for ASM SW Engineers.
• Show the graphical pictures that the hardware
folks use to describe logic diagrams.
• Show how De Morgan’s theorem can be
represented graphically.
• Explain the equations and “what is going on” in
Lab 5
Terminology
• “High” => High voltage or “true” or “1”
• “Low” => Low voltage or “false” or “0”.
• These mean the same:
=> high, true, logical “1”
=> low, false, logical “0”.
• A means: A is true when it is low, sometimes written *A
(over_scores are hard to type), not the same meaning as
!A
• AdressBit0 means that AddressBit0 is true when this
voltage is “false”, not the same in as !AddressBit0.
Voltage Levels
• There is uncertainty between what is “sent” and what
is “received”.
• This uncertainty can and does happen! This is a
problem for hardware folks.
Inversion at every gate
• This is “how the hardware works”
• This is the fastest way
• If there is an inverted and a non-inverted output, the
inverted output is faster
• Easiest example is an “open collector” output
+5V
R
Input
output
Basic Gates / Symbols
• AND
• OR
• INVERTER
• NAND
• NOR


NAND / “OR” Duality
• NAND Gate :
• If it takes two “highs” to get “low”.
Doesn’t that mean that any low on
any input input gives a high on the
output?
• So, another way to draw the exact
same hardware part is:
NOR / “AND” Duality
• NOR Gate:
• If any high gives a low, then doesn’t it
take all low’s to give a high?
• Another way to draw the exact same
physical part:
De Morgan’s Theorem
• A or B  A and B

• A and B  A or B

• Swap the bubbles and the symbols
XOR
• Many students freak out about XOR, but is it not
that complicated!
• If the 2 inputs are the same:
you get ”false”, 0, or “even”
• If the 2 inputs are different:
you get “true”, 1 or “odd”.
• It is as simple as that – think “odd” or “even”
• This simple thing can be used for numeric
addition, calculating parity (the number of even or
odd bits that are turned on in an variable), and even
polynomial division!
Half Adder
• XOR Version
A
B
Sum
Even
Carry
Odd
• Gate Version (same)
A
B
Sum
Carry
A
B sum
Carry
0
1
1
0
0
1
0
1
0
1
0
0
0
0
1
1
Sum = (A+B) and AB
Sum is “1” if A or B is true,
unless: A and B
are both true,
Then sum=0 with carry
Full Adder
• Incorporates the Carryin
• This combines 2 half-adders with an “or”
A
B
Cin
Sum
Sum=1 if
A,B,C
Are odd
Cout
Note: all Nand gates
Cout = (A and B) or ( (A xor B) and Cin))
Cout= ab + bc + ac
Cout= ab + bc + ac
• The cascading XOR’s can do the sum:
Sum is “1” if combination of (A,B,C) is
odd, else sum is “0”
• 1+1 = 0 (sum is even) with carry
1+1+1= 1 (sum is odd) with carry
• The tricky part is the carry.
• We are adding 3 bits together, if any 2
of these bits are “1”, we get a carry.
If all 3 are “1”, we still get a carry
although the sum is different.
Full Adder
Truth Table
Even => Sum =0
Odd => Sum =1
Cout if any
2 inputs are “1”
A
B
Cin Sum 2 bit Cout
sum
0
0
1
0
1
0
0
1
1
0
0
0
00
10
10
0
1
1
1
0
0
1
1
0
1
0
0
1
0
0
0
1
1
1
10
01
01
01
1
0
0
0
1
1
1
1
11
1