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sol3-1

# sol3-1 - Chapter 3 3.1(a x1 x2 x3 f 0 0 0 0 1 1 1 1(b 0 0 1...

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Chapter 3 3.1. ( a ) x 1 x 2 x 3 f 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 0 0 0 0 ( b ) #transistors = NOT gates × 2 + AND gates × 8 + OR gates = 3 × 2 + 4 × 8 + 1 × 10 = 48 3.2. ( a ) In problem 3.1 the canonical SOP for f is f = x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 This expression is equivalent to f in Figure P3.2, as derived below. x 1 x 2 x 3 x 3 x 2 x 3 x 2 x 3 + x 2 x 3 x 2 x 3 + x 1 x 2 x 3 x 1 x 2 x 3 + + x 1 x 2 x 3 x 1 x 2 x 3 + ( b ) Assuming the multiplexers are implemented using transmission gates #transistors = NOT gates × 2 + MUXes × 6 = 1 × 2 + 3 × 6 = 20 3-1

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3.3. ( a ) A SOP expression for f in Figure P3.3 is: f = ( x 1 x 2 ) x 3 = ( x 1 x 2 ) x 3 + ( x 1 x 2 ) x 3 = x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 which is equivalent to the expression derived in problem 3.2. (b) Assuming the XOR gates are implemented as shown in Figure 3.61 b #transistors = XOR gates × 8 = 2 × 8 = 16 3.4. Using the circuit The number of transistors needed is 16. 3.5. Using the circuit The number of transistors needed is 20. 3.6. ( a ) x 1 x 2 x 3 f 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 0 0 1 1 3-2
( b ) The canonical SOP expression is f = x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 + x 1 x 2 x 3 The number of transistors required using only AND, OR, and NOT gates is #transistors = NOT gates × 2 + AND gates × 8 + OR gates × 12 = 3 × 2 + 5 × 8 + 1 × 12 = 58 3.7. ( a ) x 2 x 3 x 4 f 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 0 0 0 1 x 1 0 0 0 0 0 0 0 0 1 1 1 0 x 2 x 3 x 4 f 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 x 1 1 1 1 1 1 1 1 1 1 1 1 0 ( b ) f = x 1 x 2 x 3 x 4 + x 1 x 2 x 3 x 4 + x 1 x 2 x 3 x 4 = x 1 x 3 x 4 + x 2 x 3 x 4 The number of transistors required using only AND, OR, and NOT gates is #transistors = NOT gates × 2 + AND gates × 8 + OR gates × 4 = 4 × 2 + 2 × 8 + 1 × 4 = 28 3.8. V DD V x 1 V x 2 V x 3 V f 3-3

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3.9. V x 1 V x 3 V x 4 V x 2 V f 3.10. Minimum SOP expression for f is f = x 2 x 3 + x 1 x 3 + x 2 x 4 + x 1 x 4 = ( x 1 + x 2 )( x 3 + x 4 ) which leads to the circuit V DD V x 1 V x 2 V x 3 V f V x 4 3.11. Minimum SOP expression for f is f = x 4 + x 1 x 2 x 3 which leads to the circuit 3-4
V DD V x 1 V x 3 V f V x 4 V x 2 3.12. V DD V y V z V f V x V DD 3-5

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3.13. V DD V y V z V y V z V x V f V DD V z V y V x V y V z 3.14. ( a ) Since V DS V GS - V T the NMOS transistor is operating in the saturation region: I D = 1 2 k n W L ( V GS - V T ) 2 = 10 μ A V 2 × 5 × (5 V - 1 V) 2 = 800 μ A ( b ) In this case V DS < V GS - V T , thus the NMOS transistor is operating in the triode region: I D = k n W L ( V GS - V T ) V DS - 1 2 V 2 DS = 20 μ A V 2 × 5 × (5 V - 1 V) × 0 . 2 V - 1 2 × (0 . 2 V) 2 = 78 μ A 3.15. ( a ) Since V DS V GS - V T the PMOS transistor is operating in the saturation region: I D = 1 2 k p W L ( V GS - V T ) 2 = 5 μ A V 2 × 5 × ( - 5 V + 1 V) 2 = 400 μ A ( b ) In this case V DS > V GS - V T , thus the PMOS transistor is operating in the triode region: I D = k p W L ( V GS - V T ) V DS - 1 2 V 2 DS = 10 μ A V 2 × 5 × ( - 5 V + 1 V) × ( - 0 . 2) V - 1 2 × ( - 0 . 2 V) 2 = 39 μ A 3-6
3.16. R DS = 1 / k n W L ( V GS - V T ) = 1 / 0 . 020 mA V 2 × 10 × (5 V - 1 V) = 1 . 25 k Ω 3.17.

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