Combinational Circuits: Adders, Multiplexers And More

A combinational circuit's output depends only on its present inputs, with no memory of the past. Same inputs always give the same output, like a calculator always answering 2 + 3 identically. These circuits are the workhorses doing the actual computing inside processors.

The key combinational blocks have standard jobs: adders do arithmetic, multiplexers route data, decoders translate codes into one-hot signals, encoders reverse that. Once you know these four blocks, you can read almost any digital block diagram, since complex chips are just these wired together.

Key Concepts

• Half adder: adds two bits giving Sum and Carry; cannot accept an incoming carry.
• Full adder: adds two bits plus carry-in, giving Sum and Carry-out; chained to build multi-bit adders.
• Ripple-carry adder: carry ripples from least to most significant stage; simple but slow for many bits.
• Multiplexer (MUX): data selector; a 4-to-1 MUX has 4 data inputs, 2 select lines, 1 output.
• Demultiplexer: routes one input to one of several outputs chosen by select lines.
• Decoder: n-bit code to 2n outputs with exactly one active; a 3-to-8 decoder turns 101 into line 5.
• Encoder: 2n inputs to an n-bit code; a priority encoder resolves multiple active inputs by rank.
• Seven-segment decoder: converts a 4-bit BCD digit into signals lighting a digital display.

Formulas

• Half adder: Sum = A ⊕ B, Carry = A·B.
• Full adder: Sum = A ⊕ B ⊕ Cin; Carry-out = A·B + Cin·(A ⊕ B).
• 2-to-1 MUX: Y = (NOT S)·D0 + S·D1; n select lines handle 2n inputs.

Worked Example

A full adder gets A = 1, B = 1, Cin = 1. Find Sum and Carry-out.

1. Sum = 1 ⊕ 1 ⊕ 1 = 0 ⊕ 1 = 1.
2. Carry-out = 1·1 + 1·(1 ⊕ 1) = 1 + 0 = 1. Result 112 = 3, and indeed 1 + 1 + 1 = 3.

Common Misconceptions

• Adders can only add. With 2's complement, invert B and set carry-in to 1: the same adder computes A − B. That is how ALUs reuse one circuit.
• Decoder and encoder are exact opposites. A plain encoder misbehaves with zero or multiple active inputs; the priority encoder fixes that.
• A MUX needs no select lines. Without them it cannot choose; n select lines control 2n data inputs.

Real-World Applications

• CPU arithmetic units chain full adders for 32- and 64-bit addition.
• Multiplexers route signals in communications and select register sources in processors.
• Decoders turn address lines into chip-enable signals selecting one memory device.
• Keyboard encoders convert key presses to binary; seven-segment decoders drive displays.

To recap: adders compute sums, multiplexers select inputs, decoders expand codes to one-hot outputs, encoders compress back. These four combinational blocks wired together do nearly all data crunching in digital systems.