We know binary We know how to add and subtract in binary –Same as in decimal Next up: learn how apply this knowledge Boolean and Binary Inputs

Discrete voltages represented by 1 and 0 For example: 0 = ground (GND) or 0 volts 1 = V DD or 5 volts What about 4.99 volts? Is that a 0 or a 1? What about 3.2 volts? Boolean and Binary Inputs

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

Logic Gates Perform logic functions: –inversion (NOT), AND, OR, etc. Single-input: –NOT gate, buffer Two-input: –AND, OR, etc. Boolean and Binary Inputs

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

Practical Application NM H = V OH – V IH NM L = V IL – V OL

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

Boolean Algebra 8 Boolean algebra is based on the binary number system George Boole (November 2, 1815 – December 8, 1864)

Boolean Algebra 9 Truth Tables Boolean operations can be defined using a Truth Table. Inversion:A AND: ABA·B OR: ABA+B or just AB

Boolean Algebra 10 Truth Tables Boolean operations can be defined using a Truth Table. XOR: ABABABAB NAND: ABA·B NOR: ABA+B

Boolean Algebra 11 DeMorgan’s Theorems ABA·BA·BABA + B A·B = A + BA+B = A · B Proof: ABA+BA+BABA · B

Boolean Algebra 12 Example ABCABCABBCBCABF F = AB + BC + BC + AB

Boolean Algebra 13 Another Example F = AB + BC + BC + AC ABCCABBCBCACF

Boolean Algebra 14 F = AB + BC + BC + AC ABCCABBCBCACF Is there a simpler way to determine how these inputs can produce these outputs? Probably.

Boolean Algebra 15 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 ABCDF ABCDF F = A B C D

Boolean Algebra 16 ABCDF ABCDF F = _ A B C D Karnaugh maps Karnaugh Maps - Rules of Simplification Rule 1. Groups may not include any cell containing a zero

Boolean Algebra 17 ABCDF ABCDF F = _ A B C D Karnaugh maps Karnaugh Maps - Rules of Simplification Rule 2. Groups may be horizontal or vertical, but not diagonal.

Boolean Algebra 18 ABCDF ABCDF F = A B C D Karnaugh Maps - Rules of Simplification Rule 3. Groups must contain 1, 2, 4, 8, or in general 2 n cells. That is if n = 1, a group will contain two 1's since 2 1 = 2. If n = 2, a group will contain four 1's since 2 2 = 4.

Boolean Algebra 19 ABCDF ABCDF F = A B C D 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.

Boolean Algebra 20 ABCDF ABCDF F = A B C D Karnaugh Maps - Rules of Simplification Rule 5. Groups may overlap.

Boolean Algebra 21 ABCDF ABCDF F = A B C D 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.

Boolean Algebra 22 ABCDF ABCDF F = A B C D 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.

Boolean Algebra 23 F = A B C D 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. 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. Rule 5. Groups may overlap. h?v=PA0kBrpHLM4 Rule 1. Groups may not include any cell containing a zero Rule 4. Each group should be as large as possible. Each cell containing a one must be in at least one group. Rule 3. Groups must contain 1, 2, 4, 8, or in general 2 n cells. That is if n = 1, a group will contain two 1's since 2 1 = 2. If n = 2, a group will contain four 1's since 2 2 = 4. Rule 2. Groups may be horizontal or vertical, but not diagonal.

Boolean Algebra 24 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 ABCDF ABCDF F = AC + AB + BCD ___ A B C D

Implementing Logic with Switches 25 X Y XYL XYL XOR: ABABABAB

Logic Gates 26

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 Robert Noyce ( )

Practical Application - MOS Transistors Metal oxide silicon (MOS) transistors: –Polysilicon (used to be metal) gate –Oxide (silicon dioxide) insulator –Doped silicon

Practical Application - Transistors: nMos Gate = 0 OFF (no connection between source and drain) Gate = 1 ON (channel between source and drain)

Practical Application - Transistors: pMOS pMOS transistor is opposite –ON when Gate = 0 –OFF when Gate = 1

Practical Application - Transistor Function

Practical Application - nMOS vs pMOS nMOS: pass good 0’s, so connect source to GND pMOS: pass good 1’s, so connect source to V DD V DD GND

Practical Application - CMOS Gates: nMOS AP1N1Y 0 1

Practical Application - CMOS Gates: nMOS AP1N1Y 0ONOFF1 1 ON0