We know binary

1
Boolean and Binary Inputs

• We know binary
• We know how to add and subtract in binary
• Same as in decimal
• Next up learn how apply this knowledge

2
Boolean and Binary Inputs

• Discrete voltages represented by 1 and 0
• For example
• 0 ground (GND) or 0 volts
• 1 VDD or 5 volts
• What about 4.99 volts? Is that a 0 or a 1?
• What about 3.2 volts?

3
Logic Levels

• Range of voltages for 1 and 0
• Different ranges for inputs and outputs to allow
  for noise

4
Logic Gates
Boolean and Binary Inputs

• Perform logic functions
• inversion (NOT), AND, OR, etc.
• Single-input
• NOT gate, buffer
• Two-input
• AND, OR, etc.

5
The Static Discipline

• With logically valid inputs, every circuit
  element must produce logically valid outputs
• Use limited ranges of voltages to represent
  discrete values

6
Practical Application

NMH VOH VIH NML VIL VOL
7
Practical Application - Transistors

• Logic gates built from transistors
• 3-ported voltage-controlled switch
• 2 ports connected depending on voltage of 3rd
• d and s are connected (ON) when g is 1

8
Boolean Algebra
Boolean algebra is based on the binary number
system
8
9
Boolean Algebra
Truth Tables
Boolean operations can be defined using a Truth
Table.
9
10
Boolean Algebra
Truth Tables
Boolean operations can be defined using a Truth
Table.
10
11
Boolean Algebra
DeMorgans Theorems
Proof
A B AB AB A B A B

• 0 0 0 1 1 1 1
• 0 1 0 1 1 0 1
• 0 0 1 0 1 1
• 1 1 1 0 0 0 0

11
12
Boolean Algebra
Example
F AB BC BC AB
12
13
Boolean Algebra
Another Example
F AB BC BC AC
13
14
Boolean Algebra
Probably.
14
15
Boolean Algebra
Simplifying logical expression using Boolean
algebra is not easy. Obscure identities must be
applied in clever ways (this requires LOTS of
practice).
There is a much easier (and more practical) way
Karnaugh maps
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0
0
0
F
1
0
0
0
1
0
1
1
1
1
1
1
15
16

Boolean Algebra
Karnaugh maps
Karnaugh Maps - Rules of Simplification
Rule 1. Groups may not include any cell
containing a zero
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0
0
0
1
0
0
0
1
0
1
1
1
1
1
1
16
17

Boolean Algebra
Karnaugh maps
Karnaugh Maps - Rules of Simplification
Rule 2. Groups may be horizontal or vertical,
but not diagonal.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0
0
0
1
0
0
0
1
0
1
1
1
1
1
1
17
18

Boolean Algebra
Karnaugh Maps - Rules of Simplification
Rule 3. Groups must contain 1, 2, 4, 8, or in
general 2n cells. That is if n 1, a group will
contain two 1's since 21 2. If n 2, a group
will contain four 1's since 22 4.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0
0
0
F
1
0
0
0
1
0
1
1
1
1
1
1
18
19

Boolean Algebra
Karnaugh Maps - Rules of Simplification
Rule 4. Each group should be as large as
possible. Each cell containing a one must be in
at least one group.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0
0
0
F
1
0
0
0
1
0
1
1
1
1
1
1
19
20

Boolean Algebra
Karnaugh Maps - Rules of Simplification
Rule 5. Groups may overlap.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0
0
0
F
1
0
0
0
1
0
1
1
1
1
1
1
20
21

Boolean Algebra
Karnaugh Maps - Rules of Simplification
Rule 6. Groups may wrap around the table. The
leftmost cell in a row may be grouped with the
rightmost cell and the top cell in a column may
be grouped with the bottom cell.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0
0
0
F
1
0
0
0
1
0
1
1
1
1
1
1
21
22

Boolean Algebra
Karnaugh Maps - Rules of Simplification
Rule 7. There should be as few groups as
possible, as long as this does not contradict any
of the previous rules.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0
0
0
F
1
0
0
0
1
0
1
1
1
1
1
1
22
23

Boolean Algebra
Karnaugh Maps - Rules of Simplification
Rule 1. Groups may not include any cell
containing a zero
Rule 2. Groups may be horizontal or vertical,
but not diagonal.
Rule 3. Groups must contain 1, 2, 4, 8, or in
general 2n cells. That is if n 1, a group will
contain two 1's since 21 2. If n 2, a group
will contain four 1's since 22 4.
Rule 4. Each group should be as large as
possible. Each cell containing a one must be in
at least one group.
Rule 5. Groups may overlap.
Rule 6. Groups may wrap around the table. The
leftmost cell in a row may be grouped with the
rightmost cell and the top cell in a column may
be grouped with the bottom cell.
0
0
0
0
F
1
0
0
0
1
0
1
1
Rule 7. There should be as few groups as
possible, as long as this does not contradict any
of the previous rules.
http//www.youtube.com/watch?vPA0kBrpHLM4
1
1
1
1
23
24
Boolean Algebra
Simplifying logical expression using Boolean
algebra is not easy. Obscure identities must be
applied in clever ways (this requires LOTS of
practice).
There is a much easier (and more practical) way
Karnaugh maps
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
0
0
0
1
0
0
0
1
0
1
1
1
1
1
1
24
25
Implementing Logic with Switches
25
26
Logic Gates
26
27
Robert Noyce (1927-1990)

• Nicknamed Mayor of Silicon Valley
• Cofounded Fairchild Semiconductor in 1957
• Cofounded Intel in 1968
• Co-invented the integrated circuit
• Figured out how to connect multiple transistors
  on a silicon chip

28
Practical Application - MOS Transistors

• Metal oxide silicon (MOS) transistors
• Polysilicon (used to be metal) gate
• Oxide (silicon dioxide) insulator
• Doped silicon

29
Practical Application - Transistors nMos

Gate 0 OFF (no connection between source and
drain)
Gate 1 ON (channel between source and drain)
30
Practical Application - Transistors pMOS

• pMOS transistor is opposite
• ON when Gate 0
• OFF when Gate 1

31
Practical Application - Transistor Function

32
Practical Application - nMOS vs pMOS

• nMOS pass good 0s, so connect source to GND
• pMOS pass good 1s, so connect source to VDD

VDD
GND
33
Practical Application - CMOS Gates nMOS

A P1 N1 Y
0
1
34
Practical Application - CMOS Gates nMOS

A P1 N1 Y
0 ON OFF 1
1 OFF ON 0