Engineering:AND-OR-invert
AND-OR-invert (AOI) logic and AOI gates are two-level compound (or complex) logic functions constructed from the combination of one or more AND gates followed by a NOR gate (equivalent to an OR gate through an Inverter gate, which is the "OI" part of "AOI"). Construction of AOI cells is particularly efficient using CMOS technology, where the total number of transistor gates can be compared to the same construction using NAND logic or NOR logic. The complement of AOI logic is OR-AND-invert (OAI) logic, where the OR gates precede a NAND gate.[1]
Overview
Most logic optimization result in a sum-of-products or product-of-sums logic expression.[2]
AOI is used for sum-of-products, the variables are ANDed to form minterms which are ORed together then inverted, such as:
- AB + C is known as a 2-1 AOI gate.
- AB + CD is known as a 2-2 AOI gate.[3]
- ABC + DEF is known as a 3-3 AOI gate.[4]
- ABCD + EFGH is known as a 4-4 AOI gate.[5]
- ABCDE + FGH + JK is known as a 4-3-2 AOI gate.
- and other variations.
Examples
AOI gates perform one or more AND operations followed by an OR operation then an inversion.
2-1 AOI gate
The 2-1 AOI gate can be represented by the following boolean equation and truth table:
- [math]\displaystyle{ Q = \overline{(A \wedge B) \vee C}. }[/math]
| 2-1 AOI | |||
| INPUT A B C |
OUTPUT Q | ||
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 0 |
2-2 AOI gate
Real world examples of an 2-2 AOI gate are found in the CD4085B, SN74LS51, SN5450 logic ICs (see further below).[3][4][6]
The 2-2 AOI gate can be represented by the following boolean equation and truth table:
- [math]\displaystyle{ Q = \overline{(A \wedge B) \vee (C \wedge D)}. }[/math]
| 2-2 AOI | ||||
| INPUT A B C D |
OUTPUT Q | |||
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 0 | 1 | 1 | 0 |
| 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 | 0 |
3-3 AOI gate
Real world examples of an 3-3 AOI gate is found in the SN74LS51 logic IC (see further below).[4]
The 3-3 AOI gate can be represented by the following boolean equation and truth table:
- [math]\displaystyle{ Q = \overline{(A \wedge B \wedge C) \vee (D \wedge E \wedge F)}. }[/math]
Its logic table would have 64 entries, but is not shown.
4-4 AOI gate
Real world examples of an 4-4 AOI gate is found in the CD4048B logic IC (see further below).[5]
The 4-4 AOI gate can be represented by the following boolean equation and truth table:
- [math]\displaystyle{ Q = \overline{(A \wedge B \wedge C \wedge D) \vee (E \wedge F \wedge G \wedge H)}. }[/math]
Its logic table would have 256 entries, but is not shown.
Electronic implementation
AND-OR-invert (AOI) and OAI gates can be readily implemented in CMOS circuitry. AOI gates are particularly advantaged in that the total number of transistors (or gates) is less than if the AND, NOT, and OR functions were implemented separately. This results in increased speed, reduced power, smaller area, and potentially lower fabrication cost. For example, a 2-1 AOI gate can be constructed with 6 transistors in CMOS, compared to 10 transistors using a 2-input NAND gate (4 transistors), an inverter (2 transistors), and a 2-input NOR gate (4 transistors).
In NMOS logic, the lower half of the CMOS circuit is used in combination with a load device or pull-up transistor (typically a depletion load or a dynamic load).
AOI gates are similarly efficient in transistor–transistor logic (TTL).
- Examples
CMOS 4000-series logic family:
- CD4085B = dual 2-2 AOI gate[3]
- CD4086B = single expandable 2-2-2-2 AOI gate"[7]
- CD4048B = single expandable 8-input 8-function with three-state output, 8 choices for gate type: 8 NOR / 8 OR / 8 NAND / 8 AND / 4-4 AND-OR-Invert / 4-4 AND-OR / 4-4 OR-AND-Invert / 4-4 OR-AND[5]
TTL 7400-series logic family: (in past decades, a number of AOI parts were available in the 7400 family, but currently most are obsolete (no longer manufactured))
- SN5450 = dual 2-2 AOI gate, one is expandable[6] (SN54 is military version of SN74)
- SN74LS51 = 2-2 AOI gate and 3-3 AOI gate[4]
- SN54LS54 = single 2-3-3-2 AOI gate[8]
See also
References
- ↑ Product of Sums reduction using Karnaugh Map.
- ↑ Sum Of Product (SOP) & Product Of Sum (POS).
- ↑ 3.0 3.1 3.2 "CD4085B Datasheet". 2003. https://www.ti.com/lit/gpn/cd4085b.
- ↑ 4.0 4.1 4.2 4.3 4.4 4.5 "SN74LS51 Datasheet". 1988. https://www.ti.com/lit/gpn/sn74ls51.
- ↑ 5.0 5.1 5.2 "CD4048B Datasheet". 2003. https://www.ti.com/lit/gpn/cd4048b.
- ↑ 6.0 6.1 "SN5450 Datasheet". 1988. https://www.ti.com/lit/gpn/sn5450.
- ↑ "CD4086B Datasheet". 2003. https://www.ti.com/lit/gpn/cd4086b.
- ↑ "SN54LS54 Datasheet". 1988. https://www.ti.com/lit/gpn/sn54ls54.
- Tinder, Richard F. (2000). Engineering digital design: Revised Second Edition. pp. 317–319. ISBN 0-12-691295-5. https://books.google.com/books?id=6x0pjjMKRh0C&dq=AOI+gate&pg=PT347. Retrieved 2008-07-04.
- John, Michael (1997). Application-Specific Integrated Circuits. http://iroi.seu.edu.cn/books/asics/Book2/CH02/CH02.4.htm#pgfId=799. Retrieved 2008-07-04.
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