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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.

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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:3

  • AB + C   is known as a AOI 2-1 gate.
  • AB + CD   is known as a AOI 2-2 gate.
  • ABC + DEF   is known as a AOI 3-3 gate.
  • ABCD + EFGH   is known as a AOI 4-4 gate.
  • ABCDE + FGH + JK   is known as a AOI 4-3-2 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:

Q = ( A ∧ B ) ∨ C ¯ . {\displaystyle Q={\overline {(A\wedge B)\vee C}}.}
INPUTA   B   COUTPUT Q
0001
0010
0101
0110
1001
1010
1100
1110

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).456

The 2-2 AOI gate can be represented by the following boolean equation and truth table:

Q = ( A ∧ B ) ∨ ( C ∧ D ) ¯ . {\displaystyle Q={\overline {(A\wedge B)\vee (C\wedge D)}}.}
INPUTA   B   C   DOUTPUT Q
00001
00011
00101
00110
01001
01011
01101
01110
10001
10011
10101
10110
11000
11010
11100
11110

3-3 AOI gate

Real world examples of an 3-3 AOI gate is found in the SN74LS51 logic IC (see further below).7

The 3-3 AOI gate can be represented by the following boolean equation and truth table:

Q = ( A ∧ B ∧ C ) ∨ ( D ∧ E ∧ F ) ¯ . {\displaystyle Q={\overline {(A\wedge B\wedge C)\vee (D\wedge E\wedge F)}}.}

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).8

The 4-4 AOI gate can be represented by the following boolean equation and truth table:

Q = ( A ∧ B ∧ C ∧ D ) ∨ ( E ∧ F ∧ G ∧ H ) ¯ . {\displaystyle Q={\overline {(A\wedge B\wedge C\wedge D)\vee (E\wedge F\wedge G\wedge H)}}.}

Its logic table would have 256 entries, but is not shown.

Extensions to multiple levels

It is possible to create multi-level compound gates, which combine the logic of AND-OR-Invert gates with OR-AND-invert gates.9 An example is shown below. The parts implementing the same logic have been put in boxes with the same color.

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 gate10
  • CD4086B = single expandable 2-2-2-2 AOI gate"11
  • 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-AND12

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 expandable13 (SN54 is military version of SN74)
  • SN74LS51 = 2-2 AOI gate and 3-3 AOI gate14
  • SN54LS54 = single 2-3-3-2 AOI gate15

See also

Wikimedia Commons has media related to AND-OR-NOT gates.

References

  1. Product of Sums reduction using Karnaugh Map. https://dyclassroom.com/boolean-algebra/product-of-sums-reduction-using-karnaugh-map ↩

  2. Sum Of Product (SOP) & Product Of Sum (POS). https://www.electricaltechnology.org/2018/05/sum-of-product-sop-product-of-sum-pos.html ↩

  3. AOI and OAI complex gates https://tams.informatik.uni-hamburg.de/applets/hades/webdemos/10-gates/00-gates/complex.html ↩

  4. "CD4085B Datasheet". Texas Instruments. 2003. Archived (PDF) from the original on March 5, 2019. https://www.ti.com/lit/gpn/cd4085b ↩

  5. "SN74LS51 Datasheet". Texas Instruments. 1988. Archived (PDF) from the original on November 30, 2020. https://www.ti.com/lit/gpn/sn74ls51 ↩

  6. "SN5450 Datasheet". Texas Instruments. 1988. Archived (PDF) from the original on July 26, 2018. https://www.ti.com/lit/gpn/sn5450 ↩

  7. "SN74LS51 Datasheet". Texas Instruments. 1988. Archived (PDF) from the original on November 30, 2020. https://www.ti.com/lit/gpn/sn74ls51 ↩

  8. "CD4048B Datasheet". Texas Instruments. 2003. Archived (PDF) from the original on March 5, 2019. https://www.ti.com/lit/gpn/cd4048b ↩

  9. Fischer, P. "Aussagenlogik und Gatter" (PDF). University of Heidelberg. Retrieved 2024-01-21. https://sus.ziti.uni-heidelberg.de/Lehre/WS1617_DST/DST_Fischer_03_Logik_Gatter.pptx.pdf ↩

  10. "CD4085B Datasheet". Texas Instruments. 2003. Archived (PDF) from the original on March 5, 2019. https://www.ti.com/lit/gpn/cd4085b ↩

  11. "CD4086B Datasheet". Texas Instruments. 2003. Archived (PDF) from the original on April 15, 2019. https://www.ti.com/lit/gpn/cd4086b ↩

  12. "CD4048B Datasheet". Texas Instruments. 2003. Archived (PDF) from the original on March 5, 2019. https://www.ti.com/lit/gpn/cd4048b ↩

  13. "SN5450 Datasheet". Texas Instruments. 1988. Archived (PDF) from the original on July 26, 2018. https://www.ti.com/lit/gpn/sn5450 ↩

  14. "SN74LS51 Datasheet". Texas Instruments. 1988. Archived (PDF) from the original on November 30, 2020. https://www.ti.com/lit/gpn/sn74ls51 ↩

  15. "SN54LS54 Datasheet". Texas Instruments. 1988. Archived (PDF) from the original on March 5, 2018. https://www.ti.com/lit/gpn/sn54ls54 ↩