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Modified from John Wakerly Lecture #2 and #3

Modified from John Wakerly Lecture #2 and #3. CMOS gates at the transistor level Boolean algebra Combinational-circuit analysis. CMOS NAND Gates. Use 2 n transistors for n -input gate. CMOS NAND -- switch model. CMOS NAND -- more inputs (3). Inherent inversion. Non-inverting buffer:.

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Modified from John Wakerly Lecture #2 and #3

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  1. Modified from John Wakerly Lecture #2 and #3 CMOS gates at the transistor level Boolean algebraCombinational-circuit analysis

  2. CMOS NAND Gates • Use 2n transistors for n-input gate

  3. CMOS NAND -- switch model

  4. CMOS NAND -- more inputs (3)

  5. Inherent inversion. • Non-inverting buffer:

  6. 2-input AND gate:

  7. CMOS NOR Gates • Like NAND -- 2n transistors for n-input gate

  8. NAND NOR NAND vs. NOR • For a given silicon area, PMOS transistors are “weaker” than NMOS transistors. • Result: NAND gates are preferred in CMOS.

  9. Boolean algebra • a.k.a. “switching algebra” • deals with boolean values -- 0, 1 • Positive-logic convention • analog voltages LOW, HIGH --> 0, 1 • Negative logic -- seldom used • Signal values denoted by variables(X, Y, FRED, etc.)

  10. Boolean operators • Complement: X¢ (opposite of X) • AND: X × Y • OR: X + Y • Axiomatic definition: A1-A5, A1¢-A5¢ binary operators, describedfunctionally by truth table.

  11. More definitions • Literal: a variable or its complement • X, X¢, FRED¢, CS_L • Expression: literals combined by AND, OR, parentheses, complementation • X+Y • P × Q × R • A + B × C • ((FRED × Z¢) + CS_L × A × B¢× C + Q5) × RESET¢ • Equation: Variable = expression • P = ((FRED × Z¢) + CS_L × A × B¢× C + Q5) × RESET¢

  12. Logic symbols

  13. Theorems • Proofs by perfect induction

  14. More Theorems • N.B. T8¢, T10, T11

  15. Duality • Swap 0 & 1, AND & OR • Result: Theorems still true • Why? • Each axiom (A1-A5) has a dual (A1¢-A5¢) • Counterexample:X + X × Y = X (T9)X × X + Y = X (dual)X + Y = X (T3¢)???????????? X + (X×Y) = X (T9)X× (X + Y) = X (dual)(X× X) + (X× Y) = X (T8)X+ (X× Y) = X (T3¢) parentheses,operator precedence!

  16. N-variable Theorems • Prove using finite induction • Most important: DeMorgan theorems

  17. DeMorgan Symbol Equivalence

  18. Likewise for OR • (be sure to check errata!)

  19. DeMorgan Symbols

  20. Even more definitions (Sec. 4.1.6) • Product term • Sum-of-products expression • Sum term • Product-of-sums expression • Normal term • Minterm (n variables) • Maxterm (n variables)

  21. Truth table vs. minterms & maxterms

  22. Combinational analysis

  23. Signal expressions • Multiply out:F = ((X + Y¢) × Z) + (X¢× Y × Z¢) = (X × Z) + (Y¢× Z) + (X¢× Y × Z¢)

  24. New circuit, same function

  25. “Add out” logic function • Circuit:

  26. Shortcut: Symbol substitution

  27. Different circuit, same function

  28. Another example

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