**Combinational Circuits** [Image of a block diagram of a combinational circuit with n inputs and m outputs] **Design Procedure:** * Determine the required number of inputs & outputs and assign a symbol to each input & output. * Derive the truth table. * Obtain the simplified function. * Draw the logic diagram. **Ex:** Design a circuit that produces 1 if the number of ones in a 3-bit input is odd, otherwise produce zero. | A | B | C | F | |---|---|---|---| | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 1 | | 0 | 1 | 0 | 1 | | 0 | 1 | 1 | 0 | | 1 | 0 | 0 | 1 | | 1 | 0 | 1 | 0 | | 1 | 1 | 0 | 0 | | 1 | 1 | 1 | 1 | F = Σ(1, 2, 4, 7) F = A'B'C + A'BC' + AB'C' + ABC **Design a circuit** [Image of a logic circuit diagram implementing the function F] **Ex:** Implement a function in minterm format that finds the multiplication of bits in a 3-bit input. | A | B | C | F | |---|---|---|---| | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | | 0 | 1 | 0 | 0 | | 0 | 1 | 1 | 0 | | 1 | 0 | 0 | 0 | | 1 | 0 | 1 | 1 | | 1 | 1 | 0 | 1 | | 1 | 1 | 1 | 1 | F = Σ(7) F = ABC [Image of a logic circuit diagram implementing the function F] **Q) Design a Combinational circuit that find the summation of two bits in binary system.** | X | Y | Sum | Carry | |---|---|---|---| | 0 | 0 | 0 | 0 | | 0 | 1 | 1 | 0 | | 1 | 0 | 1 | 0 | | 1 | 1 | 0 | 1 | Sum = Σ(1, 2) Carry = Σ(3) **Half Adder** [Image of a Half Adder circuit] **Q) Design a Combinational circuit that find the summation of three bits in binary system.** | A | B | C | Sum | Carry | |---|---|---|---|---| | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 1 | 0 | | 0 | 1 | 0 | 1 | 0 | | 0 | 1 | 1 | 0 | 1 | | 1 | 0 | 0 | 1 | 0 | | 1 | 0 | 1 | 0 | 1 | | 1 | 1 | 0 | 0 | 1 | | 1 | 1 | 1 | 1 | 1 | Sum = Σ(1, 2, 4, 7) Carry = Σ(3, 5, 6, 7) Sum = A'B'C + A'BC' + AB'C' + ABC Carry = A'BC + AB'C + ABC' + ABC A = (A'B')' + (A'B)' B = (AB')' + (AB)' C = (AB)' + (AB)' **Q) Design a Combinational circuit that find the subtraction of two bits in binary system.** | A | B | D | Borrow | |---|---|---|---| | 0 | 0 | 0 | 0 | | 0 | 1 | 1 | 1 | | 1 | 0 | 1 | 0 | | 1 | 1 | 0 | 0 | Difference (D) = Σ(1, 2) Borrow (B) = Σ(1) D = x'y + xy' B = x'y **Half Subtractor** [Image of a Half Subtractor circuit] **Q) Design a Combinational circuit that find the subtraction of three bits in binary system** | A | B | C | D | Borrow | |---|---|---|---|---| | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 1 | 1 | | 0 | 1 | 0 | 1 | 1 | | 0 | 1 | 1 | 0 | 1 | | 1 | 0 | 0 | 1 | 0 | | 1 | 0 | 1 | 0 | 0 | | 1 | 1 | 0 | 0 | 0 | | 1 | 1 | 1 | 1 | 0 | D = Σ(1, 2, 4, 7) Borrow = Σ(1, 2, 3, 7) D = A'B'C + A'BC' + AB'C' + ABC Borrow = A'BC + AB'C + ABC' + ABC A = (A'B')' + (A'B)' B = (AB')' + (AB)' C = (AB)' + (AB)'