Combinational Circuits Notes & Formulas

1. Definition

Circuits whose output depends only on the current input - no memory, no feedback, no clock.

2. Core Idea

Output = f(inputs only). No state. Examples: Adder, Subtractor, Multiplexer, Decoder, Encoder, Comparator.

3. Half Adder

• Sum = A ⊕ B
• Carry = A·B
• 2 inputs, 2 outputs, no carry-in

4. Full Adder (Very Important)

• Sum = A ⊕ B ⊕ Cin
• Cout = AB + BCin + ACin = AB + Cin(A⊕B)
• Can be built from 2 Half Adders + 1 OR gate

5. Subtractor

• Half Subtractor: Diff = A⊕B, Borrow = A'B
• Full Subtractor: Diff = A⊕B⊕Bin, Borrow = A'B + A'Bin + BBin

6. Comparator

For A, B (1-bit): A>B = AB', A<B = A'B, A=B = A⊕B' (XNOR)

7. Carry Propagation

• Ripple Carry Adder: Simple but slow - carry propagates serially. Delay = O(n)
• Carry Lookahead Adder (CLA): Computes carry in parallel. Delay = O(log n). Uses Generate G=AB, Propagate P=A⊕B

8. GATE Trick

Full adder gate count and delay are common. CLA vs ripple carry delay comparison is frequently tested. Know: ripple carry n-bit adder has 2n gate delays.