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From Switches to Transistors and Gates
Prof. Sirer
CS 316
Cornell University
Origins
The most amazing and likely to be most long-lived invention of the 1800’s was…
Origins
The most amazing and likely to be most long-lived invention of the 1800’s was…
The steam engine? The lightning rod? The carbonated beverage?
Origins
The most amazing and likely to be most long-lived invention of the 1800’s was…
THE ELECTRIC SWITCH
A switchA switch is a simple device that can act as a conductor or isolator
Can be used for amazing things…
Switches
Either (OR)
Both (AND)
+
-
-
One can build basic devices, but operation requires mechanical force
TransistorsWhat happens when switches are replaced with solid-state devices The most amazing invention
of the 1900s Can be used for purposes
other than switching (e.g. radios)
Two types, PNP and NPNbase
collector
emitter
NP Pcollector emitter- +
PNP base
collector
PN N
emitter
+ - NPN
NPN Transistor
Connect E to C whenbase = 1
P and N Transistors
PNP Transistor
Connect E to C whenbase = 0
E
C
E
C
B B
Can be used to build primitives from which electronic machines can be constructed
Then and Now
The first transistor, on a workbench at AT&T Bell Laboratories in 1947
An Intel Pentium has approximately 125 million transistors
Inverter
in out
+
gnd
In Out
0 1
1 0
Function: NOT
Called an inverter
Symbol:
Useful for taking the inverse of an input
in out
Truth table
NAND Gate
a
out
+
gnd
b
A B out
0 0 1
1 0 1
0 1 1
1 1 0
Function: NAND
Symbol:
ba out
NOR Gate
a
out
+
gnd
b
A B out
0 0 1
1
0 0
0 1 0
1 1 0
Function: NOR
Symbol:
Building FunctionsNOT:
AND:
OR:
NAND and NOR are universal, can implement any function using just NAND or just NOR gates
useful for manufacturing
Can specify functions by describing gates, truth tables or logic equations
ba
ba
Logic Equations
AND out = ab
OR out = a + b
NOT out = a
Identities
Identities are useful for manipulating logic equations
For purposes of optimization & ease of implementation a + a = 1 a + 0 = a a + 1 = 1 aa = 0 a0 = 0 a1 = a a(b+c) = ab + ac (a + b) = a b (a b) = a + b a + a b = a + b
Logic Manipulation
a b c a+b a+c LHS bc RHS
0 0 0 0 0 0 0 0
0 0 1 0 1 0 0 0
0 1 0 1 0 0 0 0
0 1 1 1 1 1 1 1
1 0 0 1 1 1 0 1
1 0 1 1 1 1 0 1
1 1 0 1 1 1 0 1
1 1 1 1 1 1 1 1
Can manipulate logic equations algebraicallyCan also use a truth table to prove equivalenceExample: (a+b)(a+c) = a + bc
(a+b)(a+c)= aa + ab + ac + bc= a + a(b+c) + bc= a(1 + (b+c)) + bc= a + bc
Logic MinimizationA common problem is how to implement a desired function most efficiently
One can derive the equation from the truth table
How does one find the most efficient equation? Manipulate algebraically until satisfied Use Karnaugh maps
a b c minterm
0 0 0 abc
0 0 1 abc
0 1 0 abc
0 1 1 abc
1 0 0 abc
1 0 1 abc
1 1 0 abc
1 1 1 abc
for all outputs that are 1,take the correspondingminterm,OR the minterms toobtain the result in “sum of products” form
Karnaugh maps
Encoding of the truth table where adjacent cells differ in only one bit
a b out
0 0 0
0 1 0
1 0 0
1 1 1
0 0 1 0
00 01 11 10
truth tablefor AND
Corresponding Karnaugh map
ab
Bigger Karnaugh Maps
3-inputfunc
a
b
c
y
00 01 11 10
0
1
abc
4-inputfunc
abc y
00 01 11 10
00
01
abcd
d
1110
Minimization with Karnaugh maps (1)
a b c out
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 0
Sum of minterms yields abc + abc + abc + abc
Minimization with Karnaugh maps (2)
a b c out
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 0
Sum of minterms yields abc + abc + abc + abc
Karnaugh maps identify which inputs are (ir)relevant to the output
0 0 0 1
1 1 0 1
00 01 11 10
0
1
c ab
Minimization with Karnaugh maps (2)
a b c out
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 0
Sum of minterms yields abc + abc + abc + abc
Karnaugh map minimization Cover all 1’s Group adjacent blocks of 2n 1’s
that yield a rectangular shape Encode the common features
of the rectangle out = ab + ac0 0 0 1
1 1 0 1
00 01 11 10
0
1
c ab
Karnaugh Minimization Tricks (1)
Minterms can overlap out = bc + ac + ab
Minterms can span 2, 4, 8 or more cells out = c + ab
0 1 1 1
0 0 1 0
00 01 11 10
0
1
c ab
1 1 1 1
0 0 1 0
00 01 11 10
0
1
cab
Karnaugh Minimization Tricks (2)
The map wraps around out = bd
out = bd
1 0 0 1
0 0 0 0
0 0 0 0
1 0 0 1
00 01 11 10
00
01
ab
cd
1110
0 0 0 0
1 0 0 1
1 0 0 1
0 0 0 0
00 01 11 10
00
01
ab
cd
1110
Karnaugh Minimization Tricks (3)
“Don’t care” values can be interpreted individually in whatever way is convenient
assume all x’s = 1 out = d
assume middle x’s = 0 assume 4th column x = 1 out = bd
1 0 0 x
0 x x 0
0 x x 0
1 0 0 1
00 01 11 10
00
01
ab
cd
1110
0 0 0 0
1 x x x
1 x x 1
0 0 0 0
00 01 11 10
00
01
ab
cd
1110
Multiplexors
A multiplexor selects between multiple inputs out = a, if d = 0 out = b, if d = 1
Build truth table
Build Karnaugh map
Derive logic diagram
a
b
d
0
Multiplexor Implementation
a b d out
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
Build a truth tablea
b
d
0
Multiplexor Implementation
a b d out
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
Build the Karnaugh mapa
b
d
0
0 0 1 1
0 1 1 0
00 01 11 10
0
1
d ab
Multiplexor Implementation
a b d out
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
a
b
d
0
0 0 1 1
0 1 1 0
00 01 11 10
0
1
d ab
Multiplexor Implementation
a b d out
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
a
b
d
0
0 0 1 1
0 1 1 0
00 01 11 10
0
1
d ab
d out
b
a
Summary
We can now implement any logic circuit Can do it efficiently, using Karnaugh maps to
find the minimal terms required Can use either NAND or NOR gates to
implement the logic circuit Can use P- and N-transistors to implement
NAND or NOR gates
