Unformatted text preview: CSE261 Spring 2008 Objectives:
To learn about the variety of ways in which a logic function can be represented how to analyze combinational logic circuits how to realize logic functions using available gates how to transform a word description into a logic function Combinational Analysis Review:
Logic Circuit Logic Function Truth Table Analyze a combinational logic circuit by obtaining a formal description of its logic function Outputs of a combinational logic circuit depends only on its current inputs (not on history) Kinds of combinational analysis: exhaustive (truth table) algebraic (expressions) CSE261 Combinational Logic Design #1 #1 CSE261 Combinational Logic Design #2 #2 Example: I) A B (A + B) B) F
(B C) Combinational Circuit Synthesis
II) Truth Table
1. C Logic Function Logic Circuit III)
A 0 1 2 3 4 5 6 7 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 F 0 0 1 1 1 1 1 1 F = [(A + B) (B C)] B) FA,B,C= (2,3,4,5,6,7) IV) F = A'BC'+A'BC+AB'C'+AB'C+ABC'+ABC BC' +A' BC+AB' +AB' C+ABC' Logic Function: The canonical sum expression. The canonical product expression. The canonical Implementations: ANDOR and its equivalent NANDNAND ANDNAND ORAND and its equivalent NORNOR ORNORLogic Function minimization: simplifying the logic function to reduce the number of gates. Minimization methods: Using theorems Karnaugh map
CSE261 Combinational Logic Design #4 #4 2. FA,B,C = (0,1) V)
A B VI)
F=A+B
F 3. F = (A+B+C)(A+B+C') (A+B+C)(A+B+C' 4. CSE261 Combinational Logic Design #3 #3 Combinational Logic Design 1 CSE261 Spring 2008 CombinationalCircuit Design CombinationalSometimes you can write an equation or equations directly using `logic' (the kind in your brain). logic' Example (alarm circuit):
ALARM = PANIC + ENABLE . EXITING' . SECURE' EXITING' SECURE' SECURE = WINDOW . DOOR . GARAGE ALARM = PANIC + ENABLE . EXITING' . (WINDOW . DOOR . GARAGE)' EXITING' GARAGE)' Example (cont.): Alarmcircuit transformation AlarmSumofproducts form Sum of Useful for programmable logic devices Multiply out Corresponding circuit: CSE261 Combinational Logic Design #5 #5 CSE261 Combinational Logic Design #6 #6 Sumofproducts form Sum of NANDNAND Implementation NANDF = XYZ + XYZ + XYZ
(XYZ) (X Y Z) X Z X (XYZ) (X YZ X Z)
F ANDOR AND Y Z (XYZZ (X Y ) X F = [(XYZ) (XYZ) (XYZ)] [(X Z) Z) NANDNAND NANDCSE261 Combinational Logic Design #7 #7 F = XYZ + XYZ + XYZ
CSE261 Combinational Logic Design #8 #8 Combinational Logic Design 2 CSE261 Spring 2008 Productofsums form Product of NORNOR Implementation NORF = (X + Y + Z)(X + Y' + Z)(X' + Y + Z)(X' + Y + Z')(X' + Y' + Z') (X' (X' (X' X Y OR  AND Z NOR  NOR
CSE261 Combinational Logic Design #9 #9 CSE261 Combinational Logic Design #10 #10 Minimization Example: Prime Number Detector
Truth table canonical sum (sum of minterms) Example (cont.): Minterm list
Row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 N3 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 N2 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 N1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 N0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F 0 1 1 1 0 1 0 1 0 0 0 1 0 1 0 0 canonical sum F = N3N2N1N0 (1,2,3,5,7,11,13) = N3 '. N2 '. N1 '. N0 + N 3 '. N2 '. N1 . N0 ' + N 3 '. N2 '. N1 . N0 + N 3 '. N2 . N1 '. N0 + N 3 '. N2 . N1 . N0 + N 3 . N2 '. N1 . N0 + N 3 . N2 . N1 '. N0 F = N3N2N1N0 (1,2,3,5,7,11,13) CSE261 Combinational Logic Design #11 #11 CSE261 Combinational Logic Design #12 #12 Combinational Logic Design 3 CSE261 Spring 2008 Example (cont.): Algebraic simplification
(T10) Visualizing T10  Karnaugh Maps
A representation of the truth table by a matrix of cells, where each cell corresponds to a minterm (or a maxterm) of the logic function. For nvariable function, we need 2n rows truth table and 2n cells. cells. The cell number is equivalent to the row number in the truth table. table. The truth table values are copied into their corresponding cells. cells. The cell arrangements help to identify the input variable redundancy X Y + X Y = X X Y + X Y = X F = N3N2N1N0 (1,2,3,5,7,11,13) = N3 '. N2 '. N1 '. N0 + N 3 '. N2 '. N1 . N0 + N 3 '. N2 . N1 '. N0 + N 3 '. N2 . N1 . N0 + ... = (N3'. N2'. N1'. N0 + N3'. N2'. N1. N0) + (N3'. N2. N1'. N0 + N3'. N2. N1'. N0) + ... = N 3 '. N2 '. N0 + N 3 '. N2 . N0 + ... Reduce number of gates and gate inputs CSE261 Combinational Logic Design #13 #13 CSE261 Combinational Logic Design #14 #14 Twovariable Karnaugh map TwoExample 1: F = XY + XY XY
X
Row 0 1 2 3 X 0 0 1 1 Y 0 1 0 1 F 0 0 1 1 1 Example 2: F = XY + XY
X
Row 0 X 0 0 1 1 Y 0 1 0 1 F 1 1 0 0 1 X
0 1 2 X
0 1 2 Y
0 0 Y
0 0 0
1 3 1 1
Y 1 2 3 1
1 3 0 0
Y 0 1 Simplification: F = X(Y + Y) = X1 = X X(Y Simplification: F = X(Y + Y) = X1 = X CSE261 Combinational Logic Design #15 #15 CSE261 Combinational Logic Design #16 #16 Combinational Logic Design 4 CSE261 Spring 2008 Threevariable Karnaugh map ThreeExample 3:
Row 0 1 2 3 4 5 6 7 X 0 0 0 0 1 1 1 1 Y 0 0 1 1 0 0 1 1 Karnaughmap usage Karnaugh1. F = XYZ + XYZ + XYZ + XYZ XY XY
Z 0 1 0 1 0 1 0 1 F 1 1 0 0 1 1 0 0 1 0 Plot 1s corresponding to minterms of function. Circle largest possible rectangular sets of 1s. 1s. XY Z
0 00 2 01 6 11 X
10 4 2. # of 1s in set must be power of 2 OK to cross edges Variable is 1 Variable is 0 include variable include complement of variable variable not included 1
1 3 0
7 0
5 1 1
Z 3. Read off product terms, one per circled set. 1 0
Y 0 Variable is both 0 and 1 4. 5. Circled sets and corresponding product terms are called `prime implicants' implicants' Minimum number of gates and gate inputs
CSE261 Combinational Logic Design #18 #18 Simplification: F = XY(Z + Z) + XY(Z + Z) = XY + XY = (X + X)Y= Y XY XY )Y
CSE261 Combinational Logic Design #17 #17 Example 1:
Combining Two cells: I) Cells (0,1): X=Y=0; Z=0,1 (0,1): Product term: X'Y' Cells (4,5): X=1; Y=0; Z=0,1 (4,5): Product term : XY' XY' OR: II) Cells (0,4): Y=0; Z=0; X=0,1 (0,4): Product term: Y'Z' Cells (1,5): Y=0; Z=1; X=0,1 (1,5): Product term : Y'Z Z Z XY
00 0 1 0 2 01 6 7 11 X
10 4 Example 1 (cont.):
XY Combining four cells III) Cells (0,1,4,5) (0,1,4,5) Z X=0,1; Z=0,1; Y=0 The product term: Y'
1 1 X
00 01 2 6 7 11 4 10 1
1 3 5 1 1
Y Z
0 0 1 1
3 5 1 1
Y Z 1 XY
00 0 1 0 2 01 6 7 11 4 10 The sum of products expression of F: From I) From II) From III) F =X'Y'+XY' or =X' +XY' F=Y'Z'+Y'Z or: F=Y' +Y' 1
1 3 5 1 1
CSE261 Combinational Logic Design #19 #19 1 F =Y' =Y'
CSE261 Combinational Logic Design #20 #20 Combinational Logic Design 5 CSE261 Spring 2008 Example 2:
Z Combine cells (0,2,6,4) (0,2,6,4) X=0,1; Y= 0,1; Z=0 Product Term: Z' Combine cells (4,5) (4,5) X=1; Y=0; Z=0,1 Product Term: XY' XY' F = XY'+Z' XY' +Z' XY
00 0 1 0 2 01 6 11 X
10 4 Exercise:
X
00 0 1 0 2 01 6 11 4 10 1
1 3 1
7 1
5 1 1
Z XY Z 1
1 3 1
7 5 1 1
Y Y 1 1 Z The canonical sum is: F=X'Y'Z'+XYZ'+XY'Z'+XY'Z+XYZ' F=X' +XYZ' +XY' +XY' Z+XYZ' CSE261 Combinational Logic Design #21 #21 CSE261 Combinational Logic Design #22 #22 Definitions:
1. In Example 1:
Z XY
00 0 0 2 01 6 11 X
10 4 A logic function P implies a logic function F if for every input combination for which P=1, then F=1 also. (P is an implicant of F) Any minterm or combination of minterms in the canonical sum expression is an implicant of the output function A prime implicant is a group of combined minterms that can't be can' combined with any other minterm or group of minterms minterms are unique (i.e at least one minterm is not contained in any in other prime implicant. A unique minterm is called an Essential 1Cell 1A minimal sum of a logic function is a sumofproducts expression sum offor F such that no sumofproducts expression for F has fewer product sum ofterms. 2. Implicants
X'Y'Z' XY' XY' 1
1 3 1
7 1
5 1 1
Z 1 3. Y 4. Essential prime implicant is a prime implicant in which one or more Prime Implicants
XY' XY' Z' 5. Essential Prime Implicants
XY' XY' Z'
CSE261 Combinational Logic Design #24 #24 CSE261 Combinational Logic Design #23 #23 Combinational Logic Design 6 CSE261 Spring 2008 Example 3:
Combining (0,2) Product term: X'Z' Combining (2,3) Product term: X'Y Combining (3,7) Product term: YZ XY Z
0 1 0 00 2 01 6 11 X
10 4 5 Example 3:
The prime implicants: 1 (0,2) X'Z' 2 (0,4) Y'Z' Z XY
00 0 1 0 2 01 6 11 X
10 4 1
1 3 1
7 1
Y 1 Z 1
1 3 1
7 5 1 1
Y Largest possible set Unique minterm X'Z', X'Y, and YZ are prime implicants X'Z', YZ are essential prime implicants X'Y is nonessential prime implicant (redundant) because all its nonminterms are covered in the other essential prime implicants F= X'Z'+X'Y+YZ OR: F = X'Z'+YZ (the minimal sum of F) X' X'
CSE261 Combinational Logic Design #25 #25 No ess 3 (2,3) X'Y en t ia lp r im ei 4 (3,7) YZ mp lica nt! 1 1 Z 6 (5,7) XZ 5 (4,5) XY' XY' Two possible minimal sums: 1 Using the prime implicants 1,4,and 5; F= X'Z'+YZ+XY' X' +YZ+XY' 2 Using the prime implicants 2,3,and 6; F= Y'Z'+ X'Y+XZ Y' X'
CSE261 Combinational Logic Design #26 #26 Example 4: F = (1,2,5,7)
1. Summary:
Load the minterms and maxterms into the Kmap by placing the 1's K1' and 0's in the appropriate cells. 0' Look for groups of minterms and write the corresponding product terms (the prime implicants): 2. The group size should be a power of 2. Find the largest groups of minterms first then find smaller groups groups of minterms until all groups are found and all 1cells are covered. 1 3. 4. Determine the essential prime implicants. implicants. Select all essential prime implicants and the minimal set of the remaining prime implicants that cover the remaining 1's. 1' Its possible to get more that one equally simplified expression if more than one set of the remaining prime implicants contains the same number of minterms.
CSE261 Combinational Logic Design #28 #28 5. CSE261 Combinational Logic Design #27 #27 Combinational Logic Design 7 CSE261 Spring 2008 Fourvariable Karnaugh map FourWX YZ
00 01 11 00 0 1 3 2 4 5 7 6 01 11 12 13 15 14 8 9 11 10 Example 4:
Row 0 W 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 X 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 Y 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1
CSE261 Combinational Logic Design #30 #30 F= W'X'Y'Z'+ W'X'YZ'+ YZ' W'XYZ'+ XYZ' W'XYZ+ WX'Y'Z'+ WX' WX'YZ'+ WX' YZ' WXY'Z'+ WXY' WXYZ W
10 1 2 3 4 5 6 7 8 9 10 11 12 Z Y
10 X 13 14
CSE261 Combinational Logic Design #29 #29 15 Example 4 (cont.):
WX Essential prime implicants: The product term: XZ The product term : X'Z' Y YZ
00 01 11 10 00 01 11 Example 5:
W
10 WX Essential prime implicants: The product term: X'Z Z The product term : XZ' XZ' Y YZ
00 01 11 10 00 01 11 W
10 1 1 1 1
X 1 1 1 1 1 1 1 1
X 1 1 1 1
Z F = XZ + X'Z' X'
CSE261 Combinational Logic Design #31 #31 F = X'Z + XZ' XZ' CSE261 Combinational Logic Design #32 #32 Combinational Logic Design 8 CSE261 Spring 2008 Exercise:
Row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 W 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 X 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 Y 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F 1 1 1 0 1 1 1 0 1 1 0 0 1 1 1 0
CSE261 Combinational Logic Design #33 #33 Exercise (cont.):
W
00 00 01 11 0 1 3 2 4 5 7 6 01 11 12 13 15 14 8 9 11 10 10 WX YZ
00 01 00 0 1 3 2 01 4 5 7 6 11 12 13 15 14 W
10 8 9 WX YZ 1 1 1 1 1 1 1 1
Z Z 11 11 10 Y
10 Y
10 1 1 1 X X F = Y' + W'Z' + XZ' Y' W' XZ'
CSE261 Combinational Logic Design #34 #34 Example 6:
The prime implicants: W'X' W'Y' W'Z XYZ WXY The essential prime implicants: Y W'X' W'Y' WXY Cell 7 is not covered by any of the essential prime implicants. Its covered by two nonessential prime nonimplicant. We choose the one with the less number of variables which is W'Z W' WX YZ
00 01 11 10 00 0 1 3 2 01 4 5 7 6 11 12 13 15 W
10 8 9 11 10 Exercise:
FW,X,Y,Z= (0,1,2,3,4,6,7,8,9,12,14) 1 1 1 1 1 1 1 Z 1 1 14 X F= W'X'+W'Y'+WXY+W'Z W' +W' +WXY+W' CSE261 Combinational Logic Design #35 #35 CSE261 Combinational Logic Design #36 #36 Combinational Logic Design 9 CSE261 Spring 2008 Fivevariable Karnaugh map FiveFive variable Kmap is formed using two connected fourvariable maps: Kfourmaps: V VWX YZ
00 01 11 10 0 1 3 2 000 4 5 7 6 001 011 12 13 15 14 8 9 11 10 Cells (0,2,8,10) and (16,18,24,26) form essential prime implicant: X'Z' (0,2,8,10) (16,18,24,26) implicant: Cells(5,7,13,15) and (21,23,29,31) form essential prime implicant: XZ Cells(5,7,13,15) (21,23,29,31) implicant: F=XZ+X'Z' F=XZ+X'
VWX YZ
000 00 0 4 001 011 12 13 W
010 8 100 16 101 20 21 V
111 28 29 W
010 100 16 17 19 18 101 20 21 23 22 111 28 29 31 30 W
110 W
110 24 Example 7: 24 25 27 26 1
1 3 2 5 9 1 1
7 1
17 19 18 1
25 27 26 01 11 Z 1
15 11 10 1
23 1
31 Z Y Y
10 1
6 1
14 1
22 1
30 1
X 1 1
X 1
CSE261 Combinational Logic Design #38 #38 X X
CSE261 Combinational Logic Design #37 #37 Simplifying the Product of Sums
1. 2. Example 8:
The prime implicants:
Cells (0,1,8,9): X=0, Y=0 (0,1,8,9): The sum term: X+Y Plot 0s corresponding to maxterms of function. Circle largest possible rectangular sets of 0s. 0s. WX YZ
00 01 11 00 0 1 3 2 01 4 5 7 6 11 12 13 15 14 W
10 8 9 # of 0s in set must be power of 2 OK to cross edges Variable is 0 Variable is 1 include variable include complement of variable variable not included 0 0 0 0 0
Z 3. Read off product terms, one per circled set. Cells (8,10,12,14): W=1, Z=0 (8,10,12,14): The sum term: W'+Z 11 10 Y
10 Variable is both 1 and 0 0 0 4. 5. Circled sets and corresponding product terms are called `prime implicants' implicants' Minimum number of gates and gate inputs
CSE261 Combinational Logic Design #39 #39 The two prime implicants are essential prime implicants and cover all zeros X The minimal product of sums: F = (X + Y) . (W' + Z) (W'
CSE261 Combinational Logic Design #40 #40 Combinational Logic Design 10 CSE261 Spring 2008 Exercise:
Row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 W 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 X 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 Y 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F 1 1 1 0 1 1 1 0 1 1 0 0 1 1 1 0 WX YZ
00 01 11 00 0 1 3 2 4 5 7 6 01 11 12 13 15 14 W
10 8 9 11 10 Simplifying POS  Another method
Complement the function. Use Kmap to get the minimal sum Kof the complement function. Complement the minimal sum to get the minimal product W WX 1. The function is complemented and represented using Kmap: K2. The essential prime implicants are: WZ', X'Y WZ' X' YZ
00 01 11 00 0 1 3 2 01 4 5 7 6 11 12 13 15 14 10 8 9 Z Y
10 1 1 1 1 1
Z X Essential prime implicants: (3,7,15,11): Z=1, Y=1; The sum term: Z'+Y' +Y' (10,11): W=1, X=0,Y=1; The sum term: W'+X+Y' +X+Y' CSE261 #41 F=(Z'+Y').(W'+X+Y') Combinational Logic Design #41 F=(Z'+Y' ).(W' +X+Y' 3. The minimal Sum: M = X'Y' + WZ' Y X' WZ'
10 11 10 1 1 4. F = M' = (X'Y'+WZ')' = (X+Y).(W'+Z) M' (X' +WZ' (X+Y).(W' X
CSE261 Combinational Logic Design #42 #42 Minimal Product of Sums vs. Minimal Sum of Products:
Compare the minimal product and the minimal sum designs to find the best (least # of gates) realization. In the previous example: The minimal sum is: F = W'X + W'Y + XZ + YZ W' W' The minimal product: F = (X + Y) . (W' + Z) product: (W' The minimal product implementation is cheaper. Primenumber detector (again) Prime CSE261 Combinational Logic Design #43 #43 CSE261 Combinational Logic Design #44 #44 Combinational Logic Design 11 CSE261 Spring 2008 Primenumber detector (cont.) PrimeWhen we solved algebraically, we missed one simplification  the circuit below has three less gate inputs. Example 9: CSE261 Combinational Logic Design #45 #45 CSE261 Combinational Logic Design #46 #46 Example 10: Don't Care Conditions: Don'
In some applications, the Boolean function for certain combinations of the input variables is not specified. The corresponding minterms (maxterms) are called "don't care don' minterms(maxterms)". minterms(maxterms)" In Kmap , these are represented by `d'. KSince the output function for those minterms(maxterms) is not specified, those minterms(maxterms) could be combined with the adjacent 1 cells (0cells) to get a more simplified sumof(0sum ofproducts (productofsums) expression. (product ofCSE261 Combinational Logic Design #47 #47 CSE261 Combinational Logic Design #48 #48 Combinational Logic Design 12 CSE261 Spring 2008 Example 11:
Build a logic circuit that determines if a decimal digit is 5 Example 11 (cont.): Truth Table
Row 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
CSE261 Combinational Logic Design #49 #49 W 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 X 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 Y 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 Z 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 F 0 0 0 0 0 1 1 1 1 1 d d d d d
CSE261 Combinational Logic Design #50 #50 1. 2. 3. 4. The decimal digits (0,1,2,...,9) are represented by 4 bit BCD code. code. The logic circuit should have 4 input variables and 1 output. output. There are 16 different input combinations but only 10 of them are used. The logic function should produce 0 if the number is < 5, and 1 if it is 5. 15 d Example 11 (cont.): KMap K Example 11 (cont.): Minimal Product
WX
10 8 9 WX The Minimal Sum: Combining the `1' cells only, the minimal sum is: F = WX'Y'+W'XZ+W'XY WX' +W' XZ+W' Combining the dont care minterms with the `1' cells, the minimal sum is: YZ
00 01 11 00 0 1 3 2 4 5 7 01 11 12 13 15 W W
00 01 4 11 12 10 8 9 0 The Minimal Product: YZ
00 d d d d 1 1 F= (W+X).(W+Y+Z)
Z Y
10 01 11 0
1 5 0 1 1
6 d d d d 1 1
Z 13 15 14 1 1 0
3 7 11 10 11 10 Y
10 d d 0
2 d d 6 1 14 0 1 F = W+XZ+XY X X CSE261 Combinational Logic Design #51 #51 CSE261 Combinational Logic Design #52 #52 Combinational Logic Design 13 CSE261 Spring 2008 Example 11 (cont.): Implementation
The minimal sum implementation: F = W+XZ+XY
W X Y Z F MultipleOutput Minimization MultipleMost digital applications require multiple outputs derived from the same input variables.
X Y Z Logic Circuit F1 F2 The minimal product implementation: F= (W+X).(W+Y+Z)
W X Y Z F W X Y Z F 1. Each output function could be minimized using Kmap and Krealized independently. The output functions could share one or more product terms (prime implicant) which reduces the total number of gates.
CSE261 Combinational Logic Design #54 #54 2. CSE261 Combinational Logic Design #53 #53 Example 12:
F1 = XZ + YZ' YZ' F2 = XY' + YZ' XY' YZ' F1
Z
0 1 0 1 Example 12 (cont.): Logic Diagram
F2
Z
10 4 0 1 5 0 1 XY
00 2 01 6 11 X
10 4 XY
00 2 01 6 11 X Independent realization Minimal realization F1= XZ+YZ' XZ+YZ'
Z X Y Z F1 1
3 7 1
5 1 1 1
3 7 1 1
Y
XY Z F2= XY'+YZ' XY' YZ' 1 Z Y F2 X
00 0 1 0 1 2 01 6 11 4 10 To find the common terms multiply the two functions (F1.F2) The common terms are: YZ', XY'Z YZ' XY' F1 X Y Z
CSE261 Combinational Logic Design #56 #56 1
3 7 1
5 1
Y Z F2 CSE261 Combinational Logic Design #55 #55 Combinational Logic Design 14 CSE261 Spring 2008 Timing Hazards
The Truth Table determines the Steady State behavior of a Combinational Logic Circuit Transient behavior:
1. 2. Definitions
Static1 Hazard: Two input combinations that: Static Hazard:
differ in only one variable. both produce logic 1. 1 0 Output could produce glitches when input variables change. Glitches occur when the paths between inputs and output have different delays. Timing Hazards refer to the possibility of having glitches during input transitions. possibly produce Logic 0 glitch during input variable transition Static0 Hazard: Two input combinations that Static Hazard: 3. differ in only one variable both produce logic 0 1 0 Hazards:
1. 2. 3. possibly produce Logic 1 glitch during input variable transition Definitions. Finding hazards. Eliminating hazards.
CSE261 Combinational Logic Design #57 #57 Dynamic hazards: hazards: The output could change more than once during input transitions Caused by multiple paths with different delays from input to the output
CSE261 Combinational Logic Design #58 #58 Example
F = YZ + XZ' XZ' Delay in each gate is T . Input changes from XYZ = 111 to 110 Finding Timing hazards using Kmap KANDOR Circuits: ANDStatic 0 hazards do not exist in the sumof products (ANDOR) implementation. sum(ANDStatic 1 hazards are possible The Kmap of the function F in the previous example: KCell 6 and 7 are covered in two product terms. X Y Z Z' YZ XZ' XZ' F T glitch Y Z X F Faster Y Z X 1
1 0 0 1 1 0 Z
0 1 0 1 XY X
00 2 3 01 6 11 4 10 F
1 0 1 1
7 5 1
Z 1 Static 1 hazard
CSE261 Combinational Logic Design #59 #59 1
Y 1 CSE261 Combinational Logic Design #60 #60 Combinational Logic Design 15 CSE261 Spring 2008 Timing hazards in ORAND circuits OROR  AND Circuits: Static 1 hazards do not exist in the productsof sum (ORAND) implementation. products(ORStatic 0 hazards are possible The minimal product of F = (X+Z)(Y+Z') (X+Z)(Y+Z' Cell 0 and Cell 1 are covered in two sum terms Static 0 hazard occurs when inputs switched between 000 to 001
Faster X Z Y 0
1 0 0 1 Z XY
00 0 1 0 2 01 6 11 4 5 Eliminating Timing Hazards
ANDOR Circuit ANDAdd a prime implicant that combines the two inputs that cause static 1 hazard.
XY X
00 01 2 3 6 11 4 10 0 1 Y Z X
10 1 1 0 1 1 0 F
1 0 1 1 Z
0 1 1
7 5 1
Z X 1
Y 1 F 0 1 0
1 0 0
1 3 0
7 0 0
Y 0 Z The hazardfree circuit hazard Cells 6 & 7 are combined: XY Static 0 hazard CSE261 Combinational Logic Design #61 #61 CSE261 Combinational Logic Design #62 #62 Eliminating Timing Hazards
OR  AND Circuit Add a prime implicant that combines the two inputs that cause static 0 hazard.
XY Exercise
F = W'X + Y'Z + WXYZ + WX'YZ' W' Y' WX' YZ'
1. W X Y Z W' W'X Y'Z X
00 01 2 6 11 4 5 10 0 Complete the timing diagram. (Assume all gates has same propagation delay=T ) Use Kmap to show Static0 hazards KStaticBuild a hazardfree circuit hazard X Z Y 0
1 0 0 1 Z F
1 0
0 2. 3. 0
1 0
1 3 0
7 0
0 0
Y 0 Z WZYZ WX'YZ' WX' YZ' F T The hazardfree circuit hazard Cells 0 & 1 are combined: X+Y CSE261 Combinational Logic Design #63 #63 CSE261 Combinational Logic Design #64 #64 Combinational Logic Design 16 CSE261 Spring 2008 QuineMcCluskey algorithm Quine1. RealWorld Logic Design RealLots more than 6 inputs Design correctness more important than gate minimization
Use Minimization process can be made into a program, using appropriate algorithms and data structures. guaranteed to find `minimal' solution minimal' 2. Required computation has exponential complexity (run time and storage) works well for functions with up to 812 variables, but 8quickly blows up for larger problems. `higherlevel language' to specify logic operations higherlanguage' Use programs to manipulate logic expressions and minimize logic.
PALASM, VHDL, VHDL, ABEL  developed for PLDs 3. Heuristic programs (e.g., Espresso) used for larger problems, usually give minimal results.
CSE261 Combinational Logic Design #65 #65 Verilog  developed for ASICs CSE261 Combinational Logic Design #66 #66 ABEL
Advanced Boolean Equation Language
Developed for use with programmable logic devices, which have a programmable ANDOR structure. ANDCombinational logic functions Operators: Sets: ABEL Program Structure
module module name title string device ID device deviceType; deviceType; pin declarations other declarations equations equations test_vectors test vectors end module name
CSE261 Combinational Logic Design #68 #68 AND, OR, NOT, XOR, XNOR, & # ! $ !$ XBUS = [X3,X2,X1,X0]; XBUS = [1,1,0,1]; XBUS = 13; (XBUS == YBUS) Relations: Intermediate variables: (XBUS > [1,1,0,1])
CSE261 Combinational Logic Design #67 #67 Combinational Logic Design 17 CSE261 Spring 2008 ABEL Example
Module Alarm_circuit Title `Alarm Circuit Example ALARMCKT device `P16V8C'; P16V8C' "Input pins PANIC, ENABLE, EXITING WINDOW, DOOR, GARAGE "Output pins ALARM "intermediate equation SECURE = WINDOW & DOOR & GARAGE; equations ALARM = PANIC # ENABLEA & !EXITING & !SECURE;
CSE261 Combinational Logic Design #69 #69 ABEL Example (continued)
test_vectors ([PANIC, ENABLEA,EXITING,WINDOW,DOOR,GARAGE] [ 1, 0, 0, 0, 0, 0, 0, .X., 0, 1, 1, 1, 1, 1, .X., .X., 1, 0, 0, 0, 0, .X., .X., .X., 0, .X., .X., 1, .X., .X., .X., .X., 0, .X., 1, .X.] .X.] .X.] .X.] .X.] 0] 1] [ [ [ [ [ [ [ [ALARM]) 1]; 0]; 0]; 1]; 1]; 1]; 0]; pin 1,2,3; pin 4,5,6; [ [ [ pin 11 istype `com'; com' [ [ [ End Alarm_Circuit CSE261 Combinational Logic Design #70 #70 ABEL language processor (compiler)
1. 2. 3. 4. ABEL equations file (cont.)
Equations: ALARM = (ENABLE & !EXITING & !DOOR # ENABLEA & !EXITING & !WINDOW # ENABLEA & !EXITING & !GARAGE # PANIC); ReversePolarity Equations: Reverse!ALARM = (!PANIC & WINDOWS & DOOR & GARAGE # !PANIC & EXITING # !PANIC & !ENABLEA); Checks syntax Checks devicepin capabilities deviceExpands language statements and equations Reduces equations into sumofproducts form form for sum ofprogrammable logic device (PLD) Checks test vectors against equations Checks equation requirements (# of inputs, product terms) against PLD resources Determines `fuse map' to program the PLD map'
CSE261 Combinational Logic Design #71 #71 5. 6. 7. CSE261 Combinational Logic Design #72 #72 Combinational Logic Design 18 CSE261 Spring 2008 ABEL WHEN Statements
Analysis WHEN (!A#B) THEN X1=C&!D; X1 = (!A#B)&(C&!D); WHEN (A&B) THEN X2=C#D; ELSE X2=E#F; X2=(A&B)&(C#D)#!(A&B)&(E#F); Logic Circuit Summary
Logic Function Truth Table Truth Table Synthesis Logic Function Nested WHEN Statements
WHEN (A) THEN X3=D; ELSE WHEN (B) THEN X3=E; ELSE WHEN (C) THEN X3=F; X3=(A)&(D) #!(A)&(AB)&(E) #!(A)&!(B)&(C)&(F);
CSE261 Combinational Logic Design #73 #73 Canonical sum / Canonical product Boolean algebra theorems / KMaps / Programs K The minimal sum / product implementations
CSE261 Combinational Logic Design #74 #74 Logic Circuit Next
1. Documentation Standards 2. Programmable Logic Devices 3. Decoders 4. Encoders 5. Threestate devices Three6. Multiplexers 7. XOR gates 8. Comparators 9. Adders 10. Comparators 11. Multipliers 12. Readonly memories (ROMs) ReadCSE261 Combinational Logic Design #75 #75 Combinational Logic Design 19 ...
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 Spring '08
 Ercanli
 Boolean Algebra, Logic gate

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