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Gates – Part 1. Gates are Built With Transistors. drain. drain. drain. no current flows. 3 volts. 0 volts. gate. current flows. source. source. source. nFET. nFET On. nFET Off. N-type field-effect transistor = nFet. Gates are Built With Transistors. source. source. source. - PowerPoint PPT Presentation
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04 GATES1 Page 1
ECEn 224 © 2003-2008BYU
Gates – Part 1
04 GATES1 Page 2
ECEn 224 © 2003-2008BYU
Gates are Built With Transistors
nFET
gate
drain
source
3 volts
drain
source
0 volts
drain
source
currentflows
nocurrentflows
nFET On nFET Off
N-type field-effect transistor = nFet
04 GATES1 Page 3
ECEn 224 © 2003-2008BYU
Gates are Built With Transistors
pFET pFET On pFET Off
gate
source
drain
0 volts 3 voltscurrentflows
nocurrentflows
source
drain
source
drain
P-type field-effect transistor = pFet
04 GATES1 Page 4
ECEn 224 © 2003-2008BYU
Complement
Also known as invert or not.
x x'0 11 0
x’x
This is a schematic symbol.It is a graphical representationof a circuit which implementsthe operation.
04 GATES1 Page 5
ECEn 224 © 2003-2008BYU
FET-Based Inverter
Vin Vout
Vcc = 3V
GND = 0V
3V 0V 0V 3V
off
on
on
off
Vcc = 3V
GND = 0V
Vcc = 3V
GND = 0V
“Ground”
04 GATES1 Page 6
ECEn 224 © 2003-2008BYU
AND and OR Gates
A B Q=A•B
0 0 00 1 01 0 01 1 1
A
BQ
A
BQ
A B Q=A+B
0 0 00 1 11 0 11 1 1
04 GATES1 Page 7
ECEn 224 © 2003-2008BYU
Boolean Expressions and GatesEach Boolean expression has a corresponding realization with logic gates.
A
BC
F
F = A’ + B C
04 GATES1 Page 8
ECEn 224 © 2003-2008BYU
NAND Gates
A
BQ
A B Q=(A•B)'
0 0 10 1 11 0 11 1 0
NAND
A
BQ
A
BQ
Q is true iff A AND B are true
Q is false iff A AND B are true
AND
NANDBubble means NOT
04 GATES1 Page 9
ECEn 224 © 2003-2008BYU
FET-Based NAND Gate
A
B
BA
Vcc
GND
F
Vcc
GND
1
1
11 off
on
off
on
0
Vcc
GND
1
0
01 off
on
on
off
1
04 GATES1 Page 10
ECEn 224 © 2003-2008BYU
NOR Gates
A B Q=(A+B)'
0 0 10 1 01 0 01 1 0
A
BQ
NOR
A
BQ
A
BQ
Q is true if A OR B is true
Q is false if A OR B is true
OR
NOR
04 GATES1 Page 11
ECEn 224 © 2003-2008BYU
FET-Based NOR Gate
A
B
BA
F
Can you complete the truth table?
A B F
0V 0V ?0V 5V ?5V 0V ?5V 5V ?
Vcc = 5V
GND
04 GATES1 Page 12
ECEn 224 © 2003-2008BYU
FET-Based NOR Gate
A
B
BA
F
Can you complete the truth table?
A B F
0V 0V 5V0V 5V 0V5V 0V 0V5V 5V 0V
Vcc = 5V
GND
04 GATES1 Page 13
ECEn 224 © 2003-2008BYU
FET-Based Gates
• P-type FETs must be on top of gate• N-type FETs must be on bottom of gate• Due to electrical characteristics of the two
FET types
• Output is driven to either ‘1’ or ‘0’– never both– never neither
04 GATES1 Page 14
ECEn 224 © 2003-2008BYU
Exclusive-OR (XOR)
Output is true iff inputs are different.
A
B
Q = A B = A’B + AB’
Another definition: Q is true iff A does not equal B
A B Q = A B
0 0 00 1 11 0 11 1 0
04 GATES1 Page 15
ECEn 224 © 2003-2008BYU
Exclusive-OR Theorems
X 0 = X
X 1 = X'
X X = 0
X X' = 1
X Y = Y X Commutative law
( X Y) Z = X ( Y Z ) = X Y Z Associative law
( X Y)' = X Y' = X' Y
The first 4 are important,
the others are used less frequently
04 GATES1 Page 16
ECEn 224 © 2003-2008BYU
Equivalence Operation
denotes equivalence (also written as X==Y)
Output is true iff inputs are equal
X
Y
Q = (X==Y)= X’Y’ + XY
X Y X==Y
0 0 10 1 01 0 01 1 1
04 GATES1 Page 17
ECEn 224 © 2003-2008BYU
XOR and EQUIV are Complements !!
X Y XY X==Y
0 0 0 10 1 1 01 0 1 01 1 0 1
Gate often called exclusive NOR or XNOR
Alternate equivalencesymbol
04 GATES1 Page 18
ECEn 224 © 2003-2008BYU
Multi-Input Gates
A
B
BA
Vcc
GND
F
C
C
GND
A
B
BA
Vcc
F
Which one will be slower/faster?
04 GATES1 Page 19
ECEn 224 © 2003-2008BYU
Alternative Gate Symbols
The symbolic meaning of the circuit should be clear from what
you draw...
04 GATES1 Page 20
ECEn 224 © 2003-2008BYU
Alternative Gate Symbols
Q is true iff A is false OR B is false
A
BQ
Q is true iff A is false AND B is false
A B Q
0 0 10 1 11 0 11 1 0
A B Q
0 0 10 1 01 0 01 1 0
A
BQ A
BQ
A
BQ
04 GATES1 Page 21
ECEn 224 © 2003-2008BYU
Alternative Gate Symbols
• Turn on sprinklers if it is not a holiday and it is not a weekend
or?
The problem statement uses AND, so use the AND symbol
04 GATES1 Page 22
ECEn 224 © 2003-2008BYU
Alternative Gate Symbols
• Turn off the sprinklers if it is a holiday or it is a weekend
or?
The problem statement uses OR, so use the OR symbol
04 GATES1 Page 23
ECEn 224 © 2003-2008BYU
Another Example
• Design a circuit to determine whether the bits of a 4-bit wire are all zero
This is the appropriate symbol to use…
04 GATES1 Page 24
ECEn 224 © 2003-2008BYU
Mixed Symbols
• Such a gate doesn’t likely exist• Build from AND gate and inverter• Simplifies schematics, enhances
readability
Q is true iff A is false AND B is true
A
BQ
04 GATES1 Page 25
ECEn 224 © 2003-2008BYU
Single Gate Conversion Rules
• How to change one symbol to another:– Change symbol
• AND to OR• OR to AND
– Invert all inputs and outputs• No change in behavior – merely a symbol change
Q is true iff A is false AND B is true Q is false if A is true OR B is false
A
BQ
A
BQ
04 GATES1 Page 26
ECEn 224 © 2003-2008BYU
Alternative Gate Symbols - Summary
• Use the symbol that matches the problem statement– Clarity– Documentation– Maintenance
• If function is correct but symbol is wrong then your schematic is wrong
04 GATES1 Page 27
ECEn 224 © 2003-2008BYU
Positive vs. Negative Logic
04 GATES1 Page 28
ECEn 224 © 2003-2008BYU
Positive Logic and Negative Logic
V1LogicGate
V1 V2 V3 Vout
0V 0V 0V 0V0V 0V 5V 0V0V 5V 0V 0V0V 5V 5V 0V5V 0V 0V 0V5V 0V 5V 0V5V 5V 0V 0V5V 5V 5V 5V
V2
V3Vout
04 GATES1 Page 29
ECEn 224 © 2003-2008BYU
Positive LogicLet:
0 volts => 05 volts => 1
The circuit is a logical AND gate
v1 v2 v3 vout
0 0 0 00 0 1 00 1 0 00 1 1 01 0 0 01 0 1 01 1 0 01 1 1 1
V1 V2 V3 Vout
0V 0V 0V 0V0V 0V 5V 0V0V 5V 0V 0V0V 5V 5V 0V5V 0V 0V 0V5V 0V 5V 0V5V 5V 0V 0V5V 5V 5V 5V
04 GATES1 Page 30
ECEn 224 © 2003-2008BYU
Negative LogicLet:
0 volts => 15 volts => 0
The same circuit is a logical OR gate
v1 v2 v3 vout
1 1 1 11 1 0 11 0 1 11 0 0 10 1 1 10 1 0 10 0 1 10 0 0 0
V1 V2 V3 Vout
0V 0V 0V 0V0V 0V 5V 0V0V 5V 0V 0V0V 5V 5V 0V5V 0V 0V 0V5V 0V 5V 0V5V 5V 0V 0V5V 5V 5V 5V
04 GATES1 Page 31
ECEn 224 © 2003-2008BYU
Positive/Negative Logic
• The most common mapping is positive logic:
+V 10V 0
• Different systems have used different mappings in the past
04 GATES1 Page 32
ECEn 224 © 2003-2008BYU
Multi-Level Logic
04 GATES1 Page 33
ECEn 224 © 2003-2008BYU
Levels of a Network
Maximum number of gates between an input and the output
5 Levels
3 LevelsIn general:
- the more levels, the slower the circuit
04 GATES1 Page 34
ECEn 224 © 2003-2008BYU
Number of Levels
• Number of levels can be increased by factoring
• Number of levels can be decreased by multiplying out
• Changing the number of levels affects area and speed
G = AB + ACDE + ACF = A(B+CDE+CF)
G = A(B+CDE+CF) = AB + ACDE + ACF
04 GATES1 Page 35
ECEn 224 © 2003-2008BYU
Example
G = AB + ACDE + ACFLevels = 2Gates = 4Delay = tAND4 + tOR3
Gate Inputs = 12Transistors = 24Largest gate = 4 inputs
Area Calculations: - Each input to a gate costs ~2 transistors - Area number of transistorsDelay Calculations: - Find slowest path from inputs to output
tdelay = tAND4 + tOR3
- The 4-input AND is likely slower than the other AND gates
A
B
ACDE
ACF
G
04 GATES1 Page 36
ECEn 224 © 2003-2008BYU
Change the number of levels by factoring
G = ACDE + ACF + AB = A(CDE + CF + B)
CDE
C
F
BA
G
Levels = 3Gates = 4Delay = tAND3 + tOR3 + tAND2
Inputs = 10Transistors = 20Largest gate = 3 inputs
factor
This is a 3-level circuit…
04 GATES1 Page 37
ECEn 224 © 2003-2008BYU
Factor Again
G = A(CDE + CF + B) = A[B+C(F+DE)]
A
BC
F
D
E
G
Levels = 5Gates = 5Delay = 3 x tAND2 + 2 x tOR2
Inputs = 10Transistors = 20Largest gate = 2 inputs
factor
This is a 5-level circuit…
04 GATES1 Page 38
ECEn 224 © 2003-2008BYU
Changing the number of levels
Three alternative solutions for same function…
2-level 3-level 5-level
Logic Levels 2 3 5Delay tAND4 + tOR3 tAND3 + tOR3 + tAND2 3 x tAND2 + 2 x tOR2
Gate Count 4 4 5Gate Inputs 12 10 10Transistors 24 20 20
Largest Gate 4 3 2
Each has different area/speed characteristics
04 GATES1 Page 39
ECEn 224 © 2003-2008BYU
Two-Level vs. Multi-Level
• In general:– Two-level is fastest– Multi-level can be smaller
• Exploring by hand to find just the right solution can be difficult
• We will focus on two-level– Easy to get from truth table– Minimization techniques in later chapters focus on
it