Ripple-carry delay

Also called: adder delay, carry ripple delay, worst-case adder delay

The time a ripple-carry adder needs before every output is correct, set by the carry crossing all n stages. It grows linearly: about 2 gate delays per bit.

In a ripple carry adder, each stage has to wait for the carry from the stage below. The total time is set by the longest chain of waiting: the ripple-carry delay.

In this course's gate delay model (each full adder computes Cout = , every gate takes 1 delay, all inputs arrive at time 0):

every and is ready at time 1, in all bits at once
C1 is ready at 3 (AND at 2, OR at 3), and each later carry adds 2 more: Cᵢ at 2i + 1
each sum is one XOR after its carry: Sᵢ at 2i + 2
the worst case for an n-bit adder is Cₙ at 2n + 1

Counted from C0 only, a carry needs 2 gate delays per stage, so 2n to cross n stages. Either way the delay is linear in n: double the width, double the wait. That's why wide, fast adders use carry-lookahead.

When does the worst case actually happen? When a carry is created at bit 0 (both bits 1) and passed on by every higher bit (exactly one 1 in each). A column with two 0s kills the carry and cuts the chain. A clock period must still allow for the worst case, because the outputs pass through wrong values before they settle.

Worked examples

Example

Guaranteed times in a 6-bit adder

Apply Cᵢ = 2i + 1 and Sᵢ = 2i + 2.

1.
C1 at 3, C2 at 5, C3 at 7, C4 at 9, C5 at 11, C6 at 13.
2.
S0 at 2, S1 at 4, and so on up to S5 at 12.
3.
Worst case: C6 at 13 gate delays.

Example

Watching a carry ripple through 3 bits

A = 111, B = 000, held steady so every P = 1 is already settled. C0 rises in slot 1. Each slot is one gate delay. The outputs pass through 110 and 100 before settling at 000 with C3 = 1 (7 + 0 + 1 = 8).

Common mistakes

Thinking all the sum bits settle at the same moment.
Assuming every input pattern takes the worst-case time. Kills cut the chain short.
Thinking a wider adder adds a constant amount of delay. Doubling the width doubles it.

Practice Ripple-carry delay

Interactive questions with instant feedback and a worked solution for every wrong answer.

Adders and ALUs lesson full course

Learn it step by step

Ripple-carry delay is taught in Adders and ALUs.

Adders and ALUs →

All terms