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111 lines
3.1 KiB
Markdown
111 lines
3.1 KiB
Markdown
---
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title: "02-combinatorial-logic"
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aliases:
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tags:
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- cosc204
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- lecture
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sr-due: 2022-08-16
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sr-interval: 20
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sr-ease: 250
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---
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- [slides](https://blackboard.otago.ac.nz/bbcswebdav/pid-2954102-dt-content-rid-18888626_1/courses/COSC204_S2DNI_2022/L2%20-%20Combinatorial%20Logic.pdf)
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- [transistors](notes/transistors.md)
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- [combinatorial-logic-circuit](notes/combinatorial-logic-circuit.md)
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- [boolean-equations](notes/boolean-equations.md)
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- [logic-gates](notes/logic-gates.md)
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- [demultiplexor](notes/demultiplexor.md)
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# Circuit basics:
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- current flows + to -
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- input to a unit (e.g., LED) is the + end
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# Combinatorial Logic Circuit
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> [!Definition]
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> Combinatorial Logic Circuit is a circuit whose digital outputs are dependent only on its digital inputs
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They can be described using logic expressions and therefore logic gates. We assume the outputs respond immediately^[1]
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They can be defined:
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- Using a truth table
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- Using boolean equations ($Q\ =\ A+\ B$)
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- Using graphical symbols
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- [1 Bit half adder](https://i.imgur.com/mjCVU4I.png)
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- [1 Bit full adder: (includes carry input)](https://i.imgur.com/yu6kS83.png)
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- [Ripple carry adder](https://i.imgur.com/HtEIZ5t.png)
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- 3 Bit parity Generator
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- Adds an extra bit to the input data so that the number of ones in the output is always odd
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- Used for error checking
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- [truth table](https://i.imgur.com/KDUiJbN.png)
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- [boolean equation](https://i.imgur.com/mwBpnlO.png)
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- [circuit](https://i.imgur.com/tsgDISC.png)
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- [7 segment displlay](https://i.imgur.com/qtPmtwR.png)
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# Boolean Equations
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Precedence
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- NOT is unary, so it has the highest precedence
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- AND is mulitply, so it comes next
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- OR is like plusl, to it comes last
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Creating boolean equations:
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- for each row where output is 1
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- write the equation as a function of the inputs (using AND)
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- Write the final equation, putting OR between each clause
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- [example](https://i.imgur.com/RoBTEfH.png)
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# De Morgan's Theorum
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- !(A + B) = A! & !B
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- !(A + B + C + ... + X) = !A & !B & !C & ... & !X
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- [truth table](https://i.imgur.com/QegVxkx.png)
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- any boolean function can be represented as the sum of logical products
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- All combinatorial circuits can be described using just one gate type (either nand or nor) [^2]
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# Transistors
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- B: Base ⇒ A swtich connecting C to E
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- open (C is disconnected from E) when supplied 0v
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- closed (C is connected to E) by applying +5V
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- C: Collector
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- E: Emitter
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Possible to create NAND gate using just transistors ∴ possible to create all logic gates using only transistors [^3]
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# Demultiplexor
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- Mutli way swtich where the address determines which output recieves the input
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- [2 bit address diagram and truth table](https://i.imgur.com/XQteI3j.png)
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# Logic Gates
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- NOT
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- AND
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- OR
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- NAND
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- NOR
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- XOR/EOR
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[^1] : In real circuits propagation delay must be considered, hence the clock cycle on CPUs
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[^2] : The Apollo Guidance Computer used about 5600 NOR gates and no other gate types!
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[^3] : They dont actually do it this way (see lab) |