数字设计基础双语课件(第3章)

1、 3. Designing combinational circuits 3.1 Combinational circuits 3.2Implementing Boolean expressions 3.3 Simplifying logic circuits3.4 MSI combinational logic devices1 3.1 Combinational circuitsA combinational function can be defined by Boolean expression, truth table and circuit diagram. F = AB + CT

2、ruth tableABCFCircuit diagramFor example:2 3.2 Implementing Boolean expressions1. Sum-of-product expressions （1）Product termA term consisting of variables connected with AND operators is a product term.（2）Sum-of-product expressionAn expression with product terms connected by OR operators is a sum-of

3、-product (SOP) expression.For example: F(A,B,C) = AB + AC3 3.2 Implementing Boolean expressions2. Canonical expressions （1）MintermThe product term which contains all the variables (or their complements) is called a minterm.Each minterm can be described by a binary number such that each digit in the

4、number is a 1 if the variable in the minterm is true, and a 0 if the variable is complemented.The binary number for each minterm is unique.4 3.2 Implementing Boolean expressionsThree-variable Minterm numbering5 3.2 Implementing Boolean expressions（2）Canonical sum-of-product expressionThe expression

5、with minterms connected by OR operators is a canonical sum-of-product expression.（3）Conversion into canonical formA canonical sum-of-product expression can be obtained from a non-canonical sum-of-product expression by multiplying each term by (A+A)(B+B)(), where A,B, etc. are variables not in the pr

6、oduct term.6 3.2 Implementing Boolean expressionsExp1: Expand the expression F(A,B,C)= AB+AC+ABC to the canonical sum-of-product expression.F(A, B, C) = AB + AC + ABC= AB (C+C) + A (B+B) C + ABC= ABC+ ABC+ ABC+ ABC + ABC= ABC+ ABC+ ABC+ ABC 7 3.2 Implementing Boolean expressions2. Product-of-sum exp

7、ressions （1）Sum termA term consisting of variables (or their complements) connected with OR operators is a sum term.（2）Product-of-sum expressionAn expression with sum terms connected by AND operators is a product-of-sum (POS) expression.For example: F(A,B,C) = (A+B) (A+C)8 3.2 Implementing Boolean e

8、xpressions（3）MaxtermThe sum term which contains all the variables (or their complements) is called a maxterm.Each maxterm can be described by a binary number such that each digit in the number is a 0 if the variable in the maxterm is true, and a 1 if the variable is complemented.（4）Canonical product

9、-of-sum expressionThe expression with maxterms connected by OR operators is a canonical sum-of-product expression.9 3.2 Implementing Boolean expressionsThree-variable Maxterm numbering10 3.3 Simplifying logic circuitsSimplifying an expression usually means reducing the number of variables in each te

10、rm and reducing the number of terms.1. Simplification2. Algebraic minimization methodExp2: F(A,B,C) = A + AC + ABC= A (1+ C + BC) = AAlgebraic minimization applies Boolean identities and laws to simplify expressions.11 3.3 Simplifying logic circuitsAlgebraic minimization is capable of reducing any e

11、xpression to its simplest form, however, it relies on the skill of the individual in applying the appropriate rules. Limitations for algebraic method 12 3.3 Simplifying logic circuits3. Karnaugh map minimization method（1）Karnaugh mapKarnaugh map consists of two-dimensional array squares. Each square

12、 corresponds to a minterm. The minterms of adjacent squares, either horizontally or vertically, have the same variables except only one variable different.Karnaugh map method is a visual minimization method, by which we can identify the terms that can be combined.13 3.3 Simplifying logic circuits3,4

13、 variable Karnaugh map14 3.3 Simplifying logic circuits（2）Mapping functions onto the Karnaugh mapSquares should be marked with 1s for those minterms appear in the function to be minimized.Exp3: Mapping the function expression F(A,B,C) = ABC + ABC + ABC onto a Karnaugh map.15 3.3 Simplifying logic ci

14、rcuits（3）Karnaugh map minimization methodTo the sum-of-product expressions, the basic minimization steps are:) Mark those squares with a 1 that correspond to the terms in the expression.) Form the 1s into the largest valid groups of 1s.) Read the terms from the map which correspond to the groups (co

15、ver the 1s).16 3.3 Simplifying logic circuitsExp4: Minimize the function expression by Karnaugh map: F(A,B,C) = A B C + A B C + A B C F(A,B,C) = A B + A B C Adjacent 1s can be combined into a group corresponding to a single product term, eliminating the different variable.17 3.3 Simplifying logic ci

16、rcuitsExp5: Minimize the function expression by Karnaugh map: F(A,B,C) = A B C + A B C F(A,B,C) = A C Squares on one edge are also adjacent to those on the opposite edge (top and bottom or left and right). 18 3.3 Simplifying logic circuitsLarger groupsThe final minimization term is given by those va

17、riables which are common to all the squares in the group.19 3.3 Simplifying logic circuitsTruth table description and the Karnaugh map There is a direct relationship between the squares on a Karnaugh map and the truth table description of the function as both have one entry for each minterm.20 3.3 S

18、implifying logic circuits4. Incompletely specified functions（1）dont care termThe combinations of input values that cannot occur or can occur but the output values for them are irrelevant are treated as dont care terms, and marked by X.（2）Incompletely specified functionThe function with dont care ter

19、m is called incompletely specified function.21 3.3 Simplifying logic circuits（3）Minimizing incompletely specified functionsWe can take advantage of the dont care terms in minimization. Each X can be considered as 0 or 1 whichever leads to the simplest solution.Exp6: Suppose a logic circuit is design

20、ed to generate a 1 when the BCD digit 8 appears. What will the function be? F = A D22 3.4 MSI combinational logic devices1. DecoderA decoder is designed to recognize the different patterns that can occur on a set of inputs. One output of the decoder is activated for each of the possible binary patte

21、rns that occur on the inputs.Lets take a 3-line-to-8-line decoder 74LS138 as an example:three inputs: A2, A1, A0eight outputs: Y0 Y7enable inputs: S3, S2, S1（1）Function 23 3.4 MSI combinational logic devicesTruth table of a 3-line-to-8-line decoder24 3.4 MSI combinational logic devicesThe enable inp

22、uts enable the device and allow its outputs to change with the inputs:) When S11、S2S30 ，the decoder is enabled，Y0 Y7are determined by A0 A2 ；) When S10 or S21 or S31 ，the decoder is disabled， Y0 Y7 will be limited to 1；The outputs of the decoder are active-low, which means that they are normally at

23、a logic 1 and become a logic 0 to indicate activation. An output becomes a 0 when the corresponding input pattern has been applied.25 3.4 MSI combinational logic devices（2）Applications Address decoderDecoder can be used in computer interface design to identify addresses of memory modules. Suppose th

24、e memory module address to be recognized is 010, then output Y2 is chosen to activate the memory module.Exp6:26 3.4 MSI combinational logic devicesLogic function generatorAs each output of a decoder implements one minterm, therefore, the decoder can be used to implement a combinational logic functio

25、n.Exp7: Implement the function F = ABC + ABC + ABC by using a 3-line-to-8-line decoder.Three outputs need to be selected, Y5,Y6 and Y7 which are connected to the inputs of a three-input NAND gate.27 3.4 MSI combinational logic devicesCircuit diagram 28 3.4 MSI combinational logic devices2. Multiplex

26、er (Selector)（1）Function Multiplexer is a logic circuit which allows one of a set of data inputs to be selected and fed into a single output.Lets take a 8-line-to-1-line selector or multiplexer as an example:Model of a multiplexer3 data select inputs: A2, A1, A01 data output: Y8 data inputs: D0 D729

27、 3.4 MSI combinational logic devicesMSI devices 74LS151: 8-line-to-1-line multiplexer 74LS153: dual 4-line-to-1-line multiplexer 74LS157: quad 2-line-to-1-line multiplexer 30 3.4 MSI combinational logic devices（2）Applications Address decoderExp8: Suppose the memory module address to be recognized is

28、 010.D0 D1 D2 D3 D4 D5D6D7SA2A1A00Y MultiplexerY 0100101To memory Then output Y is 0, which can be used to activate the memory module.Multiplexer can be used as an address decoder.31 3.4 MSI combinational logic devicesLogic function generatorApart from its principle function for selecting one data i

29、nput to feed to a single output, the multiplexer can be used to implement a sum-of-product expression. If a data input is set to a 1 and the data input is selected by the combination of select input values, the Y output will be a 1.Hence by connecting each data input to either a 0 or a 1, depending upon whether the associated minterm is a part of the required function that Y output will implement.32 3.4 MSI combinational logic de