Logic Design Combinational Circuits. Digital Computer Design

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1 Logic Design Combinational Circuits Digital Computer Design

2 Topics Combinational Logic Karnaugh Maps Combinational uilding locks Timing 2

3 Logic Circuit logic circuit is composed of: Inputs Outputs Functional specification Timing specification functional spec inputs outputs timing spec 3

4 Example Nodes Inputs:,, C Outputs:, Z Internal: n E n E3 Circuit elements C E2 Z E, E2, E3 Each a circuit 4

5 Types of Logic Circuits Combinational Logic Memoryless Outputs determined by current values of inputs Sequential Logic Has memory Outputs determined by previous and current values of inputs inputs functional spec timing spec outputs 5

6 Rules of Combinational Composition Every element is combinational Every node is either an input or connects to exactly one output The circuit contains no cyclic paths Example: 6

7 oolean Equations Functional specification of outputs in terms of inputs Example: S = F(,, C in ) C out = F(,, C in ) S C L C C out in S = C in C out = + C in + C in 7

8 Some Definitions Complement: variable with a bar over it,, C Literal: variable or its complement,,,, C, C Implicant: product of literals C, C, C Minterm: product that includes all input variables C, C, C Maxterm: sum that includes all input variables (++C), (++C), (++C) 8

9 Exercise Simplify each of the following oolean equations. = C+ C + C = (+ + ) +(+ ) 9

10 Sum-of-Products (SOP) Form ll equations can be written in SOP form Each row has a minterm minterm is a product (ND) of literals Each minterm is TRUE for that row (and only that row) Form function by ORing minterms where output is Thus, a sum (OR) of products (ND terms) minterm minterm name m m m 2 m 3 = F(, ) = + = Σ m (, 3)

11 Product-of-Sums (POS) Form ll oolean equations can be written in POS form Each row has a maxterm maxterm is a sum (OR) of literals Each maxterm is FLSE for that row (and only that row) Form function by NDing maxterms where output is Thus, a product (ND) of sums (OR terms) maxterm maxterm name M M M 2 M 3 = F(, ) = ( + )( + ) = Π M (, 2)

12 Exercise Q: Write a oolean equation in sum-of-products and product-of-sums canonical form for (a) (b) F(X,,Z)=X +Z+XZ 2

13 From Minterms to Gates (Two-Level Logic) Two-level logic: NDs followed by ORs Example (In SOP Form): = C + C + C Circuit without simplification: C C minterm: C minterm: C minterm: C 3

14 Multilevel Combinational Logic Some logic functions require an enormous amount of hardware when built using two-level logic. Multi-level combinational circuits may use less hardware than their twolevel counterparts. (a) 3-input XOR and its two-level implementation (b) Multilevel implementation 8-input XOR. It would be really complex with 2-level implementation 4

15 DeMorgan s Theorem Theorem 2 = Dual = 2 The complement of the product is the sum of the complements. Dual: The complement of the sum is the product of the complements. = = + = + = 5

16 ubble Pushing Using DeMorgan s Theorem ackward: ody changes dds bubbles to inputs Forward: ody changes dds bubble to output 6

17 ubble Pushing Multilevel Circuit nalysis ubble pushing is especially helpful in analyzing and designing multilevel circuits. C D Draw gates in a form so bubbles cancel! 7

18 ubble Pushing Example no output bubble C D 8

19 ubble Pushing Example no output bubble C D bubble on input and output C D 9

20 ubble Pushing Example no output bubble C D C D C D no bubble on input and output = C + D bubble on input and output 2

21 Exercise Using De Morgan equivalent gates and bubble pushing methods, redraw the circuit and write the oolean equation. 2

22 Multiple-Output Circuits Design Example: Priority Circuit Output asserted corresponding to most significant TRUE input 3 2 PRIORIT CiIRCUIT Truth Table

23 Priority Circuit Hardware (From Truth Table) Given a clear specification, simply turn the words into equations and the equations into gates. Truth Table

24 Don t Cares We use the symbol X to describe inputs that the output doesn t care about Truth Table With Don t Cares 3 2 X X X X 3 2 X X 24

25 Exercise priority encoder has 2 N inputs. It produces an N-bit binary output indicating the most significant bit of the input that is TRUE, or if none of the inputs are TRUE. It also produces an output NONE that is TRUE if none of the inputs are TRUE. Design an eight-input priority encoder with inputs 7: and outputs 2. and NONE. For example, if the input is, the output should be and NONE should be. Give a simplified oolean equation for each output and sketch a schematic. 25

26 Exercise Find a minimal oolean equation for the following function: 26

27 Karnaugh Maps (K-Maps) oolean expressions can be minimized by combining terms Reducing an equation to the fewest number of implicants, where each implicant has the fewest literals K-maps minimize equations graphically. C C C C C C C C C C C 27

28 Example: 4-Input K-Map C D CD 28

29 Example: 4-Input K-Map C D CD = C + D + C + D 29

30 Example: K-Maps with Don t Cares C D X X X X X X X CD X X X X X X X 3

31 Example: K-Maps with Don t Cares C D X X X X X X X CD X X X X X = + D + C X X 3

32 Exercise circuit has four inputs and two outputs. The inputs Α 3: represent a number from to 5. Output P should be TRUE if the number is prime ( and are not prime, but 2, 3, 5, and so on, are prime). Output D should be TRUE if the number is divisible by 3. Give simplified oolean equations for each output using K-maps and sketch a circuit. 32

33 Contention: X Contention: circuit tries to drive output to and ctual value somewhere in between Could be,, or in forbidden zone Might change with voltage, temperature, time, noise Often causes excessive power dissipation Warnings: Contention usually indicates a bug. = = = X The symbol X indicates that the circuit node has an unknown or illegal value. 33

34 Floating: Z Floating, high impedance, open, high Z Floating output might be,, or somewhere in between Tristate uffer E E Z Z When the enable is TRUE, the tristate buffer acts as a simple buffer, transferring the input value to the output. When the enable is FLSE, the output is allowed to float (Z). 34

35 Tristate usses processor en to bus Tristate buffers are commonly used on busses that connect multiple chips. Many different drivers Exactly one is active at once from bus video to bus from bus Ethernet to bus en2 en3 shared bus from bus memory en4 to bus from bus 35

36 Multiplexer (Mux) Selects between one of N inputs to connect to output log 2 N-bit select input control input Example: 2: Mux D D S D D S S D D S D D D S D = D S + D S 36

37 Logic using Multiplexers Using mux as a lookup table = 37

38 Logic using Multiplexers Reducing the size of the mux = 38

39 Exercise Write a minimized oolean equation for the function performed by the following circuit. 39

40 Decoders N inputs, 2 N outputs Each output is a minterm One-hot outputs: only one output HIGH at once 2:4 Decoder

41 Logic Using Decoders OR minterms 2:4 Decoder Minterm = + = 4

42 Reading Chapter 2 of your book From 2. to

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