QB behavior in a 2-bit ripple (asynchronous) counter: For a two-flip-flop ripple counter driven by a single clock into the first stage, which description best matches the QB waveform relative to the input clock?

Introduction / Context:
Ripple (asynchronous) counters are built by cascading flip-flops so that the output of one stage clocks the next. In a 2-bit ripple counter, the first stage (QA) toggles at half the input clock frequency, while the second stage (QB) toggles at half the QA frequency. Understanding these frequency divisions and phase relationships is essential for timing and decoding.

Given Data / Assumptions:

• Two T-like stages (or JK/D configured to toggle) are cascaded.
• Clock drives the first stage; QA output feeds the clock of the second stage.
• Idealized behavior is considered; propagation delays exist but are ignored for the frequency description.

Concept / Approach:
The first flip-flop divides the clock by 2, producing QA at f_clk/2. The second flip-flop uses QA as its clock and divides again by 2, yielding QB = f_clk/4. The duty cycle of each ideal toggling stage is 50%. Because the second stage clocks on transitions of QA, QB toggles when QA changes state, establishing a fixed phase relation behind QA.

Step-by-Step Solution:

Verification / Alternative check:
A timing diagram or quick simulation confirms QA transitions at every other clock, and QB at every other QA transition. Counting sequence 00 → 01 → 10 → 11 → 00 reinforces the divide-by-4 behavior at QB.

Why Other Options Are Wrong:

Common Pitfalls:
Misinterpreting asynchronous propagation as frequency error; ignoring that brief ripple delays can affect combinational decoding but not the fundamental divide-by-N behavior at Q outputs.

Final Answer:
A clean square wave at f_clk/4 with 50% duty cycle, toggling on QA transitions