# Logic and Computer Design Fundamentals. Chapter 2 Combinational Logic Circuits. Part 3 Additional Gates and Circuits

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1 Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 3 Additional Gates and Circuits Charles Kime & Thomas Kaminski 28 Pearson Education, Inc. (Hyperlinks are active in View Show mode) Overview Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard Forms Part 2 Circuit Optimization Two-Level Optimization Map Manipulation Practical Optimization (Espresso) Multi-Level Circuit Optimization Part 3 Additional Gates and Circuits Other Gate Types Exclusive-OR Operator and Gates High-Impedance Outputs Chapter 2 - Part 3 2

2 Other Gate Types Why? Implementation feasibility and low cost Power in implementing Boolean functions Convenient conceptual representation Gate classifications Primitive gate - a gate that can be described using a single primitive operation type (AND or OR) plus an optional inversion(s). Complex gate - a gate that requires more than one primitive operation type for its description Primitive gates will be covered first Chapter 2 - Part 3 3 Buffer A buffer is a gate with the function F = : In terms of Boolean function, a buffer is the same as a connection! So why use it? A buffer is an electronic amplifier used to improve circuit voltage levels and increase the speed of circuit operation. F Chapter 2 - Part 3 4 2

3 NAND Gate The basic NAND gate has the following symbol, illustrated for three inputs: AND-Invert (NAND) F(,, ) = NAND represents NOT AND, i. e., the AND function with a NOT applied. The symbol shown is an AND-Invert. The small circle ( bubble ) represents the invert function. Chapter 2 - Part 3 5 NAND Gates (continued) Applying DeMorgan's Law gives Invert-OR (NAND) F (,, ) = + + This NAND symbol is called Invert-OR, since inputs are inverted and then ORed together. AND-Invert and Invert-OR both represent the NAND gate. Having both makes visualization of circuit function easier. A NAND gate with one input degenerates to an inverter. Chapter 2 - Part 3 6 3

4 NAND Gates (continued) The NAND gate is the natural implementation for CMOS technology in terms of chip area and speed. Universal gate - a gate type that can implement any Boolean function. The NAND gate is a universal gate as shown in Figure 2-24 of the text. NAND usually does not have a operation symbol defined since the NAND operation is not associative, and we have difficulty dealing with non-associative mathematics! Chapter 2 - Part 3 7 NOR Gate The basic NOR gate has the following symbol, illustrated for three inputs: OR-Invert (NOR) F (,,) = + + NOR represents NOT - OR, i. e., the OR function with a NOT applied. The symbol shown is an OR-Invert. The small circle ( bubble ) represents the invert function. Chapter 2 - Part 3 8 4

5 NOR Gate (continued) Applying DeMorgan's Law gives Invert-AND (NOR) This NOR symbol is called Invert-AND, since inputs are inverted and then ANDed together. OR-Invert and Invert-AND both represent the NOR gate. Having both makes visualization of circuit function easier. A NOR gate with one input degenerates to an inverter. Chapter 2 - Part 3 9 NOR Gate (continued) The NOR gate is a natural implementation for some technologies other than CMOS in terms of chip area and speed. The NOR gate is a universal gate NOR usually does not have a defined operation symbol since the NOR operation is not associative, and we have difficulty dealing with non-associative mathematics! Chapter 2 - Part 3 5

6 Exclusive OR/ Exclusive NOR The eclusive OR (OR) function is an important Boolean function used extensively in logic circuits. The OR function may be; implemented directly as an electronic circuit (truly a gate) or implemented by interconnecting other gate types (used as a convenient representation) The eclusive NOR function is the complement of the OR function By our definition, OR and NOR gates are complex gates. Chapter 2 - Part 3 Exclusive OR/ Exclusive NOR Uses for the OR and NORs gate include: Adders/subtractors/multipliers Counters/incrementers/decrementers Parity generators/checkers Definitions The OR function is: = + The eclusive NOR (NOR) function, otherwise known as equivalence is: = + Strictly speaking, OR and NOR gates do no exist for more that two inputs. Instead, they are replaced by odd and even functions. Chapter 2 - Part 3 2 6

7 Truth Tables for OR/NOR Operator Rules: OR NOR The OR function means: OR, but NOT BOTH Why is the NOR function also known as the equivalence function, denoted by the operator? ( ) or Chapter 2 - Part 3 3 OR/NOR (Continued) The OR function can be extended to 3 or more variables. For more than 2 variables, it is called an odd function or modulo 2 sum (Mod 2 sum), not an OR: = The complement of the odd function is the even function. The OR identities: = = = = = ( ) = ( ) = Chapter 2 - Part 3 4 7

8 Symbols For OR and NOR OR symbol: NOR symbol: Shaped symbols exist only for two inputs Chapter 2 - Part 3 5 OR Implementations The simple SOP implementation uses the following structure: A NAND only implementation is: Chapter 2 - Part 3 6 8

9 Odd and Even Functions The odd and even functions on a K-map form checkerboard patterns. The s of an odd function correspond to minterms having an index with an odd number of s. The s of an even function correspond to minterms having an index with an even number of s. Implementation of odd and even functions for greater than four variables as a two-level circuit is difficult, so we use trees made up of : 2-input OR or NORs 3- or 4-input odd or even functions Chapter 2 - Part 3 7 Example: Odd Function Implementation Design a 3-input odd function F = + + with 2-input OR gates Factoring, F = ( + ) + The circuit: F Chapter 2 - Part 3 8 9

10 Example: Even Function Implementation Design a 4-input odd function F = W with 2-input OR and NOR gates Factoring, F = (W + ) + ( + ) The circuit: W F Chapter 2 - Part 3 9 Parity Generators and Checkers In Chapter, a parity bit added to n-bit code to produce an n + bit code: Add odd parity bit to generate code words with even parity Add even parity bit to generate code words with odd parity Use odd parity circuit to check code words with even parity Use even parity circuit to check code words with odd parity Example: n = 3. Generate even parity code words of length four with odd parity generator: Check even parity code words of length four with odd parity checker: Operation: (,,) = (,,) gives (,,,P) = (,,,) and E =. If changes from to between P generator and checker, then E = indicates an error. Chapter 2 - Part 3 2 P E

11 Hi-Impedance Outputs Logic gates introduced thus far have and output values, cannot have their outputs connected together, and transmit signals on connections in only one direction. Three-state logic adds a third logic value, Hi- Impedance (Hi-), giving three states:,, and Hi- on the outputs. The presence of a Hi- state makes a gate output as described above behave quite differently: and become,, and Hi- cannot becomes can, and only one becomes two Chapter 2 - Part 3 2 Hi-Impedance Outputs (continued) What is a Hi- value? The Hi- value behaves as an open circuit This means that, looking back into the circuit, the output appears to be disconnected. It is as if a switch between the internal circuitry and the output has been opened. Hi- may appear on the output of any gate, but we restrict gates to: a 3-state buffer, or Optional: a transmission gate (See Reading Supplement: More on CMOS Circuit-Level Design), each of which has one data input and one control input. Chapter 2 - Part 3 22

12 The 3-State Buffer For the symbol and truth table, IN is the data input, and EN, the control input. For EN =, regardless of the value on IN (denoted by ), the output value is Hi-. For EN =, the output value follows the input value. Variations: Data input, IN, can be inverted Control input, EN, can be inverted by addition of bubbles to signals. Symbol IN OUT EN Truth Table EN IN OUT Hi- Chapter 2 - Part 3 23 Resolving 3-State Values on a Connection Connection of two 3-state buffer outputs, B and B, to a wire, OUT Assumption: Buffer data inputs can take on any combination of values and Resulting Rule: At least one buffer output value must be Hi-. Why? How many valid buffer output combinations exist? What is the rule for n 3-state buffers connected to wire, OUT? How many valid buffer output combinations exist? Resolution Table B B OUT Hi- Hi- Hi- Hi- Hi- Hi- Hi- Chapter 2 - Part

13 3-State Logic Circuit Data Selection Function: If s =, OL = IN, else OL = IN Performing data selection with 3-state buffers: EN IN EN IN OL IN EN IN EN Since EN = S and EN = S, one of the two buffer outputs is always Hi- plus the last row of the table never occurs. S OL Chapter 2 - Part 3 25 More Complex Gates The remaining complex gates are SOP or POS structures with and without an output inverter. The names are derived using: A - AND O - OR I - Inverter Numbers of inputs on first-level gates or directly to second-level gates Chapter 2 - Part

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