dig2DeMorganE

dig2DeMorganE - Digital Circuit Engineering 2nd...

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© John Knight Dig Cir II p. 0 Revised; December 7, 2009 Digital Circuit Engineering Carleton University 2009 DIGITAL VLSI DESIGN (X + A)(X + B) = X + AB YX + X = X 2nd Distributive Simplification XY + XY = X Y + XY = X + Y DeMorgan Simple X + Y = XY X Y = X + Y Absorption DeMorgan General F F(a, b, . .. z,+, . ,0,1) F F(a , b , . .. z , . ,+,1,0) If Then Remember to bracket AND terms Slide i Carleton University © John Knight Digital Circuits II p. 1, dig2DeMorganE.fm Revised; December 7, 2009 Comment on Slide i DeMorgan’s Theorem Simple two variable forms As equations and as gates with inverted inputs Why Real Gates are NAND/NOR, not AND/OR Two symbols for NAND; two for NOR AND-OR designs are easier to think about NAND-NOR designs must be done to use real gates Design with AND-OR; Implement with NAND-NOR Change between them using DeMorgan’s Theorem AND/OR to NAND/NOR Conversions Generalized DeMorgan’s Thorem Common Errors
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© John Knight Dig Cir II p. 2 Revised; December 7, 2009 DeMorgan’s Simple Form A + B = A · B D + E = D · E D · E = D + E B 1 0 A 0 1 0 0 1 1 1 1 1 0 A + B 1 1 1 0 AB AB 0 0 0 1 A 1 1 0 0 B 0 1 1 0 E 1 0 D 0 1 0 0 1 1 0 1 0 0 D · E 0 1 0 0 D+E D E 1 1 0 0 0 1 1 0 DeMorgan’s Law Inverse The dual inverse Dual DeMorgan’s Law A B C A B C AND AND A · B = C = A + B A B K A B K D E G D E G D + E = G = D · E NOR NAND A · B = K = A + B D E F D E F D + E = F = D · E OR AND (DeM) (DeM) NOR OR NAND A · B = A + B Used To Find the Inverse of Expressions Equivalent graphical forms: Slide 2 DeMorgan’s Law Carleton University © John Knight Digital Circuits II p. 3, dig2DeMorganE.fm Revised; December 7, 2009 DeMorgan’s Law DeMorgan’s Laws on Complementing DeMorgan’s Laws on Complementing Expressions A theorem relating NANDs and NORs. An OR gate with inverted inputs is equivalent to an AND gate with an inverted output. An AND gate with inverted inputs is equivalent to an OR gate with an inverted output. Inverting inputs and outputs of an OR makes it an AND. Inverting inputs and outputs of an AND makes it an OR. EXAMPLE Convert to an expression with 3 letters and inversion bars only over single letters. 41.• PROBLEM Reduce to four letters with inversion bars over single letters only. 42.• PROBLEM Reduce to four letters with inversion bars over single letters only. Changing everything into NOT and AND gates It turns out that any logic circuit can be made from AND and NOT gates. DeMorgan’s law can be used to transform the circuits. 43.• PROBLEM Convert ((rw + t)u + r )t into a function with only AND and NOT operations. (a + b)(a + c ) (a + b)(a + c ) = (a + b) + (a + c ) (DeM1) = (a ·b ) + (a ·c ) (DeM2) = a ·b + a ·c = a (b + c) (Clear brackets) (D1) xy + xz = x(y+z) (a + b) + a·b d(de) + (de)e Comment on Slide 2
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© John Knight Dig Cir II p. 4 Revised; December 7, 2009 Real CMOS Digital Gates Smal Jolt JOLT FREE +V DD A F A F + Two switches, handles linked together Transistor NOT +V DD A=1 => Q 1 closed => F=0 Q 1 Q 1 PMOS transistor acts like: closed switch when A is “0” open switch when A is “1” NMOS transistor acts like: open switch when A is “0” closed switch when A is “1” All CMOS Gates Invert Real Gates are NAND NOR E F VDD A B A + B EF Transistor NAND
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dig2DeMorganE - Digital Circuit Engineering 2nd...

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