The main contents of the chapter consist of the following: Design digital circuit from specification, digital inputs and outputs known, need to determine logic that can transform data, start in truth table form, create k-map for each output based on function of inputs, determine minimized sum-of-product representation, draw circuit diagram.
Lecture 11 Combinational Design Procedure Overvie w ° Design digital circuit from specification ° Digital inputs and outputs known • Need to determine logic that can transform data ° Start in truth table form ° Create K-map for each output based on function of inputs ° Determine minimized sum-of-product representation ° Draw circuit diagram Design Procedure (Mano) Design a circuit from a specification Determine number of required inputs and outputs Derive truth table Obtain simplified Boolean functions Draw logic diagram and verify correctness A B C RS 0 0 0 1 0 S=A+B+C R = ABC 1 1 0 1 1 1 0 1 1 1 Previously, we have learned… ° Boolean algebra can be used to simplify expressions, but not obvious: • how to proceed at each step, or • if solution reached is minimal ° Have seen five ways to represent a function: • Boolean expression • truth table • logic circuit • minterms/maxterms • Karnaugh map Combinational logic design ° Use multiple representations of logic functions ° Use graphical representation to assist in simplification of function ° Use concept of “don’t care” conditions ° Example - encoding BCD to seven segment display ° Similar to approach used by designers in the field BCD to Seven Segment Display ° Used to display binary coded decimal (BCD) numbers using seven illuminated segments ° BCD uses 0’s and 1’s to represent decimal digits Need four bits to represent required 10 digits ° Binary coded decimal (BCD) represents each decimal digit with four bits 0 0 0 0 1 0 0 a f g e b c d BCD to seven segment display ° List the segments that should be illuminated for each digit a,b,c,d,e,f b,c a,b,d,e,g a,b,c,d,g b,c,f,g a,c,d,f,g a,c,d,e,f,g a,b,c a,b,c,d,e,f,g a,b,c,d,f,g a f g e b c d BCD to seven segment display ° Derive the truth table for the circuit ° Each output column in one circuit Inputs Outputs Dec w x y z a b c d e 0 0 1 1 0 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 BCD to seven segment display ° Find minimal sum-of-products representation for each output For segment “a” : yz wx 00 01 11 10 00 1 01 1 11 10 1 Note: Have only filled in ten squares, corresponding to the ten numerical digits we wish to represent Don’t care conditions (BCD display) ° Fill in don’t cares for undefined outputs • Note that these combinations of inputs should never happen ° Leads to a reduced implementation For segment “a” : yz 00 01 11 10 wx 00 1 01 1 11 X X X X 10 1 Put in “X” (don’t care), and interpret as either 1 or 0 as desired … X X 10 For segment “b” : yz wx 00 01 11 10 00 1 1 01 1 11 X X X X X X 10 Fb = YZ 21 For segment “b” : yz wx 00 01 11 10 00 1 1 01 1 11 X X X X X X 10 F = W + Y Z + W X + YZ F = W + W X + YZ + Y Z F = W + X +Y ⊕ Z 22 BCD-to-Excess-3 Code converter ° BCD is a code for the decimal digits 0-9 ° Excess-3 is also a code for the decimal digits 23 Spec ifica °tion Inputs: a BCD input, A,B,C,D with A as the most of significant bit and D as the least significant bit BC °D Outputs: an Excess3 output W,X,Y,Z that corresponds to the BCD input to °Exc Internal operation – circuit to do the conversion in ess3 combinational logic 24 Formulation of BCD-to-Excess-3 ° Excess-3 code is easily formed by adding a binary to the binary or BCD for the digit ° There are 16 possible inputs for both BCD and Excess-3 ° It can be assumed that only valid BCD inputs will appear so the six combinations not used can be treated as don’t cares 25 Optimization – BCD-to-Excess-3 ° Lay out K-maps for each output, W X Y Z ° A step in the digital circuit design process 26 Placing on K-maps ° Where are the minterms located on a K-Map? 27 Expressions for W X Y Z ° W(A,B,C,D) = Σm(5,6,7,8,9) +d(10,11,12,13,14,15) ° X(A,B,C,D) = Σm(1,2,3,4,9) +d(10,11,12,13,14,15) ° Y(A,B,C,D) = Σm(0,3,4,7,8) +d(10,11,12,13,14,15) ° Z(A,B,C,D) = Σm(0,2,4,6,8) +d(10,11,12,13,14,15) 28 Minimize K-Maps ° W minimization ° Find W = A + BC + BD 29 Minimize K-Maps ° X minimization ° Find X = BC’D’+B’C+B’D 30 Minimize K-Maps ° Y minimization ° Find Y = CD + C’D’ 31 Minimize K-Maps ° Z minimization ° Find Z = D’ 32 Two level circuit implementation ° Have equations • W = A + BC + BD = A + B(C+D) • X = B’C + B’D + BC’D’ = B’(C+D) + BC’D’ • Y = CD + C’D’ • Z = D’ ° Factoring out (C+D) and call it T ° Then T’ = (C+D)’ = C’D’ • W = A + BT • X = B’T + BT’ • Y = CD + T’ • Z = D’ 33 Create the digital circuit ° Implementing the second set of equations where T=C+D results in a lower gate count ° This gate has a fanout of 34 34 Summary ° Need to formulate circuits from problem descriptions Determine number of inputs and outputs Determine truth table format Determine K-map Determine minimal SOP o There may be multiple outputs per design o Solve each output separately o Current approach doesn’t have memory o This will be covered next week 35 ... °Exc Internal operation – circuit to do the conversion in ess3 combinational logic 24 Formulation of BCD-to-Excess-3 ° Excess-3 code is easily formed by adding a binary to the binary or BCD... output based on function of inputs ° Determine minimized sum-of-product representation ° Draw circuit diagram Design Procedure (Mano) Design a circuit from a specification Determine number of required... function: • Boolean expression • truth table • logic circuit • minterms/maxterms • Karnaugh map Combinational logic design ° Use multiple representations of logic functions ° Use graphical representation